Physics Reports 431 (2006) 1 – 38 www.elsevier.com/locate/physrep
X-ray magnetic circular dichroism at rare-earth L23 absorption edges in various compounds and alloys J.C. Parlebasa,∗ , K. Asakurab, c , A. Fujiwarad, e , I. Haradad , A. Kotanif, g a IPCMS, UM 7504 ULP-CNRS, 23 rue du Loess, BP 43, 67034 Strasbourg, France b CREST, Japan Science and Technology Agency, 4-1-8, Honcho Kawaguchi, Saitama 332-0012, Japan c Synchrotron Radiation Research Center, Japan Atomic Energy Research Inst., Hyogo 679-5148, Japan d Graduate School of National Science & Technology, Okayama Univ., 3-1-1 Tsushima-naka, Okayama 700-8530, Japan e Ryoka Systems Inc., 1-28-38 Shinkawa, Chuo-ku, Tokyo 104-0033, Japan f RIKEN, Soft X-ray Spectroscopy Laboratory, 1-1-1 Kouto, Sayo, Hyogo 679-5148, Japan g Photon Factory, Institute of Materials Structure Science, 1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan
Accepted 5 May 2006 Available online 27 June 2006 editor: D.L. Mills
Abstract Theoretical interpretations of X-ray magnetic circular dichroism (XMCD) at rare-earth (called R hereafter) L23 absorption edges are reviewed using differing models, depending on the material under investigation. In the first chapter, we present an overview of recent developments for XMCD in XAS with a few general remarks, especially at R atom absorption edges. In Section 2, we first describe basic mechanism of XMCD at L23 edges of R systems, and then we essentially discuss two examples of XMCD spectra in: (i) R2 Fe14 B metallic compounds, with the help of a cluster model, and (ii) RFe2 Laves-phase compounds, using a tight-binding approximation for R 5d and Fe 3d conducting states. A good agreement between theory and experiment for R2 Fe14 B suggests that a cluster model provides a valuable method to quantitatively calculate XMCD spectra of R systems, even with quite complicated atomic arrangements. For RFe2 systems the XMCD spectral shape, especially for the L2 edge of heavy R elements, is more complicated than those of R2 Fe14 B systems, and this is explained by the competition of some different XMCD mechanisms. In Section 3, we focus on special series of Ce systems, related to XAS and XMCD studies at the Ce L23 edges. Two clearly differing cases are interpreted: (i) A well localized 4f1 system, i.e. CeRu2 Ge2 ; (ii) A less localized 4f1 system, i.e. CeFe2 , with a 3d partner. Then, from a more experimental point of view, we investigate the influence of substitution on the low temperature properties of CePd3 compounds, i.e. Ce(Pd1−x Mnx )3 alloys where x is about 0.03, giving rise to (CePd3 )8 Mn ordered structure. We give another example: Ce(Pd1−x Nix )3 alloys with x taken up to about 0.25. Also the Ce L23 XMCD signal measured in pure CePd3 demonstrates that in the Ce based dense Kondo materials, only the 4f1 channel yields a magnetic response. © 2006 Elsevier B.V. All rights reserved.
Abbreviations: AES Auger electron spectroscopy; CF crystal field; DOS density of states; ED electric dipole; EF Fermi energy; EQ electric quadrupole; Eexc exchange splitting energy; FL Fermi liquid; FWHM full width at half maximum; MCD magnetic circular dichroism; MV mixed valence; NM non-magnetic; R rare earth; RE reduction factor of 4f–5d exchange energy; Rm reduction factor of Slater integrals; RXES resonant X-ray emission spectroscopy; RIXS resonant inelastic X-ray scattering spectroscopy; SIAM single impurity Anderson model; Tc Curie temperature; Tcom Compensation temperature; TK Kondo temperature; TM Transition metal; VF valence fluctuation; XPS X-ray photoelectron spectroscopy; XAS X-ray absorption spectroscopy; XES X-ray emission spectroscopy; XMCD X-ray magnetic circular dichroism; XPS X-ray photoelectron spectroscopy ∗ Corresponding author. Tel.: +33 3 88 10 70 74; fax: +33 3 88 10 72 49. E-mail addresses:
[email protected],
[email protected] (J.C. Parlebas). 0370-1573/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physrep.2006.05.002
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J.C. Parlebas et al. / Physics Reports 431 (2006) 1 – 38
PACS: 71.20.Eh; 71.20.Lp; 75.20.Hr; 75.30.Mb; 78.70.Dm; 78.70En Keywords: X-ray absorption spectra (XAS); X-ray magnetic circular dichroism (XMCD); Rare-earth compounds and alloys; Ce-systems
Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Rare earth series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Basic mechanism of XMCD at L23 absorption edges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1. Effect of 4f–5d exchange interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2. 4f–5d exchange interaction and enhancement of the ED transition intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3. Contribution of EQ transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.4. 5d–3d hybridization between R(5d) and TM(3d) electronic states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.5. Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. R2 Fe14 B metallic compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1. R2 Fe14 B cluster model electronic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2. Lu2 Fe14 B compound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3. Sm2 Fe14 B compound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4. Gd2 Fe14 B compound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.5. Er 2 Fe14 B compound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.6. Discussion on the comparison with experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. RFe2 type Laves-phase compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1. Laves-phase electronic bands structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2. RFe2 Laves-phase compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3. Role of R 5d–R 5d exchange interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4. LuFe2 Laves-phase compound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.5. RAl2 Laves-phase compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.6. Comparison between SmFe2 , LuFe2 , SmAl2 compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.7. Temperature dependence: case of ErFe2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.8. Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Ce systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. CeRu2 Ge2 : A well localized 4f 1 system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1. General context and proposed model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2. Numerical results and comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3. Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. CeFe2 : a less localized 4f 1 system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1. Brief presentation of the theoretical model and experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2. Numerical results and comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3. MCD in resonant X-ray emission spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. CePd3 : a dense Kondo material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1. Ce(Pd1−x Mnx )3 : experimental results for Mn substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2. Ce(Pd1−x Nix )3 : experimental results for Ni substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3. Pure CePd3 : experimental situation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 6 6 6 6 9 10 10 10 10 13 14 15 15 15 18 18 19 21 21 22 22 23 25 25 25 25 26 26 26 26 29 31 33 33 34 34 35 36 37
1. Introduction X-ray magnetic circular dichroism (XMCD) represents the dependence of the spectral intensity on the relative orientation of the magnetization and photon helicity. Since the early 1990s this phenomenon has been widely used in a variety of X-ray spectroscopies for the element specific analysis of magnetic systems (Ebert and Schütz, 1996). XMCD is usually understood as the change in absorption when the photon helicity is reversed in a magnetic material. In the case of X-ray absorption spectroscopy (XAS) one could call this “MCD in XAS,” although the term XMCD is also widely accepted and we shall use it hereafter in the present paper. Before proceeding further, let us just also recall
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the existence of MCD in photoemission (XPS) which reflects the change in XPS spectrum when the photon helicity is reversed. Similarly MCD in Auger electron spectroscopy (AES) and X-ray emission spectroscopy (XES) is the change in the AES and XES spectra, respectively, when the photon helicity is reversed. However, in the present review, we shall focus essentially on XMCD, as defined above i.e. MCD in XAS. In the case of CeFe2 , we shall also report some results for MCD in resonant XES, i.e. RXES. Actually XMCD is one of the most promising techniques to reveal local electronic and magnetic properties in a variety of magnetic materials, essentially because of its element as well as shell selectivities. In fact, with the help of circular polarized X-rays of good quality, arising from third-generation synchrotron facilities, a lot of data have been accumulated up to now on XAS and XMCD of transition-metal (TM) compounds and of rare-earth (called R hereafter) compounds. XMCD was first observed in X-ray absorption at the K threshold of Fe metal (Schütz et al., 1987). It has gained huge popularity due to the discovery of the sum rules (Thole et al., 1992; Carra et al., 1993a,b; Altarelli, 1993; Jo, 1993) which can be applied to obtain the ground state orbital and spin magnetic moments. In the present paper we review R inter-metallic systems (compounds and alloys) which contain mostly TM partners but not only. Also we focus on XMCD at the R L23 edges, i.e. essentially 2p → 5d electronic transitions. Why do we consider these edges? Actually, XMCD at L23 edges is an important tool to provide information on the R 5d electrons, separately from other shells, so that there are a lot of experimental data available. This kind of information is very difficult to obtain by other methods and thus crucial in order to understand the magnetic properties of the considered R based compounds. In the first part of this paper (Section 2) we review rare earth compound series, mostly containing TM atoms, especially iron atoms. Some of them are good candidates for permanent magnets and exhibit a variety of magnetic properties. First of all we present the basic mechanism involved in XMCD spectra at L23 edges and its various components which have to be taken into account to be able to interpret experiment properly (Section 2.1). For instance, during quite a long time, it has not been easy to draw instructive information from XMCD data (see for example, Baudelet et al., 1993): even the sign of XMCD intensities could not be predicted by a naive theory. After many challenges, Matsuyama et al. (1997) and Van Veenendaal et al. (1997) independently succeeded in explaining XMCD at R L23 edges by introducing a novel role of the intra-atomic exchange interaction between R 5d and 4f electrons. One example of compounds is provided by the R2 Fe14 B metallic series (Section 2.2). Using a cluster model for the considered series of inter-metallic systems we underline the important roles of: (i) 4f–5d exchange interaction which includes the novel effect of enhancement of electric dipole (ED) intensity related to 2p → 5d transitions; and (ii) hybridization of R 5d states with spin polarized Fe 3d states of surrounding ions. For a quantitative comparison with XMCD data of these series of compounds, it was pointed out that this effect of the spin-polarized Fe 3d states has to be included. However and furthermore the electric quadrupole (EQ) transition (2p → 4f) has also to be accounted for (Fukui et al., 2001; Asakura et al., 2002). Another example of rare earth compound series is provided by RFe2 type compounds in the Laves phase (Section 2.3). In the series of RFe2 compounds (Section 2.3.2), the R 5d electrons exhibit a subtle exchange process between a major magnetic moment, i.e. the Fe 3d moment, and an additional one, i.e. the R 4f moment (Harada et al., 2004). The latter plays special roles in applications, in connection with magnetic anisotropy. In the present review we also consider the RAl2 series (Section 2.3.5), especially when R = Sm still represents a light element of the series, i.e. SmAl2 , in order to point out the role of the 4f exchange field when Fe 3d states are replaced by nonmagnetic Al 3p states. More generally, it is instructive to consider the following cases, each of which is a limiting case for the mechanism of XMCD at R L23 edges: (i) SmAl2 (Section 2.3.5), (ii) LuFe2 (Section 2.3.4), (iii) RFe2 (Section 2.3.2) (especially, R = Gd to Tm which are the heavy elements of the series). The case (i) is a limiting case where only the effect of intra-atomic exchange interaction between 5d and 4f electrons is responsible for XMCD whereas in case (ii) the hybridization between R 5d and Fe 3d electrons is the only source to yield XMCD. In case (iii), both effects contribute to XMCD and give rise to a variety of XMCD shapes (Dartyge et al., 1998; Fujiwara et al., 2003). As far as comparison to experiment is concerned, it is to be noted that EQ contributions to the XMCD spectra ought to be also taken into account in addition to the usual ED ones. It is also to be mentioned that for XMCD at the L2 edge of RFe2 with heavy R elements all of three contributions, i.e. the ED contribution due to 5d–4f exchange interaction, the ED contribution due to R 5d–Fe 3d hybridization, and the EQ contribution, are weak and comparable in intensity, giving rise to a complicated spectral shape and its complicated temperature dependence. As mentioned above, in order to understand the relationship between the XMCD spectra for R L23 edges and the electronic and magnetic properties of R 5d, R 4f and Fe 3d states of R2 Fe14 B and RFe2 , it is essential to take into account the enhancement effect of the R 2p–5d ED transition intensity, as well as the EQ transition and the effect of R 5d–Fe 3d hybridization. On the other hand, the sum rules mentioned before cannot be applied to these XMCD spectra,
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because the enhancement effect of the ED transition intensity breaks the sum rules. Therefore, no more discussion on the sum rules will be given in the present paper. In the second part of the present review paper (Section 3) we focus on XAS and its magnetic counterpart, the XMCD at the Ce L23 edges, in order to study the specific case of Ce based systems. Especially magnetism of Ce intermetallic compounds is known to depend on the degree of localization of the 4f states and to exhibit a wide variety of behaviours from Kondo to mixed valence systems and from itinerant ferromagnets to strongly localized magnets. For many years these subjects have attracted a lot of interest of both experimentalists and theoreticians (see for example: Sereni et al., 1993; Degiorgi and Wachter, 1997; Razafimandimby et al., 1999). A very illustrative example of a manifold behaviour is magnetism of and phases in Ce metal (for theoretical works see Zölfl et al., 2001; Held et al., 2001). Other intriguing cases are Ce–Pd intermetallics. Let us give two examples: (i) A typical mixed valence system CePd7 (Beaurepaire et al., 1993) with a Kondo temperature TK of about 1000 K and a Ce–Ce inter-atomic distance of 5.67 A. Let us just recall that TK is the temperature below which the local moment (4f in case of Ce) is compensated by the conduction electrons. (ii) A differing example is CePd3 which has a much lower TK of about 240 K and a shorter Ce–Ce distance of 4.13 A. In this compound 4f states are less hybridized with conduction electrons. However now, a non-negligible Ce–Ce inter-site magnetic exchange coupling is present, that could lead to a magnetic ordering of Ce moments under certain conditions. Actually and first of all, we exhibit two clearly typical cases both from experimental and theoretical points of views: (i) A well localized 4f1 system, i.e. CeRu2 Ge2 (Section 3.1), and a phenomenological interpretation of the corresponding XMCD spectra (Fukui et al., 1999): one negative peak at L3 edge and another positive peak at L2 edge. This kind of systems is characterized by nf ≈ 1 and generally magnetic ordering is developed at low temperature (T < 10 K). (ii) A less localized 4f system, i.e. CeFe2 (Section 3.2) with a 3d partner (here it is Fe), and a theoretical interpretation of its XMCD spectra (Asakura et al., 2004a,b, 2005): a double peak structure (a replica of the edge). Moreover the sign of the dichroic signal is opposite to the preceding case. This kind of systems is characterized by nf < 1 and by ferromagnetic order mainly due to the 3d TM electrons with a higher Curie temperature. Furthermore, theoretical calculations have also been performed by Asakura et al. (2005) for MCD in RXES in addition to the previously considered MCD in XAS for the mixed valence ferromagnetic compound CeFe2 . It was shown that the excitation spectrum of Ce L3 –M5 MCD in RXES is close to a less broadened version of the Ce L3 MCD in XAS, which reveals the structure originating from the Ce 4f2 configuration. Actually this structure is not visible by conventional Ce L3 MCD in XAS measurements. Then (Section 3.3) we focus on CePd3 compounds which are indeed the most extensively studied Ce-based dense Kondo materials and we also study the alloying of the considered compounds with other substitutional TM elements (Mn or Ni), instead of Pd. It is generally accepted that the ground state of CePd3 can be described as a non-magnetic (NM) Fermi-liquid (FL). The electric resistivity as a function of temperature shows a broad peak around 120 K, which is very close to the maximum in magnetic susceptibility. At lower temperatures, resistivity decreases, indicating the formation of a coherent FL state (Lawrence et al., 1996). This and other measurements (Arts et al., 1985; Kanai et al., 1998) suggest that the Kondo temperature of CePd3 is about 240 K, as already stated above. Moreover, resonant inverse photoemission spectra at the Ce 4d threshold (Kanai et al., 1998) show significant changes in 4f peak intensity in temperature range from 20 to 290 K. This observation indicates the tendency for the 4f electrons to become more localized at high temperatures, even though they are still itinerant around TK . Moreover here, we point out (Section 3.3.3) the remarkable Ce L23 XMCD signal measured in pure CePd3 which corresponds to a magnetic field induced effect due to the 5d paramagnetic susceptibility. It demonstrates the following result (Kappler et al., 2004): in the Ce based dense Kondo materials, only the 4f1 channel gives rise to a magnetic response. Also we investigate the influence of substitution (Singh and Dhar, 2003; Kappler et al., 2004) on the low temperature properties of CePd3 (Sections 3.3.1 and 3.3.2). At first we give an example: Ce(Pd1−x Mnx )3 (Section 3.3.1) where x is about 0.03 giving rise to (CePd3 )8 Mn ordered compounds (Singh and Dhar, 2003). This special system is characterized by two phenomena: (i) Mn sublattice undergoes a ferromagnetic transition around 35 K; and (ii) Ce ions form a dense Kondo lattice and are in a paramagnetic state at least down to 1.5 K. Unfortunately up to now there is neither any XMCD data to confirm the previously considered results nor any corresponding theoretical analysis. A long time ago, Veenhuizen et al. (1987) already investigated the influence of substitutions on the low temperature properties of CePd3 , especially in Ce(Pd1−x Nix )3 alloys. On the basis of the temperature dependent susceptibility measurements the authors suggested that Ce has a tendency to a more localized 4f1 ground state with increasing Ni content up to 0.2. Magnetization measurements of Ce(Pd1−x Nix )3 alloys recorded at 4.2 K in field up to 35 T was described by the following equation: M(B) = M0 + intr B + bB 3 , where intr is the intrinsic susceptibility of CePd3
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5
Fig. 1. Definition of the intensities F ± ().
and M0 is the magnetization of the “Ni impurity” in saturation. It means that the intrinsic susceptibility of CePd3 is not suppressed through substitution of Pd by Ni. However, Veenhuizen et al. do not give any indication about possible transition to a ferromagnetic phase in these systems. This kind of transitions has already been observed several years ago in CePd1−x Nix alloys, in which the Ce sublattice remains basically unchanged with the same CrB structure (Kappler et al., 1997). Furthermore the highly localized 4f1 state of CePd leads to a ferromagnetic ordering below the Curie temperature Tc = 6.5 K, whereas CeNi shows the characteristic behaviour of a Kondo system. Thus, starting from CePd, Tc (x) increases from 6.5 K, in the 0 < x < 0.5 concentration range, to reach a plateau of about 10.5 K, in the 0.5 < x < 0.8 range. Then, at higher Ni concentration, the Tc values decrease and further steeply drop to zero near x = 0.95 (Nieva et al., 1988; Stewart, 2001). This system has been cited in Stewart’s review (2001) for its non-FL behaviour with Tc suppressed to zero via Ni doping. Let us recall that, in general, the FL provides a good description of the low temperature measurable parameters of a metal as long as the electron interactions, while T tends to zero, become temperature independent and are short ranged in both space and time. On the opposite, a non-Fermi-liquid behaviour is exhibited for a system with electron–electron interactions that are too strong to permit entry into the FL ground state at low temperatures. For the CePd0.05 Ni0.95 system, an obvious explanation is probably given by the nearness to a magnetic instability in the phase diagram. In the present paper we would like to further investigate the low temperature properties of Ce(Pd1−x Nix )3 alloys (Kappler et al., 2004), when x varies from 0 to 0.25 (Section 3.3.2). These systems have been characterized by magnetization measurements up to 4 T at low temperatures down to T = 50 mK. Temperature dependent specific heat measurements have been performed in the 0.5–20 K temperature range. Special attention is devoted to the Ce L23 absorption edges (XAS) and the corresponding XMCD spectra in order to precise the electronic and magnetic ground state of Ce. Actually in Ce(Pd1−x Nix )3 , the Ce L23 absorption edges and XMCD study (a negative sign at L3 and a positive one at L2 ) reveals indeed a coexistence of a strong 4f hybridization and a ferromagnetic order. In Ce(Pd1−x Nix )3 alloys the main source of magnetism should be due to Ce atoms as in CeRu2 Ge2 . Section 4 is devoted to a few concluding remarks on the series of rare earth compounds from both light and heavy sides, as far as the XMCD is concerned, with a special emphasis on the particular case of Ce compounds. Before closing this chapter, we would like to make clear the definition of the sign of XMCD. As mentioned by Baudelet et al. (1993), the following definition of the sign of XMCD is important to avoid any confusion: we take the axis of quantization as +z direction, which is also the direction of the photons wave vector, and the magnetic field B is applied in the −z direction to align the magnetization, as shown in Fig. 1. Then, the XMCD spectrum F ()
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is defined by F () ≡ F + (, B) − F − (, B),
(1.1)
where F + (F − ) is the XAS spectrum for the incident photon with +(−) helicity, and is the incident photon energy. In the present paper, we use this convention. For the inverted magnetic field −B, F () can also be written as F () = F − (, −B) − F − (, B) = F + (, B) − F + (, −B) = F − (, −B) − F + (, −B).
(1.2)
In the present paper, we write F ± (, B) simply as F ± (), unless it causes any confusion. 2. Rare earth series 2.1. Basic mechanism of XMCD at L23 absorption edges 2.1.1. Effect of 4f–5d exchange interaction In this review, we focus our attention on XMCD spectra observed at L23 edges of R elements (Fischer et al., 1990; Baudelet et al., 1993). The main absorption originates through the ED transition from 2p core level to 5d states. It is to be noted that the 2p hole state left behind splits into 2p1/2 and 2p3/2 states due to a relatively large spin–orbit interaction. These states correspond to the L2 and L3 edges, respectively, which are supposed to be sufficiently far from each other to be discussed separately. Jo and Imada (1993) gave the first interpretation for a systematic trend of XMCD in L23 XAS edges of R atoms from Ce to Tm, assuming a 2p → 5d ED transition and a 4f state of trivalent R ion in the Hund’s rule ground state. For the series, the electron configuration is 2p6 4f n 5d1 (n = 1.13) in the ground state and 2p5 4f n 5d2 in the final state, where the 5d electrons are in the conduction band with a semi-elliptical density of states. In this situation, Jo and Imada took into account the spin and orbital moments of the R 5d state induced through the intraatomic 4f–5d exchange interaction within a molecular field approximation. Based on these assumptions, the energy of the 5d state, specified by the z components of the orbital and spin quantum numbers, md and sd , respectively, is given by Ed ≡ E(md , sd ) = − |ck (2md , 3mf )|2 Gk n(mf , sf )(sd , sf ), (2.1) k=1,3,5 mf ,sf
where denotes the combined indices of md and sd ; ck is proportional to the Clebsch–Gordan coefficient; Gk represents the 4f–5d Slater integrals; n is the number of 4f electrons corresponding to mf and sf ; is the Kronecker delta function. We note that the energy Ed depends on the number n of 4f electrons. Assuming a 5d band width of several eV, Jo and Imada calculated the integrated XMCD spectra. The calculated result was mostly consistent with experiments in the systematic variation of the integrated XMCD amplitude over the R elements, but the result failed to recover the exact sign for the intensity of the observed results (see for example Baudelet et al., 1993).1 As a consequence there was a need left for a more sophisticated interpretation of the spectra. 2.1.2. 4f–5d exchange interaction and enhancement of the ED transition intensity In the case of Gd metal, in which the orbital magnetic moment is completely quenched, it has been pointed out that, according to band structure calculations, the spin dependent enhancement of the ED matrix element is crucial for predicting the correct XMCD sign (Carra et al., 1991; König et al., 1994). This is a consequence of the contraction of the radial part of the 5d orbital due to 4f–5d exchange interaction. This contraction effect in Gd was pointed out within band structure calculations (Harmon and Freeman, 1974), but applied to the XMCD of the Gd L23 edge was first done by Carra et al. (1991). Then, it is conceivable that, in other R elements having an open 4f shell, the magnitude of the enhancement depends not only on the 5d spin moment (sd ) but also on the 5d orbital moment (md ) through the exchange energy Ed . Van Veenendaal et al. (1997) investigated the present problem assuming empty 5d states in the 1 Actually Jo and Imada (1993) claimed that their result agreed with experimental data including the sign of XMCD, but their definition of the sign of XMCD was opposite to the convention mentioned in Section 1.
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7
Table 1 The ED transition intensity I ± (pj ; d) = jz |M ± (pjj z ; d)|2 md
2
1
0
−1
−2
(a) I + (p3/2; d) Sd = 1/2 −1/2
1 1/3
1/3 1/3
1/18 1/6
0 0
0 0
(b) I − (p3/2; d) Sd = 1/2 −1/2
0 0
0 0
1/6 1/18
1/3 1/3
1/3 1
(c) I + (p1/2; d) Sd = 1/2 −1/2
0 2/3
1/6 1/6
1/9 0
0 0
0 0
(d) I − (p1/2; d) Sd = 1/2 −1/2
0 0
0 0
0 1/9
1/6 1/6
2/3 0
ground state. Also Matsuyama et al. (1997) phenomenologically introduced the enhancement effect by an enhancement factor : W +Ed |M ± (pjj z ; d)|2 (1 − Ed ) d d ()L( + Epj − ), (2.2) Fj± () = EF
,jz
where Fj± () represents the XAS spectrum with ± helicity for the angular momentum j (=3/2, 1/2) of the 2p core state, M ± (pjj z ; d) is the electric–dipole-transition matrix element for ± helicity before taking into account the enhancement effect, d () is the semi-elliptical density of state of the 5d (d) state with the band width W expressed as 2 d () = W 2 − ( − Ed )2 , (2.3) W 2 EF is the FL, and L(x) is the Lorentzian L(x) =
/
(2.4)
x 2 + 2
with the lifetime broadening of the 2p core hole, . XAS and XMCD spectra are given, respectively, by Fj+ ()+Fj− (), and its magnetic counterpart Fj+ () − Fj− (), (Fig. 1). It is to be noted that M ± (pjj z ; d) is expressed, apart from a constant factor, as 1 M ± (pjj z ; d) = 2md 1 ± 1|1mp 1mp sp |jj z (sd , sp ), (2.5) 2 m ,s p
p
where j1 m1 j2 m2 |J M is the Wigner coefficients. For discussions given below, we list in Table 1 the ED transition intensity I ± (pj ; d) = |M ± (pjj z ; d)|2 (2.6) jz
for the ± helicity lights and for the states from j (=3/2 or 1/2) to = (md , sd ). Matsuyama et al. (1997) calculated the integrated intensity of the XMCD spectrum (normalized by the integrated XAS intensity) I pj , for the entire series of trivalent R elements (Ce–Yb). The calculated results for = 0.0, 0.4 and 0.6 (1/eV) are shown in Fig. 2. The result for = 0.0 almost coincides with that by Jo and Imada (1993),
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Fig. 2. Calculated integrated intensities of XMCD spectra at L2,3 edges for the entire series of rare earth elements. The parameter values of is set to 0.0, 0.4 and 0.6.
but by introducing = 0 (e.g. =0.4 or 0.6 1/eV) the sign of the integrated XMCD changes in accord with experimental observations. Let us first consider the case of Gd. The sign of I pj is positive and negative for j = 3/2 and 1/2, respectively, by taking into account the enhancement effect. This is easily understood as mentioned in the following: with our convention, the magnetic field B is in the −z direction, so that all of seven 4f electrons occupy ↑ spin states in the Hund’s rule ground state, where the spin quantization axis is in the +z direction. Therefore, by the 4f–5d exchange interaction the energy of 5d ↑ spin states is lower than that of 5d ↓ spin states, and the electric dipole transition amplitude is enhanced for the 2p to 5d ↑ transition, instead of the 2p to 5d ↓ transition. As seen from Table 1, the sign of XMCD is determined for L3 edge (j = 3/2) mainly by the competition between the 2p to 5d ↑ transition (the transition to (md , sd ) = (2, ↑) is most dominant) by + helicity and the 2p to 5d ↓ transition (most dominant is to (−2, ↓)) by − helicity, and for L2 edge (j = 1/2) mainly by the competition between the 2p to 5d ↓ transition (most dominant is to (2, ↓)) by + helicity and the 2p to 5d ↑ transition (most dominant is to (−2, ↑)) by − helicity. Therefore, by the enhancement of the 2p to 5d ↑ transition matrix element, the transition by + and − helicity become dominant for L3 and L2 edges, respectively, giving rise to the positive and negative signs of XMCD. This is exactly the same situation as that shown by Carra et al. (1991) through the energy band calculation.2 From comparison with the result by Carra et al., the value of is estimated to be 0.4–0.6 1/eV. In the case of = 0.0, we have no integrated intensity of XMCD if we disregard the 5d electron occupation in the ground state (as in the model by Van Veenendaal et al., 1997 and Goedkoop et al., 1997). But if we take into account the 5d electron occupation (as in the model by Jo and Imada, 1993 and by Matsuyama et al., 1997), the intensity of the 2p to 5d ↑ transition becomes weaker than the 2p to 5d ↓ transition because the 5d electron population is larger in the 5d ↑ band due to the gain of the exchange energy Ed . Therefore, the sign of XMCD becomes negative and positive, respectively, for L3 and L2 edges, as shown in Fig. 2. For other rare earth elements, it is seen from the results of = 0.4 and 0.6 (1/eV) in Fig. 2 that I pj has large positive values for L3 (j = 3/2) of heavy rare earths (Tb–Yb) and for L2 (j = 1/2) of light rare earths (Ce–Sm) , but it 2 Carra et al. (1991) published their paper before the papers by Jo and Imada (1993) and by Baudelet et al. (1993), and actually Jo and Imada, as well as Baudelet et al., referred the paper by Carra et al., but they did not notice the importance of the enhancement effect of the electric dipole transition intensity. Baudelet et al. stated “More importantly we obtain the wrong direction for the 5d magnetic moment.” on the sign of XMCD of Gd.
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9
Fig. 3. Integrated intensities of R L2,3 XAS and its XMCD by the 2p–4f EQ transition. The incident angle is taken to be 45◦ , and the intensities are normalized at the values of La XAS for L2 (closed circles) and L3 (open circles), respectively. From Fig. 3 of Fukui et al. (2001), but an error for the value of MCD in L2 of Sm is corrected.
has much smaller values for L3 of light rare earths (Ce–Sm) and for L2 of heavy rare earths (Tb–Yb). The mechanism of this behaviour is more complicated than that of Gd, but the trend is understood in the following way. Looking at Table 1, we confine ourselves, for simplicity, to the following cases of largest ED transition intensity: for the L3 edge the transition to (md , sd ) = (2, ↑) by + helicity light and that to (−2, ↓) by − helicity one, and for the L2 edge the transition to (2, ↓) by + helicity light and that to (−2, ↑) by − helicity one. For light rare earths, the dipole transition intensity to (md , sd ) = (2, ↓) is strongly enhanced by the exchange interaction with 4f states in the Hund’s rule ground state, but the dipole transition intensities to (−2, ↓), (−2, ↑) and (2, ↑) are not much enhanced.3 Therefore, in the L2 edge the XAS intensity by the + helicity light is much larger than that by the − helicity light, and the XMCD has a large positive value, but in the L3 edge the XAS intensities by the + and − helicities are comparable and the sign of XMCD is determined by higher order effects. For heavy rare earths, the dipole transition intensity to (md , sd ) = (2, ↑) is strongly enhanced, but those to (−2, ↑), (2, ↓) and (−2, ↓) are not much enhanced. Therefore, the absorption intensity by + helicity light is much stronger than − helicity light for L3 , but the absorption intensities by + and − helicity lights are comparable for L2 . As a conclusion it was stated that in addition to the previously considered spin and orbital polarization effect of the 5d states due to the 4f–5d exchange interaction, there was a novel effect of contraction in the radial part of the considered 5d wave functions which led to the enhancement of the 2p–5d ED matrix element. 2.1.3. Contribution of EQ transition Furthermore the EQ contribution is also inevitable for a quantitative comparison of the calculated spectra with experiment. Then, Fukui et al. (2001) determined it using Cowan’s program (1981) based on an atomic model. The calculated integrated intensities of XAS and XMCD are shown in Fig. 3. The atomic calculation is a reasonable method since the 4f electrons, directly concerned with the EQ process, i.e. the initial state 2p6 4f n and the final state 2p5 4f n+1 , 3 In the Hund’s rule ground state of Ce3+ , for instance, the 4f electron with (m , s ) = (3, ↓) is occupied under the magnetic field B in the −z f f direction, so that the ED transition intensity to the state (2, ↓) is much enhanced due to the gain in the exchange energy (Eq. (2.1)), but the transition to (−2, ↑) is not enhanced.
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are well localized. Many-body effects are crucial in this process, especially the interaction between the photo-excited 4f electron and the core hole left behind as well as between the other 4f electrons. However the lifetime effect of the 2p core hole smears out the detailed structure of the spectra. The contribution of the EQ transition to XMCD is generally weaker than that of the ED transition mentioned in Section 2.1.2. However, the EQ contribution is somewhat lower in energy than the ED contribution, and especially, for the L3 XMCD of light rare earths and for the L2 XMCD of heavy rare earths, the EQ contribution can be important because the ED contribution is rather weak, as seen from Fig. 2. 2.1.4. 5d–3d hybridization between R(5d) and TM(3d) electronic states Now, in order to determine XMCD spectra of R elements in intermetallic compounds containing TM, like for example R2 Fe14 B (Section 2.2) or RFe2 (Section 2.3), it is important to take into account another polarization effect of the 5d states due to the hybridization with spin polarized 3d states of surrounding TM ions (Fe in Sections 2.2 and 2.3), which occurs through the 2p–5d ED transition. Actually it is well known that the magnetic coupling between Fe 3d spin and R 4f spin always presents an antiferromagnetic alignment (via the R 5d conduction electrons) which makes them either ferromagnetic compounds for light rare earths (Ce–Sm) or ferrimagnetic compounds for half-filled (Gd) and heavy rare earths (Tb–Yb), because the R 4f spin moment is antiparallel and parallel, respectively, to the R 4f orbital moment (which is larger than the R 4f spin moment) for light and heavy rare earth elements. On the contribution of R 5d–Fe 3d hybridization to XMCD, this effect must dominate the spectra for La or Lu compound, since there is no effect from the 4f electrons. For other rare earth elements, this effect is generally weaker than that of the ED contribution due to the R 4f–5d interaction mentioned in Section 2.1.2. However, for the L3 XMCD of light rare earths and for the L2 XMCD of heavy rare earths, this contribution can be important because the ED contribution by 4f–5d interaction is rather weak, as seen from Fig. 2. In the case of R2 Fe14 B, where Fe constitutes the majority of the magnetic moment, the contribution of R 5d–Fe 3d hybridization is large and determines almost the total L2 XMCD of heavy rare earth elements as will be shown in Section 2.2.6. Since the R 4f and Fe 3d magnetic moments order ferromagnetically for light rare earth elements, the R 4f magnetization always aligns parallel to the external magnetic field, and hence the sign of the XMCD contributions from the ED contribution by R 4f–5d interaction, the EQ contribution and the ED contribution by R 5d–Fe 3d hybridization are easily determined, for instance as shown in Figs. 2 and 3. However, for heavy rare earth elements where the R 4f and Fe 3d magnetic moments are in the antiferromagnetic ordering, we have to pay attention to the following points: for R2 Fe14 B, where Fe constitutes the majority of the magnetic moment, (1) the sign of the ED contribution by R 4f–5d interaction is opposite to that in Fig. 2, (2) the sign of the EQ contribution is also opposite to that in Fig. 3, and (3) the sign of the ED contribution by R 5d–Fe 3d hybridization is the same as that of La2 Fe14 B. On the other hand, for RFe2 , where R constitutes the majority of the magnetic moment; (1) the sign of the ED contribution by R 4f–5d interaction is the same as that in Fig. 2; (2) the sign of the EQ contribution is also the same as that in Fig. 3; and (3) the sign of the ED contribution by R 5d–Fe 3d hybridization is opposite to that of La2 Fe14 B. 2.1.5. Concluding remarks Now, we are ready to review XMCD spectra in more details depending on the case under investigation. We shall report on recent theoretical works in relation with experimental spectra for a series of R elements in R2 Fe14 B (Section 2.2) and RFe2 compounds (Section 2.3). In contrast to RFe2 compounds, experimental data for almost all the R series in R2 Fe14 B are available. Also, the XMCD spectral structures in the R2 Fe14 B spectra are easier to understand as compared to RFe2 compounds. Furthermore less unsolved problems are left over. Therefore we shall next focus on R2 Fe14 B and afterwards on RFe2 compounds. 2.2. R2 Fe14 B metallic compounds 2.2.1. R2 Fe14 B cluster model electronic structure Quantitative and systematic calculations of XMCD in the series of R2 Fe14 B metallic compounds were first performed by Fukui et al. (2001) by including both ED and EQ transitions. First of all, for ED contribution, as already explained above, Fukui et al. took into account the 4f–5d exchange interaction which causes the magnetic polarization of 5d bands as well as the enhancement of 2p → 5d transition matrix element. Moreover, they pointed out the importance of the magnetic polarization of 5d bands due to the hybridization of R 5d states with spin-polarized Fe 3d states
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11
Fig. 4. Cluster model consisting of 10 R atoms (dark spheres) and 16 Fe atoms (light spheres).
(Section 2.1.3). Their results were in rather good agreement with experiment for almost the entire series of R2 Fe14 B compounds. However, Fukui et al.’s model was too much simplified in the following two points: (i) The density of states of R 5d bands was assumed to be given by a simple semi-elliptic function. (ii) The XMCD due to hybridization between R 5d and Fe 3d bands has not been explicitly calculated. Instead, Fukui et al. simply added the XMCD spectrum observed experimentally for La2 Fe14 B to the calculated XMCD spectrum of each R compound without the hybridization effect. Afterwards a unified model was proposed by Asakura et al. (2002) to reproduce, in a more consistent and quantitative fashion, the XMCD spectra observed at L23 edges of the R2 Fe14 B series, improving the theory by Fukui et al. (2001). To this end, Asakura et al. adopted a cluster model, in which the band effects of Fe 3d and R 5d states are simulated. The mixing effect between 5d and 3d states was taken into account microscopically within the cluster model. Also taken into account in a mean-field approximation were the intra-atomic exchange effects on 5d states from 4f electrons as well as on 3d states from other 3d electrons. In the calculated spectra, the enhancement of the dipole matrix element due to 4f–5d exchange interaction was included in the way that was presented in Section 2.1.2. It is known that a given R2 Fe14 B compound has tetragonal symmetry and its unit cell contains 68 atoms (Herbst, 1991). Then, a cluster consisting of 10 R atoms including the central R (on the atomic site 0) and 16 Fe atoms surrounding the site 0 was adopted by Asakura et al. (2002) as shown in Fig. 4, where the R and Fe atoms are depicted with dark and light spheres respectively. Actually the considered cluster contains all R atoms up to the fifth nearest neighbours from site 0 and all Fe atoms up to the sixth nearest neighbours from site 0. The Hamiltonian of the cluster model
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(a)
(b)
Fig. 5. Calculated partial density of states for: (a) Lu2 Fe14 B; and (b) Sm2 Fe14 B. The solid curves are the rare earth 5d states the magnitude of which is multiplied by 0.3, and the dashed curves are the Fe 3d states.
is expressed as H= 5d ()d+,i d,i + 3d ()D+,j D,j + [ti,i (, )d+,i d ,t + h.c.] i
+
j =j v,v
j
[tj,j (, )D+,j D ,j + h.c.] +
i=j ,
i=i ,
[t (, )d+,i D,j + h.c.],
(2.7)
+ where the operators d+,i (i = 0.9) and Dv,j (j = 1.16) represent, respectively, the creation of an electron in the 5d state of the R ith site and in the 3d state of the Fe jth site. The indices and denote (md , sd ) of R 5d and Fe 3d states, respectively. The 5d energy level 5d is essentially the same as the exchange energy Ed in Eq. (2.1) but now we take into account the reduction factor RE of the Slater integral Gk :
5d () = RE Ed .
(2.8)
On the other hand, the 3d energy level is expressed as 3d () = − Eexc s3d ,
(2.9)
where is the Fe 3d–R 5d energy separation in nonmagnetic state and Eexc is the exchange splitting of the Fe 3d state. These values are taken = −2.6 eV and Eexc = 1.1 eV, and the Fermi energy is fixed at −2.0 eV with respect to the center of the 5d levels. The electron transfer integrals, ti,i (, ), tj,j (, ) and ti,j (, ) are obtained from the Slater–Koster integrals given by the empirical formula by Pettifor (1977) (for more details, see Asakura et al., 2002). In the case of Lu2 Fe14 B we show the calculated one particle density of states (DOS) in Fig. 5, where the partial DOSs are depicted for Lu 5d (projected on the central Lu site) and Fe 3d (averaged over all Fe sites) states with up (majority) and down (minority) spin components. The origin of the energy is taken at the Fermi level. The original discrete DOSs are broadened appropriately to give continuous bands. Here, let us briefly explain how Asakura et al.’s cluster calculation works. Firstly, we note that the overall shape of the DOS, especially of Fe 3d states shown in Fig. 5a, is similar to that calculated by the recursion method for Y2 Fe14 B (Inoue and Shimizu, 1986), except for fine structures. This is one of the main reasons why Asakura et al. (2002) found reasonable values of n3d , n5d and Ms within a set of reasonable band parameters. Secondly, we would like to recall the fact that the hybridization between R 5d states and spin polarized Fe 3d states gives rise to a reasonable spin polarization in the R 5d bands, as will be discussed in the following. The effect of hybridization is seen most clearly in the Lu based compound, since in this case the polarization in Lu 5d bands comes only from the hybridization with Fe 3d states. Also, Asakura et al. (2002) considered the case where 4f electrons are partially filled, with Sm2 Fe14 B as an example. Now the energy of 5d levels depends on m5d and s5d because of the exchange potential of 4f states, so that the partial DOSs are somewhat modified as compared to Lu2 Fe14 B, as shown in Fig. 5b. Asakura et al.’s cluster calculation (2002) was performed as usual at zero temperature. Their subsequent calculated XMCD spectra also reflected situation at zero temperature. Actually, temperature dependence will be discussed later on in Section 2.2.6. Let us come back qualitatively on the hybridization effect. It is worth noticing that the calculated
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(a)
13
(b)
Fig. 6. (a) Calculated XAS and XMCD spectra (solid curves) at L3 and L2 edges of Lu2 Fe14 B by taking into account the ED contribution only. The results are compared with the experimental ones (× and + for XAS and XMCD, respectively). (b) Calculated XAS and XMCD spectra (solid curves) at L3 and L2 edges of Sm2 Fe14 B by taking into account the ED contribution only. The results are compared with the experimental ones (× and + for XAS and XMCD, respectively).
energy bands reasonably yield an antiferromagnetic coupling between Fe 3d and R 5d spins. As is seen from the DOS of Fig. 5a, this result is thought to be a consequence of the fact that the hybridization between the spin polarized 3d and 5d states is stronger for the down spin band (minority) than for the up spin band (majority). As a matter of fact, 3d and 5d bands are energetically closer located in the former case than in the latter case. Then, the energy of a 5d electron having a down spin is pulled down more below the Fermi energy than when it has an up spin. Hence the number of the former is larger than that of the latter. Now the effect of 5d–3d hybridization on XMCD is obvious. The unoccupied 5d states are spin-polarized due to hybridization with 3d states, so that 2p → 5d XAS depends on the helicity of incident X-ray. Actually the absorption spectrum for X-rays with positive and negative helicities has been calculated at the central Lu site of the cluster. Let us also comment on the case where 4f electrons are partially filled, i.e. Sm2 Fe14 B. In this case the energy of 5d levels depends on m5d and s5d because of the exchange potential of 4f states, so that the partial DOSs are changed with respect to Lu2 Fe14 B, as already stated. In this calculation, Asakura et al. (2002) used the values of the 4f–5d Slater integrals Gk obtained by the atomic Hartree–Fock calculations, and a reduction factor RE of 0.8 for the overall spectrum. In the calculation of XMCD, Asakura et al. took into account the enhancement of the ED transition matrix element by a factor taken to be 0.4 (1/eV). 2.2.2. Lu2 Fe14 B compound The calculated XAS and XMCD for L2 and L3 edges of Lu2 Fe14 B compounds (Asakura et al., 2002) are shown with solid curves in Fig. 6a. In these calculations only ED process was assumed. Because of 2p core hole lifetime, a Lorentzian function was used to calculate the considered spectra. The corresponding spectral broadening (FWHM) due to lifetime effect was taken to be 4.0 eV. Furthermore each spectrum was then further convoluted with a Gaussian function of 1.5 eV width, simulating the instrumental resolution, in order to fit the actual experimental data. For XAS, a background was also assumed in an arctangent form to directly compare the result with experiment. This background, of course, does not affect XMCD. The experimental data are also shown in Fig. 6a with “×” for XAS and “+” for
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Fig. 7. (a) XMCD spectra of the Sm L23 XAS in Sm2 Fe14 B. Solid curves represent the calculated results consisting of the ED contribution (dashed curves) and the EQ contribution (dotted–dashed curves), while crosses represent the experimental data. (b) XMCD spectra of the Sm L23 XAS in Sm2 Fe14 B. Solid curves are the sum of the three contributions: the ED contribution due to the 4f–5d exchange interaction (dashed curves), the ED contribution due to the Sm 5d–Fe 3d hybridization (dotted curves) and the EQ contribution (dotted–dashed curves).
XMCD and the calculated spectral shape of XMCD is in rather good agreement with experiment. It is interesting here to point out that the hybridization effect yields a correct sign of the XMCD spectrum, negative for L2 edge and positive for L3 edge, which is reasonable when it is considered that the hybridization effect polarizes the Lu 5d electrons antiferromagnetically with the Fe 3d electrons . However, the calculated amplitude of XMCD, which is defined by a percent value with respect to the maximum intensity of XAS, is much larger than the experimental one. Thus notice that the calculated XMCD is multiplied by 0.2 in Fig. 6a. 2.2.3. Sm2 Fe14 B compound In the calculation of the XMCD spectrum for this compound, Asakura et al. (2002) took into account the effects of polarization of 5d states due to 4f–5d exchange coupling as well as 5d–3d hybridization (Sections 2.1.1 and 2.1.4). Moreover the enhancement effect of ED transition amplitude was also accounted for by an enhancement factor defined in Section 2.1.2 and taken to be 0.4 (1/eV) according to Fukui et al. (2001). The calculated results for XAS and XMCD spectra for a typical example of light R elements, Sm2 Fe14 B, are shown in Fig. 6b, where the effect of EQ transitions is not taken into account. In Fig. 6b, the calculated XMCD has been again multiplied by a factor 0.22 to best fit to the experiment. The remaining contribution to XMCD arises from EQ transitions (Section 2.1.3). In Fig. 7 the corresponding EQ contribution is shown as a dotted–dashed curve and it was obtained through a straightforward atomic full-multiplet calculation by Fukui et al. (2001). However the relative intensity between EQ and ED transitions was treated as an
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adjustable parameter. In Fukui et al. (2001) assumed that the ED contributions of R 4f–5d exchange interaction and R 5d–Fe 3d hybridization can be treated to be independent, but in Figs. 6b and 7a these contributions should not be exactly independent because they are automatically correlated through the calculated electronic structure of the Sm 5d states. However, Asakura et al. (2002) showed that their correlation is not very large. They calculated the following three contributions independently: (i) the ED contribution from Sm 4f–5d exchange interaction by putting Eexc = 0, (ii) that from the Sm 5d–Fe 3d hybridization by putting RE = 0, and (iii) the EQ contribution. Then they compared the spectrum of (i) + (ii) + (iii) with the result shown in Fig. 7a, where the spectra (i), (ii), (iii) and (i) + (ii) + (iii) are shown in Fig. 7b with dashed, dotted, dash–dotted and solid curves, respectively. From comparison of the solid curves in Figs. 7a and b, we find that they almost coincide with each other. Therefore, the contribution to XMCD from the Sm 4f–5d exchange interaction is almost independent of that from the Sm 5d–Fe 3d hybridization, and they can be almost additive. Furthermore, if we compare the solid curves of XMCD in Fig. 6a with the dotted curves in Fig. 7b, we find that they almost coincide with each other. These results indicate that the assumptions made by Fukui et al. (2001) are almost valid. From Fig. 6b, it is seen that the relative intensity of (ii) vs. (iii) is comparable for both L3 and L2 edges and the relative intensities of L3 vs. L2 edges for (ii) and (iii) are also comparable. On the other hand, the intensity of (i) is much larger in the L2 edge than in the L3 edge, the reason of which has already been mentioned in Section 2.1.2 (see also Fig. 2). As a result, the intensities of (i), (ii) and (iii) are all comparable in the L3 edge, while the intensity of the L2 edge is mainly determined by (i). 2.2.4. Gd 2 Fe14 B compound Similar calculations have been performed for all other R elements of the series with fixed values of (RE , )=(0.8, 0.4). The result for Gd2 Fe14 B, as an example, is shown in Fig. 8a. From Fig. 2, we see that the XMCD of Gd has considerably large positive value for L3 and very large negative value for L2 . However, for Gd2 Fe14 B the Fe magnetic moment constitutes the majority of the total magnetic moment, so that the signs of XMCD should be inverted from those in Fig. 2. Thus we understand the considerably large negative XMCD for L3 and the very large positive XMCD for L2 which are shown in Fig. 8a. Here, most of XMCD spectra are determined by (i), and we see much smaller contributions from (iii), where the signs of the contributions (iii) are also inverted from those in Fig. 3. In Fig. 8a, we do not show the decomposition of the dashed curves to the contributions from (i) and (ii), but the contribution from (ii) is comparable with the MCD of Fig. 6a and it is much weaker than that from (i). 2.2.5. Er 2 Fe14 B compound Another example is given in Fig. 8b for a typical example of heavy R elements, Er2 Fe14 B (Asakura et al., 2002). Once more the calculated XMCD spectrum is in fair agreement with experimental data, although there are some discrepancies. In order to understand these results, we note that the signs of (i) and (iii) should be opposite to those in Figs. 2 and 3, because of the dominant Fe magnetic moment, while the sign of (ii) is the same as that of La2 Fe14 B. Actually, the L2 XMCD spectrum of Er2 Fe14 B is almost the same as that of La2 Fe14 B (see Fig. 5a), because the contribution from (ii) is much larger than (i) and (iii) (although (i) and (ii) are not decomposed in Fig. 8b). Here, it is to be noted that the contribution from (ii) is considerably large due to the large Fe magnetic moment of this material (compare with that of RFe2 in Section 2.3) and that from (i) is small as seen from Fig. 2. On the other hand, the L3 XMCD spectrum consists of (i), (ii) and (iii) with comparable contributions. We remark that the contribution (i) in Er is much larger for L3 than for L2 (see Fig. 1) and the contribution (iii) in Er is also much larger for L3 than for L2 (see Fig. 3). 2.2.6. Discussion on the comparison with experimental results In the preceding subsections, we have recalled Asakura et al.’s result (2002). Actually their calculated XMCD amplitude at zero temperature is much larger than the experimental results, although the calculated XMCD spectral shape is rather in good agreement with experiment. One of the reasons for this discrepancy in XMCD amplitude is that the experiment was performed at room temperature, so that R and Fe magnetic moments have been reduced due to thermal fluctuations. Another reason for the XMCD reduction would be the fluctuation in the magnetization directions for different magnetic domains. In Asakura et al.’s model, the Fe magnetic moment is controlled by the exchange parameter Eexc . Also the effect of the reduced 4f moment on 5d states can be effectively treated by decreasing the reduction factor RE as done by Fukui et al. (2001). Therefore, in order to compare with experiment at room temperature
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Fig. 8. (a) Similar to Fig. 7a, but for the Gd L23 edges of Gd2 Fe14 B. (b) Similar to Fig. 7a, but for the Er L23 edges of Er2 Fe14 B.
Table 2 The reduction factor (RE ) of the 4f–5d exchange energy R
La
Pr
Nd
Sm
Gd
Tb
Dy
Ho
Er
Tm
Yb
Lu
RE
–
0.12
0.12
0.16
0.40
0.12
0.12
0.20
0.12
0.08
0.04
–
more systematically, calculations of the preceding subsections were used but by reducing Eexc and RE . Actually the value of Eexc was fixed at 0.2 eV whereas the values of RE was taken as shown in Table 2. The calculated XAS and XMCD for L3 and L2 edges are shown in Figs. 9(a) and (b), respectively, together with the experimental data (Chaboy et al., 1996). The overall agreement between calculated and experimental results is quite satisfactory in both XMCD shape and amplitude. The calculated XMCD involves the contributions of ED transitions with the effects of inter-atomic hybridization and intra-atomic exchange interaction, as well as EQ contributions. We note that, since each contribution to XMCD spectra is sometimes different in shape and sign, the total XMCD spectra show a variety of shapes. The mechanism of XMCD spectra for light (R = Pr .Sm) and heavy (R = Tb.Yb) rare earths is essentially the same as that of Sm2 Fe14 B and Er2 Fe14 B, respectively, which was explained in detail in the preceding subsection.
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Fig. 9. Calculated results (solid curves) and experimental ones (crosses) of XAS and XMCD at: (a) L3 ; and (b) L2 edges of R2 Fe14 B.
Let us comment a little bit on the reduction factor RE of the exchange energy in Table 2 with respect to the calculated results of Figs. 9a and b. There are two different origins for RE : (i) One is the correction from Hartree–Fock calculation of the exchange interaction between 4f and 5d states, namely the correction by intra-atomic configuration interaction, which results in the value of RE (≈ 0.8) at zero temperature. (ii) The other one is the reduction of exchange energy by thermal fluctuations of 4f magnetic moment. In the comparison of calculated XMCD spectra with experiment at room temperature, this effect is important because the magnetic polarization of 5d states due to 4f–5d exchange interaction is reduced in proportion to the reduction of 4f magnetization. Actually, the magnetization of R2 Fe14 B at room temperature is mainly carried by Fe 3d electrons. Also the magnetization of R 4f electrons is strongly reduced by thermal fluctuations (Herbst, 1991), except for Gd. In the case of Yb and Tm, the spectral shape of XMCD is similar to that of Lu which has no 4f magnetic moment. Therefore Asakura et al. suggested a drastic reduction of 4f magnetization and extremely small values of RE (0.04 and 0.08). For Gd, on the other hand, a considerable contribution from 4f magnetization to XMCD was seen, reflected in the reduction factor 0.4. It is to be noted that the rough trend of reduction factor RE (which takes a maximum at Gd and decreases in going away from Gd) is in qualitative agreement with the behaviour of Tc in R2 Fe14 B (Herbst, 1991). Of course, Asakura et al.’s treatment of RE is still phenomenological. Especially there is room for studying the XMCD temperature dependence from a microscopic point of view (Section 2.3.6). Nevertheless in the present section, we have reported a valuable extension of Fukui et al.’s theory (2001) by Asakura et al. (2002), which brings essentially two improvements: (1) a more realistic cluster model as well as (2) explicit calculations of XMCD. On the point (1), the DOSs of 5d states obtained in Fig. 5 are actually not semi-elliptic as assumed by Fukui et al. (2001). However, in XAS and XMCD spectra, the DOS structures are mostly smeared out by the large spectral broadening due to the 2p core hole lifetime. If the lifetime broadening effect is artificially reduced, the XMCD spectrum would reflect the 5d DOS structures. We would like to point out that experimental measurements of such detailed XMCD spectra with spectral broadening smaller than 2p lifetime width should be possible, in principle nowadays, especially with the help of Hämäläinen et al.’s technique (1991). These authors already succeeded in observing the pre-edge structure of the Dy L3 XAS of Dy compounds without 2p core lifetime broadening by measuring the excitation spectrum of 3d to 2p resonant X-ray emission spectroscopy (RXES). This technique can also be applied to measure the XMCD spectrum (see Section 3.2.3). Then the present method of calculating XMCD structures, related to the detailed 5d DOS structures, should become more and more important. Actually, Asakura et al. (2004) have also performed theoretical calculations of XMCD in R2 Fe14 B with the use of the R 5d and Fe 3d DOS obtained with the tight-binding energy-band model, instead of the cluster model which we have treated in the present section. They have shown that the results of XMCD calculated with the tight-binding model coincide almost perfectly with those calculated with the
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Fig. 10. Calculated partial DOS in (a) SmFe2 and (b) SmAl2 . The vertical dashed line denotes the Fermi energy. Up and down mean the spin directions.
cluster model, if we take into account the spectral broadening due to the 2p core hole lifetime. At the same time, they have shown that some more details of the R 5d band structure obtained with the tight-binding model should be reflected in the XMCD spectra, if the spectral broadening could be suppressed. On the point (2), Asakura et al. (2002) succeeded in calculating XMCD spectra for Lu or La compounds on the basis of microscopic electronic states. Generally speaking, there is no reason why the contribution to XMCD from 4f–5d exchange interaction would be independent of that from 5d–3d hybridization, since both contributions are correlated to each other through 5d electronic structure. As far as Asakura et al.’s calculation is concerned, however, the two contributions can be shown to be almost independent as demonstrated for Sm2 Fe14 B (see Figs. 7a and b). Now we turn to the RFe2 series of compounds which present some new interesting aspects in addition to the previously considered features for R2 Fe14 B. 2.3. RFe2 type Laves-phase compounds 2.3.1. Laves-phase electronic bands structure In order to further discuss XMCD, we next report the electronic structure of RFe2 Laves-phase type compounds calculated by Harada et al. (2004). To this end, these authors used a tight-binding method (Yamada et al., 1984) for the R 5d and Fe 3d states of RFe2 , which is known to provide reasonable electronic structure, including the hybridization between them. The effect of the 2p core hole was neglected for simplicity, since the photo-excited 5d electron is rather extended. The tight-binding Hamiltonian describing these states was then diagonalized using linear combinations of R 5d and Fe 3d states which are fairly hybridized. Also the effect of exchange field due to 4f electrons in the Hund’s rule ground state has been taken into account within the energies of 5d levels as well as the exchange field due to 3d–3d exchange interaction within the energies of 3d levels. The density of states (DOS) thus obtained is presented in Fig. 10a, at least, as far as one example of the Laves phase series (with a light element, R = Sm) is considered, i.e. in the case of SmFe2 . Harada et al. (2004) also considered the counterpart example of the RAl2 series, i.e. SmAl2 , in order to extract the role of 4f exchange field, when Fe 3d states happened to be replaced by nonmagnetic Al 3p states (Fig. 10b). In such a case, they assumed another set of bands structure parameters adapted to the new compound under consideration. At last, they reported the calculated DOS of a compound at the end of the lanthanides series, i.e. LuFe2 (Fig. 11).
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Fig. 11. Calculated partial DOS in LuFe2 . The vertical dashed line denotes the Fermi energy. Up and down mean the spin directions.
From all these results, Harada et al. (2004) stated the following remarks: (i) The DOS of R 5d bands show a characteristic peak at high energies for all three cases. (ii) In SmAl2 the hybridization between R 5d and Al 3p states is not significant, although the considered states are located relatively close in energy. This may be due to the symmetry of these states. Also 5d bands are polarized over the whole energy region by 4f exchange field. (iii) In SmFe2 and LuFe2 the hybridization between R 5d and Fe 3d states is relatively strong, especially for the minority-spin band. As a consequence, the number of 5d electrons having down spin is larger than that of the electrons having up spin. This yields an antiferromagnetic spin arrangement of 5d and 3d moments. Thus, the tight-binding model reproduces a delicate energy dependence of the polarization in R 5d states, which directly affects XMCD spectra. (iv) As far as the DOS of Sm 5d states in SmAl2 compounds is concerned (Fig. 10b) it is rather different from that in SmFe2 (Fig. 10a), not only near the Fermi energy but also along the whole energy region. The former is apparently due to the absence of spin polarization in Al 3p band while the latter originates from characteristics of Al 3p band (relative energy position as well as symmetry). It is to be noted that the states peaked in the highest energy region are polarized by intra-atomic exchange interaction. Thus, the polarization in the whole energy region of 5d bands will produce a broad XMCD spectrum. (v) In LuFe2 the spin polarization of Lu 5d bands is only due to hybridization with spin-polarized Fe 3d bands, so that the spin polarization of empty 5d bands is rather localized near the Fermi level with almost no spin polarization in the high energy region of 5d bands. Once the electronic bands structures have been obtained, Harada et al. (2004) were able to calculate the XAS spectrum for each incident X-ray polarization. The value of (see Section 2.1.2) was assumed to be 0.6 (1/eV) for all R elements in the following. Whenever comparison was made between the calculated spectra and those observed from experiment, it was necessary to add the EQ contribution calculated using an atomic model (see Section 2.1.3). 2.3.2. RFe2 Laves-phase compounds Harada et al. (2004) calculated XMCD spectra of RFe2 at the R atom L23 edges for all trivalent R element series to see the trend of XMCD. Unfortunately, the experimental data for the light rare earths (La, Pr, Nd, Eu) compounds, as well as for Yb compound, are not available, in contrast to the preceding R2 Fe14 B series for which almost all experimental spectra are at hand. Then, they compared in Fig. 12(a) and (b) the calculated spectra for Gd and for the next heavy R (Tb, Dy, Ho, Er, Tm) compounds with the experimental ones (Dartyge et al., 1998). It is to be noted that the XMCD intensity calculated for each compound is much larger than the experimental results. Then, the previously considered intensities were reduced by a reduction factor 0.2 for all compounds, except for GdFe2 where it was 0.6. Possible origins of that reduction were partially discussed in Section 2.2.6 (Fukui et al., 2001; Asakura et al., 2002). From the calculated results of the series of heavy trivalent R elements (Fig. 12), a variety of spectral shapes was obtained depending on the R element and the considered absorption edge: roughly speaking, for heavy R compounds, the L3 edge spectra consist of an intense positive peak at higher energies, coming from the ED contribution, and a weak negative peak at lower energies, arising from the EQ contribution, while the L2 edge spectra are weak (except for Gd) and have a complicated shape. These characteristics are consistent with the experimental results, as shown in Fig. 12.
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Fig. 12. Calculated (solid curve) and experimental (dotted curve) results of XMCD spectra at: (a) L3 and (b) L2 edges in RFe2 . The calculated XMCD intensity is reduced by a factor 0.2 except in the case of GdFe2 where it is reduced by 0.6.
It is interesting to compare these results with those of R2 Fe14 B shown in Section 2.2. In the L3 edge of heavy R elements, the behaviour of XMCD is similar for RFe2 and R2 Fe14 B, if we invert the sign of XMCD, taking into account that the dominant magnetic moment is due to the R 4f and Fe 3d contributions, respectively, for RFe2 and R2 Fe14 B. This similarity of the XMCD behaviour is understood from the fact that the ED contribution (the contribution (i)) from the R 5d–4f exchange interaction is dominant (as seen from Fig. 2) and much larger than the ED contribution from the R 5d–Fe3d hybridization (the contribution (ii)). On the other hand, in the L2 edge of heavy R elements, both the ED contribution (i) and the EQ contribution (contribution (iii)) are very small (see Figs. 2 and 3). Therefore, for RFe2 all the contributions (i), (ii) and (iii) become comparable with each other, so that the spectral shape of XMCD are complicated, as a characteristic feature of RFe2 . However, for R2 Fe14 B the XMCD spectral shape is dominated by the contribution (ii), which is much larger than that of RFe2 because of the much larger weight of Fe in the atomic
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Fig. 13. Roles of 5d–5d exchange interaction on XMCD spectra at the L2 edge of RFe2 (R = Dy, Ho, Er and Tm). The thin solid, dotted and thick solid curves are the calculated results with the exchange splitting of the 5d band Eexc (5d) = 0.0, 0.1 and 0.3 (eV) for Dy (and scaled in proportion to the 4f magnetic moment for other elements). The experimental results are shown with the dashed curve.
composition. In the case of RFe2 with light R elements, the XMCD spectra at the L2 edge show a strong positive peak coming from (i), while those at the L3 edge exhibit a variety of shapes due to the small and comparable contributions (i) , (ii) and (iii). As shown in Fig. 12, the agreement between calculated and experimental spectra is fairly good if we except some discrepancies at the positive high-energy peak in L2 spectra for Dy, Ho, Er and Tm (as well as the MCD amplitude). By inspection, it is found that the high-energy peak in the corresponding DOS could yield a relatively stronger positive peak at high energies, if an additional exchange splitting of the 5d bands would be introduced. A plausible origin of this additional splitting is the 5d–5d exchange interaction neglected so far, and a trial to improve the XMCD spectra by this interaction will be discussed in the next subsection. 2.3.3. Role of R 5d–R 5d exchange interactions In the previous analyses, the effect of 5d–5d exchange interaction has been neglected. Although an overall trend of calculated XMCD spectra are fairly in agreement with those observed, we cannot reproduce a positive high energy peak, observed by Dartyge et al. (1998), at the L2 edge of Dy, Ho, Er and Tm compounds. In a recent paper, Fujiwara et al. (2006) have tried to improve XMCD spectra, taking into account 5d–5d exchange interactions within a mean-field approximation: they assumed the exchange field is proportional to the 4f magnetic moment for each R. The result is shown in Fig. 13. The improvement is evident although exchange field is rather weak. The reason why the weak 5d–5d exchange interaction is however appreciable in XMCD spectra is that the sharp peak in 5d DOS at high energies is a characteristic of Laves phase compounds. Furthermore, Fujiwara et al. stated that the effect is not so significant in other cases (L3 edge of these RFe2 and L2 and L3 edges of R2 Fe14 B). It is to be noted that in the XMCD spectra at the L2 edge of heavy RFe2 the three contributions (i), (ii) and (iii) are all weak and hence relatively minor effects (such as the weak 5d–5d exchange interaction) comes up as an appreciable effect. We again stress that this is a consequence of the combined effect of 5d–5d exchange interaction and the sharp peak in 5d DOS at higher energies. 2.3.4. LuFe2 Laves-phase compound In order to clarify the variety of XMCD spectra in Fig. 12 and hence their origin, we investigate in the following rather simple cases. Firstly, we consider LuFe2 , where the R 4f shell is completely filled. In that case, since there is no effect from 4f electrons the only source to polarize the R 5d states is provided by hybridization with the spinpolarized Fe 3d states (see the partial DOS shown in Fig. 11). We note that the spin polarization of R 5d electrons is energy-dependent and is most prominent near the Fermi energy (see also the calculated result shown in Fig. 16). This explains the fact observed in LuFe2 that the peak of the XMCD spectrum is located at lower energy than that of the corresponding XAS spectrum (Dartyge et al., 1998). Thus, XMCD is sensitive to the energy dependent polarization of the R 5d states, although a detailed structure is smeared out considerably by the lifetime broadening of the 2p core hole. As will be discussed in Section 3.2, the difference in energy of the XMCD and XAS peaks plays an important role in CeFe2 .
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Fig. 14. Calculated XMCD spectra from ED transition only (no EQ taken into account) at (a) L3 and (b) L2 edges in SmAl2 for various values of = 0.0, 0.2, 0.4, 0.6, 0.8 and 1.0 (1/eV), respectively.
2.3.5. RAl 2 Laves-phase compounds In this subsection, we report the XMCD for one example of the RAl2 series with R = Sm, a light element of the series, i.e. SmAl2 (Harada et al., 2004; Fujiwara et al., 2005). In this compound there is an effect of Sm 4f electrons on Sm 5d states through the intra-atomic exchange interactions. As was discussed above, the effect of intra-atomic exchange interactions between 5d and 4f electrons generally dominate XMCD spectra (even in SmFe2 ). In certain cases, the energy-dependent enhancement of the ED matrix element changes the sign of XMCD spectra. This is seen in Fig. 14, where the XMCD spectra for SmAl2 are shown for various values of . At the L2 edge the sign of XMCD intensity changes with increasing , while at the L3 edge, XMCD spectra are affected rather mildly. This fact remains true for all other light R elements, but the situation is changed for the heavy R elements, for which the effect of is more serious at the L3 than at the L2 edge. In Fig. 15, we show the calculated XMCD spectra at the L3 edge of SmFe2 and SmAl2 , and compare them with experimental results (Mizumaki and Nakamura, 2004). The value of is taken to be 0.6 1/eV, and the contributions (ii) and (iii) for SmFe2 (contribution (iii) for SmAl2 ) are also taken into account. The lowest-energy negative peak for both materials is due to the EQ contribution (iii), which is located at the lower energy region with respect to the Fermi energy EF , chosen to be at zero energy. The other structures are due to the ED contribution (i) from the R 5d–4f exchange interaction even for SmFe2 . For the L3 edge of Sm2 Fe14 B, we have shown that all the contributions (i), (ii) and (iii) are comparable in weight (see Section 2.2.3), but for SmFe2 the contribution (ii) from the R 5d–Fe3d hybridization is weaker than the contributions (i) and (iii) because of the smaller weight of Fe in the atomic composition. The difference in the XMCD spectral shapes for SmFe2 and SmAl2 is caused by the different 5d band structure, instead by the contribution (ii) in SmFe2 . We can say that the agreement of the calculated and experimental results is quite good although the amplitudes of the XMCD spectra are reduced by 0.3 and 0.03 in SmFe2 and in SmAl2 , respectively. Possible reasons for the discrepancy were discussed in Section 2.2.6. Thus, it is quite instructive to discuss the XMCD spectrum dividing the contributions, although these are not simply additive. 2.3.6. Comparison between SmFe2 , LuFe2 , SmAl 2 compounds Keeping the above mentioned results in mind, the calculated results of SmFe2 are compared with those of SmAl2 and of LuFe2 in Fig. 16 in more details. Discussion here is restricted to the ED contributions (i) and (ii). As was mentioned above, the XMCD spectrum of LuFe2 reflects the spin polarization due to hybridization with the spin-polarized Fe 3d states (ii), being effective near the Fermi energy while that of SmAl2 is the consequence of the spin and orbital polarization due to exchange field by R 4f electrons (i), spreading over the vacant 5d band. As is seen in Fig. 16, the XMCD spectrum of SmFe2 seems to be the sum of both contributions (i) and (ii), although the positive peak is more intense than the sum. This discrepancy is probably due to a non-linear correlation of both polarization effects in the electronic structure of 5d states.
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Fig. 15. Calculated XMCD spectra of SmFe2 and SmAl2 at L3 edge. Experimental data are indicated by dots from M. Mizumaki and T. Nakamura (unpublished data).
Fig. 16. Comparison of calculated results of XMCD spectra at L3 edge for SmFe2 , LuFe2 and SmAl2 .
2.3.7. Temperature dependence: case of ErFe2 So far, we have considered XMCD at zero-temperature. Now, it is relevant to investigate the temperature dependence of XMCD spectra in RFe2 compounds because the mechanism of XMCD spectra at L23 edges is quite subtle as we reported above (especially at the L2 edge of heavy R elements) and each contribution to XMCD has its own temperature dependence.
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Rm=1 0.8 0.6
MCD (%)
0.4
0
-0.5
-10
0 Relative Energy (eV)
10
Fig. 17. Calculated temperature dependence of the XMCD spectrum at the L2 edge of ErFe2 . The thick solid, thin solid, dashed and dotted–dashed curves denote, respectively, the spectra with the reduction factor Rm for the Slater integral Gk , 1.0, 0.8, 0.6 and 0.4, each of which corresponds to different temperature (decrease in Rm corresponds to increase in temperature).
Recently, Giorgetti et al. (2004) pointed out quite a new temperature dependent XMCD spectrum in the case of ErFe2 at L2 absorption edge. With increasing temperature, one of the peak intensity increases while the other peaks decrease monotonically. Let us recall that ErFe2 is a ferri-magnetic compound with a compensation temperature Tcom = 486 K and a Curie temperature Tc =587 K. Actually it is quite natural to expect a decreasing XMCD intensity versus increasing temperature since the intensity is proportional to the magnetization. However, in the case under investigation, there are differing contributions: (i) ED due to R 5d–4f exchange interaction, (ii) ED due to R 5d–Fe 3d hybridization and (iii) EQ, each contribution having its own sign and temperature dependence. Fujiwara et al. (2006) pointed out the following remarks for the XMCD spectrum at L2 edge of ErFe2 (see Fig. 12(b)). The contributions (i)–(iii) have the same order of magnitude, and at the lower energy region (i) and (ii) have opposite sign and a considerable cancellation occurs (the sign of XMCD for (i) is negative and positive in low and high energy regions, respectively, that for (ii) is mainly positive, and that for (iii) is negative). At low temperatures the Er 4f magnetization is larger than the Fe 3d magnetization, but with increasing temperature towards Tcom , the Er 4f magnetization decreases more rapidly than the Fe 3d magnetization, so that the amplitude of (i) decreases more rapidly than (ii). Thus, it is possible to have the increase of the positive contribution (ii) with increasing temperature due to the decrease of the cancellation with the negative contribution (i). In Fig. 17, we show a result of preliminary calculations by Fujiwara et al. (2006) for the temperature dependence of the XMCD spectrum at the L2 edge of ErFe2 . Some band parameters are modified, to some extent, to obtain a better agreement with the experimental XMCD spectra. In order to simulate the increase in temperature, the contributions of (i) and (iii) are reduced by introducing the reduction factor Rm , while no reduction is made for (ii). In the XMCD spectrum, the lowest energy peak with negative sign is due to (iii), the second-lowest-energy positive peak is mainly due to (ii) but overlaps with (i), the second-highest-energy negative peak is mainly due to (i) but overlaps with (ii), and the highest-energy positive peak is due to (i). This result is in considerable agreement with the experimental result (Giorgetti et al., 2004). Finally, we would like to mention that it is quite promising to investigate the temperature dependence of the XMCD spectra experimentally and theoretically for obtaining more detailed information on magnetic materials.
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2.3.8. Concluding remarks In Section 2.3, we reviewed the mechanism of XMCD spectra at R L23 edges in R inter-metallic compounds, RFe2 in the Laves phase, by using a rather elaborated model. The band nature of R 5d and Fe 3d states, including the hybridization between them, was treated by a tight-binding approximation whereas the intra-atomic exchange interaction between 4f and 5d states was described by a mean-field arising from atomic 4f electrons in the Hund’s rule ground state. Then, the XMCD spectra happened to consist of both ED and EQ contributions: the former depends on the detailed polarization of R 5d states while the latter is dominated by rather localized 4f states. Beside the weak EQ contribution, we now comment on the ED contribution. For that contribution, the R intra-atomic exchange interaction generally dominates the polarization of R 5d electrons and hence the XMCD spectra depend on R 4f states. Once more it is important to realize the novel role of intra-atomic exchange interaction, i.e. the enhancement of 2p → 5d dipole matrix element, which is indispensable for a proper interpretation of XMCD at R L23 edges. Especially for XMCD spectra of light R elements at L2 edge and of heavy R elements at L3 edge, this enhancement effect plays a crucial role. Here we would like to focus on another important ED contribution arising from the hybridization of R 5d states with spin-polarized Fe 3d states, which is straightforward taken into account in tight-binding approximation. Especially for XMCD at L2 edge of RFe2 with heavy R elements, this ED contribution becomes comparable with other ED and EQ contributions, all of which are very weak, and the competition of the three contributions give rise to complicated XMCD spectral shape and its complicated temperature dependence. This situation is very interesting and characteristic of RFe2 systems, in strong contrast to the case of R2 Fe14 B where the XMCD at L2 edge of heavy R elements is determined only by R 5d–Fe 3d hybridization contribution. We can also see relatively detailed effects in the spectrum, leading to the information on relatively minor interactions in materials such as the effect of 5d–5d exchange interaction combined with the sharp peak in 5d DOS at higher energies. Before concluding this section, we would like to stress again that it is quite promising to investigate the temperature dependence of XMCD spectra experimentally and theoretically in order not only to study more detailed information on magnetic materials but also to confirm the subtle mechanism of XMCD. We are now able to apply the present method to the analysis of XMCD spectra for various other R compounds and to discuss in details the role played by 5d electrons as far as their magnetic properties are concerned. Especially for R systems with complicated atomic arrangements, the present method will provide new powerful means to calculate XMCD spectra quantitatively. There is an interesting problem left for the future, i.e. to modify the present theory so as to apply it to insulating R compounds. 3. Ce systems 3.1. CeRu2 Ge2 : A well localized 4f 1 system 3.1.1. General context and proposed model These last decades, a variety of XMCD experiments have been performed, especially at L23 and M45 absorption edges of cerium, in various ferromagnetic inter-metallic compounds (Baudelet et al., 1990; Schütz, 1991; Carra et al., 1991; Giorgetti et al., 1993; Finazzi et al., 1996). These are motivated by interesting phenomena of 4f valence electrons of Ce; valence fluctuation (VF) or mixed valence (MV) phenomena. In the present subsection we first focus on XMCD in L23 XAS of CeRu2 Ge2 , as a typical example of trivalent Ce compounds (Rietschel et al., 1988; Severing et al., 1989; Jo and Imada, 1993). Other examples would be CePd, CeAl3 , CeRh3 B2 and so on. In such cases, the theory developed in the previous sections is applicable, assuming one 5d electron to exist in the ground state: the XMCD spectra are dominated by the 2p–5d ED transition combined with the effect of 4f–5d exchange interaction, where the 4f electrons are within Hund’s rule ground state. We notice that an enhancement of 2p–5d dipole matrix element (see Section 2.1.2) due to the effect of contraction in the radial part of 5d orbitals should be included in the corresponding calculation. Although the theoretical results are, in general, quite consistent with the experimental ones for other R elements (see Section 2), there still remain exceptional discrepancies in XMCD spectra of Ce compounds to be explained separately. In this subsection, we study one of such examples, the Ce XMCD spectra of CeRu2 Ge2 . Within the simplest Ce3+ ionic model, a strong positive and a weak positive spectra are suggested by the theory, respectively, at L2 and L3 edges, while a strong positive and a weak negative spectra are observed, respectively, at L2 and L3 edges. As was mentioned before, at L2 edge the effect of enhancement of ED transition matrix element is crucial while at L3 edge, where the integrated XMCD intensity is weak, the cancellation occurs more or less among contributions from different 5d states,
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which are affected by 4f electron. Hence the result at L3 edge may be sensitive to the 4f electron state. In this sense, the assumption of the Hund’s rule ground state for 4f electrons adopted in former works seems to be too crude and more detailed information on the 4f electron states is needed in this case. It is known that, in the case of Ce compounds, isostructural with CeRu2 Ge2 , the three doublets in the J = 5/2 manifold due to the relatively large spin–orbit coupling are furthermore separated by a tetragonal crystal field (CF) (Thole et al., 1992). Within the J = 5/2 manifold, the tetragonal CF may yield the following ground state: |4f = a| − 5/2 − b|3/2,
(3.1)
where a ratio of two parameters a and b is determined by the strength of the tetragonal CF. Based on this 4f ground state, we try to reproduce the XMCD spectra by taking into account the 4f interactions and find out that the tetragonal CF plays an important role in solving the discrepancy in XMCD spectra of Ce3+ in CeRu2 Ge2 . It is interesting to note that even in this simplest case of Ce3+ compounds corresponding to 4f1 configuration, our analysis needs detailed information on 4f states, which affect indirectly the ED transition (2p → 5d) through 4f–5d exchange interaction. 3.1.2. Numerical results and comments We performed a series of calculation of XMCD spectra for a reasonable value of taken to be √ 0.6 1/eV (Matsuyama et al., 1997), and for a variable ratio b/a (of the above introduced CF parameters) taken from 5 (a cubic value) to 1.0. In Fig. 18 we show the results for b/a = 1.0 (a), 1.1 (b) and 1.2 (c), and it comes out that b/a = 1.1 (b) is the most appropriate choice to fit experimental spectra of CeRu2 Ge2 . Actually, in Fig. 18(b), the signs as well as the relative intensities of XMCD at L2 and at L3 appear to be consistent with experiment. The value of b/a determined above seems to be reasonable since it is also reproduced when using a set of tetragonal CF parameters (see Fukui et al., 1999 for details) which yield the characteristic splittings, i.e. 43 meV and 65 meV for the first and second excited states, respectively, according to the specific heat measurements (Rietschel et al., 1988) of CeRu2 Ge2 . We note that this set of tetragonal CF parameters is not unique since the XMCD data alone are not sufficient to determine it. 3.1.3. Concluding remarks So far we have seen that in the case of a trivalent Ce XMCD spectra are affected by details of 4f electron states, which, in ED transition (2p → 5d), give rise not only to the polarization effect on the 5d ground state, but also to an effect of contraction in the radial part of 5d orbitals and hence an enhancement of the transition dipole matrix element through the 4f–5d exchange interaction. Also we have found that, in the case of Ce L3 edge, more detailed information than Hund’s coupling on the 4f electron state is required. Especially to obtain a negative XMCD spectrum at L3 edge it is essential to take into account the tetragonal CF, which mixes the states, Jz = | − 5/2 and |3/2 in the J = 5/2 manifold split by the relatively large spin–orbit coupling, although the 4f electron is not directly concerned with the 2p–5d transition. The sum rule (Thole et al., 1992) for XMCD is known to be useful to obtain information about orbital states. These orbital states are actually important in understanding magnetic anisotropy of rare-earth compounds. Unfortunately, sum rule is not straightforward applicable in our case, since the radial part of the transition matrix element depends on the z-component of the azimuthal quantum number as well as on the spin quantum number of the 5d electron. Discussions so far are restricted to the case of Ce3+ . It is well known that there are many Ce compounds showing VF or MV phenomena, and their XMCD spectra have also been observed (Baudelet et al., 1990, 1993). In order to study this problem, we are now reporting on a calculation based on a many-electron theory, taking into account the inter-atomic configuration interaction. The result is reported in the next subsection. 3.2. CeFe2 : a less localized 4f 1 system 3.2.1. Brief presentation of the theoretical model and experimental results Let us consider such a MV inter-metallic compound, CeFe2 , as an example. Other examples would be CeCo5 , Ce/Fe multi-layers and so on. CeFe2 behaves as a ferromagnetic material but has anomalously small magnetic moment and a low Curie temperature (Tc = 230 K) as compared to other RFe2 (see Section 2.1). However it is higher as compared to CeRu2 Ge2 where Tc = 9 K. We ascribe the considered anomalous behaviour to the MV electronic state of Ce which can be satisfactorily described by the well known extended SIAM (single impurity Anderson (1961) model)(Kotani et al., 1988; Gehring, 2002), including various Coulomb interactions, i.e. Uff , −Ufc , Ufd , −Udc
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Fig. 18. Calculated XMCD spectra of Ce3+ L2 and L3 edges for the 4f electron state characterized by the ratio of the mixing coefficients b/a between |Jz = −5/2 and |Jz = 3/2, (a) b/a = 1.0, (b) 1.1 and (c) 1.2. The inset in (b) shows the experimental result for CeRu2 Ge2 .
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f
g
h i
Ce L3 XAS
j
e
k l m
d
Intensity (arb. units)
c
b
a
MCD* 150
-10
-5
0 5 10 Relative Energy (eV)
15
20
Fig. 19. Calculated results (solid curves) and experimental ones (crosses) of XAS and XMCD at Ce L3 edge of CeFe2 . Dashed curve represents the background contribution assumed in the calculation.
(‘c’ means ‘core hole’). Here we combine it with a LCAO (Ce17 Fe12 ) cluster calculation for Ce 5d and Fe 3d conduction states. Thanks to that cluster approach, the band nature of Fe 3d and Ce 5d states is well treated, as well as hybridization between both types of states. Also let us recall that in the considered model, we handle the three following electronic configurations: 4f0 , 4f1 and 4f2 . Let us also precise that, in CeFe2 , the hybridization of Ce 5d band with its Fe 3d partner is the origin of spin polarization of 5d states and then it happens to be crucial for XMCD at L23 edges. Furthermore, similarly to our calculation of Ce L3 edge and 3d RXPS in CeRh3 (Parlebas and Kotani, 2004), the effects of Ufd and −Udc are also explicitly taken into account in the corresponding Hamiltonian (Asakura et al., 2004b, 2005), as already stated above. Experimental results of XAS (×) and XMCD (+) for Ce L3 and L2 edges of CeFe2 are shown in Figs. 19 and 20, respectively (Giorgetti et al., 1993; Schille et al., 1994). The double-peak structure of XAS is a characteristic feature of MV Ce compounds and usually explained by 4f0 (higher energy peak) and 4f1 (lower energy peak) components in the final state. Corresponding to them XMCD spectra also exhibit a double-peak structure. The sign of XMCD is positive for L3 and negative for L2 , similar to that in LuFe2 , so that we expect that XMCD in CeFe2 is caused mainly by the spin polarization of Ce 5d bands due to hybridization with spin-polarized Fe 3d bands, as in the case of LuFe2 . This is reasonable because the spin and orbital polarization of Ce 4f states would be very small due to the MV character. However, there remain some features to be solved theoretically: (1) The energy positions of the two peaks in XAS are
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Fig. 20. Similar to Fig. 19, but at Ce L2 edge of CeFe2 .
different from those of XMCD. The difference is larger for the lower energy peak (4f1 peak) than the higher energy one (4f0 ). (2) The widths of the two XMCD peaks are different; the lower energy peak is broader than the high energy one. (3) It is not clear where we can see the contribution from the 4f2 configuration. Theoretical calculations have been made to solve these problems (Asakura et al., 2004b, 2005). 3.2.2. Numerical results and comments We consider a Ce17 Fe12 cluster model to describe Ce 5d and Fe 3d states, because the cluster with finite size is more convenient to be combined with SIAM. The Hamiltonian of the cluster is given by Eq. (2.7), and diagonalized in the form + Hc = v av av , (3.2)
by the transformation + av = , i|a+i + , j |a+j . ,i
(3.3)
,j
The mixed Ce 5d–Fe 3d states are combined with Ce 4f states in the frame of an extended SIAM. The Hamiltonian of the extended SIAM in the initial state of XAS is written as + + Hg = 4f af+ af + Uff af+ af af+ af + v av av + (V av af + h.c.), (3.4)
>
,
where represents the combined spin and orbital quantum numbers of the Ce 4f state, Uff is the Coulomb interaction of Ce 4f states and V is the hybridization between 4f ( ) and v() states. In the final state of XAS, the 2p core electron
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Fig. 21. Calculated results of XAS and XMCD at L3 edge for various values of Udc in the limit of 4f0 configuration. The peak position of XMCD is taken as the origin of the relative energy.
is excited to the 5d band and the 4f level is pulled down due to the core hole potential −Ufc . Furthermore, 5d electrons on the core hole site (j = 0) interact with the 4f electron through the Coulomb interaction Ufd and the core hole through −Udc . Therefore, the Hamiltonian in the final state is given by + + Hf = Hg − Ufc af+ af − Udc Ck,k avk avk + Ufd Ck,k af+ af avk avk , (3.5)
where Ck,k =
k|v, 0, 0|k .
k,k
k,k ,
(3.6)
The Hamiltonians Hg and Hf are diagonalized by taking into account the three configurations 4f0 , 4f1 and 4f2 . In diagonalizing Hf it is assumed that only a photoexcited electron is affected by the Coulomb interactions −Udc and Ufd . By the calculation CeFe2 is shown to be in the MV state with an average 4f electron number in the ground state nf = 0.64. The calculated results of XAS and XMCD are shown in Figs. 19 and 20 by solid curves, where the dashed curve is the background contribution to XAS. The values of Udc and Ufd are taken to be 2.0 and 1.7 eV, respectively. The agreement between calculated and experimental results is satisfactory both for XAS and XMCD at both L2 and L3 edges. In order to see the mechanism determining the relative peak positions of XAS and XMCD, the effect of the core-hole potential −Udc on XAS and XMCD is calculated for 4f0 peak, disregarding 4f1 and 4f2 contributions. The results are shown in Fig. 21. In the case of Udc = 0, the peak of XMCD is shifted by about 4 eV towards lower photon energy with
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respect to that of XAS, and its width is smaller. The reason for this is as follows: since the XAS peak occurs by optical transition of the 2p3/2 core electron to the Ce 5d conduction bands above the Fermi level, the position and width of the XAS peak correspond to the center and the width of the unoccupied part of Ce 5d bands. On the other hand, XMCD is caused only by the spin polarized part of the unoccupied Ce 5d bands. The spin polarization of Ce 5d bands is induced by Fe 3d spin polarization through hybridization between Ce 5d and Fe 3d states, but Fe 3d bands are mainly located at the lower energy side of Ce 5d bands, so that the spin polarization of the empty states of Ce 5d bands is limited to the states near the FL. With increasing Udc , on the other hand, the oscillator strength of XAS is transferred to the lower energy side (toward the FL, which is the peak position of XMCD), and therefore the energy difference of XAS and XMCD peaks decreases. This explains that the positions of the higher energy peak (4f0 peak) of XAS and XMCD are close to each other in Fig. 19. For the lower energy peak (mainly 4f1 ), on the other hand, we have to take into account the effect of Coulomb interaction between 4f and 5d states Ufd , too. Then, the effects of attractive −Udc and repulsive Ufd almost cancelled each other, so that the situation becomes similar to the case of Udc = 0 in Fig. 21. Therefore, the effect of −Udc and Ufd is essential in understanding the relative positions of the two peaks in XAS and XMCD in CeFe2 . More generally we notice that our parameter values are comparable to those used for 3d XPS spectra (Konishi et al., 2000). The difference in the spectral widths of the two XMCD peaks, as well as the difference in the spectral widths of XAS and XMCD peaks, is also caused mainly by the effect of −Udc and Ufd , as seen from Figs. 19 and 21. For the contribution from the 4f2 configuration, a detailed discussion will be given in the next subsection. 3.2.3. MCD in resonant X-ray emission spectra Here we consider resonant X-ray emission spectroscopy (RXES) and its MCD (MCD-RXES) for CeFe2 , where the Ce 3d5/2 to 2p3/2 radiative transition occurs following the Ce 2p3/2 to 5d radiative excitation. Through the calculations of RXES and MCD-RXES, we would like to discuss where is the 4f2 contribution in the Ce L3 XAS of CeFe2 (Kotani et al., 2005). The RXES spectrum is given by ( 1 , 2 ) FRXES (, ) =
2 j |T 2 |ii|T 1 |g
M / 2 1 , Eg + − Ei + i L (Eg + − Ej − )2 + 2M j
(3.8)
i
where and are energies of the incident and emitted photons with helicities 1 and 2 , respectively, |g, |i and |j are the ground, intermediate and final states with energies Eg , Ei and Ej , respectively, and T1 1 and T1 2 are the operators of optical transitions. MCD-RXES is a newly developed field of spectroscopy, which gives important information on the magnetic polarization of electronic states in ferromagnetic materials. MCD-RXES is defined by the difference of RXES spectra for the circular polarized incident X-ray with + and − helicities, where the polarization of the emitted X-ray is not detected. Therefore, the MCD-RXES spectrum FRXES (, ) is described by FRXES (, ) ≡
(+, ) (−, ) [FRXES2 (, ) − FRXES2 (, )].
(3.9)
2
In the calculations presented in this section, we assume that the incident X-ray beam is antiparallel to the magnetization of CeFe2 and the angle between the emitted and incident X-ray directions is fixed at 54.7◦ (magic angle). As seen from the expression of RXES spectrum given above, the spectral broadening of RXES (as well as MCDRXES) is given by M , the lifetime broadening of the Ce 3d core hole in the final state of RXES. Actually M is much smaller than L , the lifetime broadening of the Ce 2p core hole in the final state of XAS (intermediate state of RXES). The value of M is estimated to be about 0.7 eV, while L is about 3.0 eV from the fitting of the experimental Ce L3 XAS spectrum in Fig. 19. Therefore, we expect that RXES (and MCD-RXES) spectra exhibit more precise structures than XAS (and XMCD). The calculations of RXES and MCD-RXES are performed with the extended SIAM for the incident X-ray energy tuned to the positions a to m in Fig. 19, and the results are shown in Fig. 22 with the solid (RXES) and dashed (MCD-RXES) curves, where the background contribution to RXES is disregarded. The calculated MCD-RXES spectra for the excitation energies a–e exhibit two peaks, which correspond to the 4f2 (higher energy peak) and 4f1 (lower energy one) configurations. For e to m, another peak occurs on the lower photon
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Fig. 22. Calculated results of RXES (solid curves) and MCD-RXES (dashed curves) for CeFe2 . The incident X-ray energies are taken at the positions a-m in Fig. 19.
energy side, corresponding to the 4f0 configuration. Therefore, we obtain the signal of the 4f2 configuration in the MCD-RXES spectrum. A more direct way to detect 4f2 contribution in L3 edge of XAS is to calculate the excitation spectrum of RXES (especially MCD-RXES) for a fixed emitted X-ray energy at the energy difference of the 3d5/2 and 2p3/2 core levels, dp . Namely, the amplitude of RXES (MCD-RXES) is calculated by fixing the emitted X-ray energy at dp , which is taken as the origin of the abscissa in Fig. 22, as a function of the incident X-ray energy. The results of the excitation spectra are shown in Fig. 23. By comparing the calculated results in Figs. 19 and 23, we note that the results in Fig. 23 correspond to a less broadened version of those in Fig. 19. Actually, by calculating the XAS and XMCD spectra with smaller spectral broadening M = 0.7 eV, we can check that the result coincides almost completely with those in Fig. 23. Then, owing to the reduced spectral broadening, we can find the 4f2 contribution which is hidden in the conventional L3 XAS. Namely, we find that the lower energy peak of XMCD split into two peaks in Fig. 23, which correspond to 4f1 (higher photon energy one) and 4f2 (lower photon energy one) contributions, although the splitting is not very clear in XAS. We would like to emphasize that the excitation spectrum of MCD-RXES is a powerful tool to detect the contribution of Ce 4f2 configuration in Ce L3 edge of the X-ray spectroscopy.
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Fig. 23. Calculated excitation spectra of RXES (solid curve) and MCD-RXES (dashed curve) for CeFe2 .
In summary we have shown theoretically that the technique of the excitation MCD-RXES spectrum is very useful to observe fine structures of XMCD beyond the lifetime broadening of the L3 core hole. This technique is an extension of the technique of the excitation spectrum of RXES by Hämäläinen et al. (1991) to MCD-RXES. Hämäläinen et al. succeeded in measuring the signal of weak electric quadrupole excitation at L3 edge of Dy compounds by observing the excitation spectrum of RXES. After the experiments, Tanaka et al. (1994) explained that the excitation spectrum of RXES corresponds to a less broadened version of XAS. It is highly desirable that the present theoretical prediction of observing the Ce 4f2 signal by the excitation spectrum of MCD-RXES will be confirmed by experimental observations, and that the present technique will be used more generally in order to get high resolution MCD spectra. Let us just add that quite recently, the detection of 4f2 contribution in L3 XAS for Ce–Th and Ce–Sc alloys has been reported by Rueff et al. (2004) taking advantage of resonant inelastic X-ray scattering spectroscopy (RIXS). 3.3. CePd 3 : a dense Kondo material We have so far discussed the XMCD behaviour of two typical classes of Ce compounds; a typical trivalence Ce compound CeRu2 Ge2 and a typical mixed-valence Ce compound CeFe2 . In this section we describe experimental results of XMCD behaviour of another class of Ce compounds, CePd3 and its alloy with TM elements, with no theoretical interpretation up to now. 3.3.1. Ce(Pd 1−x Mnx )3 : experimental results for Mn substitution Before treating the case of Ni impurities in CePd3 in more details, we would like to mention another example of TM impurities in the same compound, i.e. Mn impurities. Unfortunately from the available literature (Singh and Dhar, 2003), only a concentration of about 3% of Mn has been considered. At that concentration the Mn ions form ˚ and give rise to an ordered compound the chemical formula of a sublattice with a large Mn–Mn distance of 8.5 A which is (CePd3 )8 Mn. From the study of a field dependent magnetization M(H), Singh and Dhar could not find any saturation, even at low temperature (2.8 K) and high magnetic field (120 KOe). Instead they got an almost linear variation of M(H) with respect to H, which they attributed to the contribution of Ce atoms in a paramagnetic state. Also from Arrots plots, due to the presence of Mn impurities, they could estimate a Curie temperature Tc between 30 and 40 K. Finally from the study of the temperature dependent specific heat of (CePd3 )8 Mn, Singh and Dhar (2003) exhibited a peak at 32 K (i) in accord with the Tc estimated from the previously considered Arrots plots and (ii) in relation with a ferro-magnetic order of the Mn sublattice. Furthermore from their estimation of a specific heat coefficient as high as 275 mJ/mol K2 , Singh and Dhar concluded that (CePd3 )8 Mn behaves as a dense Kondo lattice. Of course this
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study should be completed by selective XMCD measurements as well as a complementary investigation of a series of Ce(Pd1−x Mnx )3 alloys with various x contents. Such a detailed experimental work has been done for Ni impurities in CePd3 (Kappler et al., 2004) and is reported in the next subsection. 3.3.2. Ce(Pd 1−x Nix )3 : experimental results for Ni substitution First of all Kappler et al. (2004) made a full study of lattice parameters of various Ce(Pd1−x TMx )3 alloys with TM = Ni, Ag: (i) as a function of Ni concentration x, with x taken up to 25 at % Ni, the solubility limit of Ni in CePd3 ; (ii) as a function of Ag concentration x, with x taken up to 13 at% Ag. At low concentration, up to x = 0.05, the Ni substitution drives the Ce atoms towards the 4f1 state limit. The lattice parameter reaches a broad maximum around x = 0.05 and then decreases with increasing concentration, showing a complex behaviour of the Ce–4f configuration. Of course in the case of Ag substitution the concentration dependent lattice parameter is almost linear and there is no maximum at all. Also magnetization was studied by Kappler et al. for x < 0.20 in the 2–20 K temperature range and for x 0.20 down to T = 50 mK in fields up to 4 T. Especially they showed magnetization curves measured for Ce(Pd0.75 Ni0.25 )3 . As the temperature is lower than 2 K the saturation of the magnetization is observed with a corresponding opening of an hysteresis loop with a coercive field of about 0.1 T. These results evidence the existence of a ferromagnetic state below the Curie temperature Tc ≈ 2 K. The saturated magnetic moment extracted from the magnetization measurements at 50 mK is about MS = 0.15 B /fu. This low MS value can be explained by the Kondo effect acting on the Ce ground state, reducing the moment of the 7 state in a cubic crystalline field, expected to be equal to 0.7 B . A similar magnetic behaviour is also observed for a solid solution with x = 0.20. In addition Kappler et al. studied the overall behaviour of the temperature dependent specific heat CM (T ) in the 0.5–20 K temperature range for the series of Ce(Pd1−x Nix )3 samples and in comparison to a Ce(Pd0.87 Ag0.13 )3 alloy. Within their temperature dependent specific heat measurements a very interesting feature is the almost fixed and pronounced peak which appeared at T = 1.45 K for the various CM (T ) curves of all samples with x 0.1. The amplitude of this peak increased with Ni concentration but was absent in pure CePd3 . Thus it should be directly related to the Ni effect on a CePd3 compound and more precisely to a magnetic transition, as illustrated by the magnetization measurements at low temperatures. Compared to CePd3 one may conclude that by increasing the Ce–Ce spacing a ˚ for CePd3 to a maximum of 4.140 A ˚ for Ce(Pd0.95 Ni0.05 )3 does not prevent the disappearance little bit from 4.126 A of the nonmagnetic FL behaviour and the creation, instead, of a magnetically ordered state. Probably the d conduction band as well as the 4 f–d hybridization change sufficiently from a Ce–Pd to a Ce–Ni environment to give rise to these different behaviours. Kappler et al. also presented CM (T ) results for a Ce(Pd0.87 Ag0.13 )3 alloy. The significant increase of the electronic specific heat coefficient term, defined as the limit (C/T )T →0 , (about 150 mJ/K 2 mol for this alloy, compared to 38 mJ/K2 mol for CePd3 ) suggests that a heavy fermion state characterized Ce(Pd0.87 Ag0.13 )3 in contrast to the specific effect of Ni impurities in CePd3 . The magnetic transition, with almost constant lattice parameters suggests that this transition comes more from an electronic effect rather than from a chemical pressure effect. The Ce L23 XAS spectra in CePd3 (see Section 3.3.3) and in Ce(Pd0.75 Ni0.25 )3 are reproduced in Fig. 24. As expected they exhibit a double peak structure which is characteristic of highly correlated Ce systems: the ground state is a mixture of |4f 1 and |4f 0 states. For the Ni doped alloy it is remarkable to note that, despite the magnetic state of this sample, the double peak structure is still observed. However, as illustrated in Fig. 24, a decrease of 4f0 /4f1 intensity ratio leads to a nf increasing of about 5%, when going from CePd3 to an alloy with x = 0.25. In both L23 edges of Ce(Pd0.75 Ni0.25 )3 alloy the magnetic dichroism consists of a main structure and possibly some tiny structures at high energy (see Fig. 24). Each main peak points 2 eV before the edge maximum, with a line width (≈ 5 eV) much narrower than the edge line (≈ 10 eV). The sign of the dichroic response is positive (negative) at the L2 (L3 ), which corresponds to a 5d moment parallel to the applied magnetic field. The normalized amplitudes of the dichroic signals are about 2 × 10−3 and 8 × 10−3 at L3 and L2 edges, respectively. The integrated XMCD signals at L23 edges give a 5d-XMCD branching ratio close to −4. This reproduces the expected behaviour for a well localized 4f1 system, as in CeRu2 Ge2 compounds, and consequently demonstrates that the main source of magnetism, in the Ce(Pd1−x Nix )3 system, is due to Ce atoms. 3.3.3. Pure CePd 3 : experimental situation The observation (Fig. 24) at Ce L2 edge of a dichroic signal in a pure CePd3 compound shows the extreme sensitivity of XMCD technique. The signal corresponds to a magnetic field induced effect only due to the 5d paramagnetic
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Fig. 24. XAS and XMCD spectra at Ce L23 edges in CePd3 (open circles) and Ce(Pd0.75 Ni0.25 )3 (line) samples, at T = 4.5 K and H = 7 T. The CePd3 XMCD spectrum is shifted to facilitate comparison. Inset: magnetic field dependence of the maximum XMCD signal.
susceptibility which is a part of the total susceptibility. The small amplitude (only 1.8 × 10−3 ) at L2 edge explains the lack of signal at L3 edge. Indeed if we suppose a branching ratio of about −4, as observed for Ce(Pd0.75 Ni0.25 )3 , the amplitude of L3 signal should be approximately −0.5 × 10−3 . To detect such a tiny signal would require by far much more beam time than used for the present experiment. The magnetic field dependence of the maximum dichroic signal, presented in the inset of Fig. 24, demonstrates, if needed, the validity of these experiments, as they well reproduce the magnetic character of the different samples, specially the linear variation for CePd3 . Let us highlight that the observation of only one magnetic peak corresponding to the main 4f1 final state points out that only this channel leads to a magnetic response. It has been already discussed that the sum rules (Carra et al., 1993a) do not apply in the case of 5d bands of R systems, due to 4f exchange. However, it is possible to make a rough comparison of the dichroic signals with those of CeRu2 Ge2 where the M5d (4K) moment is known to be ≈ 0.1B . As the ratio between the two dichroic signals is about 10, M5d is approximately 0.01B in Ce(Pd0.75 Ni0.25 )3 and 4 × 10−3 B in CePd3 . Summarizing, the XAS spectrum of Ce(Pd1−x Nix )3 and CePd3 systems has a double-peak structure corresponding to 4f1 and 4f0 configurations but their XMCD spectrum exhibits almost a single-peak structure corresponding to 4f1 configuration. This is in strong contrast to trivalence compounds like CeRu2 Ge2 and the mixed valence compounds like CeFe2 , where both XAS and XMCD spectra display a single peak (4f1 peak) for the trivalent compounds and double peaks (4f1 and 4f0 ) for the mixed valence compounds. Theoretical interpretations for the characteristic behaviours of XMCD in Ce(Pd1−x Nix )3 and CePd3 are left to future investigations. 4. Summary In the present paper, theoretical interpretations of XMCD at rare-earth L23 absorption edges were reviewed using differing models, depending on the material under investigation. In the first chapter, as an introduction, some recent developments were briefly recalled for XMCD at R atom L23 XAS edges with a few general remarks. Then (Section 2), after having presented an overview of the basic mechanism of XMCD at L23 absorption edges and after having clarified the definition of XMCD sign, we essentially discussed two examples of XMCD spectra in: (i) R2 Fe14 B metallic compounds, with the help of a cluster model; and (ii) RFe2 Laves-phase compounds, using a tight-binding approximation for R 5d and Fe 3d conducting states. A good agreement between theory and experiment for R2 Fe14 B compounds suggests that a cluster model provides a valuable method to quantitatively calculate XMCD spectra of R systems, even with quite complicated atomic
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arrangements as it is the case for R2 Fe14 B. Actually, for the considered series of R2 Fe14 B metallic systems we underlined the important roles of (i) 5d–4f exchange interaction which includes the novel effect of enhancement of ED intensity related to 2p → 5d transitions and (ii) hybridization of R 5d states with spin polarized Fe 3d states of surrounding ions. From a quantitative comparison with XMCD data of these series of compounds, it was pointed out that the effect of spin-polarized Fe 3d states had to be included. However and furthermore the EQ transition (2p → 4f) had to be accounted for as well. Also, there is no reason why the contribution to XMCD from 5d–4f exchange interaction would be independent of that from 5d–3d hybridization, since both contributions are correlated to each other through 5d electronic structure. However, the two contributions have been shown to be almost independent, as it is the case for Sm2 Fe14 B. In the series of RFe2 compounds, it appeared some new interesting features in addition to the previously considered aspects of R2 Fe14 B systems. The most interesting feature in RFe2 is that the ED contribution due to the R 5d–4f exchange interaction, that due to the R 5d–Fe 3d hybridization and the EQ contribution to the XMCD at the L2 edge of heavy R elements are all weak and comparable in intensity, so that by their competition, the XMCD spectra exhibit complicated shapes and complicated temperature-dependence. In this situation, we can see the effect of relatively minor interactions in materials such as the 5d–5d exchange interaction. It is also interesting to compare the XMCD features for RFe2 and RAl2 . In Section 3, we focused on the special case of Ce systems, related to XAS and XMCD studies at the Ce L23 edges. Two clearly differing cases were interpreted: (i) A well localized 4f1 system, i.e. CeRu2 Ge2 ; (ii) A less localized 4f1 system, i.e. CeFe2 , with a 3d partner. In the case of a trivalence Ce, as it is the case for CeRu2 Ge2 , the XMCD spectra was affected by details of 4f electron states, which, in the ED transition (2p → 5d), gave rise not only to the polarization effect on the 5d ground state, but also to an effect of contraction in the radial part of 5d orbitals and hence an enhancement of the transition dipole matrix element through 4f–5d exchange interaction. Already in previous works the latter effect has been introduced by a phenomenological parameter . Also we have recalled that, in the case of Ce, more detailed information than Hund’s rule coupling on 4f electron states is required. Especially to obtain a negative XMCD spectrum at L3 edge it was essential to take into account the tetragonal CF. As a matter of fact, the 4f orbital state is sensitively reflected in XMCD spectra, although the 4f electron is not directly concerned with the 2p → 5d transition. In the case of a less localized 4f1 systems, like CeFe2 , the effect of hybridization between the Ce 5d and spin polarized Fe 3d states plays a main role in the XMCD spectrum. Then the effects of attractive −Udc and repulsive Ufd interactions have been shown to be essential in understanding the relative positions of the two peaks in the corresponding XAS and XMCD spectra. Also the difference in the spectral widths of the two XMCD peaks, as well as the difference in the spectral width of XAS and XMCD peaks were also caused mainly by the effect of −Udc and Ufd . For the contribution from the 4f2 configuration, a detailed discussion could be provided through MCD in RXES. Theoretically we have shown that the technique of the excitation spectrum of MCD-RXES is very useful to observe fine structures of XMCD beyond the lifetime broadening of the L3 core hole. This technique is actually an extension of the technique of RXES excitation spectrum by Hämäläinen et al. to MCD-RXES. Then, from a more experimental point of view, we reported the influence of impurity substitution on the low temperature properties of CePd3 compounds, i.e. Ce(Pd1−x Mnx )3 alloys where x is about 0.03, giving rise to a (CePd3 )8 Mn ordered structure. In this system Mn sublattice undergoes a ferromagnetic transition whereas Ce ions form a dense Kondo lattice and are in a paramagnetic state. We gave another differing example: Ce(Pd1−x Nix )3 alloys with x taken up to about 0.25. In this alloy series thermal and magnetization measurements exhibited a transition from a non-magnetic state (pure CePd3 ) to a ferromagnetic state for x > 0.05, with a Curie temperature Tc ≈ 2 K. The Ce L23 absorption edges and magnetic circular dichroism studies revealed the coexistence of strong 4f hybridization and ferromagnetic order essentially due to Ce atoms. Also the Ce L23 XMCD signal measured in pure CePd3 pointed out that in Ce based dense Kondo materials, only the 4f1 channel yields a magnetic response. Acknowledgments Some parts of the present work were done with a partial support from the French CNRS-PICS No. 1632 with Japan. One of the authors (J.C.P.) would like to acknowledge a kind hospitality at Okayama University as well as at Spring8 during the Summer 2005. A large number of collaborating people have been helpful in discussing various problems mentioned in the present review: H. Ogasawara, K. Okada, H. Matsuyama, K. Fukui, J. Nakahara, H. Maruyama, N.
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Physics Reports 431 (2006) 39 – 86 www.elsevier.com/locate/physrep
Non-compact string backgrounds and non-rational CFT Volker Schomerusa, b,∗,1 a DESY Theory Group, Notkestrasse 85, D-22603 Hamburg, Germany b Service de Physique Théorique, CEA Saclay, F-91191 Gif-sur-Yvette, France
Accepted 8 May 2006 Available online 7 July 2006 editor: A. Schwimmer
Abstract This is an introduction to the microscopic techniques of non-rational bulk and boundary conformal field theory which are needed to describe strings moving in non-compact curved backgrounds. The latter arise, e.g. in the context of AdS/CFT-like dualities and for studies of time-dependent processes. After a general outline of the central concepts, we focus on two specific but rather prototypical models: Liouville field theory and the 2D cigar. Rather than following the historical path, the presentation attempts to be systematic and self-contained. © 2006 Elsevier B.V. All rights reserved. PACS: 11.25.−w; 11.25.Hf; 11.10.Kk
Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. 2D boundary conformal field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Chiral algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1. Representation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2. Example: the U(1) theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. The bootstrap program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1. Bulk fields and bulk OPE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2. Crossing symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3. Example: the free boson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. The boundary bootstrap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1. Branes—the microscopic setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2. Ward identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3. One-point functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4. The cluster property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.5. Example: the free boson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Boundary states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1. Example: the free boson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∗ DESY Theory Group, Notkestrasse 85, D-22603 Hamburg, Germany.
E-mail address:
[email protected]. 1 Permanent address after May 2005.
0370-1573/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physrep.2006.05.001
40 41 42 42 44 45 45 45 46 46 46 47 48 49 50 50 51
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V. Schomerus / Physics Reports 431 (2006) 39 – 86
2.5. The modular bootstrap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1. The boundary spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Bulk Liouville field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. The minisuperspace analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. The path integral approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Degenerate fields and Teschner’s trick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1. Equations of motion and degenerate fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2. Crossing symmetry and shift equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. The exact (DOZZ) solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Branes in the Liouville model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Localized (ZZ) branes in Liouville theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1. The one-point coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2. The open string spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3. Application to 2D string theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Extended (FZZT) branes in Liouville theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1. The one-point coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2. The open string spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3. The c = 1 limit and tachyon condensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Strings in the semi-infinite cigar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Remarks on the bulk theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1. The minisuperspace model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2. The stringy corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. From branes to bulk—D0 branes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1. The inverse procedure—some preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2. D0-branes in the 2D cigar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. D1 and D2 branes in the cigar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1. D1-branes in the 2D cigar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2. D2-branes in the 2D cigar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Conclusions and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A. Dotsenko–Fateev integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix B. Elements of the fusing matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51 51 53 54 55 56 56 58 58 60 60 60 62 63 63 63 65 67 68 70 70 70 72 73 74 77 77 79 80 81 81 82 83
Lectures presented at Spring School on Superstring Theory, Trieste, March 16–23, 2004, at the ESI workshop on String theory in curved backgrounds and bCFT, Vienna, April 1–June 30, 2004, and at the Summer School on Strings, Gravity & Cosmology, Vancouver, August 3–13, 2004. 1. Introduction The microscopic techniques of (boundary) conformal field theory have played a key role in our understanding of string theory and a lot of technology has been developed in this field over the last 20 years. Most of the powerful results apply to strings moving in compact spaces. This is partly explained by the fact that, for at least one decade, progress of world-sheet methods was mainly driven by our need to understand non-trivial string compactifications. In addition, compact target spaces are simply more easy to deal with. Note in particular that compactness renders the spectra of the underlying world-sheet models discrete. This is a crucial feature of the associated “rational conformal field theories” which allows to solve them using only algebraic tools (see e.g. [1–5]). More recently, however, several profound problems of string theory urge us to consider models with continuous spectra. The aim of these lectures is therefore to explain how conformal field theory may be extended beyond the rational cases, to a description of closed and open strings moving in non-compact target spaces. There are several motivations from string theory to address such issues. One of the main reasons for studies of nonrational conformal field theory comes from AdS/CFT-like dualities, i.e. from strong/weak coupling dualities between closed string theories in AdS-geometries and gauge theory on the boundary (see e.g. [6]). If we want to use such a dual string theory description to learn something about gauge theory at finite ‘t Hooft coupling, we have to solve string theory in AdS, i.e. in a curved and non-compact space. Similarly, the study of little string theory, i.e. of the string theory
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on NS5-branes, involves a dual theory of closed strings moving in a non-trivial non-compact target space [7]. Finally, all studies of time-dependent processes in string theory, such as, e.g. the decay of tachyons [8], necessarily involve non-compact target space–time since time is not compact. Both, the applications to some of the most interesting string dualities and to time-dependent backgrounds certainly provide sufficient motivation to develop non-rational conformal field theory. For compact backgrounds there are many model-independent results, i.e. solutions or partial solutions that apply to a large class of models regardless of their geometry. The situation is quite different with non-compact target spaces. In fact, so far all studies have been restricted to just a few fundamental models. This will reflect itself in the content of these lectures as we will mainly look at two different examples. The first is known as Liouville theory and it describes strings moving in an exponential potential (with a non-constant dilaton). Our second example is the SL2 (R)/U(1)-coset theory. Its target space is a Euclidean version of the famous 2D black hole solution of string theory. Fortunately, these two models have quite a few interesting applications already (which we shall only sketch in passing). From a more fundamental point of view, our examples are the non-compact analogues of the minimal models and the coset SU(2)/U(1) which have been crucial for the development of rational conformal field theory and its applications to string theory. As we shall review below, Liouville theory is e.g. used to build an important exactly solvable 2D toy model of string theory and it is believed to have applications to the study of tachyon condensation. The coset SL2 (R)/U(1), on the other hand, appears as part of the transverse geometry of NS5-branes. Moreover, since the space AdS3 is the universal cover of group manifold SL2 (R), one expects the coset space SL2 (R)/U(1) to participate in an interesting low-dimensional version of the AdS/CFT correspondence. The material of these lectures will be presented in four parts. In the first lecture, we shall review some basic elements of (boundary) conformal field theory. Our main goal is to list the quantities that characterize an exact conformal field theory solution and to explain how they are determined. This part is entirely model-independent and it can be skipped by readers who have some acquaintance with boundary conformal field theory and who are more interested in the specific problems of non-rational models. In lectures 2 and 3 we shall then focus on the solution of Liouville theory. Our discussion will begin with the closed string background. Branes and open strings in Liouville theory are the subject of the third lecture. Finally, we shall move on to the coset SL2 (R)/U(1). There, our presentation will rely in parts on the experience from Liouville theory and it will stress the most interesting new aspects of the coset model.
2. 2D boundary conformal field theory The world-sheet description of strings moving in any target space of dimension d with background metric g is based on the following 2D field theory for a d-component bosonic field X , = 1, . . . , d, 1 S[X] = d2 z g jX jX + · · · . 4
(2.1)
In addition to the term we have shown, many further terms can appear and are necessary to describe the effect of nontrivial background B-fields, dilatons, tachyons or gauge fields when we are dealing with open strings. For superstring backgrounds, the world-sheet theory also contains fermionic fields . We shall see some of the extra terms in our examples below. The world-sheet that we integrate over in Eq. (2.1) will be either the entire complex plane (in the case of closed strings) or the upper half-plane (in the case of open strings). In the complex coordinates z, z¯ that we use throughout these notes, lines of constant Euclidean time are (half-)circles around the origin of the complex plane. The origin itself can be thought of as the infinite past. When we are given any string background, our central task is to compute string amplitudes, e.g. for the joining of two closed strings or the absorption of a closed string mode by some brane etc. These quantities are directly related to various correlation functions in the 2D world-sheet theory (2.1) and they may be computed, at least in principle, using path-integral techniques. The remarkable success of 2D conformal field theory, however, was mainly based on a different approach that systematically exploits the representation theory of certain infinite dimensional symmetries which are known as chiral- or W-algebras. We will explain some of the underlying ideas and concepts momentarily.
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One example of such a symmetry algebra already arises for strings moving in flat space. In this case, the equations off motion for the fields X require that X = jj¯ X = 0. Hence, the field n z−n−1 J (z) := jX (z, z¯ ) = n
depends holomorphically on z so that we can expand it in terms of Fourier modes n . The canonical commutation relations for the bosonic fields X are easily seen to imply that the modes n obey the following relations:
[n , m ] = n n,−m .
(2.2)
This is a very simple, infinite dimensional algebra which is known as U(1) current algebra. There exists a second commuting copy of this algebra that is constructed from the anti-holomorphic field J¯ = j¯ X (z, z¯ ). Even though the operators n are certainly useful in describing oscillation modes of closed and open strings, their algebraic structure is not really needed to solve the 2D free field theory underlying string theory in flat space. Things change drastically when the background in curved, i.e. when the background fields depend on the coordinates X . In fact, whenever this happens, action (2.1) ceases to be quadratic in the fields and hence its integration can no longer be reduced to the computation of Gaussian integrals. In such more intricate situations, it becomes crucial to find and exploit generalizations of the U(1) current algebra. We will see this in more detail below after a few introductory comments on chiral algebras. 2.1. Chiral algebras Chiral algebras can be considered as symmetries of 2D conformal field theory. Since they play such a crucial role for all exact solutions, we shall briefly go through the most important notions in the representation theory of chiral algebras. These include the set J of representations, modular transformations, the fusion of representations and the fusing matrix F . The general concepts are illustrated in the case of the U(1) current algebra. Readers feeling familiar with the aforementioned notions may safely skip this subsection. 2.1.1. Representation theory Chiral- or W-algebras are mimic the role played by Lie algebras in atomic physics. Recall that transition amplitudes in atomic physics can be expressed as products of Clebsch–Gordan coefficients and so-called reduced matrix elements. While the former are purely representation-theoretic data which depend only on the symmetry of the theory, the latter contain all the information about the physics of the specific system. Similarly, amplitudes in conformal field theory are built from representation theoretic data of W-algebras along with structure constants of the various operator product expansions, the latter being the reduced matrix elements of conformal field theory. In the conformal bootstrap, the structure constants are determined as solutions of certain algebraic equations which arise as factorization constraints and we will have to say a lot more about such equations as we proceed. Constructing the representation-theoretic data, on the other hand, is essentially a mathematical problem which is the same for all models that possess the same chiral symmetry. Throughout most of the following text we shall not be concerned with this part of the analysis and simply use the known results. But it will be useful to have a few elementary notions in mind. By definition, a chiral- or W-algebra is generated by a Virasoro field T (z) along with a finite number of bosonic chiral primary fields W (z). Modes Ln of the Virasoro field satisfy the usual Virasoro relations for central charge c, c [Ln , Lm ] = (n − m)Ln+m + n(n2 − 1)n+m,0 . 12 The commutation relations of the generators Ln with the Laurent modes Wn of W (z) are assumed to be of the form . [Ln , Wm ] = (n(h − 1) − m)Wn+m
(2.3)
This expresses that the fields W (z) are primary fields of conformal weight h . In addition, the modes of the generating chiral fields W (z) also possess commutation relations among each other which need not be linear in the modes. The algebra generated by the modes Wn is the chiral or W-algebra W (for a precise definition and examples see [9] and in particular [10]). We shall also demand that W comes equipped with a ∗-operation which preserves the algebraic relations between its generators.
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Sectors Vi of the chiral algebra are irreducible (unitary) representations of W for which the spectrum of L0 is bounded from below. Our requirement on the spectrum of L0 along with the commutation relations (2.3) implies that any Vi contains a sub-space Vi0 of ground states which are annihilated by all modes Wn such that n > 0. The spaces Vi0 carry an irreducible representation of the zero mode algebra W0 , i.e. of the algebra that is generated by the zero modes W0 , and we can use the operators Wn , n < 0, to create the whole sector Vi out of states in Vi0 . Unitarity of the sectors means that the space Vi may be equipped with a non-negative bilinear form which is compatible with the ∗-operation on W. This requirement imposes a constraint on the allowed representations of the zero mode algebra on ground states. Hence, one can associate a representation Vi0 of the zero mode algebra to every sector Vi , but for most chiral algebras the converse is not true. In other words, the sectors Vi of W are labeled by elements i taken from a subset J within the set of all irreducible (unitary) representations of the zero mode algebra. For a given sector Vi let us denote by hi the lowest eigenvalue of the Virasoro mode L0 . Furthermore, we introduce the character i (q) = tr Vi (q L0 −c/24 ). In the examples arising from rational conformal field theories, the full set of these characters i , i ∈ J, has the remarkable property to close under modular conjugation, i.e. there exists a complex valued matrix S = (Sij ) such that i (q) ˜ = Sij j (q),
(2.4)
where q˜ = exp(−2i/ ) for q = exp(2i ), as usual. Just as in the representation theory of Lie-algebras, there exists a product ◦ of sectors which is known as the fusion product. Its definition is based on the following family of homomorphisms (see e.g. [11] for a motivation of this formula) z (Wn ) := e−zL−1 Wn ezL−1 ⊗ 1 + 1 ⊗ Wn ∞ h + n − 1 n+h −1−m = W1+m−h ⊗ 1 + 1 ⊗ Wn z m
(2.5)
m=0
which is defined for n > − h . The condition on n guarantees that the sum on the right-hand side terminates after a finite number of terms. Suppose now that we are given two sectors Vj and Vi . With the help of z we define an action of the modes Wn , n > − h , on their product. This action can be used to search for ground states and hence for sectors k in the fusion product j ◦ i. To any three such labels j, i, k there is assigned an intertwiner j V (z) : Vj ⊗ Vi → Vk k i which intertwines between the action z on the product and the usual action on Vk . If we pick an orthonormal basis {|j, } of vectors in Vj we can represent the intertwiner V as an infinite set of operators j, j V (z) := V [|j, ; ·](z) : Vi → Vk . k i k i Up to normalization, these operators are uniquely determined by the intertwining property mentioned above. The latter also restricts their operator product expansions to be of the form j1 , j2 j 1 s,
j2 , V V (z2 ) Frs (z1 )V (z2 ) = i k k i k r r i s,
j2 , × s, V (z12 ) j1 , , s j1 where z12 = z1 − z2 . The coefficients F that appear in this expansion form the fusing matrix of the chiral algebra W. Once the operators V have been constructed for all ground states |j, , the fusing matrix can be read off from the leading terms in the expansion of their products. Explicit formulas can be found in the literature, at least for some chiral algebras. We also note that the defining relation for the fusing matrix admits a nice pictorial presentation (see Fig. 1). It presents the fusing matrix as a close relative of the 6J -symbols which are known from the representation theory of finite-dimensional Lie algebras.
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j2
j2
j1
j1
r F
i
k
s
i
k
Fig. 1. Graphical description of the fusing matrix. All the lines are directed as shown in the picture. Reversal of the orientation can be compensated by conjugation of the label. Note that in our conventions, one of the external legs is oriented outwards. This will simplify some of the formulas below.
2.1.2. Example: the U(1) theory The chiral algebra of a single free bosonic field is known as U(1) current algebra. It is generated by the modes n of the current J (z) with relations (2.2) and the reality condition ∗n = −n . Let also recall that the Virasoro element of this theory is obtained through the famous Sugawara construction, Ln =
∞ : n−m m : 2 m=∞
where the symbol : · : is used to denote normal ordering. There is only one real zero mode 0 = ∗0 so that the zero mode algebra W0 is the abelian u(1) Lie algebra. Hence, all its irreducible representations are 1D and there is one such representation for each real number k. The vector that spans the corresponding 1D space Vk0 is denoted by |k. It is easy to see that the space Vk which we generate out of |k by the creation operators −n admits a positive definite bilinear form for any choice of k. Hence, J = R coincides with the set of irreducible representations of the zero mode algebra in this special case. The character k of the sector Vk with conformal weight hk = k 2 /2 is given by k (q) =
1 k 2 /2 . q (q)
√ Along with the well-known property (q) ˜ = −i (q), the computation of a simple Gaussian integral shows that √ √ k (q) ˜ = dk e2i kk k (q) =: dk Skk k (q). (2.6) This means that the entries of the S-matrix are phases, i.e. Skk = exp(2i kk ). Furthermore, it is not too difficult to determine the fusion of two sectors. In fact, the action of z on the zero mode 0 is given by z (0 ) = 0 ⊗ 1 + 1 ⊗ 0 since the current J has conformal weight h = 1. This shows that the fusion product amounts to adding the momenta, i.e. k1 ◦ k2 = k1 + k2 . In other words, the product of two sectors k1 and k2 contains a single sector k1 + k2 . In this case even the Fusing matrix is rather easy to compute. In fact, we can write down an explicit formula for the intertwining operators V . They are given by the normal ordered exponentials exp ikX of the chiral field
z X(z) = dz J (z ), restricted to the spaces Vk . When the operator product of two such exponentials with momenta k1 and k2 is expanded in the distance z1 − z2 , we find an exponential with momentum k1 + k2 . The coefficient in front of this term is trivial, implying triviality for the fusing matrix.
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2.2. The bootstrap program Having reviewed some elements from the representation theory of chiral algebras we will now turn to discuss the “reduced matrix elements” that are most important for the exact solution of closed and open string backgrounds. These include the coupling of three closed strings and the coupling of closed strings to branes in the background. We shall find that these data are constrained by certain non-linear (integral) equations. Two of these conditions, the crossing symmetry for the bulk couplings and the cluster property for the coupling of closed strings to branes will be worked out explicitly. 2.2.1. Bulk fields and bulk OPE Among the bulk fields (z, z¯ ) of a conformal field theory on the complex plane we have already singled out the so-called chiral fields which depend on only one of the coordinates z or z¯ so that they are either holomorphic, W =W (z), or anti-holomorphic, W¯ = W¯ (¯z). The most important of these chiral fields, the Virasoro fields T (z) and T¯ (z), come with the stress tensor and hence they are present in any conformal field theory. But in most models there exist further (anti-)holomorphic fields W (z) whose Laurent modes give rise to two commuting chiral algebras. These chiral fields W (z) and W¯ (¯z) along with all other fields (z, z¯ ) that might be present in the theory can be considered as operators on the state space H(P ) of the model. The latter admits a decomposition into irreducible ¯ ™¯ of the two commuting chiral algebras, representations Vi and V ¯ ™¯ . Vi ⊗ V (2.7) H(P ) = i,¯™
While writing a sum over i and ¯™ we should keep in mind that for non-compact backgrounds the “momenta” i are typically continuous, though there can appear discrete contributions in the spectrum as well. Throughout this general introduction we shall stick to summations rather than writing integrals. Let us finally also agree to reserve the label i = 0 for the vacuum representation V0 of the chiral algebra. To each state in the space H(P ) we can assign a (normalizable) field. Fields associated with ground states of H(P ) are particularly important. We shall denote them by i,¯™ (z, z¯ ) and refer to these fields as primary fields. In this way, we single out one field for each summand in decomposition (2.7). All other fields in the theory can be obtained by multiplying the primary fields with chiral fields and their derivatives. So far, we have merely talked about the space of bulk fields. But more data is needed to characterize a closed string background. These are encoded in the short distance singularities of correlation functions or, equivalently, in the structure constants of the operator product expansions h −h −h h¯ −h¯ −h¯ i,¯™ (z1 , z¯ 1 ) j,¯E (z2 , z¯ 2 ) = Ci¯™;j E¯ nn¯ z12n i j z¯ 12n i j n,n¯ (z2 , z¯ 2 ) + · · · . (2.8) nn¯
Here, z12 =z1 −z2 and hi , h¯ ™¯ denote the conformal weights of the field i,¯™ , i.e. the values of L0 and L¯ 0 on Vi0 ⊗ V¯i0 . The numbers C describe the scattering amplitude for two closed string modes combining into a single one (“pant diagram”). Since all higher scattering diagrams can be cut into such three-point vertices, the couplings C should encode the full information about our closed string background. This is indeed the case. 2.2.2. Crossing symmetry Obviously, the possible closed string couplings of a consistent string background must be very strongly constrained. The basic condition on the couplings C arises from the investigation of four-point amplitudes. Fig. 2 encodes two ways to decompose a diagram with four external closed string states into three-point vertices. Accordingly, there exist two ways to express the amplitude through products of couplings C. Since both cutting patterns must ultimately lead to the same answer, consistency of the four-point amplitude gives rise to a quadratic equation for the three-point couplings. A more detailed investigation shows that the coefficients in this equation are determined by the Fusing matrix F , j k ¯E k¯ ¯ l l¯ Fp¯ q¯ Ci¯™,j E¯ pp¯ Cpp,k Fpq = Cj E¯,k k¯ q q¯ Ci¯™,q q¯ l l . (2.9) ¯ k¯ i l ¯™ l¯ pp¯
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kk
j j
kk
jj C
pp C
qq
C
C i i
l l
ii
l l
Fig. 2. Graphical representation of the crossing symmetry conditions. The double lines represent closed string modes and remind us of the two commuting chiral algebras (bared and unbarred) in a bulk theory.
This constraint on the three-point couplings C of closed strings is known as crossing symmetry condition. The construction of a consistent closed string background is essentially equivalent to finding a solution of Eq. (2.9). 2.2.3. Example: the free boson Let us once more pause for a moment and illustrate the general concepts in the example of a single free boson. In this case, the state space of the bulk theory is given by ¯ k. (2.10) H(P ) = dkVk ⊗ V As long as we do not compactify the theory, there is a continuum of sectors parametrized by i = k = ¯™. Formula (2.10) provides a decomposition of the space of bulk fields into irreducible representations of the chiral algebra that is ¯ k are used to describe all the modes of a closed string generated by the modes n and ¯ n (see above). States in Vk ⊗ V that moves with center of mass momentum k through the flat space. Associated with the ground states |k ⊗ |k we have bulk field k,k (z, z¯ ) for each momentum k. These fields are the familiar normal ordered closed string vertex operators, k,k (z, z¯ )= : exp(ikX(z, z¯ )) : . Their correlation functions are rather easy to compute (see e.g. [12]). From such expressions one can read off the following short distance expansion 2 2 2 k1 ,k1 (z1 , z¯ 1 ) k2 ,k2 (z2 , z¯ 2 ) ∼ dk(k1 + k2 − k)|z1 − z2 | (k1 +k2 −k ) k,k (z2 , z¯ 2 ) + · · · . Comparison with our general form (2.8) of the operator products shows that the coefficients C are simply given by Ck
1 k¯1 ,k2 k¯2
k k¯
= Ck1 k2 k = (k1 + k2 − k).
(2.11)
Note that the exponent and the coefficient of the short distance singularity are a direct consequence of the equation of motion X(z, z¯ ) = 0 for the free bosonic field. In fact, the equation implies that correlators of X itself possess the usual logarithmic singularity when two coordinates approach each other. After exponentiation, this gives rise to the leading term in the operator product expansion of the fields k,k . In this sense, the short distance singularity encodes the dynamics of the bulk field and hence characterizes the background of the model. Finally, the reader is invited to verify that the couplings C satisfy the crossing symmetry condition (2.9) with a trivial fusing matrix. 2.3. The boundary bootstrap 2.3.1. Branes—the microscopic setup With some basic notations for the (“parent”) bulk theory set up, we can begin our analysis of associated boundary theories (“open descendants”). These are conformal field theories on the upper half-plane H = {Im z 0} which, in the
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interior Im z > 0, are locally equivalent (see below) to the given bulk theory. The state space H(H ) of the boundary conformal field theory is equipped with the action of a Hamiltonian H (H ) and of bulk fields (z, z¯ ) = (z, z¯ )(H ) which are well-defined for Im z > 0. While the space of these fields is the same as in the bulk theory, they are mathematically different objects since they act on different state spaces. Throughout most of our discussion below we shall neglect such subtleties and omit the extra super script (H ). Our first important condition on the boundary theory is that all the leading terms in the OPEs of bulk fields coincide with the OPEs (2.8) in the bulk theory, i.e. for the fields i,¯™ one has h −h −h h¯ −h¯ −h¯ Ci¯™;j E¯ nn¯ z12n i j z¯ 12n i j n,n¯ (z2 , z¯ 2 ) + · · · (2.12) i,¯™ (z1 , z¯ 1 ) j,¯E (z2 , z¯ 2 ) = nn¯
These relations express the condition that our brane is placed into the given closed string background.2 At the example of the free bosonic field we have discussed that the structure of the short distance expansion encodes the world-sheet dynamics. Having the same singularities as in the bulk theory simply reflects that the boundary conditions do not affect the equations of motion in the bulk. In addition, we must require the boundary theory to be conformal. This is guaranteed if the Virasoro field obeys the following gluing condition: T (z) = T¯ (¯z)
for z = z¯ .
(2.13)
In the 2D field theory, this condition ensures that there is no momentum flow across the boundary. Note that Eq. (2.13) is indeed satisfied for the Virasoro fields T ∼ (jX)2 and T¯ ∼ (j¯ X)2 in the flat space theory, both for Dirichlet and Neumann boundary conditions, i.e. when jX = ±j¯ X. Considering all possible conformal boundary theories associated to a bulk theory whose chiral algebra is a true extension of the Virasoro algebra is, at present, too difficult a problem to be addressed systematically (see however [13–16] for some recent progress). For the moment, we restrict our considerations to maximally symmetric boundary theories, i.e. to the class of boundary conditions which leave the whole symmetry algebra W unbroken. More precisely, we assume that there exists an automorphism —called the gluing map—of the chiral algebra W such that [17] W (z) = W¯ (¯z)
for z = z¯ .
(2.14)
Condition (2.13) is included in Eq. (2.14) if we require to act trivially on the Virasoro field. The freedom incorporated in the choice of is necessary to accommodate the standard Dirichlet and Neumann boundary conditions for strings in flat space. Recall that in this case, the left and right moving currents must satisfy J (z) = ±J¯(¯z) all along the boundary. The trivial gluing automorphism N = id in this case corresponds to Neumann boundary conditions while we have to choose D = −id when we want to impose Dirichlet boundary conditions. For later use let us remark that the gluing map on the chiral algebra induces a map on the set of sectors. In fact, since acts trivially on the Virasoro modes, and in particular on L0 , it may be restricted to an automorphism of the zero modes in the theory. If we pick any representation j of the zero mode algebra we can obtain a new representation (j ) by composition with the automorphism . This construction lifts from the representations of W0 on ground states to the full W-sectors. As a simple example consider the U(1) theory with the Dirichlet gluing map D (n ) = − √ n . We k of restrict the latter to the zero mode 0 . As we have explained above, different sectors are labeled by the value √ 0 on the ground state |k. If we compose the action of 0 with the gluing map , we find D (0 )|k = − k|k. This imitates the action of 0 on | − k. Hence, the map D is given by D (k) = −k. 2.3.2. Ward identities As an aside, we shall discuss some more technical consequences that our assumption on the existence of the gluing map brings about. To begin with, it gives rise to an action of one chiral algebra W on the state space H ≡ H(H ) of (H ) the boundary theory. Explicitly, the modes Wn = Wn of a chiral field W dimension h are given by 1 1 Wn := zn+h−1 W (z) dz + z¯ n+h−1 W¯ (¯z) d¯z, 2i C 2i C 2 At tree level in string theory, the backreaction of branes on the bulk geometry is suppressed.
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3 2 1
3
n
2
1
n
Σ
Fig. 3. With the help of operator product expansions in the bulk, the computation of n-point functions in a boundary theory can be reduced to computing one-point functions on the half-plane. Consequently, the latter must contain all information about the boundary condition.
where C is a half-circle around z = 0 in the upper half-plane. The operators Wn on the state space H are easily seen to obey the defining relations of the chiral algebra W. Note that there is just one such action of W constructed out of the two chiral bulk fields W (z) and W¯ (¯z). In the usual way, the representation of W on H leads to Ward identities for correlation functions of the boundary theory. They follow directly from the singular parts of the operator product expansions of the fields W, W¯ with the bulk fields (z, z¯ ). These expansions are fixed by our requirement of local equivalence between the bulk theory and the bulk of the boundary theory. Rather than explaining the general form of these Ward identities, we shall only give one special example, namely the relations that arise from the Virasoro field. In this case one find that h¯ h j j¯ (T (w) (z, z¯ ))sing = + (z, z¯ ). (2.15) + + w − z (w − z¯ )2 w − z¯ (w − z)2 The subscript “sing” reminds us that we only look at the singular part of the operator product expansion. Let us remark that the first two terms in the brackets are well known from the Ward identities in the bulk theory. The other two terms can be interpreted as arising from a “mirror charge” that is located at the point w = z¯ in the lower half-plane. 2.3.3. One-point functions So far we have formalized what it means in world-sheet terms to place a brane in a given background (the principle of “local equivalence”) and how to control its symmetries through gluing conditions (2.14) for chiral fields. Now it is time to derive some consequences and, in particular, to show that a rational boundary theory is fully characterized by just a family of numbers. Using the Ward identities described in the previous paragraph together with the OPE (2.12) in the bulk, we can reduce the computation of correlators involving n bulk fields to the evaluation of one-point functions i,¯™ for the bulk primaries (see Fig. 3). Here, the subscript has been introduced to label different boundary theories that can appear for given gluing map . To control the remaining freedom, we notice that the transformation properties of i,¯™ with respect to Ln , n = 0, ±1, [Ln , i,¯™ (z, z¯ )] = zn (zj + hi (n + 1)) i,¯™ (z, z¯ ) + z¯ n (¯zj¯ + h¯ ™¯ (n + 1)) i,¯™ (z, z¯ ) determine the one-point functions up to scalar factors. Indeed, an elementary computation using the invariance of the vacuum state reveals that the vacuum expectation values i,¯™ must be of the form i,¯™ (z, z¯ ) =
Ai¯™ . |z − z¯ |hi +h¯™
(2.16)
Ward identities for the Virasoro field and other chiral fields, should they exist, also imply ¯™ = (i + ) as a necessary condition for a non-vanishing one-point function (i + denotes the representation conjugate to i), i.e. Ai¯™ = Ai ™¯,(i)+ .
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i
j
49
i
j Cij k
Ai
ω (i)
0
Aj
ω (j)
Ak
k
ω (i)
ω (j)
Fig. 4. Eqs. (2.18), (2.19) are derived by comparing two limits of the two-point function. The dashed line represents the boundary of the world-sheet and we have drawn the left moving sector in the lower half-plane (doubling trick).
Since hi = hi + = h(i) we can put hi + h™¯ = 2hi in the exponent of Eq. (2.16). Our arguments above have reduced the description of a boundary condition to a family of scalar parameters Ai in the one-point functions. Once we know their values, we have specified the boundary theory. This agrees with our intuition that a brane should be completely characterized by its couplings to closed string modes such as the mass and RR charges. 2.3.4. The cluster property We are certainly not free to choose the remaining parameters Ai in the one-point functions arbitrarily. In fact, there exist strong sewing constraints on them that have been worked out by several authors [18–22]. These can be derived from the following cluster property of the two-point functions lim i,¯™ (z1 , z¯ 1 ) j,¯E (z2 + a, z¯ 2 + a) = i,¯™ (z1 , z¯ 1 ) j,¯E (z2 , z¯ 2 ) .
a→∞
(2.17)
Here, a is a real parameter, and the field j,¯E on the right-hand side can be placed at (z2 , z¯ 2 ) since the whole theory is invariant under translations parallel to the boundary. Let us now see how the cluster property restricts the choice of possible one-point functions. We consider the two-point function of the two bulk fields as in Eq. (2.17). There are two different ways to evaluate this function. On the one hand, we can go into a regime where the two bulk fields are very far from each other in the direction along the boundary. By the cluster property, the result can be expressed as a product of two one-point functions and it involves the product of the couplings Ai and Aj . Alternatively, we can pass into a regime in which the two bulk fields are very close to each other and then employ the operator product (2.12) to reduce their two-point function to a sum over one-point functions. Comparison of the two procedures provides the following important relation: Ai Aj = kij A0 Ak . (2.18) k
It follows from our derivation that the coefficient kij can be expressed as a combination j (j ) k k k¯ ij = Ci¯™;j E¯ Fk0 i (i)+
(2.19)
of the coefficients C in the bulk OPE and of the fusing matrix. The latter arises when we pass from the regime in which the bulk fields are far apart to the regime in which they are close together (see Fig. 4). The importance of Eq. (2.18) for a classification of boundary conformal field theories has been stressed in a number of publications [22–24] and is further supported by their close relationship with algebraic structures that entered the classification of bulk conformal field theories already some time ago (see e.g. [25–27]). The algebraic relations (2.18) typically possess several solutions which are distinguished by our index . Hence, maximally symmetric boundary conditions are labeled by pairs ( , ). The automorphism is used to glue holomorphic and anti-holomorphic fields along the boundary and the consistent choices for are rather easy to classify. Once has been fixed, it determines the set of bulk fields that can have a non-vanishing one-point function. For each gluing automorphism , the non-zero one-point functions are constrained by algebraic equations (2.18) with coefficients
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which are determined by the closed string background. A complete list of solutions is available in a large number of cases with compact target spaces and we shall see a few solutions in our non-compact examples later on. Right now, however, it is most important to explain how one can reconstruct further information on the brane from the couplings Ai . In particular, we will be able to recover the entire spectrum of open string excitations. Since the derivation makes use of boundary states, we need to introduce this concept in the next subsection. 2.3.5. Example: the free boson We wish to close this subsection with a short discussion of boundary conditions in the theory of a single free boson. The corresponding bulk model has been discussed earlier so that we know its bulk couplings (2.11). Let us analyze Dirichlet boundary conditions jX = −j¯ X = D (j¯ X) first. In this case, all bulk tachyon vertex operators possess a non-vanishing coupling to the boundary since the momentum k and the associated label D (k) = −k of its mirror image add up to k + (k) = 0 for all k. Using the triviality of the Fusing matrix, we are now able to spell out the cluster condition (2.18) for the couplings AD , D, D, AD, A = dk3 (k1 + k2 − k3 )AD, k1 k2 k3 = Ak1 +k2 . Solutions to this equation are parametrized by a single real parameter = x0 and possess the form 0 AD,x = e2ikx 0 . k
Obviously, the parameter = x0 is interpreted as the transverse position of a point-like brane. For Neumann boundary conditions N = id and hence only the identity field 0 can have a non-vanishing one-point coupling. The factorization constraint for the latter is certainly trivial. 2.4. Boundary states It is possible to store all information about the couplings Ai in a single object, the so-called boundary state. To some extent, such a boundary state can be considered as the wave function of a closed string that is sent off from the brane ( , ). It is a special linear combination of generalized coherent states (the so-called Ishibashi states). The coefficients in this combination are essentially the closed string couplings Ai . One way to introduce boundary states is to equate correlators of bulk fields on the half-plane and on the complement of the unit disk in the plane. With z, z¯ as before, we introduce coordinates , ¯ on the complement of the unit disk by =
1 − iz 1 + iz
and
1 + i¯z ¯ = . 1 − i¯z
(2.20)
If we use |0 to denote the vacuum of the bulk conformal field theory, then the boundary state | = | can be uniquely characterized by [18,17] h ¯ h¯ d d (H ) ¯ · 0| (P ) (, )| (2.21) (z, z¯ ) = dz d¯z ¯ Note that all quantities on the right-hand side are defined in the bulk for primaries with conformal weights (h, h). conformal field theory (super script P ), while objects on the left-hand side live on the half-plane (super script H ). ¯ on the gluing condition (2.14) to obtain In particular, we can apply the coordinate transformation from (z, z¯ ) to (, ) ¯ W () = (−1)h ¯ W¯ () 2h
along the boundary at ¯ = 1. Expanding this into modes, we see that the gluing condition (2.14) for chiral fields translates into the following linear constraints for the boundary state, [Wn − (−1)hW W¯ −n ]| = 0.
(2.22)
These constraints posses a linear space of solutions. It is spanned by generalized coherent (or Ishibashi) states |i. Given the gluing automorphism , there exists one such solution for each pair (i, (i + )) of irreducibles that occur in
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the bulk Hilbert space [28]. |i is unique up to a scalar factor which can be used to normalize the Ishibashi states such that
j |q˜
(P )
L0 −c/24
|i = i,j i (q). ˜
(2.23)
Following an idea in [28], it is easy to write down an expression for the generalized coherent states (see e.g. [17]), but the formula is fairly abstract. Only for strings in a flat background their constructions can be made very explicit (see below). Full boundary states | ≡ |( , ) are given as certain linear combinations of Ishibashi states, | = Bi |i . (2.24) i
With the help of (2.21), one can show [18,17] that the coefficients Bi are related to the one-point functions of the boundary theory by +
Ai = Bi .
(2.25)
The decomposition of a boundary state into Ishibashi states contains the same information as the set of one-point functions and therefore specifies the “descendant” boundary conformal field theory of a given bulk conformal field theory completely. 2.4.1. Example: the free boson Here we want to spell out explicit formulas for the boundary states in the theory of a single free boson. Let us first discuss this for Dirichlet boundary conditions, i.e. the case when the U(1) currents of the model satisfy the gluing condition (2.14) with D J¯ = −J¯. Since k + = −k (recall that fusion of sectors is given by adding momenta) and D (k) = −k, we have D (k)+ = k and so there exists a coherent state for each sector in the bulk theory. These states are given by
∞ 1 |kD = exp −n ¯ −n |k ⊗ |k. n n=1
Using the commutation relation of n and ¯ n it is easy to check that |kD is annihilated by n − ¯ −n as we required in Eq. (2.22). Since one-point functions of closed string vertex operators for Dirichlet boundary conditions have the form 0 (2.16) with AD,x = exp(i2kx 0 ), we obtain the following boundary state: k √ |x0 D = dk e−i2kx 0 |kD . For Neumann boundary conditions the analysis is different. Here we have to use the trivial gluing map N = id and a simple computation reveals that the condition N (k) = k + is only solved by k = 0. This means that we can only construct one coherent state,
∞ 1 B |0N = exp − −n ¯ −n |0 ⊗ |0. n n=1
This coherent state coincides with the boundary state |0N = |0N for Neumann boundary conditions. 2.5. The modular bootstrap 2.5.1. The boundary spectrum While the one-point functions (or boundary states) uniquely characterize a boundary conformal field theory, there exist more quantities we are interested in. In particular, we shall now see how the coefficients of the boundary states determine the spectrum of open string vertex operators that can be inserted along the boundary of the world-sheet.
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β Z αβ
α
τ
β
α Fig. 5. The open string partition function Z can be computed by world-sheet duality. In the figure, the time runs upwards so that the left-hand side is interpreted as an open string one-loop diagram while the right-hand side is a closed string tree diagram.
Our aim is to determine the spectrum of open string modes which can stretch between two branes labeled by and , both being of the same type . In world-sheet terms, the quantity we want to compute is the partition function on a strip with boundary conditions and imposed along the two sides. This is depicted on the left-hand side of Fig. 5. The figure also illustrates the main idea of the calculation. In fact, world-sheet duality allows to exchange space and time and hence to turn the one loop open string diagram on the left-hand side into a closed string tree diagram which is depicted on the right-hand side. The latter corresponds to a process in which a closed string is created on the brane and propagates until it gets absorbed by the brane . Since creation and absorption are controlled by the amplitudes Ai
and Aj , the right-hand side—and hence the partition function on the left-hand side—is determined by the one-point functions of bulk fields. Let us now become a bit more precise and derive the exact relation between the couplings A and the partition function. Reversing the above sketch of the calculation, we begin on the left-hand side of Fig. 5 and compute |q˜ H
(P )
| =
(P ) Aj + Aj j + |q˜ L0 −c/24 |j + = Aj + Aj j + (q). ˜ j
j
Here we have dropped all subscripts since all the boundary and generalized coherent states are assumed to be of the same type. The symbol denotes the world-sheet CPT operator in the bulk theory. It is a anti-linear map which sends sectors to their conjugate, i.e.
Aj + |j = (Aj + )∗ |j + . Having explained these notations, we can describe the steps we performed in the above short computation. To begin with we inserted expansion (2.24),(2.25) of the boundary states in terms of Ishibashi states and the formula H (P ) = 1/2(L0 + L¯ 0 ) − c/24 for the Hamiltonian on the plane. With the help of the linear relation (2.22) we then traded L¯ 0 for L0 before we finally employed formula (2.23). At this point we need to recall property (2.4) of characters to arrive at (P ) |q˜ H | = Aj + Aj Sj + i i (q) =: Z (q). (2.26) j
As argued above, the quantity we have computed should be interpreted as a boundary partition function and hence as a trace of the operator exp(2i H (H ) ) over some space H of states for the system on a strip with boundary conditions and imposed along the boundaries. If at least one of the two branes is compact, we expect to find a discrete open string spectrum. In this case, our computation leads to a powerful constraint on the numbers Ai . In fact, since our boundary conditions preserve the chiral symmetry, the partition function is guaranteed to decompose into a sum of the associated characters. If this sum
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is discrete, i.e. not an integral, the coefficients in this expansion must be positive integers and so we conclude Z (q) =
ni i (q) where ni =
i
Aj + Aj Sj + i ∈ N.
(2.27)
j
Although there exists no general proof, it is believed that every solution of the factorization constraints (2.18) gives rise to a consistent spectrum with integer coefficients ni . A priori, the integrality of the numbers ni provides a strong constraint, known as the Cardy condition, on the set of boundary states and it has often been used instead of Eqs. (2.18) to determine the coefficients Ai . Note that the Cardy conditions are easier to write down since they only involve the modular S-matrix. To spell out the factorization constraints (2.18), on the other hand, one needs explicit formulas for the fusing matrix and the bulk operator product expansion. There is one fundamental difference between the Cardy condition (2.27) and the factorization constraints (2.18) that is worth pointing out. Suppose that we are given a set of solutions of the Cardy constraint. Then every non-negative integer linear combination of the corresponding boundary states defines another Cardy-consistent boundary theory. In other words, solutions of the Cardy condition form a cone over the integers. The factorization constraints (2.18) do not share this property. Geometrically, this is easy to understand: we know that it is possible to construct new brane configurations from arbitrary superpositions of branes in the background (though they are often unstable). These brane configurations possess a consistent open string spectrum but they are not elementary. As long as we are solving the Cardy condition, we look for such configurations of branes. The factorization constraints (2.18) were derived from the cluster property which ensures the system to be in a “pure phase”. Hence, by solving Eqs. (2.18) we search systematically for elementary brane configurations that cannot be decomposed any further. Whenever the coefficients are known, solving the factorization constraints is clearly the preferable strategy, but sometimes the required information is just hard to come by. In such cases, one can still learn a lot about possible brane configurations by studying Cardy’s conditions. Let us stress again, however, that the derivation of the Cardy condition required compactness of at least one of the branes. If both branes are non-compact, the open string spectrum will contain continuous parts which involve an a priori unknown spectral density function rather than integer coefficients. We will come back to such issues later. 3. Bulk Liouville field theory Our goal for this lecture is to present the solution of the Liouville bulk theory. This model describes the motion of closed strings in a 1D exponential potential and it can be considered as the minimal model of non-rational conformal field theory. Before we explain how to determine the bulk spectrum and the exact couplings, we would like to make a few more comments on the model and its applications. On a 2D world-sheet with metric ab and curvature R, the action of Liouville theory takes the form SL [X] =
√ d2 (ab ja Xjb X + RQX + e2bX ),
(3.1)
where and b are two real parameters of the model. The third quantity Q, on the other hand, must be adjusted to the choice of b in order to ensure conformal invariance of the model (see below). We observe that the second term in this action describes a linear dilaton and that such a term would render perturbative string theory invalid if the strong coupling region was not screened by the third term. In fact, the exponential potential has the effect to keep closed strings away from the strong coupling region of the model. Liouville theory should be considered as a marginal deformation of the free linear dilaton theory, SLD [X] =
√ d2 (ab ja Xjb X + RQX).
The Virasoro field of a linear dilaton theory is given by the familiar expression T = (jX)2 + Qj2 X.
(3.2)
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The modes of this field form a Virasoro algebra with central charge c = 1 + 6Q2 . Furthermore, the usual closed string vertex operators = : exp 2X :
have h = (Q − ) = h¯ .
Note the conformal weights h, h¯ are real if is of the form = Q/2 + iP . We interpret the real parameter P as the momentum of the closed string tachyon scattering state that is created by the above vertex operator. In order for the exponential potential in the Liouville action to be marginal, i.e. (hb , h¯ b ) = (1, 1), we must now also adjust the parameter b to the choice of Q in such a way that Q = b + b−1 . As one may easily check, Weyl invariance of the classical action SL leads to the relation Qc =b−1 . Quantum corrections deform this correspondence such that Q = Qc + b. The extra term, which certainly becomes small in the semi-classical limit b → 0, has a remarkable consequence: It renders the parameter Q (and hence the central charge) invariant under the replacement b → b−1 . We will have a lot more to say about this interesting quantum symmetry of Liouville theory. After this preparation we are able to describe two interesting application of Liouville theory. The first one comes with the observation that Liouville theory manages to contribute a value c 25 to the central charge even though it involves only a single dimension. Hence, in order to obtain a consistent string background with c = 26, it suffices to add one more direction with central charge c = 1 or less. Geometrically, this would then describe a string background with D 2. These theories have indeed been studied extensively in the past and there exist many results, mostly due to the existence of a dual matrix model description. The second application is much more recent and also less well tested: it has been proposed that time-like Liouville theory at c = 1 describes the homogeneous condensation of a closed string tachyon. A short look at classical actions makes this proposal seem rather plausible. In fact, the condensation of a closed string tachyon is described on the world-sheet by adding the following term to the action of some static background, 0 S[X] = d2 z e X (z, z¯ ). (3.3)
Here, X0 is a time-like free field and the bulk field must be a relevant field in the conformal field theory of a spatial slice of the static background. describes the profile of the tachyon. If we want the tachyon to have a constant profile we must choose = =const. The parameter is then forced to be = 2 so that the interaction term (3.3) becomes scale invariant. If we now Wick-rotate the field X0 = iX the interaction term looks formally like the interaction term in Liouville theory, only that the parameter b assumes the unusual value b = i. At this point, the central charge of the model is c = 1 and hence we recover the content of a proposal formulated in [29]: The rolling tachyon background is a Lorentzian c = 1 Liouville theory. 3.1. The minisuperspace analysis In order to prepare for the analysis of exact conformal field theories it is usually a good idea to first study the particle limit → 0 in which all higher string modes are decoupled. Here, sending to zero is equivalent to sending b to zero after rescaling both the coordinate X = b−1 x and the coupling = b−2 . What we end up with is the theory of a particle moving in an exponential potential. The stationary Schroedinger equation for this system is given by j2 HL := − 2 + e2x (x) = 42 (x). jx
(3.4)
This differential equation is immediately recognized as Bessel’s equation. Hence, its solutions are linear combinations of the Bessel functions of first kind, √ x ± (x) = J±2i (i e ).
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These functions ± describe an incoming/outgoing plane wave in the region x → −∞, but they are both unbounded as can be seen from the asymptotic behavior at x → ∞, √ −x/2 x . (x) − → e cosh e ∓ + i ± 4 There is only one particular linear combination of these two solutions that stays finite for x → ∞. This is given by √ (3.5) (x) = (/4)−i −1 (−2i)K−2i ( ex ), where K (z) := J (iz) − exp(i)J− (iz) is known as modified Bessel function. We have also fixed an the overall normalization such that the incoming plane wave has unit coefficient. Here we are mainly interested in the three-point function, since in the full bulk conformal field theory this quantity encodes all the information about the exact solution. Its counterpart in the minisuperspace model can be evaluated through the following integral over a product of Bessel functions (see e.g. [45, Section 6.576]), ∞ 1 |e2 |3 := dx1 (x)e2i2 x 3 (x) (3.6) −∞
(−(−1)j 2i ˜ j) 1 = (/4)−2i˜ (2i) ˜ , j 2 ((−1) 2ij ) j =1 3
(3.7)
where ˜ = 21 (1 + 2 + 3 ),
˜i = ˜ − i .
(3.8)
The formula that was used to compute the integral can be found in standard mathematical tables. Let us remark already that the exact answer in the 2D field theory will have a very similar form, only that the function gets replaced by a ˜ and zeroes whenever more complicated special function (see below). Observe also that the result has poles at ˜ i =0= one of the frequencies ˜ i vanishes. From the three-point function it is not hard to extract the two-point function of our toy model in the limit 2 =ε → 0. In particular, after taking the limit, the poles at 1 ∓ 3 ± ε = 0 produce terms of the form (1 ∓ 3 ). More precisely, we obtain lim 1 |e2 |3 ∼ (1 + 3 ) + R0 (1 )(1 − 3 )
2 →0
with R0 () = −2i
(2i) . (−2i)
(3.9)
The quantity R0 is known as the reflection amplitude. It describes the phase shift of the wave function upon reflection in the Liouville potential as can be seen from the following expansion of the wave functions for x → −∞, (x) ∼ e2ix + −2i
(2i) −2ix e . (−2i)
We conclude that the two coefficients of the -functions in the two-point function encode both the normalization of the incoming plane wave and the phase shift that appears during its reflection. 3.2. The path integral approach In an attempt to obtain formulas for the exact correlation function of closed string vertex operators in Liouville theory one might try to evaluate their path integral, n e2r X(zr ,¯zr ) . e21 X(z1 ,¯z1 ) · · · e2n X(zn ,¯zn ) = DXe−SL [X] r=1
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The first step of the evaluation is to split the field X into its constant zero mode x0 and a fluctuation field X˜ around the zero mode. Once this split is performed, we calculate the integral over the zero mode x0 using a simple integral formula for the function ∞ (x) = dt exp(xt − et ). −∞
After executing these steps, we obtain the following expression for the correlator of the tachyon vertex operators, n s n √ ˜ 2r X(z ˜ ˜ r ,¯zr ) (−s) 2r X(zr ,¯zr ) ˜ −SLD [X] d2 e2bX , = DXe e e (3.10) 2b r=1
r=1
where 1 j s=Q − b b n
(3.11)
r=1
and SLD is the linear dilaton action defined in Eq. (3.2) There are quite a few observations we can make about this result. To begin with, we see that the expression on the right-hand side can only be evaluated when s is a non-negative integer since it determines the number of insertions of the interaction term. Along with Eq. (3.11), such a condition on s provides a strong constraint for the parameters r that will not be satisfied for generic choices of the momentum labels r . But if s is integer then we can evaluate the integral rather easily because the action for the fluctuation field X˜ is the action of the linear dilaton, i.e. of a free field theory. The resulting integrals over the position of the field insertions are not that easy to compute, but they have been solved many years ago and are known to be expressible through functions (see [30] and Appendix A). Let us furthermore observe that in the case of non-negative integer s, the coefficient (−s) diverges. Hence, the correlators whose computation we sketched in the previous paragraph all contain a divergent factor.A more mathematical interpretation of this observation is not hard to find. Note that even our semi-classical couplings (3.6) possess poles, e.g. at points where 2i ˜ is a negative integer. The exact couplings are not expected to behave any better, i.e. the correlation functions of Liouville theory have poles at points in momentum space at which the expression s becomes a non-negative integer. Only the residue at these poles can be computed directly through the free field computation above. The presence of singularities in the correlators admits a simple explanation. In fact, in Liouville theory, the interaction is switched off as we send x → −∞ so that the theory becomes free in this region. The infinities that we see in the correlators are associated with the fact that the region of small interaction is infinite, leading to contributions which diverge with the volume of the space. Finally, with singularities arising from the weakly coupled region, it is no longer surprising that the coefficients of the residues can be computed through free field computations. Even though our attempt to compute the path integral has lead to some valuable insights, it certainly falls short of providing an exact solution of the theory. After all we want to find the three-point couplings for all triples 1 , 2 , 3 , not just for a subset thereof. To proceed further we need to understand a few other features of Liouville theory. 3.3. Degenerate fields and Teschner’s trick 3.3.1. Equations of motion and degenerate fields Let us first study in more detail the equations of motion for the Liouville field. In the classical theory, these can be derived easily from the action, jj¯ Xc = b exp 2bXc .
(3.12)
2 There exits an interesting way to rewrite this equation. As a preparation, we compute the second derivatives j2 and j¯ c of the classical field −b/2 = exp(−bX c ), e.g.
j2 e−bXc (z,¯z) = b2 Tc (z)e−bXc (z,¯z) .
(3.13)
The fields Tc and T¯c that appear on the right-hand side of this equation and its anti-holomorphic counterpart are the classical analogues of the Virasoro fields. It is now easy to verify that the classical equation of motion (3.12) is equivalent to the (anti-)holomorphicity of Tc (T¯c ).
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In its new form, it appears straightforward to come up with a generalization of the equations of motion to the full quantum theory. The proposal is to demand that3 j2 −b/2 = −b2 : T (z) −b/2 : .
(3.14)
If we evaluate this equation at the point z = 0 and apply it naively to the vacuum state, we find that b 2 2 |+ := (L−1 + b L−2 ) − = 0. 2 In order to understand this equation better, let us note that |+ is a so-called singular vector in the sector V−b/2 of the Virasoro algebra, i.e. a vector that is annihilated by all modes Ln , n > 0, of the Virasoro algebra (but is not the ground state of the sector). Such singular vectors can be set to zero consistently and our arguments above suggest that this is what happens in Liouville theory. In other words, decoupling the singular vector of the sector V−b/2 implements the quantum equation of motion of Liouville theory. It is worthwhile pointing out that the origin for the decoupling of singular vectors in Liouville theory is different from rational conformal field theories, where it is a consequence of unitarity. Here, the singular vector is not among the normalizable states of the model and hence it does not belong to the physical spectrum anyway. In fact, the only way to reach the point = −b/2 from the set Q/2 + iR is through analytic continuation. The presence of such unphysical singular vectors would not be in conflict with unitarity. Once we have accepted the decoupling of |+ , we observe that there exists another sector of the Virasoro algebra that has a singular vector on the second level. This is the sector V−b−1 /2 and its singular vector has the form 1 2 −2 |− := (L−1 + b L−2 ) − . 2b Note that this second singular vector is obtained from the first through the substitution b → b−1 . Earlier on, we noted that, e.g. the relation between Q and b received quantum corrections which rendered it invariant under a replacement of b by its inverse. It is therefore tempting to conjecture that in the exact Liouville theory, the second singular vector |− decouples as well. If we accept this proposal, we end up with two different fields in the theory that both satisfy a second-order differential equation of form (3.14), namely the field −b/2 and the dual field −1/2b . Such fields are called degenerate fields and we shall see in a moment that their presence has very important consequences for the structure constants of Liouville theory. Let us point out that a relation of form (3.14) imposes strong constraints on the operator product expansion of the fields −b±1 /2 with any other field in the theory. Since momentum is not conserved in our background, the expansion of two generic bulk fields contains a continuum of bulk fields. But if we replace one of the two fields on the left-hand side by one of our degenerate fields −b±1 /2 , then the whole operator product expansion must satisfy a second-order differential equation and hence only two terms can possibly arise on the right-hand side, e.g. ¯ −b/2 (z, z¯ ) = (w, w)
±
cb± () ∓b/2 (z, z¯ ) + · · · , |z − w|h±
(3.15)
where h± = ∓b + Q(−b/2 ∓ b/2). A similar expansion for the second degenerate field is obtained through our usual replacement b → b−1 . We can even be more specific about the operator expansions of degenerate fields and compute the coefficients c± . Note that the labels 1 = −b/2 and 2 = of the fields on the left-hand side along with the conjugate labels 3 = ( ∓ b/2)∗ = Q − ∓ b/2 of the fields that appear in the operator product obey s+ = 1 and s− = 0 (the quantity s was defined in Eq. (3.11)). Hence, c± can be determined through a free field computation in the linear dilaton background. With the help of our above formulas we find that c+ = 1 and − cb () = − d2 z −b/2 (0, 0) (1, 1) b (z, z¯ ) Q−b/2− (∞, ∞)LD = −
(1 + b2 )(1 − 2b) , (2 + b2 − 2b)
3 Normal ordering instructs us to move annihilation modes L , n > 0, of the Virasoro field to the right. n
(3.16)
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where we have introduced (x) = (x)/(1 − x). Factor (−1) in front of the integral is the contribution from the residue of factor (−s) at s = 1. Our result in the second line is obtained using the explicit integral formulas that were derived by Dotsenko and Fateev in [30] (see also Appendix A). 3.3.2. Crossing symmetry and shift equations Now it is time to combine all our recent insights into Liouville theory with the general strategy we have outlined in the first lecture and to re-address the construction of the exact bulk three-point couplings. Recall that these couplings may be obtained as solutions to the crossing symmetry condition (2.9). Unfortunately, in its original form, the latter involves four external states with momentum labels i ∈ Q/2 + iR. Consequently, there is a continuum of closed string modes that can be exchanged in the intermediate channels and hence the crossing symmetry requires solving a rather complicated integral equation. To overcome this difficulty, Teschner [31] suggested a continuation of one external label, e.g. the label 2 , to one of the values 2 = −b±1 /2. The corresponding field is then degenerate and it possesses an operator product consisting of two terms only. Teschner’s trick converts the crossing symmetry condition into a much simpler algebraic condition. Moreover, since we have already computed the coefficients of operator products with degenerate fields, the crossing symmetry equation is in fact linear in the unknown generic three-point couplings. One component of these conditions for the degenerate field −b/2 reads as follows
b b − −− 0 = C 1 + , 3 , 4 cb (1 )P+− + C 1 − , 3 , 4 cb+ (1 )P++ +− , 2 2
(3.17)
where P±± +−
= F1 ∓b/2,3 −b/2
−b/2 1
3 −b/2 F1 ∓b/2,3 +b/2 4 1
3 . 4
Note that the combination on the right-hand side must vanish because in a consistent model, the off-diagonal bulk mode (4 − b/2, 4 + b/2) does not exist and hence it cannot propagate in the intermediate channel. The required special entries of the Fusing matrix were computed in [31] and they can be expressed through a combination of functions (see Appendix B for explicit formulas). Once the expressions for c± and P are inserted (note that they only involve functions), the crossing symmetry condition may be written as follows: C(1 + b, 2 , 3 ) (b(21 + b))(2b1 ) (b(2˜1 − b)) =− C(1 , 2 , 3 ) (1 + b2 )(b(2˜ − Q)) (2b˜ 2 )(2b˜ 3 )
(3.18)
with our usual shorthand (x) = (x)/(1 − x) and ˜ , ˜ i defined as in Eq. (3.8). The constraint takes the form of a shift equation that describes how the coupling changes if one of its arguments is shifted by b. Clearly, this one equation alone cannot fix the coupling C. But now we recall that we have a second degenerate field in the theory which is related to the first degenerate field by the substitution b → b−1 . This second degenerate field provides another shift equation that encodes how the three-point coupling behaves under shifts by b−1 . The equation is simply obtained by performing the substitution b → b−1 in Eq. (3.18). For irrational values of b, the two shift equations determine the couplings completely, at least if we require that they are analytic in the momenta. Once we have found the unique analytic solution for irrational b, we shall see that it is also analytic in the parameter b. Hence, there is a unique solution to the shift equations that is both analytic in b and in the momenta. 3.4. The exact (DOZZ) solution It now remains to solve the shift (3.18). For this purpose it is useful to introduce Barnes’ double -function b (y). It may be defined through the following integral representation,
∞
ln b (y) = 0
d
(Q/2 − y)2 − Q/2 − y e−y − e−Q /2 − e − (1 − e−b )(1 − e− /b ) 2
(3.19)
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for all b ∈ R. The integral exists when 0 < Re(y) and it defines an analytic function which may be extended onto the entire complex y-plane. Under shifts by b±1 , the function b behaves according to b (y + b) =
√
2
bby−1/2 b (y), (by)
b (y + b−1 ) =
√
2
b−y/b+1/2 b (y). (b−1 y)
(3.20)
These shift equations let b appear as an interesting generalization of the usual function which may also be characterized through its behavior under shifts of the argument. But in contrast to the ordinary function, Barnes’ double function satisfies two such equations which are independent if b is not rational. We furthermore deduce from Eqs. (3.20) that b has poles at yn,m = −nb − mb−1
for n, m = 0, 1, 2, . . . .
(3.21)
From Branes’ double Gamma function one may construct the basic building block of our exact solution, Υb () := b ()−1 b (Q − )−1 .
(3.22)
The integral formula (3.19) for the double -function implies that Υb possesses the following integral representation: 2 ∞ dt sinh2 (Q/2 − y)t/2 Q −t ln Υb (y) = −y e − . (3.23) t 2 sinh bt/2 sinh t/2b 0 Moreover, we deduce from the two shift properties (3.20) of the double -function that Υb (y + b−1 ) = (b−1 y)b−1+2b
Υb (y + b) = (by)b1−2by Υb (y),
−1 y
Υb (y).
(3.24)
Note that the second equation can be obtained from the first with the help of the self-duality property Υb (y) = Υb−1 (y). Now we are prepared to solve our shift equations (3.18). In fact, it is easy to see that their solution is provided by the following combination of Υb functions [32,33], C(1 , 2 , 3 ) := [(b2 )b2−2b ](Q−2˜ )/b 2
3 Υb (0) Υb (2j ) , Υb (2˜ − Q) Υb (2˜j )
(3.25)
j =1
where ˜ and ˜ j are the linear combinations of j which are introduced just as in Eqs. (3.8) of the previous subsection. Solution (3.25) was first proposed several years ago by Dorn and Otto [32] and by Zamolodchikov and Al. Zamolodchikov [33], based on extensive earlier work by many authors (see e.g. the reviews [34,35] for references). The derivation we presented here has been proposed by Teschner in [31]. Full crossing symmetry of the conjectured three-point function (not just for the special case that involves one degenerate field) was then checked analytically in two steps by Ponsot and Teschner [36] and by Teschner [35,37]. It is quite instructive to see how the minisuperspace result (3.6) can be recovered from the exact answer. To this end, we choose the parameters i to be of the form 1 =
Q + ib1 , 2
2 = b2 ,
3 =
Q + ib3 2
and perform the limit b → 0 with the help of the following formula for the asymptotics of the function Υb (see e.g. [38]) b→0
Υb (by) ∼ bΥb (0)b−b
2 y 2 +(b2 −1)y
−1 (y) + · · · .
(3.26)
Recall that Barnes’ double Gamma function possesses a double series (3.21) of poles. In our limit, most of these poles move out to infinity and we are just left with the poles of the ordinary function. Not only does this observation explain formula (3.26), it also makes Υb appear as the most natural replacement of the functions in Eq. (3.6) that is consistent with the quantum symmetry b ↔ 1/b of Liouville theory. Let us furthermore stress that expression (3.25) can be analytically continued into the entire complex -plane. Even though the corresponding fields with ∈ / Q/2 + iR do not correspond to normalizable states of the model, they
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may be considered as well-defined but non-normalizable fields. It is tempting to identify the identity field with the limit lim→0 . This identification can indeed be confirmed by computing the corresponding limit of the coefficients C which is given by lim C(1 , 2 , 3 ) = 2(i(1 + 3 − Q)) + R(1 )2(i(1 − 3 )),
2 →0
where R() = ((b2 ))(Q−2)/b
b−2 (2b − b2 ) (2 − 2b−1 + b−2 )
(3.27)
for all 1,3 = Q/2 + iP1,3 with Pi ∈ R. The -functions again emerge from the singularities of C, just as in the minisuperspace example. We would like to point out that the reflection amplitude R() may also be obtained directly from the three-point coupling without ever performing a limit 2 → 0. In fact, in Liouville theory the labels =Q/2+iP and Q − = Q/2 − iP do not correspond to two independent fields. This is intuitively obvious because there exits only one asymptotic infinity so that wave functions are parametrized by the half-line P 0. Correspondingly, the three-point couplings C possess the following simple reflection property, C(1 , 2 , 3 ) = R(1 )C(Q − 1 , 2 , 3 ) with the same function R that we found in the two-point function. Similar relations hold for reflections in the other two momentum labels. This observation implies that the fields and Q− itself can be identified up to a multiplication with the reflection amplitude. As a preparation for later discussions we would finally like to point out that the expression (3.27) for our exact reflection amplitude may be rewritten in terms of its semi-classical analogue (3.9), −2i (1 + 2b2 i) (b2 ) R(Q/2 + ib) = R0 () . (1 − 2b2 i) b2
(3.28)
Here, we have also inserted the relation b2 = between the coupling constant of the minisuperspace theory and our parameter . The formula shows how finite b corrections to the semi-classical result are simply encoded in a new multiplicative factor. 4. Branes in the Liouville model In this lecture we will study branes in Liouville theory. It turns out that there exist two different types of branes. The first consists of branes that are localized in the strong coupling region and possess a discrete open string spectrum. Branes in the other class are 1D and extend all the way to x = −∞. These extended branes possess a continuous spectrum of open strings. 4.1. Localized (ZZ) branes in Liouville theory 4.1.1. The one-point coupling As we have explained in the first lecture, branes are uniquely characterized by the one-point couplings A of the bulk vertex operators . These couplings are strongly constrained by the cluster condition (2.18). Our aim therefore is to come up with solutions to this factorization constraint for the specific choice of the coefficients that is determined by the three-point coupling (3.25) of the Liouville bulk theory. In the most direct approach we would replace the labels (i, ¯™) and (j, ¯E) of the bulk fields by the parameters and , respectively, and then take the latter from the set Q/2 + iP that labels the normalizable states of the bulk model. But as we discussed above, this choice would leave us with a complicated integral equation in which we integrate over all possible closed string momenta ∈ Q/2 + iR. To avoid such an integral equation, we shall follow the same approach (“Teschner’s trick”) that we applied so successfully when we determined the three-point couplings in the bulk model: we shall evaluate the cluster property
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in two cases in which one of the bulk fields is degenerate, i.e. for = −b/2 and = −1/2b . An argument similar to the one explained in the last lecture then gives the constraint, b b A(−b/2) b + b − − 2 − b2 − 2 − b2 A() = A − c ()F− b ,0 c ()F+b/2,0 +A + (4.1) 2 A(0) 2 b 2 b and a second equation of the same type with b → b−1 . The functions cb± () are the same that appeared in Eq. (3.15) and we have computed them before (see Eq. (3.16)). Furthermore, the special elements of the Fusing matrix that appear in this version of the cluster condition are also easy to calculate (see Appendix B for explicit formulas). Note that all these quantities can be expressed in terms of ordinary functions. Once they are inserted, condition (4.1) becomes (−b2 ) b A − A()A(0)−1 (−1 − 2b2 ) 2 (2b − b2 ) b (2b − b2 − 1) b = A − − A + (4.2) (2b − 2b2 − 1) 2 (−b2 ) (2b) 2 and there is a dual equation with b → b−1 . Solutions to these equations were found in [39]. They are parametrized through two integers n, m = 1, 2, . . . and read as follows4 A(n,m) (Q/2 + iP ) =
sin bQ sin 2nP b sin b−1 Q sin 2mP b−1 A(1,1) (Q/2 + iP ) sin bnQ sin 2P b sin b−1 mQ sin 2P b−1
A(1,1) (Q/2 + iP ) = ((b2 ))−iP /b
21/4 4iP . (1 − 2iP b−1 )(1 − 2iP b)
(4.3)
According to our general discussion in the first lecture, we have thereby constructed branes in the Liouville model, though later we shall argue that only one of them, namely the one with (n, m) = (1, 1) is actually “physical”. The general algebraic procedure we used to determine the closed string couplings A to these branes remains so abstract that it is rather reassuring to discover that at least the branes with label (n, 1) possess a semi-classical limit and therefore some nice geometric interpretation. In fact, it is not difficult to see that these branes are localized in the strong coupling region of the model. We check this assertion by sending the parameter b to zero after rescaling the momentum P = b, A(n,1) ∼ −i −1 (−2i) = lim N (x0 ) (x0 ). b→0
x0 →∞
(4.4)
Here, the function on the right-hand side is the minisuperspace wave function (3.5) of a particle that moves in an exponential potential and N (x0 ) is an appropriate normalization that is independent of the momentum and that can be read off from the asymptotics of the Bessel function K (y) at large y. Note that the first term of such an expansion does not depend on the index . We see from Eq. (4.4) that the semi-classical coupling is entirely determined by the value of the semi-classical wave function at one point x0 in the strong coupling limit x0 → ∞ of the theory. This means that the branes with labels (n, 1) are point-like localized. It is not so surprising that point-like branes prefer to sit in the string coupling region of the model. Recall that the mass of a brane is proportional to the inverse of the string coupling. Since we are dealing with a linear dilaton background in which the string coupling grows exponentially from left to right, branes will tend to reduce their energy my moving into the region where the string coupling is largest, i.e. to the very far right. Let us also remark that the semi-classical coupling of closed strings is independent of the parameter n. This implies that we cannot interprete the (n, 1) branes as an n-fold super-position of (1, 1) branes. The absence of a good geometrical interpretation for the parameter n might seem a bit disturbing, but it is certainly not yet sufficient to discard solutions with n = 1 from the list of branes in Liouville theory. 4 Note that the cluster condition (4.2) does not determine the overall normalization A (n,m) (0) of the structure constants. Here, the normalization of the coefficients A(n,m) () is chosen to ensure consistency with the modular bootstrap below.
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As in our brief discussion of the reflection amplitude (3.28), we would like to split our expression for the coupling A(1,1) into a semi-classical factor A0(1,1) (see Eq. (4.4)) and its stringy correction, A(1,1) (Q/2 + ib) = NZZ (b)A0(1,1) ()
−i (b2 ) 1 . (1 − 2ib2 ) b2
(4.5)
Here, A0(1,1) () = −i −1 (−2i), the normalization factor N is independent of the momentum and of = b2 . 4.1.2. The open string spectrum We emphasized before that the closed string couplings A contain all the information about the corresponding branes. In particular, it should now be possible to determine the spectrum of open string modes that live on the ZZ branes. We shall achieve this in the same way that we sketched in the first lecture, using world-sheet duality (modular bootstrap). In view of our previous remarks and in order to simplify our task a bit, let us restrict to the case in which the boundary conditions on both sides of the cylinder are taken to be (n, m) = (1, 1), i.e. to open string excitations on a single (1, 1)-brane. On the closed string side, the amplitude reads ∞ Z(1,1) (q) = Z(1,1)(1,1) (q) = dP P (q) ˜ sinh 2P b sinh 2P b−1 , (4.6) 0
where P (q) = −1 (q)q P . 2
(4.7)
In order to calculate the right-hand side we had to evaluate the product A(1,1) (P )A∗(1,1) (P ) with the help of the standard relation (x)(1 − x) = / sin x. Now we can rewrite the partition sum using a simple trigonometric identity, ∞ Z(1,1) (q) = dP P (q)(cosh ˜ 2P Q − cosh 2P (b − b−1 )) 0
and then insert the usual formula for the modular transformation of the characters P , ∞ 3/2 P (q) = 2 dP cosh 4P P P (q). ˜ −∞
The final result of this short computation is Z(1,1) (q) = −1 (q)(q −Q
2 /4
− q −Q
2 /4+1
) = q − 24 (1 + q 2 + q 3 + · · ·). c
There are a few comments one can make about this answer. To begin with, the spectrum is clearly discrete, i.e. it does not involve any continuous open string momentum in target space. This is certainly consistent with our interpretation of the (1, 1)-brane as a localized object in the strong coupling region. We can appreciate another important feature of our result by contrasting it with the corresponding expression for point-like branes in a flat 1D background, free (q) = q −1/24 (1 + q + 2q 2 + 3q 3 . . .). ZD0
Here, the first term in the brackets signals the presence of an open string tachyon5 on the point-like brane while the second term corresponds to the massless scalar field. All higher terms are associated with massive modes in the spectrum of brane excitations. The presence of a scalar field on the point-like brane is directly linked to the modulus that describes the transverse position of such a brane in flat space. Note now that the corresponding term is missing in the spectrum on the ZZ brane. We conclude that our ZZ brane does not possess moduli, i.e. that it is pinned down at x0 → ∞, in perfect agreement with our geometric arguments above. 5 Recall that the space–time mass M of a string mode is related to the world-sheet conformal weight h of the corresponding vertex operator through M 2 = h − 1.
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}
63
g-1 s
Fig. 6. In the double scaling limit, the hermitian matrix model can be mapped to a system of non-interacting fermions moving through an inverse oscillator potential, one side of which has been filled up to a Fermi level at = − c ∼ gs−1 .
4.1.3. Application to 2D string theory Let us conclude this subsection on the ZZ branes with a few remarks on some recent applications. As we pointed out earlier, Liouville theory is a building block for 2D string theory. The latter is dual to the following model of matrix quantum mechanics, 1 SMQM ∼ − dt (jt M(t))2 + V (M(t)) , 2 where M(t) are hermitian N × N matrices and V is a cubic potential. To be more precise, the duality involves taking N and to infinity while keeping their ratio = N/ fixed and close to some critical value c . In this double scaling limit, the matrix model can be mapped to a system of non-interacting fermions moving through an inverse oscillator potential, one side of which has been filled up to a Fermi level at =−c ∼ gs−1 . With a quick glance at Fig. 6, we conclude that the model must be non-perturbatively unstable against tunneling of Fermions from the left to the right. This instability is reflected in the asymptotic expansion of the partition sum and even quantitative predictions for the mass m ∼ a/gs of the instantons were obtained. The general dependence of brane masses on the string coupling gs along with the specific form of the coupling (4.3) have been used recently to identify the instanton of matrix quantum mechanics with the localized brane in the Liouville model [40–43]. In this sense, branes had been seen through investigations of matrix quantum mechanics more than 10 years ago, i.e. long before their central role for string theory was fully appreciated. 4.2. Extended (FZZT) branes in Liouville theory 4.2.1. The one-point coupling In a flat 1D space one can impose Dirichlet and Neumann boundary conditions on X and thereby describe both point-like and extended branes. It is therefore natural to expect that Liouville theory admits extended brane solutions as well. When equipped with an exponential potential for the ends of open strings, the action of extended branes should take the form SBL [X] = SL [X] + du B ebX(u) . j
In the case of free field theory, we switch from point-like to extended branes by changing the way in which we glue left and right moving chiral currents J and J¯. But unlike for flat space, the symmetry of Liouville theory is generated by the Virasoro field alone and the gluing between left and right moving components of the stress-energy tensor is fixed to be trivial. Hence, we cannot switch between different brane geometries simply by changing a gluing map , as in flat space. At this point one may wonder whether our previous analysis has been complete and we should conclude that there are only localized branes. We will see in a moment, however, that this conclusion is incorrect and that at one point in our analysis of branes above we have implicitly assumed that they possessed a discrete open string spectrum.
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To understand the issue, let us review our usual derivation of the cluster condition. We should always start from the cluster condition for two fields within the spectrum of normalizable states in the theory, i.e. with , that belong to the set Q/2 + iR. As we have seen in the previous lecture, operator product expansions of such fields involve a continuum of modes. This applies not only to the modes that emerge when two closed strings collide, but also to the absorption of closed strings by a brane. More precisely, a closed string mode with label = Q/2 + iP can excite a continuum of open string states on the brane, provided such a continuum of states exists, i.e. that the brane is non-compact. In formal terms, the process is captured by the following bulk-boundary operator product expansion z→x dB(, ) (x). (4.8) (z, z¯ ) ∼ Q/2+iR
Here, with ∈ Q 2 + iR are open string vertex operators and the operator product coefficients B may depend on the choice of boundary condition. In evaluating all our factorization constraints, we try to avoid any integrals over intermediate closed or open string modes. This is why we employ “Teschner’s trick” (see lecture 2). It instructs us to analytically continue the variable to the values = −b±1 /2, i.e. to the labels of degenerate fields. Representation theory of the Virasoro algebra then ensures that only two open string vertex operators can occur on the right-hand side of the operator product expansion (4.8), namely the terms with = 0 and = −b±1 . Hence, we are tempted to conclude that the coefficients A(−b±1 /2) that appear on the left-hand side of the cluster condition are simply given by the unknown quantity B(−b±1 /2, 0). Such a conclusion, however, would be a bit too naive. As we shall argue in a moment, B(, 0) is actually singular at = −b±1 /2. In order to see this we have to understand how exactly the theory manages to pass from an expansion of form (4.8) to a discrete bulk-boundary operator product expansion at = −b±1 . In the following argument we consider B(, ) as a family of functions in the parameter . If there exist branes that extend all the way to the weak coupling region, the corresponding functions B () = B(, ) are expected to possess singularities, for the same reasons that e.g. the bulk 3-point function displays poles (see lecture 2). As we change the parameter to reach = −b±1 /2, the position of the poles of B will change and some of these poles can actually cross the contour of integration in Eq. (4.8), thereby producing discrete contributions on the right-hand side of the bulk-boundary operator product expansion. At generic points in the -plane, these discrete parts are accompanied by a continuous contribution. But when we reach the degenerate fields, the continuous parts must vanish and we remain with the two discrete terms that are consistent with the fusion of degenerate representations. There is one crucial observation we can take out of this discussion: The coefficients in front of the discrete fields in the bulk boundary operator product of degenerate bulk fields are not given through an evaluation of the function B at some special points but rather through the residues of B at certain poles. Since we understand the origin of such singularities as coming from the infinite region with weak interaction, we know that we can compute the coefficients using free field calculations. In the case at hand we find that ∞ i i res=−b/2 (B(, 0)) = − B ,− Q (∞)b (u) du −b/2 2 2 −∞ LD 2B (−1 − 2b2 ) . (4.9) = − 2 (−b2 ) It is this quantity rather than the value B(−b/2, 0) ∼ A(−b/2) that appears in the cluster condition for extended branes. In other words, we obtain the cluster condition for extended branes in Liouville theory by replacing the quantity A(−b/2) in Eq. (4.2) through the result of computation (4.9), 2B (2b − b2 ) b (2b − b2 − 1) b − A() = A − − A + . (4.10) (−b2 ) (2b − 2b2 − 1) 2 (−b2 ) (2b) 2 As usual, there is a dual equation with b → b−1 . Observe that the cluster conditions for extended branes are linear rather than quadratic in the desired couplings. Solutions to these equations were found in [44]. They are given by the following expression: As (Q/2 + iP ) = 21/4 ((b2 ))−iP /b
cos 2sP (1 + 2ib−1 P )(1 + 2ibP ), −2iP
(4.11)
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where √
cosh sb = B sin b2 .
65
(4.12)
Here we parametrize the couplings A through the new parameter s instead of the coupling constant B which appears in the boundary term of the action. Both parameters are related through Eq. (4.12). Let us again compare this answer to the semi-classical expectation. Evaluation of the coupling A in the limit b → 0 gives As (Q/2 + ib) ∼ A0s () := (/4)−i (2i) cos(2 ) ∞ x dxe−B e (P ). = b→0
(4.13)
−∞
Before taking the limit, we have rescaled the momentum P = b, the bulk (boundary) couplings b2 = (B b2 = B ) and the boundary parameter s = b−1 so that relation (4.12) becomes cosh2 = 2B . In the second line we have rewritten the answer6 to show that it reproduces the semi-classical limit for the coupling of closed strings to an extended brane with boundary potential VB (x) = B exp(x). Once more, we can split the coupling to FZZT branes into the semi-classical part A0s and a stringy correction factor, As (Q/2 + ib) = NFZZT A0s ()(1 + 2ib2 )
(b2 ) b2
−i .
(4.14)
As before, we have used that = b2 and we inserted = b−1 s. The quantity A0s has been defined in Eq. (4.13) above. Let us conclude this subsection with a short comment on the brane parameter s. It was introduced above as a convenient way to encode the dependence of A on the boundary coupling B . There is, however, much more one can say about the reparametrization of FZZT branes in terms of s. In particular, it is rather tempting to extend s beyond the real line and to allow for imaginary values. But in the complex plane, each value of B can be represented by infinitely many values of s and one may wonder about possible relations between branes whose s parameters differ by multiples of s = 2i/b. With the help of our explicit formulas it is not difficult to verify, e.g. that AiQ () = AiQ−2i/b () + A(1,1) ().
(4.15)
This interesting relation between FZZT and ZZ branes was first observed in [41] and then beautifully interpreted in the context of minimal string theory [47]. Geometrically, we may picture Eq. (4.15) as follows: Let us begin by considering large real values of s. These correspond to large values of B and hence to a brane whose density decreases fast toward the strong coupling regime, as can be seen from the semi-classical limit above. While we lower s, mass is moved further to the right. This process continues as we move along the imaginary axis and reach the value s = iQ. At this point, part of the brane’s mass is sucked into the strong coupling regime where it forms a ZZ brane. The shift in the parameter s to back to s = iQ − 2i/b (we assume b > 1) may be visualized as a retraction of the remaining extended brane. 4.2.2. The open string spectrum Our analysis of the open string spectrum on extended branes requires a few introductory remarks. For concreteness, let us consider a 1D quantum system with a positive potential V (x) which vanishes at x → −∞ and diverges as we approach x = ∞. Such a system has a continuous spectrum which is bounded from below by E = 0 and, under some mild assumptions, the set of possible eigenvalues does not depend at all on details of the potential V (see e.g. [48]). There is much more dynamical information stored in the reflection amplitude of the system, i.e. in the phase shift R(p) that plane waves undergo upon reflection at the potential V . R(p) is a functional of the potential which is very sensitive to small changes of V . In fact, it even encodes enough data to reconstruct the entire potential. From the reflection amplitude R(p) we can extract a spectral density function . To this end, let us regularize the system by placing a reflecting wall at x = −L, with large positive L. Later we will remove the cutoff L, i.e. send it to infinity. But as long as L is finite, our system has a discrete spectrum so that we can count the number of energy or 6 Use, e.g. formula 6.62 (3) of [45] to express the integral in the second line through a hypergeometric function. Then apply formula 15.1.19 of [46] to evaluate the latter.
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momentum levels in each interval of some fixed size and thereby we define a density of the spectrum. Its expansion around L = ∞ starts with the following two terms:
L (p) =
1 j L + ln R(p) + · · · 2i jp
(4.16)
where the first one diverges for L → ∞. This divergence is associated with the infinite region of large x in which the whole system approximates a free theory and consequently it is universal, i.e. independent of the potential V (x). The sub-leading term, however, is much more interesting. We can extract it from the regularized theory, e.g. by computing relative spectral densities before taking the limit L → ∞. It is not difficult to transfer these observations from quantum mechanics to the investigation of non-compact branes. We are thereby lead to expect that the annulus amplitude Z diverges for open strings stretching between two noncompact branes. This divergence, however, must be universal and there should also appear an interesting sub-leading contribution which is related to the phase shift that arises when open strings are reflected by the Liouville potential. All these features of the annulus amplitude can be confirmed by explicit computation. Once more, we try to compute the annulus amplitude from the couplings (4.11) of closed strings to extended branes. For real s, the result is7 ∞ ∞ cos 2sP Zss (q) = dP P (q) ˜ = dP ss (P )P (q), sinh 2P b sinh 2P b−1 −∞ −∞ where
ss (P ) =
∞ −∞
dt cos2 st . 2 sinh tb sinh tb−1
(4.17)
As we have predicted before, the spectral density ss (P ) diverges. The divergence arises from the double pole at t = 0 in the integral representation of ss . The coefficient of the double pole, and hence of the divergent term, does not depend on the boundary parameter s, i.e. it is universal. If we consider annulus amplitudes relative to some fixed reference brane with parameter s ∗ , however, we obtain an interesting finite answer and hence, according to our formula (4.16), a prediction for the reflection amplitude of open strings. The latter appears in the two-point function of open string vertex operators, 1 (u1 )2 (u2 )s ∼ (2(1 + 2 − Q) + (1 − 1 )R(1 |s))
1 |u1 − u2 |2h1
.
(4.18)
In order to turn our computation of the annulus amplitude into an independent test of the couplings (4.11), we are therefore left with the problem of finding an expression for the boundary two-point function, or, more generally, the couplings for open strings on extended branes in Liouville theory. Formulas for the two-point couplings have indeed been found using factorization constraints [44] and they are consistent with the modular bootstrap. Let us briefly mention that even general expressions for three-point couplings of open strings on extended Liouville branes are known [50]. The same is true for the exact bulk-boundary structure constants B (see Eq. (4.8)), both for extended branes [51] and for ZZ branes [52]. The methods that are used to obtain such additional data are essentially the same that we have used several times throughout our analysis. The interested reader is referred to the original literature (see also [53] for a very extensive list of references). While the modular bootstrap on non-compact branes alone does not lead to constraints on the one-point couplings As , at least not without further analysis of open string data (see remark after Eq. (4.18)), one may test our formulas (4.14) by studying the annulus amplitude for open strings that stretch between the discrete and extended branes of Liouville theory. We shall simply quote the final result of this straightforward computation, Z(m,n),s (q) =
m−1 n−1
(s+i(k/b+lb))/2 (q).
(4.19)
k=1−m l=1−n 7 When s is becomes complex, there exist issues with the convergence of integrals. These can ultimately lead to extra discrete contributions in the open string spectrum [49]. Note that such discrete terms are in perfect agreement with the observation (4.15).
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Here, denotes a summation in steps of two. We observe that the spectrum of open strings is discrete, just as one would expect for the setup we consider. But whenever (n, m) = (1, 1), we encounter complex exponents of q on the right-hand side. This is inconsistent with our interpretation of the quantity Z as a partition function and therefore it suggests that ZZ branes with (n, m) = (1, 1) are unphysical. Before we accept such a conclusion we might ask ourselves why the couplings with (n, m) = (1, 1) did show up when we solved the cluster condition (4.2). It turns out that there is a good reason. Let us observe that the coefficients of our cluster condition are analytic in b and there is formally no problem to continue these equations to arbitrary complex values of b. When b becomes purely imaginary, the central charge c assumes values c 1 which are realized in minimal models. For the latter, the existence of a two-parameter set of non-trivial and physical discrete branes is well established. These solutions had to show up in our analysis simply because the constraints we analyzed were analytic in b. But while such branes are consistent for c 1, there is no reason for them to remain so after continuation back to c 25. And indeed we have seen in the modular bootstrap that they are not! Needless to stress that the problem with complex exponents in Eq. (4.19) disappears for imaginary b. In the corresponding models with c 1 the brane parameters m and n also possess a nice geometric interpretation related to a position and extension along a 1D line [54]. When we try to continue back into Liouville theory, these parameters become imaginary. All this clearly supports our proposal to discard solutions with n = 1 or m = 1. 4.2.3. The c = 1 limit and tachyon condensation Before we conclude our discussion of boundary Liouville theory, we would like to briefly comment on its possible applications to the condensation of tachyons. Let us recall from our introductory remarks in the second lecture that we need to take the central charge to c = 1 or, equivalently, our parameter b to b = i,
1 c=1 2 SBL [X] = d zjX j¯ X + exp 2bX(z, z¯ ) + duB exp bX(u) . (4.20) 4 j b=i
Here we have allowed for an additional boundary term so as to capture the condensation of both open and closed string tachyons. It is important to keep in mind that any application of Liouville theory to time-dependent processes also requires a Wick rotation, i.e. we need to consider correlation functions with imaginary rather than real momenta P . We shall argue below that the two steps of this program, the limit c → 1 and the Wick rotation, meet quite significant technical difficulties. Nevertheless, there exists at least one quantity that we can compute easily from Liouville theory [29] and that we can even compare with results from a more direct calculation in the rolling tachyon background. It concerns the case in which merely open strings condense, i.e. in which = 0. Because of relation (4.12), switching off the bulk coupling while keeping the boundary coupling B fixed is equivalent to considering the limit s → ∞. The corresponding limit of the one-point coupling (4.11) is straightforward to compute and it is analytic in both b and P so that neither the continuation to b = i nor the Wick rotation pose any problem. The resulting expression for the one-point coupling is exp(iEX0 (z, z¯ )) ∼ ()iE
1 . sinh E
This answer from Liouville theory may be checked directly [55–57] through perturbative computations in free field theory [58–61]. Unfortunately, other quantities in the rolling tachyon background have a much more singular behavior at b = i. Barnes’ double -function, which appears as a building block for many couplings in Liouville theory (see e.g. Eq. (3.25)), is a well-defined analytic function as long as Re b = 0. If we send b → i, on the other hand, b becomes singular as one may infer, e.g. from the integral formula (3.19). In fact, the integrand has double poles along the integration contour whenever b becomes imaginary. A careful analysis reveals that the limit may still be well defined, but it is a distribution and not an analytic function. Rather than discussing any of the mathematical details of the limit procedure (see [62]) we would like to sketch a more physical argument that provides some insight into the origin of the problem and the structure of the solution. For simplicity, let us begin with the pure bulk theory. Recall from ordinary Liouville theory that it has a trivial dependence on the coupling constant . Since any changes in the coupling can be absorbed in a shift of the zero mode, one cannot vary the strength of the interaction. This feature of Liouville theory persists when the parameter b moves away from the real axis into the complex plane. As we reach the point b = i, our model seems to change quite drastically: at this
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point, the “Liouville wall” disappears and the potential becomes periodic. Standard intuition therefore suggests that the spectrum of closed string modes develops gaps at b = i. Since the strength of the interaction cannot be tuned in the bulk theory, the band gaps must be point-like. The emerging band gaps explain both the difficulties with the b = i limit and the non-analyticity of the resulting couplings. Though our argument here was based on properties of the classical action which we cannot fully trust, the point-like band-gaps are indeed a characteristic property of the c = 1 bulk theory. It was shown in [62] that the bulk three-point couplings of Liouville theory possess a b = i limit which is well-defined for real momenta of the participating closed strings. The resulting model turns out to coincide with the c = 1 limit of unitary minimal models which was constructed by Runkel and Watts in [63]. Since the couplings cease to be analytic in the momenta, the model cannot be Wick-rotated directly. Nevertheless, it is possible to construct the Lorentzian background. To this end, the Wick rotation is performed before sending the central charge to c = 1. The corresponding couplings with imaginary momenta were constructed in [62], correcting an earlier proposal of [64]. On the other hand, this Lorentzian c = 1 limit depends on the path along which b is sent to b = i. It is tempting to relate this non-uniqueness to a choice of boundary conditions at x0 = ∞ (see [65] a related minisuperspace toy model), but this issue certainly deserves further study. A similar investigation of the c = 1 boundary model (4.20) was recently carried out in [66] for Euclidean signature. The properties of this model are similar to the bulk case, only that the band gaps in the boundary spectrum can now have finite width. In the presence of a boundary, Liouville theory contains a second coupling constant B which controls the strength of an exponential interaction on the boundary of the world-sheet. B is a real parameter of the model since the freedom of shifting the zero mode can only be used to renormalize one of the couplings or B . Once more, the boundary potential becomes periodic at b = i and hence the open string spectrum develops gaps, as in the case of the bulk model. But this time, the width of these gaps can be tuned by changes of the parameter B . All these rather non-trivial properties were confirmed in [66] through an exact constructions of the spectrum and various couplings of these novel conformal field theories. So far, the Wick-rotated model has not been obtained from Liouville theory, thought there exist recent predictions for some of its structure constants [67] (see also [68–70] for related studies). 5. Strings in the semi-infinite cigar In the previous lectures we have analyzed Liouville theory mainly because it is the simplest non-trivial example of a model with non-compact target space. As we have reviewed in the introduction, however, many interesting applications of non-rational conformal field theory, in particular those that arise from the usual AdS/CFT correspondence, employ higher dimensional curved backgrounds such as AdS3 or AdS5 . The aim of this final lecture is to provide some overview over results in this direction. As we proceed, we shall start to appreciate how valuable the lessons are that we have learned from Liouville theory. Ultimately, one would certainly like to address strings moving in AdS5 . But unfortunately, this goes far beyond our present technology, mainly because consistency of the AdS5 background requires to turn on a RR 5-form field. The situation is somewhat better for AdS3 . In this case, consistency may be achieved by switching on a NSNS 3-form H . For reasons that we shall not explain here, such pure NSNS backgrounds are much easier to deal with in boundary conformal field theory. In cylindrical coordinates ( , , ) and with the string tension being fixed to 1/ = 2, the non-trivial background fields of this geometry read k ds 2 = (d 2 − cosh2 d 2 + sinh2 d2 ), 2 H=
k sinh 2 d ∧ d ∧ d . 2
(5.1) (5.2)
We wish to point out that these are the background fields of a WZW model on the universal covering space of the group manifold SL2 (R). In our cylindrical coordinates, background (5.1, 5.2) is manifestly invariant under shifts of the time coordinate . We can use this symmetry to pass to the 2D coset space SL2 (R)/U(1) (see e.g. [72–74] for details), k ds 2 = (d 2 + tanh2 d2 ), 2
(5.3)
exp = exp 0 cosh .
(5.4)
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θ
k 2
ρ
0 X Fig. 7. The cigar is parametrized by two coordinates ∈ [0, ∞] and ∈ [0, 2]. It comes equipped with a non-constant dilaton that vanishes at
= ∞ and assumes its largest value 0 at the tip = 0 of the cigar.
Obviously, the 2D coset space cannot carry any NSNS 3-form H . Instead, it comes equipped with a non-trivial dilaton field . The latter arises because the orbits of -translations possess different lengths. Long orbits at large values of
correspond to regions of small string coupling. Let us observe that the dilaton field becomes linear as we send
→ ∞. In this sense, the -coordinate of the coset geometry is similar to the Liouville direction. The cigar geometry also avoids the strong coupling problem of a linear dilaton background. Only the mechanism is somewhat different from the scenario in Liouville theory: On the cigar, the dilaton field itself is deformed so that the string coupling stays finite throughout the entire target space. The background (5.3, 5.4) and its Lorentzian counterpart were first described in [75,76,73] (Fig. 7). The close relation with AdS3 makes the coset SL2 (R)/U(1) an interesting background to consider. Additional motivations arise from the study of little string theory and the near horizon geometry of NS5-branes. If we place N such branes on top of each other, the near horizon geometry is given by 3 XNS5 R(1|5) × R+ Q × SN .
(5.5)
Here, the three factors are associated with the six directions along, the radial distance from and the three spheres surrounding the NS5-branes. The second factor stands for a linear dilaton with background charge Q2 = 1/N . The last factor represents a well-studied rational conformal field theory, namely the (N = 1 supersymmetric) SU(2) WZW model at level k = N − 2. Since the dilaton field in the background (5.5) is unbounded, string perturbation theory cannot be trusted. To overcome this issue, it has been suggested to separate the N NS5-branes and to place them along a circle in a 2D plane transverse to their world-volume [77]. Such a geometric picture is associated with a deformation of the corresponding world-sheet theory. In order to guess the appropriate deformation, we make use of the following well-known fact 3 SN = SU(2)N = (U(1)N × SU(2)N /U(1)N )/ZN ,
(5.6)
where U(1)N denotes a compactified free boson and we orbifold with a group ZN of simple currents. After inserting Eq. (5.6) into the NS5-brane background (5.5), we can combine the linear dilaton from the latter with the U(1)N of the three sphere. In this way the NS5 brane background is seen to involve a non-compact half-infinite cylinder R+ Q × U(1)N that we can deform into our cigar geometry, thereby resolving our strong coupling problem, (1|5) XNS5 −→ XNS5 × (SL2 (R)N /U(1)N × SU(2)N /U(1)N )/ZN , def = R
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with k = N + 2. Hence, the interest in NS5-branes provides strong motivation to investigate string theory on the cigar. Let us also note that there exists a rich and interesting class of compactifications of NS5-branes on Calabi–Yau spaces that, by the same line of reasoning, involve the cigar as a central building block [78] (see also [7]). We should finally note that these backgrounds certainly involve some amount of supersymmetry which we suppress in our discussion here. As in the case of the analogous compact coset theories, adding supersymmetry has relatively minor effects on the world-sheet theory. Since we are more interested in the qualitative features of our non-compact coset model, we shall neglect the corrections that supersymmetry brings about, even though they are certainly important in concrete applications. For treatments of the supersymmetric models, we can refer the reader to a number of interesting recent publications [79–89]. 5.1. Remarks on the bulk theory 5.1.1. The minisuperspace model As in the case of Liouville theory we can get some intuition into the spectrum of closed string modes and their couplings from the minisuperspace approximation. To this end, we are looking for eigenfunctions of the Laplacian on the cigar, = −
e−2
1 √
j e−2 det gg j
det g 2 2 = − [j + (coth + tanh )j + coth2 j2 ]. k
(5.7)
The -function normalizable eigenfunctions of this operator can be expressed in terms of hypergeometric functions through |n| 2 −j + |n| |n| 2 j n0 ( , ) = − ein sinh|n| F j + 1 + (5.8) , −j + , |n| + 1; −sinh2 , (|n| + 1)(−2j − 1) 2 2 where j ∈ −1/2 + iR describes the momentum along the -direction of the cigar and n ∈ Z is the angular momentum for rotations around the tip (shifts of ). For the associated eigenvalues one finds j
n0 = −
2j (j + 1) n2 + . k 2k j
(5.9)
j
In the symbols n0 and n0 we have inserted an index “0” which will be associated with a new stringy degree of freedom to be discussed below. Our motivation will become clear once we start to look into the spectrum of the bulk conformal field theory. Let us also stress that there are no L2 -normalizable eigenfunction of the Laplacian on the cigar. Such eigenfunctions would correspond to discrete states living near the tip, but in the minisuperspace approximation one finds exclusively continuous states which behave like plane waves at → ∞. From our explicit formula for wave functions and general properties of hypergeometric functions it is not hard to read off the reflection amplitude of the particle toy model. It is given by 2 −j + n j (2j + 1) ( , ) 2 . R0 (j, n) = −jn0−1 (5.10) = n 2 n0 ( , ) (−2j − 1) j + 1 + 2 One may also compute a particle analog of the three-point couplings. Since the result is very similar to the corresponding formula in Liouville theory, we do not present more details here. 5.1.2. The stringy corrections Instead, let us now try to guess how the results of the minisuperspace toy model get modified in the full quantum field theory. Each of the wave functions (5.8) in the minisuperspace theory lifts to a primary field in the conformal field theory. But the full story must be a bit more complicated. In fact, at → ∞, the cigar looks like an infinite cylinder
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and for the latter we know that primary fields are labeled by momentum n and winding w around the compact circle, in addition to the continuous momentum iP = j + 21 along the uncompactified direction. Hence, we expect that the j full conformal field theory on the cigar has primary fields nw (z, z¯ ) which are labeled by three quantum numbers j, n and w. The exact couplings for these fields are known. In particular, the bulk reflection amplitude is given by n + kw n − kw −j + −j + (2j + 1) 2 2 R(j, n, w) = n − kw n + kw (−2j − 1) j +1+ j +1+ 2 2 2 (2j + 1)) (1 + b × ((b2 )b2 )−2j −1 , (1 − b2 (2j + 1))
(5.11)
where we used b2 = 1/(k − 2). Even without any more detailed analysis of factorization constraints, this formula is rather easy to understand. Let us first note that, for the modes with zero winding number w, we have R(j, n, w = 0) = R0 (j, n)
(1 + b2 (2j + 1)) ((b2 )b2 )−2j −1 . (1 − b2 (2j + 1))
(5.12)
Here, we inserted the semi-classical reflection amplitude (5.10). Relation (5.12) should be compared to the analogous Eq. (3.28) in Liouville theory. Remarkably, the stringy correction factors in both formulas coincide if we identify 2j +1 with i, at least up to a simple renormalization of the bulk fields. Such an agreement should not come as a complete surprise since the reflection amplitude concerns the momentum in the non-compact direction of the cigar which—at
→ ∞—approximates a linear dilaton background, just as the region x → −∞ of Liouville theory. The dependence of the exact reflection amplitude on the winding number w is also rather easy to argue for. Since it is associated with the angular direction, one would expect that the parameter w enters through the same rules as in the theory of a single compactified boson, i.e. by using the replacement n → n ± kw. If we apply this prescription to the reflection amplitude R(j, n, w = 0) we obtain the correct formula (5.11). Note also that n enters the reflection amplitude only through the semi-classical factor R0 . Having gained some confidence into formula (5.11), we wish to look briefly at some properties of R. It is well known that bound states of a scattering potential cause singularities in the reflection amplitude. The converse, however, is not true, i.e. a reflection amplitude can possess singularities that are not linked to any bound states. Our reflection amplitude (5.11) has several different series of poles. Singularities in the w = 0 sector, i.e. of expression (5.12), are either associated with the semi-classical reflection amplitude (5.10) or the stringy correction factor. Since we did neither find bound states in the semi-classical model nor in Liouville theory, we are ready to discard these poles from our list of possible bound states in the cigar. The situation changes for w = 0. These sectors of winding strings did not exist in the semi-classical model and hence the corresponding poles of R could very well signal bound states that are localized near the tip of the cigar. We shall see later that this is indeed the case, at least for a subset thereof. The existence of stringy bound states near the tip of the cigar has been suspected for a long time [90], but the reasoning and the counting of these bound states stood on shaky grounds. A satisfactory derivation was only given a few years ago through a computation of the path integral for the partition function of the system. It turned out that, in addition to the expected continuous series with j ∈ − 21 + iR+ 0,
n ∈ Z, w ∈ Z,
there is also a discrete series of primary fields for with w, n ∈ Z such that |kw| > |n| and j ∈ Jdnw :=
1−k 1 1 1 ,− ∩ N − |kw| + |n| . 2 2 2 2
(5.13)
The correct list of bound states was found in [91] (see also [92] for a closely related results in the case of strings on SL2 (R)) and it differs slightly from the predictions in [90]. Later we shall confirm these findings quite beautifully
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through world-sheet duality involving open strings. Let us also point out that for w = 0, the discrete series is empty, in agreement with the minisuperspace analysis. The primary fields of these two series have conformal weights given by j
hnw = −
j (j + 1) (n + kw)2 + k−2 4k
and
j (j + 1) (n − kw)2 j h¯ nw = − + . k−2 4k
(5.14)
Notice that these conformal weights are all positive, as for any Euclidean unitary conformal field theory. In the limit j j of large level k, the sum hn0 + h¯ n0 of the left and right conformal weights with w = 0 reproduces spectrum (5.9) of the minisuperspace Laplacian. 5.2. From branes to bulk—D0 branes In the first lecture we have outlined a general procedure that allows us to construct closed and open string backgrounds. The first step of this general recipe was to find a consistent spectrum of closed string modes and their three-point couplings C. Consistency required that the latter obey the crossing symmetry condition (2.9). In our second step we searched for possible D-branes by solving the cluster condition (2.18, 2.19) for the couplings A of closed string modes to the brane. Finally, we showed how to recover the spectrum of open string modes from the couplings A through the so-called modular bootstrap (see Eq. (2.26)). Looking back at the central equations (2.9, 2.18, 2.19, 2.26) of our program one may have the idea to reverse the entire procedure and to start from the end, i.e. from the spectrum of open string modes on some brane. Suppose we were able to guess somehow an annulus amplitude Z. Then we could try to recover the couplings A from it through formula (2.26). If this was successful, it could also teach us about possible closed string modes since Eq. (2.26) involves a sum (or integral) over closed string modes from the spectrum of the model. Moreover, using Eqs. (2.18, 2.19), we could even recover closed string couplings C, provided we knew the fusing matrix of the chiral algebra. In this simple form, the reversed program we have sketched here has many problems and we shall comment on some of them later on. Nevertheless, there exist a few good reasons to believe that ultimately some refined version of this procedure could be quite successful. In fact, knowing the entire open string spectrum on some brane, including all the massive modes, provides us with a lot of information not only on small fluctuations of open string, and hence on the dimension of the background, but also, e.g. on the non-trivial cycles through the associated open string winding modes. We can illustrate some of these very general remarks with two rather simple examples before we come back to the study of the 2D cigar geometry. Let us begin with a point-like (D0) brane in a flat 1D target space R. Its boundary partition function is given by ∞ 2 q˜ p R −1 ZD0 (q) = (q) = dp . (q) ˜ −∞ The -function on the left hand contains all the oscillation modes of open strings on our point-like brane and there are no zero modes. When transformed to the closed string picture we recover a continuum of modes that are parametrized by the momentum p along with the usual tower of string oscillations. Hence, we have recovered all the closed string modes that exits in a 1D flat space. Obviously, this example is quite trivial and it may not be sufficient to support our reversed program. But after some small modification, our analysis starts to look a bit more interesting. In fact, let us now consider a point-like (D0) brane that sits at the singular point of the half-line R/Z2 . Here, the non-trivial element of Z2 acts by reflection x → −x at the origin x = 0. It is rather easy to find the annulus amplitude for such a brane. All we have to do is to count the number of states that are created by an even number of bosonic oscillators an from the ground state |0. The result of this simple combinatorial problem is given by 2 1 (2m)2 1 R/Z (q − q (2m+1) ) = (1 + ϑ4 (q 2 )). ZD0 2 = (q) 2(q) m0
Here, ϑ4 is one of Jacobi’s ϑ-functions. After modular transformation to the closed string picture, we obtain ∞ 2 q˜ 1/48 q˜ p 1 R/Z . dp +√ ZD0 2 = (q) ˜ 2 n 1 (1 − q˜ n−1/2 ) 0
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The interpretation of the first term is essentially the same as in our first example. It arises from closed string modes that move along the half-line. Of course, the origin of the second term is also well known: It comes with closed string states from the twisted sector that is characteristic for any orbifold geometry. In this sense, the modular bootstrap has predicted the existence of a twisted closed string sector. Even though our second example is still rather trivial, we now begin to grasp that sometimes it may be much easier to guess an annulus amplitude for open strings on a brane than the spectrum of closed strings. In a moment we shall see this idea at work is a much less trivial case: For open strings on a point-like brane in our cigar background the whole issue of bound states (see our discussion at the end Section 5.1) does not arise simply because there is no continuous open string spectrum. Hence, it should be rather straightforward to come up with a good Ansatz for an open string partition function Z on such branes. According to our general ideas, this Z contains information on the bulk spectrum, in particular on the predicted closed string bound states, that we are able to decipher through modular transformation. 5.2.1. The inverse procedure—some preparation We have argued above that the reconstruction of an entire model from no more than the theory of open strings on a single brane has quite realistic chances. At least for rational models (i.e. for compact targets) the program can be made much more precise [93,94] (see also [95–98]). The input it requires is two-fold. First of all, we need complete knowledge about the symmetry algebra and its representation theory, including the fusion rules, the modular S-matrix and the Fusing matrix. Throughout this text we have always assumed that this information is provided somehow. The second important input into the reconstruction program consists of data on the open strings. What we need to know is the spectrum of open strings on the brane and the operator products of the associated open string vertex operators. The latter is often uniquely fixed by the former. In this subsection we would like to address the first part of the input by listing relevant facts on the representation theory of the symmetry algebra of cigar background. It will suffice to list its unitary representations and explicit formulas for the corresponding characters. The chiral algebra in question is the so-called coset algebra SL2 (R)/U(1). Its description is a bit indirect through the SL2 (R) current algebra. The latter is generated by the modes of three chiral currents J a (z), a = 0, 1, 2, with the standard relations, c [Jna , Jmb ] = f ab c Jn+m − 21 knab n,−m ,
where f denotes the structure constants and of the Lie algebra sl2 . Each of the currents J a by itself generates a U(1) current algebra. We select one of these U(1) current algebras, say the one that is generated by J 0 , and then form the coset chiral algebra from all the fields that may be constructed out of SL2 (R) currents and that commute with Jn0 . Note that this algebra does not contain fields of dimension h = 1 since there is no SL2 (R) current whose modes commute with all Jn0 . But there exists one field of dimension h = 2, namely the Virasoro field T SL/U (z) = T SL (z) − T U (z), where the two Virasoro fields on the right-hand side denote the Sugawara bilinears in the SL2 (R) and the U(1) current algebra, respectively. It is not difficult to check that the field T cos commutes with Jn0 and that it generates a Virasoro algebra with central charge c = cSL/U = cSL − cU =
3k − 1. k−2
Representations of the coset chiral algebra can be obtained through decomposition from representations of the SL2 (R) current algebra. The latter has three different types of irreducible unitary representations. Representation spaces Vc(j,) of the continuous series representations are labeled by a spin j ∈ Q/2 + iR and a real parameter ∈ [0, 1[. For the discrete series, we denote the representation spaces by Vj d with j real. Finally, the vacuum representation is V00 . It is the only unitary representation with a finite dimensional space of ground states. From each of these representations we may prepare an infinite number of SL2 (R)/U(1) representations. Vectors in the corresponding representation spaces possess a fixed U(1) charge, i.e. the same eigenvalues of the zero mode J00 , and they are annihilated by all the modes Jn0 with n > 0. With the help of this simple characterization one can work out
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formulas for the characters of all the sectors of the coset chiral algebra. Here we only reproduce the results (see e.g. [99] for a derivation). For the continuous series, the chiral characters read as follows c(j,) (q) = Tr (j,) q L0 −c/24
=
q−
(j + 21 )2 k −2
2
+ k
(q)2
,
(5.15)
where is some real number parametrizing the SL2 (R)/U(1) chiral algebra representations descending from a continuous representation of SL2 (R) with spin j = − 21 + iP . In the case of the discrete series, the expressions for chiral characters are a bit more complicated, d(j,−j ) (q) = where
=
1 −1
q
−
(j + 21 )2 k −2
+
(j −)2 k
(q)2
∞ s=0
s (1/2)s(s+2|+ 21 |)
(−1) q
− 1 − , 2
(5.16)
if 0, if − 1
and l is an integer that labels different SL2 (R)/U(1) sectors that derive from one of the two discrete series representation of SL2 (R) with spin j . Finally, we obtain one series of sectors from the vacuum representation of the SL2 (R) current algebra. Its characters are given by 0(0,n) (q) = d(0,n) (q) − d(−1,n+1) (q).
(5.17)
The integer n ∈ Z that parametrizes sectors in this series refers to the U(1) charge of the corresponding states in the j = 0 sector of the parent SL2 (R) theory. 5.2.2. D0-branes in the 2D cigar Our first aim is to construct the annulus amplitude of some brane in the cigar. Geometrical intuition suggests that there should exist a point-like brane at the tip of the cigar, i.e. at the point where the string coupling assumes its largest value. It is not too hard to guess the annulus amplitude for such a brane. To this end, we recall that our cigar background may be considered as a coset SL2 (R)/U(1). Open string modes on a point-like brane in SL2 (R) are rather easy to count. They are generated from a ground state |0 by the modes Jna , a = 0, 1, 2, SL n1 ...ns a1 as HD0 = a1 ...as Jn1 . . . Jns |0|nr < 0 . s ai ,ni
States | of open strings in the coset geometry form a subspace of HD0 which may be characterized by the condition Jn0 | = 0
for all n1.
U It is in principle straightforward to count the number of solutions to these equations for each eigenvalue of L0 =LSL 0 −L0 . This counting problems leads directly to the following partition sum for open strings on our point-like brane (we shall explain the subscript 1, 1 at the end of this section) SL (R)/U(1)
Z1,12
∞ (q) = q −c/24 1 + 2q 1+1/k + q 2 + · · · = 0(0,n) (q).
(5.18)
n=−∞
The first few terms which we displayed explicitly are very easy to check. The full annulus amplitude is composed from an infinite sum of characters rather than the vacuum character 0(0,0) of our coset chiral algebra. This may seem a bit unusual at first, but there are good reasons for such a behavior. In fact, when we expand the character 0(0,0) (q) in powers of q we find the following first few terms, 0(0,0) (q) = q −c/24 (1 + q 2 + · · ·).
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Hence, we are, e.g. missing the term 2q 1+1/k from the correct answer in Eq. (5.18). But a quick look at the geometry tells us that such a term must be present in the annulus amplitude we consider. Recall that point-like branes in a flat 2D space possess two moduli of transverse displacement. This means that the annulus amplitude for such branes contains a term 2q which is indeed present in the flat space limit k → ∞ of our annulus amplitude (5.18), but not in 0(0,0) . Transverse displacements cease to be moduli of the curved space model because of the varying string coupling which pins the D0 brane down at the tip of the cigar. Correspondingly, the exponent of the second term in the expansion of Z must be deformed at finite k. Our term 2q 1+1/k in the proposed annulus amplitude has exactly the expected properties. It can only come from the characters (0,±1) (q). But once we have added these two characters to (0,0) , closure of the space of open string modes with respect to fusion forces us to sum over all the U(1) charges n ∈ Z. Our argument here provides very strong evidence for the annulus amplitude (5.18) in addition to the derivation we gave above. Our next aim is to modular transform expression (5.18). The result of a short and rather direct computation is given by i 2 k 2 2 tanh P SL2 (R)/U(1) n b (P − 2b2 n) + 4 n 2 (q) = −2 b k dP sinh 2b2 P (−1) q ˜ . (5.19) Z1,1 (q) ˜ 2 n∈Z On the right-hand side we have some series containing powers of the parameter q, ˜ but the exponents are complex. This spoils a direct interpretation of these exponents as energies of closed string states. To cure the problem, we exchange the summation of n with the integration over P and we substitute P by the new variable Pn = P −
i n 2b2
in each of the summands. Pn is integrated along the line R − in/2b2 . The crucial idea now is to shift all the different integration contours back to the real line. This will give contributions associated with the continuous part of the boundary state. But while we shift the contours, we pick up residues from the singularities. The latter lead to terms associated with the discrete series. To work out the details, we split the partition function into a continuous and a discrete piece, c d (q) + Z1,1 (q). Z1,1 (q) = Z1,1
Note that this split is defined with respect to the closed string modes. In terms of open string modes, our partition function contains only discrete contributions. According to our description above, the continuous part of the partition function reads as follows: q˜ b2 P 2 + k4 w2 sinh 2b2 P sinh 2P c Z1,1 . (5.20) (q) = −2 b2 k dP (q) ˜ 2 cosh 2P + cos kw w∈Z
In writing this formula we have renamed the summation index from n to w. It is not difficult to see that this part of our partition function can be expressed as follows c c(j,kw/2) (q)A ˜ 1 (j, n, w)A1 (j, n, w)∗ Z1,1 (q) = dP w∈Z
where
kw kw 2 1/4 −j + −j − j +1 √ sin b2 kb 2 2 (1 + b2 )b . A1 (j, n, w) = n,0 w 2 2 (−1) (−2j − 1) 2(1 − b (2j + 1))
(5.21)
We claim that the coefficients A1 that we have introduced here are the couplings of closed strings to the point-like brane on the cigar. In the last step of our short computation, however, we had to take the square root of the coefficients that appear in front of the characters. The choice we made here is difficult to justify as long as we try to stay within the modular bootstrap of the cigar (this was also remarked in [80]). But once we allow our experience from Liouville theory to enter, the remaining freedom is essentially removed.
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In order to present the argument, we compare the proposed couplings (5.21) with their semi-classical analogue. In the j minisuperspace model we only have wave functions n0 corresponding to closed string modes with vanishing winding number w = 0. For these particular fields, the semi-classical limit of our couplings reads j
( n0 D0 )k→∞ = −
(−j )2 j n,0 = n0 ( = 0), (−2j − 1)
(5.22)
j
i.e. it is given by the value of the function n0 at the point = 0, in perfect agreement with our geometric picture. Note that due to the rotational symmetry of the point-like brane, only the modes with angular momentum n = 0 have a non-vanishing coupling.The semi-classical result suggest to introduce kw kw −j + −j − 2 2 n,0 A01 (j, n, w) := −(−1)w (−2j − 1) and to rewrite the exact answer in the form A1 (j, n, w) = ND0 A01 (j, n, w)
((b2 )b2 )−j −1/2 , (1 − b2 (2j + 1))
(5.23)
where we have combined several momentum independent factors into a constant ND0 . Hence, the stringy improvement factor is exactly the same as for ZZ branes in Liouville theory. As in our discussion of the bulk reflection amplitude, we have incorporated the rather simple string effects (winding) for the -direction into the definition of the “semi-classical” amplitude A0 . Formula (5.23) strongly supports our expression (5.21) for the couplings A1 , and in particular the choice of square root we have made. After this first success we now turn to the calculation of the discrete piece of the amplitude. It is clear from (5.19) that the residues we pick up while shifting the contours give the following discrete contribution to the partition function ∞
1 E( k−2 2 n− 2 )
n=1
m=0
k 2 1 d (−1)n q˜ 4 n Z1,1 (q) = −2 b2 k 2
2(m + 21 ) sin k−2
1
1
q˜ − k−2 (m+ 2 − (q) ˜ 2
k−2 2 2 n)
.
(5.24)
Here, E(x) denotes the integer part of x. A careful study of the energies which appear in this partition sum shows that the contributing states can be mapped to discrete closed string states with zero momentum. The latter are parametrized by their winding number w and by their spin j or, equivalently, by the label = kw/2 + j ∈ Z, and the level number s as in the character (5.16). The map between the parameters (w, , s) and the summation indices (m, n) of formula (5.24) is m = − w,
n = s + w.
(5.25)
It can easily be inverted to compute the labels (w, , s) in terms of (m, n), 2m + 1 2m + 1 2m + 1 , =m+E , s=n−E . w=E k−2 k−2 k−2 In terms of j and w, the partition function (5.24) may now be rewritten as follows: (2j + 1) d 2 sin ˜ (j, kw ) (q). Z1,1 (q) = −2 b k 2 k−2 d w∈Z j ∈J
(5.26)
(5.27)
0w
It remains to verify that the coefficients of the characters coincide with those derived from the boundary state (5.21). In our normalization, the boundary coefficients A1 (j, n, w) are given through the same expression (5.21) as for the continuous series but we have to divide each term in the annulus amplitude by the non-trivial value of the bulk two-point
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function of discrete closed string states, i.e. Zd1,1 (q) =
A1 (j, n = 0, w)A1 (j, n = 0, w)∗ w∈Z j ∈Jd
0w
=
Resx=j
w∈Z j ∈Jd
j j 0w 0w
˜ d(j, kw ) (q),
A1 (x, 0, w)A1 (x, 0, w)∗ R(x, n = 0, w)
(5.28)
2
˜ d(x, kw ) (q).
(5.29)
2
0w
The second line provides a more precise version of what we mean in the first line using the reflection amplitude (5.11) rather than the two-point function. Recall that the bulk two-point correlator contains a -function which arises because of the infinite volume divergence. If we drop this -function in the denominator by passing to the reflection amplitude, the quotient has poles and the physical quantities are to be read off from their residues. A short explicit computation shows that the argument of the Res-operation in Eq. (5.29) indeed has simple poles at x ∈ Jd0w and that the residues agree exactly with the coefficients in formula (5.27), just as required by world-sheet duality. Our calculation therefore provides clear evidence for the existence of closed string bound states, as we have anticipated. They are labeled by elements of the set Jd , in agreement with the findings of [91]. Our input, however, was no more than a well motivated and simple Ansatz (5.18) for the partition function of a D0 brane at the tip of the cigar. Before we conclude, we would like to add a few comments on further localized branes in the cigar geometry. Recall that the ZZ branes in Liouville theory were labeled by two discrete parameters n, m1, though we have argued that branes with (n, m) = (1, 1) do in some sense not belong into Liouville theory. In complete analogy, one can find a discrete family of point-like branes on the cigar which is parametrized by one integer n1. The form of the corresponding couplings An and their annulus amplitudes Zn,n can be found in [99]. These branes descent from a similar discrete family of compact branes in the Euclidean AdS3 (see [100]). The latter have been interpreted as objects which are localized along spheres with an imaginary radius. We have argued in [99] that localized branes with n = 1 are unphysical in the cigar background. 5.3. D1 and D2 branes in the cigar Besides the compact branes that we have studied in great detail in the last section, there exist two families of noncompact branes on the cigar. One of them consists of branes which are localized along lines (D1-branes), members of the other family are volume filling (D2-branes). We would like to discuss briefly at least some of their properties. 5.3.1. D1-branes in the 2D cigar D1-branes in the cigar are most easily studied using a new coordinate u = sinh along with the usual angle . We have u0 and u = 0 corresponds to the point at the tip of the cigar. In the new coordinate system, the background fields read ds 2 =
k du2 + u2 d2 , 2 1 + u2
e =
e0 (1 + u2 )1/2
.
When we insert these background data into the Born-Infeld action for 1D branes we obtain SBI ∝ dy u2 + u2 2 ,
(5.30)
(5.31)
where the primes denote derivatives with respect to the world-volume coordinate y on the D1-brane. It is now easy to read off that D1-branes are straight lines in the plane (u = sinh , ). These are parametrized by two quantities, one being their slope, the other the transverse distance from the origin. In our original coordinates , , this two-parameter family of 1D branes is characterized by the equations sinh sin( − 0 ) = sinh r.
(5.32)
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θ
ρ
θ u
x
0
x
Fig. 8. D1-branes on the cigar extend all the way to two opposite points on the circle at = ∞. The position of these points is parametrized by 0 . In the -direction they cover all values r down to some parameter r.
Note that the brane passes through the tip if we fix the parameter r to r = 0. All branes reach the circle at infinity ( = ∞) at two opposite points. The positions 0 and 0 + of the latter depend on the second parameter 0 (see Fig. 8). Now that we have some idea about the surfaces along which our branes are localized we can calculate their coupling to closed string modes in the semi-classical limit. This is done by integrating the minisuperspace wave functions (5.8) of closed string modes over the 1D surfaces (5.32). The result of this straightforward computation provides a prediction for the semi-classical limit of the exact one-point couplings AD1 , j
0;D1 ( nw D1 r )k→∞ = A(r, ) (j, n, w) 0
= w,0 ein0
(2j + 1) −r(2j +1) + (−1)n er(2j +1) ). n n (e 1+j + 1+j − 2 2
(5.33)
The minisuperspace model of the cigar does not include any states associated with closed string modes of non-vanishing winding number. But in the case of the D1-branes, experience from the analysis of branes on a 1D infinite cylinder teaches us that closed string modes with w = 0 do not couple at all. Since the discrete closed string modes only appear at w = 0, they are likewise not expected to couple to the D1-branes. Consequently, our formula (5.33) predicts the semi-classical limit of all non-vanishing couplings in the theory. The exact one-point couplings of the D1 branes are rather straightforward to extrapolate from the semi-classical result (5.33) and formula (4.14) for FZZT branes in Liouville theory. We claim that the exact solution is parametrized by two continuous parameters r and 0 , just as in the semi-classical limit, and that the associated couplings are given by 0;D1 2 2 2 −j −1/2 AD1 . (r,0 ) (j, n, w) = ND1 A(r, ) (j, n, w)(1 + b (2j + 1))((b )b ) 0
(5.34)
These couplings were first proposed in [100] where one can also find an explicit formula for the constant ND1 . Their consistency with world-sheet duality was established in [99] (see also [101]). Let us also remark that our D1-branes are very close relatives of the so-called hairpin brane [102]. The latter is a curved brane in a flat 2D target. It is localized along two parallel lines at infinity and then bends away from these lines into a smooth curve (semi-infinite hairpin). Brane dynamics attempts to straighten all curved branes. But the shape of the hairpin brane is chosen in such a way that it stays invariant so that dynamical effects on the brane merely cause a rigid translation of the entire brane. Such rigid translations of the background can be compensated by introducing a linear dilaton. Hence, we can alternatively think of the hairpin brane as a 1D brane which is pending between two points at infinity, bending deeply into the 2D plane in order to reduce its mass. This alternative description of the hairpin brane shows the close relation with our D1 branes. Needless to say that the structure of the boundary states is essentially identical. Following [103], a Lorentzian version of this boundary state has been studied to describe a time-dependent process in which a D-brane falls into an NS5 branes (see e.g. [104–107]).
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5.3.2. D2-branes in the 2D cigar The main new feature that distinguishes the D2-branes from the branes we have discussed above is that they can carry a world-volume two-form gauge field F = F d ∧ d. In the presence of the latter, the Born-Infeld action for a D2-brane on the cigar becomes 2 . SBI ∝ d d cosh tanh2 + F (5.35) We shall choose a gauge in which the component A of the gauge field vanishes so that we can write F = j A . A short computation shows that the equation of motion for the one-form gauge potential A is equivalent to 2 F =
2 tanh2
cosh2 − 2
.
(5.36)
If the integration constant is greater than one, then the D2-brane is localized in the region cosh , i.e. it does not reach the tip of the cigar. We will exclude this case in our semi-classical discussion and assume that = sin 1. The corresponding D2-branes cover the whole cigar. When the parameter tends to = /2, the F -field on the brane blows up. We should thus consider = /2 as a physical bound for . Let us point out that the F -field we found here vanishes at = ∞. In other words, it is concentrated near the tip of the cigar. By the usual arguments, the presence of a non-vanishing F -field implies that our D2-branes carry a D0-brane charge which is given by the integral of the F -field. Like the F -field itself, the D0-brane charge is localized near the tip of the cigar, i.e. in a compact subset of the 2D background. Hence, one expects the D0-brane charge, and therefore the parameter of the D2-brane, to be quantized. We shall explain below how such a quantization of the brane parameter emerges from the conformal field theory treatment of D2 branes. The exact one point couplings for these D2 branes were first proposed in [99]. They are parametrized by a quantity ∈ 0, (1 + b2 ) . (5.37) 2 Note that this interval shrinks to its semi-classical analogue as we send b to zero. For the associated one-point functions of closed string modes we found 0;D2 AD2 + b2 (2j + 1))((b2 )b2 )−j −1/2 , (j, n, w) = ND2 A (j, n, w)(1 ⎞ ⎛ kw kw −j + −j − ⎟ ⎜ 2 2 ⎜ ei(2j +1) + e−i(2j +1) ⎟ A0;D2 (j, n, w) = n,0 (2j + 1) ⎝ ⎠ kw kw j +1+ j +1− 2 2
(5.38)
with a constant ND2 that may be found in [99]. The semi-classical coupling A0 can be derived from our geometric description of the D2 branes. In the full field theory, it receives the same correction as the FZZT branes in the Liouville model. Formula (5.38) holds for closed string modes from the continuous series. It also encodes all information about the couplings of discrete modes, but they have to be read off carefully because of the infinite factors (see the discussion in the case of D0-branes). Following our usual steps, we could now compute the spectrum of open strings that stretch between two D2 branes with parameters and . We do not want to present the calculation here (it can be found in [99]). But some qualitative aspects are quite interesting. Our experience with closed string modes on the cigar suggests that open strings on D2 branes could possess bound states near the tip of the cigar. World-sheet duality confirms this expectation. In other words, the annulus amplitudes computed from Eq. (5.38) contain both continuous and discrete contributions. While the continuous parts involve some complicated spectral density, the discrete parts must be expressible as a sum of coset characters with integer coefficients. The latter will depend smoothly on the choice of branes. Integrality of the coefficients then provides a condition on the brane labels and which reads − = 2
m , k−2
m ∈ Z.
(5.39)
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This is the quantization of the brane label that we have argued for above. In addition, the computation of the partition function Z also shows that the density of continuous open string states diverges when ( + )/2 reaches the upper bound /2(1 + b2 ). The classical version of this bound on also appeared in our discussion of the geometry. This concludes our discussion of branes in the cigar geometry. Let us mention that a few additional boundary states have been suggested in the literature, including a 2D non-compact brane which does not reach the tip of the cigar [86] and various compact branes including some with non-vanishing couplings to closed string modes of momentum n = 0 [85,109]. Even in the absence of a fully satisfactory conformal field theory analysis8 it seems very plausible that such branes do indeed exist. In the case of the D2 branes, evidence comes simply from a semi-classical treatment and from the Hosomichi’s bootstrap analysis of the N = 2 Liouville theory [87]. For the additional point-like branes, the minisuperspace approximation is insufficient. On the other hand, certain ring-shaped branes in flat space are known to collapse into point-like objects which cannot be identified with D0 branes [108] if the collapsing brane comes equipped with a non-trivial Wilson line. There is no reason to doubt that similar processes can appear on the cigar. Furthermore, there exists a dual matrix model [110] with a non-perturbative instability that does not seem to be associated with the D0 branes we studied above, thereby also pointing towards the existence of new localized brane solutions. 6. Conclusions and outlook In these notes we have explained the main techniques that are involved in solving non-rational conformal field theories. At least in our discussion of Liouville theory we have tried to follow as closely as possible the usual conformal bootstrap that was developed mainly in the context of rational conformal field theory. Among the new features we highlighted Teschner’s trick and the use of free field computations. Let us recall that Teschner’s trick exploited the existence of so-called degenerate fields that are not part of the physical spectrum but can be obtained by analytic continuation. It turned out that the resulting bootstrap equations were sufficiently restrictive to determine the solution uniquely. Let us also recall that these equations are typically linear (“shift equations”) in the couplings of physical fields (see e.g. Eqs. (3.14), (4.10)). The coefficients of such special bootstrap equations involve couplings of degenerate fields which we have been able to calculate through free field computations. Intuitively, we understood the relevance of an associated free field theory (the linear dilaton in the case of Liouville theory) from the fact that the interaction of the investigated models is falling off at infinity. Let us mention that one can avoid all free field calculations and obtain the required couplings of degenerate fields through a “degenerate bootstrap”. Since degenerate fields in a non-rational model behave very much like fields of a rational theory, a bootstrap for couplings of degenerate fields is similar to the usual bootstrap in rational models (see [31,118] for more details). In the final lecture, we have attempted to reverse the usual bootstrap procedure and to place the modular bootstrap in the center of the programme. Even though it remains to be seen whether such an approach can be developed into a systematic technique for solving non-rational models, its application to the cigar conformal field theory was quite successful. Nevertheless, it is important to keep in mind that our success heavily relied on our experience with Liouville theory. Actually, Liouville theory models the radial direction of the cigar background so perfectly that only the semiclassical factors in the various couplings have to be replaced when passing to the cigar.9 Once the cigar conformal field theory is well understood, it is not difficult to lift the results to the group manifold SL2 (R) or its Euclidean counterpart H3+ SL2 (C)/SU(2). Historically, the latter was addressed more directly, following step by step the bootstrap program we carried out for Liouville theory in the second and third lecture. Minisuperspace computations for the various bulk and boundary spectra and couplings can be found at several places in the literature [112–117,100]. The shift equations for the bulk bootstrap were derived and solved in [118,119] building on prior work on the spectrum of the theory [112]. Full consistency has been established through an interesting relation of the bulk correlators with those of Liouville theory [111] (see also [120] for an earlier and more technical proof). As indicated above, Teschner did not use any free field computations and relied entirely on bootstrap procedures. Nevertheless, it is certainly possible to employ free field techniques (see [121,122]). The boundary bootstrap was carried out in [100] 8 Such an analysis might require the use of factorization constraints similar to the ones we discussed in the context of Liouville theory (see [100,87] for some steps in this direction). 9 For a more precise formulation of the relation between Liouville theory and the Euclidean AdS background see also the recent papers [111,109] 3 and references therein.
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(see also [123] for a previous attempt and [124] for a partial discussion). The Wick rotation from the Euclidean to the Lorentzian bulk model was worked out in [131] (see also [125–128] for free field computations) and aspects of the boundary theory were addressed more recently in [132]. Even though the solution of Liouville theory and closely related models has certainly been a major success in non-rational model building, there remain many challenging problems to address. In these lectures, we have used potential applications to AdS/CFT like dualities as our main motivation. In spite of the progress we have described, the constructing string theory on AdS5 still appears as a rather distant goal for now. Note that the string equations of motions require a non-vanishing RR background when we are dealing with the metric of AdS5 . Switching on RR fields tends to reduce the chiral symmetry algebras of the involved world-sheet theories [133] and hence it makes such backgrounds very difficult to tackle. In this context, the example of AdS3 might turn out to provide an interesting intermediate step. All the developments we sketched in the previous paragraph concern the special case in which the string equations of motion are satisfied through a non-vanishing NSNS three-form H . Beyond this point, there exists a whole family of models with non-zero RR three-form flux (see [134,133] for more explanations). It seems likely that at least some of these models may be solved using tools of non-rational conformal field theory. Concerning the basic mathematical structures of non-rational conformal field theory, the whole field is still in its infancy. The study of strings in compact backgrounds can draw on a rich pool of formulas which hold regardless of the concrete geometry. In fact, for large classes of models, solutions of the factorization constraints may be constructed from the representation theoretic quantities (modular S-matrix, Fusing matrix etc.) of the underlying symmetry. Similar results in non-rational models are not known.10 We hope that these lectures may contribute drawing some attention to this vast and interesting field which remains to be explored. Acknowledgments I wish to thank the participants of the three events at which I delivered these lectures for many good questions, their interest and feedback. Special thanks are also due to the organizers of the schools in Trieste and Vancouver and to my co-organizers at the ESI in Vienna for (co-)organizing such stimulating meetings. I am grateful to Vasilis Niarchos, Ari Pakman, Valentina Petkova, Nikolaos Prezas and Jörg Teschner for comments on the manuscript and in particular to Sylvain Ribault very helpful remarks and careful proof-reading. This work was partially supported by the EU Research Training Network grants “Euclid”, contract number HPRN-CT-2002-00325, “Superstring Theory”, contract number MRTN-CT-2004-512194, and “ForcesUniverse”, contract number MRTN-CT-2004-005104. Appendix A. Dotsenko–Fateev integrals We have argued in Section 3.2, residues of correlation functions in non-compact backgrounds can be computed through free field theory calculations. The latter involve integrations over the insertion points of special bulk or boundary fields. In the case of bulk fields, such integrations can be carried out with the help of the following formula: k d2 zj |zj |2 |1 − zj |2 |zj − zj |4
j <j
j =1
= k!k
k−1 j =0
((j + 1) ) (1 + + j )(1 + + j ) . ( ) (2 + + + (k − 1 + j ))
(A.1)
This is a complex version of the original Dotsenko–Fateev integral formulas [30], 1 x1 xk−1 k dx1 dx2 . . . dxk xj (1 − xj ) (xj − xj )2
0
0
=
k−1 j =0
0
j =1
j <j
((j + 1) ) (1 + + j )(1 + + j ) . ( ) (2 + + + (k − 1 + j ))
10 Note, however, that in the special case of Liouville theory related structures have been partially uncovered, see e.g. [50], Eq. (53).
(A.2)
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In Liouville theory, Dotsenko–Fateev integrals emerge after the evaluation of correlators in a linear dilaton background (see e.g. Eqs. (3.13) or (4.9)), either on a full plane (P ) or on a half-plane (H ). These are given by (P )
1 |z − zj |8i j i>j i
1 (z1 , z¯ 1 ) . . . n (zn , z¯ n )LD = n
(A.3)
and (H )
1 (z1 , z¯ 1 ) . . . n (zn , z¯ n )1 (x1 ) . . . m (xm )LD n −22i −4i r i,r |zi − xr | i=1 |zi − z¯ i | = m , n 4i j 2 r s r<s |xr − xs | i>j |(zi − zj )(zi − z¯ j )| where (x) =: exp X(x): are the boundary vertex operators of the linear dilaton theory. From the above formulas we can in particular compute the non-trivial constant c− that appears in the operator product expansions of Section 3.2, (1 + b2 )(1 − 2b) 2 . cb− () = d2 z|z|2b |1 − z|−4b = (2 + b2 − 2b) A similar free field theory computation was used in Section 4.2 to determine the residue of the bulk boundary structure constants B(, ) at = −b/2, res=−b/2 (B(, 0)) = −B
2b2 i −2B (−1 − 2b2 ) . du − u = 2 2 (−b2 ) −∞ ∞
Free field theory calculations of this type can be performed for other singular points of correlation functions and they provide a rather non-trivial test for the proposed exact couplings. Appendix B. Elements of the fusing matrix In the first lecture we encountered the construction of the so-called Fusing matrix as a problem in the representation theory of chiral algebras. When we deal with Liouville theory, the relevant algebraic structure is the Virasoro algebra with central charge c 25. Given the importance of the Virasoro field for 2D conformal field theory it may seem quite surprising that its Fusing matrix was only obtained a few years ago by Ponsot and Teschner. Even though the construction itself is rather involved, it leads to elegant expressions through an integral over a product of Barnes’ double functions [36]. When specialized to cases in which at least one external field is degenerate11 the general formulas simplify significantly. In fact, such special elements of the Fusing matrix can be written in terms of ordinary functions. Rather than spelling out the general expression for the Fusing matrix and then specializing it to the required cases, we shall pursue another, more direct route which exploits the presence of the degenerate field from the very beginning. Since this procedure is similar to analogous constructions in rational theories, we will only give a brief sketch here. To this end, let us look at the following four-point conformal block G(z) = V4 (∞)V3 (1)V2 (z)V1 (0). Here, V denote chiral vertex operators for the Virasoro algebra where, in contrast to Section 2.1, we have not specified their source and target spaces. Since we are only interested in elements of the Fusing matrix with 2 = −b±1 /2, we can specialize to the case where V2 is degenerate of order two. Consequently, it satisfies an equation of form (3.14). With the help of the intertwining properties of chiral vertex operators (see Section 2.1) or, equivalently, a chiral version of the Ward identities (2.15), one can then derive second order differential equations for the conformal blocks G, 1 h1 h3 + h2 + h1 − h4 1 d2 1 d h3 − + − 2 2+ + − G(z) = 0, (B.1) b dz z − 1 z dz (z − 1)2 z2 z(z − 1) 11 Recall that these are the only matrix elements that we need in Eqs. (3.17) and (4.1).
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where hi = i (Q − i ) and 2 = −b±1 /2. For definiteness, let us concentrate on 2 = −b/2. Two linearly independent solutions of the differential equations (B.1) can be expressed in term of the hypergeometric function F (A, B; C; z) as follows: G± (z) = z± (1 − z) F (A± , B± ; C± ; z) with ± = h1 ±b/2 − h2 − h1 ,
= h3 −b/2 − h3 − h2
and A± = ∓b(1 − Q/2) + b(3 + 4 − b) − 1/2, B± = A± − 2b4 + b2 + 1,
C± = 1 ∓ b(21 − Q).
A standard identity for the hypergeometric function F , F (A, B; C; z) =
(C)(C − A − B) F (A, B; A + B − C + 1, 1 − z) (C − B)(C − A) (C)(A + B − C) + (1 − z)C−A−B F (C − A, C − B; C − A − B + 1, 1 − z) (A)(B)
˜ ± in the space of solutions of the differential equation (B.1), allows to expand G± in terms a second basis G b − 3 ˜ Gs (z) = Gt (1 − z), F1 −s b ,3 −t b 2 2 2 1 4 t=±
(B.2)
˜ ± can be found in the literature. We only list the coefficients, with s = ±. Formulas for G b (b(21 − b))(b(b − 23 ) + 1) − 3 , F1 − b ,3 − b = 2 2 2 b b 1 4 b 1 − 3 − 4 + + 1 b 1 − 3 + 4 − 2 2 b (b(21 − b))(b(23 − b) − 1) − 3 , = F1 − b ,3 + b 2 2 2 3b b 1 4 − 1 b 1 + 3 − 4 − b 1 + 3 + 4 − 2 2 b (2 − b(21 − b))(b(b − 23 ) + 1) − 3 , F1 + b ,3 − b = 2 2 2 3b b 1 4 1 − b 1 + 3 − 4 − 2 − b 1 + 3 + 4 − 2 2 b (2 − b(21 − b))(b(23 − b) − 1) − 3 . F1 + b ,3 + b = 2 2 2 b b 1 4 b −1 + 3 + 4 − b −1 + 3 − 4 + +1 2 2 Let us stress once more that this simple construction of the fusing matrix does only work for elements with one degenerate external label. In more general cases, explicit formulas for the conformal blocks G are not available so that one has to resort to more indirect methods of finding the Fusing matrix (see [36,37]). References [1] P. Di Francesco, P. Mathieu, D. Senechal, Conformal Field Theory, Springer, Berlin, 1997 and references therein. [2] V.B. Petkova, J.B. Zuber, Conformal boundary conditions and what they teach us, arXiv:hep-th/0103007. [3] V. Schomerus, Lectures on branes in curved backgrounds, Class. Quant. Grav. 19 (2002) 5781 [arXiv:hep-th/0209241]. [4] C. Angelantonj, A. Sagnotti, Open strings, Phys. Rep. 371 (1) (2002) [Erratum-ibid. 376 (2003) 339] [arXiv:hep-th/0204089]. [5] Y.S. Stanev, Two dimensional conformal field theory on open and unoriented surfaces [arXiv:hep-th/0112222].
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Physics Reports 431 (2006) 87 – 172 www.elsevier.com/locate/physrep
Enhanced nonlinear optical responses of materials: Composite effects J.P. Huanga,∗ , K.W. Yub a Surface Physics Laboratory (National key laboratory) and Department of Physics, Fudan University, Shanghai 200433, China b Department of Physics and Institute of Theoretical Physics, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong
Accepted 19 May 2006 Available online 7 July 2006 editor: A.A. Maradudin
Abstract We review recent theoretical progress in understanding physical processes of composite effects on enhanced third-order nonlinear optical responses of various kinds of the recently-proposed nonlinear optical materials, namely, colloidal nanocrystals with inhomogeneous metallodielectric particles or a graded-index host, metallic films with inhomogeneous microstructures adjusted by ion doping or temperature gradient, composites with compositional gradation or graded particles, and magneto-controlled ferrofluidbased nonlinear optical materials. © 2006 Published by Elsevier B.V. PACS: 42.70.−a; 77.84.Lf; 42.65.An; 42.65.−k Keywords: Enhanced nonlinear optical responses; Composite effects; Nonlinear optical materials; Graded composite materials
Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 2. Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 2.1. The Maxwell–Garnett theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 2.2. The Bruggeman theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 2.3. The Bergman–Milton spectral representation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 2.3.1. Original derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 2.3.2. An easy-to-understand illustration of the Bergman–Milton spectral representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 2.4. The differential effective dipole theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 2.4.1. Dipole moment of a graded spherical particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 2.4.2. Dipole moment of a coated spherical particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 2.4.3. Expression for the differential effective dipole theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 2.4.4. Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 3. Colloidal nanocrystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 3.1. With inhomogeneous metallodielectric particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 3.2. With a graded-index host . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 ∗ Corresponding author. Tel.: +86 21 55664928; fax: +86 21 65104949.
E-mail address:
[email protected] (J.P. Huang). 0370-1573/$ - see front matter © 2006 Published by Elsevier B.V. doi:10.1016/j.physrep.2006.05.004
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4. Metallic films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Inhomogeneous metallic films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Multilayer metallic films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Graded composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Compositionally graded metal–dielectric composite films of anisotropic particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1. Responses of a layer inside the graded film: ¯ (z, ) and ¯ (z, ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2. Overall responses of the graded film: ¯ () and ¯ () . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Compositionally graded metal–dielectric films: effects of microstructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Composite media of graded spherical particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1. Model and definition of effective linear and nonlinear responses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2. Nonlinear differential effective dipole approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3. Exact solution for power-law gradation profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4. Composites of graded particles with dielectric anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1. Model and definition of effective linear dielectric constant and third-order nonlinear susceptibility . . . . . . . . . . . . . . . . . . . . . . 5.4.2. Nonlinear anisotropic differential effective dipole approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.3. Exact solution for linear gradation profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5. Spectral representation for understanding the effective dielectric constant of graded and multilayer composites . . . . . . . . . . . . . . . . . . 5.5.1. Spectral density function of a graded film . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2. Spectral density function of a graded sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.3. Spectral density function of multilayer composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Magneto-controlled nonlinear optical materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
107 108 111 114 114 117 118 126 133 133 134 136 142 142 143 146 153 157 159 159 163 169 170 170
1. Introduction The field of nonlinear optics traces its beginning to 1961, when a ruby laser was first used to generate the secondharmonic radiation inside a crystal [1]. Since then, nonlinear optics have become the basis of all the fledgling photonics technologies, where light works, or even replaces, electrons in applications traditionally carried out by microelectronics. The interaction between lights (electromagnetic fields) and matters is described by the Maxwell equations. The realization of all-optical switching, modulating and computing devices is an important goal in modern optical technology. The suitability of a material for these device applications requires a large magnitude of third-order nonlinear optical susceptibilities. In general, many applications of nonlinear optics that have been demonstrated under controlled laboratory conditions could become practical for technological uses if such materials were available. Thus, finding nonlinear optical materials with large nonlinear optical susceptibilities is up to now a challenge [2–10]. Many authors (e.g., see Refs. [3,4,6,7,9,11]) have devoted themselves to obtaining a large nonlinearity enhancement or optimal figure of merit (FOM) of bulk composites by taking into account various elements, such as the surface-plasmon resonance in metal–dielectric composites, structural information, etc. In the present review, the FOM denotes the ratio of a thirdorder nonlinear susceptibility to an optical absorption [12]. If there were a larger third-order nonlinear susceptibility and/or a smaller linear optical absorption under certain conditions, the corresponding FOM should be relatively larger, thus being attractive. The most common way to achieve materials with a large nonlinear susceptibility is to use composite materials in which the constituent components possess large intrinsic nonlinear responses. Owing to composite effects, composite materials can have larger nonlinear susceptibilities at zero and finite frequencies than those of ordinary bulk materials or constituent materials from which the composite is constructed, which are thus called enhanced nonlinear optical responses. The formation of composite materials thus constitutes a means for engineering new materials with desired nonlinear optical properties [13]. In fact, the composite effects arise from strong enhancement or fluctuations of the local fields. One [14] studied nonlinear optical properties of fractal aggregates and showed that the aggregation of initially isolated particles into fractal clusters results in a huge enhancement of the nonlinear response within the spectral range of collective dipolar resonances like surface plasmon resonances. Basically, the response of a nonlinear composite can be tuned by controlling the volume fraction and morphology of constitutes. The latter can be purposefully adjusted by using external electric/magnetic field, or by adding dielectric gradation profile to the system of interest. In other
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words, depending on different microstructure, it is expected to achieve various kinds of composite effects on enhanced nonlinear optical responses. When particles have a size that is much less than the wavelength of an incident light, the quasi-static approximation can be used to describe the response of an individual particle. (It should be remarked that all the approximations discussed in the article are valid in the quasi-static limit.) Typically, the particle size ranges from tens to hundreds of nanometers. The particles are embedded in a host material and can be aggregated into chains or clusters, under different conditions. It is important to keep in mind that, because of the symmetry requirement, all even order susceptibilities vanish when the material is centrosymmetric. In other words, odd order optical susceptibilities are possible in all systems. Noble metal (typically gold, silver and copper) is often chosen as an ingredient due to their extremely large and fast nonlinear optical responses. Many different microstructures have been exploited in an attempt to access the intrinsic optical nonlinearity of metals, for example, the random metallodielectric composites [15–17], fractal films [16,17], and alternative bilayers [4,5,18], etc. They basically rely on the enhanced local fields in space or on the effectively lengthened scale of the interactions between the matter and the incident light field. There is also a great demand for particular optical materials in devices applications, which would benefit from additional tunability of the optical properties. For example, recently we studied graded composites, which provided an extra degree of freedom for controlling the nonlinear optical properties of these materials by choosing appropriate gradation profiles [19–23]. In fact, there exist in Nature abundant graded materials, such as biological cells [24] and liquid crystal droplets [25]. Furthermore, many artificially-graded-index optical metamaterials and elements have been fabricated nowadays [26]. Composite effects are always expected to open a fascinating field of new phenomena in nonlinear optics, and the effects on the optical nonlinearity enhancement are different for various microstructures. This review presents our recent research aimed at understanding of the physical processes that determine the enhanced nonlinear optical properties of graded composite materials. This is an original, comprehensive, and first-handed review of the state-of-the-art development of the field of enhanced nonlinear optical responses in the materials. Regarding the previous development in the field, please refer to the excellent reviews by Shalaev and his coauthors [16,17], in which graded composites were not touched. We shall review theories and models that have recently been devised to study the composite effect on enhanced nonlinear optical responses of various kinds of the recently-proposed nonlinear optical materials, namely, colloidal nanocrystals with inhomogeneous metallodielectric particles or a graded-index host (Section 3), metallic films with inhomogeneous microstructures adjusted by ion doping or temperature gradient (Section 4), composites with compositional gradation or graded particles (Section 5), and magneto-controlled ferrofluid-based nonlinear optical materials (Section 6). In addition, Section 2 also presents some fundamental theories that are to be used in the other sections. This review ends with a summary in Section 7.
2. Fundamentals Composites often contain a macroscopic scale of inhomogeneity. In such a material, there are small, yet much larger than atomic, regions where macroscopic homogeneity prevails and where the foregoing macroscopic parameters suffices to characterize the physics, but different regions may have quite different values for those parameters. If we are interested in the physical properties at scales that are much larger than those regions and at which the material appears to be homogeneous, then the macroscopic behavior can again be characterized by bulk effective values, e.g. effective dielectric constant e . In what follows, we shall review three typical theories for calculating e , namely, the Maxwell–Garnett theory (Section 2.1), the Bruggeman theory (Section 2.2) as well as the Bergman–Milton spectral representation theory (Section 2.3). Graded materials, whose material properties can vary continuously in space, are abundant in nature. With the advent of fabrication techniques, however, these materials may also be produced in laboratory to tailor their properties for specific need. Moreover, composites of graded inclusions can be more useful and interesting than those of homogeneous inclusions. However, the established theory for homogeneous inclusions cannot be applied directly. It is thus necessary to develop a new theory to study the effective properties of graded composite materials under externally applied fields. For this purpose, we shall develop an effective dipole theory (Section 2.4).
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Fig. 1. Schematic graph showing an asymmetrical microstructure for the Maxwell–Garnett theory in which component 1 with dielectric constant 1 is embedded in component 2 with 2 .
2.1. The Maxwell–Garnett theory The Maxwell–Garnett theory [27,28] (or Maxwell–Garnett approximation) is also known as the Clausius–Mossotti theory. Regarding how to derive the equation for the Maxwell–Garnett theory, there are several approaches. Here we would like to start from the view of effective local electric field. Let us discuss a two-component composite where many particles of the dielectric constant 1 and the volume fraction p are randomly embedded in a host medium of 2 , in the presence of an external electric field E0 along the z-axis. Then, we denote the local electric field inside the particle by E1 , and that inside the host medium by E2 . Solving a fundamental equation in electrodynamics yields E1 =
32 E2 . 1 + 22
(1)
It is known that the average electric field E(≡ pE1 + (1 − p)E2 ) inside the composite should be equal to the external electric field, namely, E = E0 . On the other hand, the effective dielectric constant e may be given by the ratio of the average displacement D to the average electric field E inside the composite, namely, e =
D D . = E E0
(2)
Hence we obtain e =
p1 E1 + (1 − p)2 E2 . pE1 + (1 − p)E2
(3)
To this end, we obtain the expression for the Maxwell–Garnett theory as e = 2
1 (1 + 2p) + 22 (1 − p) . 1 (1 − p) + 2 (2 + p)
(4)
We re-express Eq. (4) in a commonly-used form as e − 2 1 − 2 =p . e + 22 1 + 22
(5)
Obviously, the Maxwell–Garnett theory is an asymmetrical theory (see Fig. 1), namely the physical property of the composite can be changed if one exchanges the notations 1 and 2. Finally, for the Maxwell–Garnett theory the extension to multi-component composite is straightforward as can be easily done on the same footing. 2.2. The Bruggeman theory Another approach to calculating e for a two-component composite similar to the above was introduced by Bruggeman [29]. Thus, this approach is called the Bruggeman theory (also called the effective medium theory or Bruggeman approximation). Its predictions are usually sensible and physically offer a means of quick insight into some problems that are difficult to attack by other approaches.
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Fig. 2. Schematic graph showing a symmetrical microstructure for the Bruggeman theory in which two components are mixed.
We would derive the expression for the Bruggeman theory by considering the fact that the effective dipole factor b of the composite should be zero, namely b = 0.
(6)
On the other hand, we have b = pb1 + (1 − p)b2 ,
(7)
where b1 and b2 are, respectively, the dipole factors of the particle and the host medium in the effective medium, and they are given by b1 =
1 − e 1 + 2e
and
b2 =
2 − e . 2 + 2e
(8)
In view of Eqs. (6)–(8), we have directly the expression for the Bruggeman theory p
2 − e 1 − e + (1 − p) = 0. 1 + 2e 2 + 2e
(9)
Obviously, the Bruggeman theory is a symmetrical theory (see Fig. 2), namely the physical property of the composite can keep unchanged if one exchanges the notations 1 and 2. In addition, for the Bruggeman theory the extension to multi-component composite is also straightforward which can be readily done on the same footing as above. 2.3. The Bergman–Milton spectral representation theory 2.3.1. Original derivation For the sake of convenience, we let E0 = −ˆez . Below we briefly review the Bergman–Milton spectral representation theory for the effective dielectric constant of a two-component composite. The problem is initiated by solving the differential equation [30] 1 (10) ∇ · 1 − (r) ∇(r) = 0, s where s = 2 /(2 − 1 ) denotes the relevant material parameter and (r) is the characteristic function of the composite, having value 1 for r in the embedding medium and 0 otherwise. The electric potential (r) can be solved formally 1 dr (r )∇ G0 (r − r ) · ∇ (r ), (11) (r) = z + s where G0 (r − r ) = |r − r |/4 is the free space Green’s function. By denoting an integral-differential operator (12) = dr (r )∇ G0 (r − r ) · ∇ ,
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and the corresponding inner product | = dr (r)∇∗ · ∇.
(13)
It is easy to show that is a Hermitian operator. Let sn and n (r) be the n-th eigenvalue and eigenfunction of the operator, respectively, then we obtain the effective dielectric constant e in the Bergman–Milton spectral representation 1 e = − dV (r)Ez V 1 j 1 dV 2 1 − (r) = V s jz 1 |n |z|2 = 2 1 − V n s − sn
= 2 1 −
n
Fn s − sn
.
(14)
The parameters sn and Fn satisfy simple properties that 0 sn 1 and Fn =p [30]. Eq. (14) is just the effective dielectric constant of a two-component system in the Bergman–Milton spectral representation. Moreover, after introducing F (s) as a function of s as F (s) ≡
n
Fn , s − sn
(15)
we may readily obtain the spectral structure of the composite. In doing so, we may further represent F (s) as 1 (x) F (s) = dx , s−x 0
(16)
where the spectral function (x) is a crucial parameter which contains the information about the spectral structure, and is thus given by 1 (x) = − Im F (x + i0+ ).
(17)
Obviously, we may observe that the Bergman–Milton spectral representation is a rigorous mathematical formalism for the effective dielectric constant of a two-phase composite material [30]. It offers the advantage of the separation of materials parameters (namely the dielectric constant or conductivity) from the particle structure information (see Eq. (14)), thus simplifying the study. For a better understanding of the Bergman–Milton spectral representation, below we would like to represent the above-mentioned Maxwell–Garnett and Bruggeman theories in the Bergman–Milton spectral representation. As a result, for the Maxwell–Garnett theory we respectively have the F (s) function and the spectral function as F (s) =
p s − (1 − p)/3
and
(x) = p [x − (1 − p)/3].
On the other hand, for the Bruggeman theory we have the F (s) function as 1
F (s) = −1 + 3p + 3s − 3 (s − x1 )(s − x2 ) , 4s where x1 and x2 are given by solving (1 − 3p)2 − 6(1 + p)x + 9x 2 = 0,
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Fig. 3. Schematic graph showing a parallel-plate capacitor of total thickness h = h1 + h2 that contains a dielectric slab of dielectric constant 1 and thickness h1 , as well as a dielectric of 2 and thickness h2 (both of the same area A).
hence, x1 =
1
1 + p − 2 2p(1 − p) 3
and
x2 =
1
1 + p + 2 2p(1 − p) . 3
In this case, the spectral function should be (x) =
3 3p − 1 (x − x1 )(x2 − x)
(3p − 1) + 2 4x
as x1 < x < x2 , and (x) =
3p − 1
(3p − 1), 2
otherwise. We would also like to mention that the extension of the Bergman–Milton spectral representation to the threecomponent composite can be made by taking into account various approaches [31–33]. 2.3.2. An easy-to-understand illustration of the Bergman–Milton spectral representation The essence of the spectral representation is to define the following transformations. If we denote a material parameter 1 −1 s= 1− , (18) 2 then the reduced effective dielectric constant e w(s) = 1 − , 2 can be written as w(s) = n
Fn , s − sn
(19)
(20)
where n is a positive integer, i.e., n = 1, 2, . . . , and Fn and sn , are the n-th microstructure parameters of the composite materials [30]. In Eq. (20), 0 sn < 1 is a real number, while Fn satisfies a sum rule [30] Fn = p. (21) n
In what follows, we illustrate the spectral representation by the capacitance of simple geometry [34]. In particular, a parallel-plate capacitor is considered as an example. We will discuss two cases, namely, the series combination (Fig. 3) and the parallel combination (Fig. 4).
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Fig. 4. Schematic graph showing a parallel-plate capacitor of total area w = w1 + w2 that contains a dielectric slab of dielectric constant 1 and area w1 , as well as a dielectric of 2 and area w2 (both of the same thickness h).
In the first case (see Fig. 3), if one inserts a dielectric slab of dielectric constant 1 and thickness h1 , as well as a dielectric of 2 and thickness h2 (both of the same area A), into a parallel-plate capacitor of total thickness h = h1 + h2 , the overall capacitance C is given by C −1 = C1−1 + C2−1 , where C1 = 1 A/ h1 and C2 = 2 A/ h2 . On the other hand, we may define the equivalent capacitance as C = e A/ h, where e is the effective dielectric constant. That is, we see the composite dielectric as a homogeneous dielectric of dielectric constant e . Let 1 = 2 (1 − 1/s), we can express C in the spectral representation, C=
A2 h1 / h2 A2 − . h s − h2 / h
In accord with the spectral representation, one may introduce w(s) = 1 − e /2 , which is in fact the same as w(s) = 1 − C/C0 , where C0 is the capacitance when the plates are all filled with a dielectric material of 2 , namely C0 = 2 A/ h. Thus we obtain w(s) =
h1 / h . s − h2 / h
from which we find that the material parameter is separated from the geometric parameter. The comparison of w(s) with Eq. (20) yields F1 = h1 / h,
s1 = h2 / h.
It is worth noting that F1 obtained herein is just equal to the volume fraction of the dielectric of 1 , and that s1 satisfies 0 s1 < 1, as required by the spectral representation theory. Next, we consider the parallel combination (see Fig. 4). If one inserts a material with dielectric constant 1 and area w1 as well as a dielectric with 2 and area w2 (both of the same thickness h), into a parallel-plate capacitor of total area A = w1 + w2 , the overall capacitance C is given by C = C 1 + C2 , where C1 = 1 w1 / h and C2 = 2 w2 / h. Similarly, after introducing the effective dielectric constant e , we may define the overall capacitance as C = e A/ h.
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Fig. 5. Schematic graph showing a graded spherical particle with a dielectric gradation profile (r) at radius r, embedded in a host medium of dielectric constant 2 .
Again, in the spectral representation, let s = (1 − 1 /2 )−1 , then C=
2 A 2 w1 − . h hs
Writing w(s) = 1 − C/C0 , we obtain w1 /A . s From this equation, the material parameter is also found to be separated from the geometric parameter. It is clear that F1 = w1 /A, i.e., the volume fraction of the dielectric of 1 , and s1 = 0. w(s) =
2.4. The differential effective dipole theory 2.4.1. Dipole moment of a graded spherical particle We consider an inhomogeneous spherical inclusion of radius a, with a dielectric gradation profile (r), embedded in a host medium of dielectric constant 2 (see Fig. 5). In a uniform applied field E0 , the potential functions in the inclusion and host regions can be obtained by solving the Laplace equation with the appropriate boundary conditions. For a power-law dielectric profile (r) = A2 r n , where A, n are constants, the potential admits an exact solution [19]: (r) = − 1 E0 r s cos , (r) = − E0 r cos +
r < a,
2 E0 cos , r2
r > a,
(22)
where the coefficients 1 and 2 have the form 1 =
3a 1−s , sAa n + 2
2 =
sAa n − 1 3 a , sAa n + 2
and s is given by the positive root of a quadratic equation [19]: 1 s= 9 + 2n + n2 − (1 + n) . 2
(23)
(24)
The local electric field can be computed by the gradient of the potential E = −∇. From the local electric field, we can integrate it over the volume of the inclusion to obtain the dipole moment of a graded inclusion. Thus we find an exact analytic expression for the dipole moment in the power-law profile [19]: p0 = 2 bE 0 a 3 , where b is the dipole factor, which measures the degree of polarization of the inclusion in an external field, s+2 Aa n 1 b= − . sAa n + 2 s + n + 2 s + 2
(25)
(26)
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Note that when n → 0, the dielectric profile becomes homogeneous. In this limit s → 1, Eq. (26) reduces to the well-known result for the dipole factor of a homogeneous spherical inclusion of dielectric constant 1 embedded in a host medium of 2 : b0 =
1 − 2 . 1 + 22
(27)
2.4.2. Dipole moment of a coated spherical particle If we add to the homogeneous inclusion a spherical shell of dielectric constant , to make a coated sphere [35] of overall radius a1 . By solving the Laplace equation with appropriate boundary conditions on the interfaces, the potential in the inclusion becomes [36] (r) = − E0 r cos , r < a, 4 (r) = − E0 3 r − 2 cos , a < r < a1 , r
(28)
where , 3 and 4 are coefficients in terms of the dielectric constants [36]. The local electric field is obtained and hence yields the dipole moment of a single coated spherical inclusion: p1 = 2 b1 E0 a13 ,
(29)
where b1 is the dipole factor of the coated sphere [37], b1 =
( − 2 ) + (2 + 2) 1 y , ( + 22 ) + 2( − 2 ) 1 y
(30)
where 1 is given by 1 =
1 − , 1 + 2
(31)
and y = (a/a1 )3 .
(32)
In Eq. (31) 1 denotes the dielectric constant of the core inside the coated spherical particle. The consideration can be extended to more shells of different dielectric constants, at the expense of more complicated expressions [37–39]. It is easy to check that b1 reduces to b0 when = 1 . Thus, the dipole factor remains unchanged if one adds a spherical shell of the same dielectric constant. We should remark that these exact results are only available for a few simple dielectric profiles, and it is our objective in the following section to develop a theory for any arbitrary profiles. 2.4.3. Expression for the differential effective dipole theory Now we develop the differential effective dipole theory (DEDT) for spherical particles of graded materials and hence compute the effective dielectric response in the dilute limit. To establish the differential effective dipole theory, we mimic the graded profile by a multi-shell construction, i.e., we build up the dielectric profile gradually by adding shells. We start with an infinitesimal spherical core of dielectric constant (0) and keep on adding spherical shells of dielectric constant given by (r) at radius r, until r = a is reached. At radius r, we have an inhomogeneous sphere whose induced dipole moment is given by p(r). Certainly p(r) is proportional to E0 , but the exact expression is lacking. We further replace the inhomogeneous sphere by a homogeneous sphere of the same dipole moment and the graded profile is replaced by an effective dielectric constant ¯ (r). Thus, p(r) = 2 b(r)E0 r 3 ,
(33)
where b(r) =
¯ (r) − 2 . ¯ (r) + 22
(34)
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Next, we add to the sphere a spherical shell of infinitesimal thickness dr, of dielectric constant (r). The dipole factor will change according to Eq. (30). Of course, the effective dielectric constant ¯ (r), being related to b(r), should also change by the same token. Let us write b1 = b + db, and take the limit dr → 0, we obtain a differential equation: 1 db =− [(1 + 2b)2 − (1 − b)(r)][(1 + 2b)2 + 2(1 − b)(r)]. dr 3r2 (r)
(35)
Thus the dipole factor of a graded spherical particle can be calculated by solving the above differential equation with a given graded profile (r). The nonlinear first-order differential equation can be integrated, at least numerically, if we are given the graded profile (r) and the initial condition b(r = 0). When b(r = a) is calculated, we can compute the dipole moment of individual spherical particles and hence the effective dielectric constant either in the dilute limit e = 2 + 3f 2 b,
(36)
or by the Clausius–Mossotti approximation e − 2 = f b, e + 22
(37)
where f is the volume fraction of spherical particles. 2.4.4. Numerical results In this section, we evaluate the DEDT for some graded profiles. We performed numerical calculations for two model profiles: (a) power-law profile (r) = Ar n , and (b) linear profile (r) = A + Br. Without loss of generality, we have set 2 = 1 and a = 1. The numerical integration has been done by the fourth-order Runge–Kutta algorithm with a step size r = 0.01, starting with a small core radius r = 0.01. In Fig. 6(a), we plot the dipole factor b versus A for various index n > 0. It is clear that b increases monotonically as the dielectric contrast A increases, while it decreases with the index n. It is attributed to the fact that ¯ decreases as n increases. Similarly, in Fig. 6(b) we plot b versus A for various slope B. We obtained similar behavior as in Fig. 6(a). It is instructive to compare the exact results with the DEDT results. In Fig. 7, we compared the exact results and the DEDT results for a spherical inclusion with a power-law graded profile. The agreement is excellent for positive index n. However, we doubted if the comparison remains good for negative index n because the dielectric constant diverges at the origin. To our surprise, the agreement is still excellent for negative index n (results not shown here). In fact, the DEDT is exact for the case of a spherical inclusion with a power-law profile. We have checked the exact solution [Eq. (26)] against the DEDT. We found that the differential equation [Eq. (35)] as well as the boundary condition at a small radius are satisfied by Eq. (26). Thus, the DEDT is exact for power-law graded spherical inclusions. The DEDT is better motivated by the following considerations. In the case of a single-shell inclusion of core dielectric constant 1 , coated by a shell of dielectric constant , with the core to whole volume ratio y (as the one considered in Section 3), the effective dielectric constant of the inclusion (¯) may be determined by the Maxwell–Garnett approximation. ¯ − 1 − =y . ¯ + 2 1 + 2 Then, we assume the effective inclusion of dielectric constant ¯ to be embedded in a host medium 2 . In this way, the dipole factor of the effective inclusion can be calculated from b=
¯ − 2 . ¯ + 22
For a single-shell inclusion, this consideration is exact, in the sense that Eq. (30) is reproduced exactly, while for multi-shell inclusions, this can be a good approximation. Moreover, the substitution of Eq. (34) into Eq. (35) yields a differential equation for ¯ (r): d [¯(r)]2 [r ¯ (r)] + = 2(r), dr (r)
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1.0
n = 0.1 n =1.0 n = 2.0
b
0.5
0.0
-0.5
0
1
2
(a)
3
4
5
3
4
5
A 1.0 B = 0.1 B =1.0 B = 2.0
b
0.5
0.0
-0.5 (b)
0
1
2 A
Fig. 6. The dipole factor b plotted versus A for two model profiles for: (a) various index n in the power-law profile; and (b) various slope B in the linear profile.
which coincides with the Tartar formula [39] derived for assemblages of spheres with varying radial and tangential conductivity. We should remark that the DEDT is also able to offer a good agreement with the exact result of a linear profile [19]. Most importantly, we have recently demonstrated that the DEDT is indeed exact for graded spherical particles of arbitrary dielectric gradation profiles [40]. 3. Colloidal nanocrystals Colloidal crystalline is extensively studied in nanomaterials engineering and its potential applications range from nanophotonics to chemistry and biomedicine [41]. Colloidal crystals can be prepared via templated sedimentation, methods based on capillary forces, and electric fields [42–44]. They exhibit body centered tetragonal (bct), body centered cubic (bcc) and face-centered cubic (fcc) structures, depending on the lattice constants and hence the volume fraction of colloidal particles. These structures can be investigated by using static and dynamic light scattering techniques [45,46].
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1.0 n = 0.1 (DEDT) n = 0.1 (Exact) n =1.0 (DEDT) n =1.0 (Exact) n = 2.0 (DEDT) n = 2.0 (Exact)
b
0.5
0.0
-0.5
0
1
2
3
4
5
A Fig. 7. Similar to Fig. 6(a), but for comparison of the exact results (lines) with differential effective dipole theory results (symbols) for the dipole factor in the power-law profile.
Fig. 8. Schematic graph to show the location of two colloidal nanoparticles in a tetragonal unit cell [49].
So far, colloidal-based optical sensors [47] (and photonic-band-gap materials based on inverse opaline structures [48]) have been made possible by these fabrication techniques. This section describes a class of colloidal-crystal-based nonlinear optical materials, which are made of graded metallodielectric nanoparticles (namely, a graded metallic core plus a dielectric shell), or a graded-index host. 3.1. With inhomogeneous metallodielectric particles Let us start by considering a tetragonal unit cell which has a basis of two colloidal nanoparticles each of which is fixed with an induced point dipole at its center. One of the two nanoparticles is located at a corner and the other one at the body center of the cell (Fig. 8). Its lattice constants are denoted by c1 (=c2 ) = q −1/2 and c3 = q along x(y) and z axes, respectively. In this case, the uniaxial anisotropic axis is directed along z axis. The degree of anisotropy of the periodic lattice is measured by how q deviates from unity. In particular, q = 0.87358, 1.0 and 21/3 represents the bct, bcc and fcc lattice, respectively. In general, for a colloidal crystal, the individual colloidal nanoparticles should be
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touching. In fact, a colloidal crystal without the particles’ touching can also be made if the colloidal nanoparticles are charged and stabilized by electrostatic forces. Below we shall investigate colloidal crystals with the particles’ touching. With recent advancements in the fabrication of nanoshells [50,51], we are allowed to use a dielectric surface layer with thickness d on an inhomogeneous (graded) metallic core with radius a, in order to activate repulsive (or attractive) force between the nanoparticles. This is also a crucial requirement because otherwise multipolar interaction between the metallic cores can be important. The dielectric constant 1 (r) (r a) of the metallic core should be a radial function, because of a radial gradation. The dielectric constant s of the surface layer can be the same as that 2 of the host fluid, as to be used in the following. In this regard, the surface layer contributes to the geometric constraint c12 + c22 + c32 = 16(a + d)2 ,
(38)
rather than the effective optical responses. Owing to this constraint, it is found that the smallest q occurs at the bct lattice while the largest q occurs at the fcc. Meanwhile, we obtain a relation between q and the volume fraction p of the metallic component, p=
3 3/2 q +2 , 24t 3 q
(39)
with thickness parameter t = (a + d)/a. When an external electric field E0 is applied along x axis, the induced dipole moment P are perpendicular to the uniaxial anisotropic axis. Then, the local field EL at the lattice point R = 0 can be determined by using the Ewald–Kornfeld formulation [52–54], EL = P
2
[− 1 (Rj ) + xj2 q 2 2 (Rj )] −
j =1 R =0
2 2 3 4P Gx exp −G + 4P√ . (G) 2 2 Vc G 4 3
(40)
=0 G
Here we have taken into account the influence of the local-field effect arising from all the other particles in the lattice (lattice effect). In Eq. (40), 1 and 2 are two coefficients, given by √
1 (r) = [erfc(r)/r 3 ] + (2/ r 2 ) exp(−2 r 2 ), (41) √ √
2 (r) = [3erfc(r)/r 5 ] + [43 /( r 2 ) + 6/( r 4 )] exp(−2 r 2 ), (42) where erfc(r) is the complementary error function, and an adjustable parameter making the summation converge rapidly. R and G denote, respectively, the lattice vector and the reciprocal lattice vector, R = (q −1/2 l xˆ + q −1/2 myˆ + qnˆz),
(43)
= (2/)(q 1/2 uxˆ + q 1/2 v yˆ + q −1 w zˆ ), G
(44)
where l, m, n, u, v, and w are integers. In addition, xj and Rj are given by xj = l − (j − 1)/2,
(45)
Rj = |R − [(j − 1)/2](a xˆ + a yˆ + cˆz)|,
(46)
= 1 + exp[i(u + v + w)/]. Now we define a local field factor in transverse field cases, and the structure factor (G) ⊥ = 3Vc EL /(4P ). It is worth noting that ⊥ is a function of a single variable q. Also, there is a sum rule [52,55] 2⊥ + = 3,
(47)
where denotes the local field factor in longitudinal field cases. Here the longitudinal (or transverse) field case corresponds to the fact that the E field of the incident light is parallel (or perpendicular) to the uniaxial anisotropic z axis. For the bct, bcc and fcc lattices, we obtain ⊥ = 0.95351, 1.0, and 1.0 (or alternatively = 1.09298, 1.0, and 1.0), respectively. If there are no special instructions, we shall use to denote both ⊥ and in the following. Next, for obtaining the effective dielectric constant e (which also indicates both ⊥ e and e , the effective dielectric
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constants in transverse and longitudinal field cases, respectively) of the colloidal crystal, we resort to the anisotropic Maxwell–Garnett formula [9] with a high degree of accuracy [56] due to the explicit determination of , e − 2 ¯ 1 − 2 =p , e + (3 − )2 ¯ 1 + 22
(48)
where the equivalent dielectric constant ¯ 1 ≡ ¯ 1 (r = a) for the graded metallic core can be obtained [20,39] by solving d¯1 (r)/dr = [1 (r) − ¯ 1 (r)][¯1 (r) + 21 (r)]/[r1 (r)],
(49)
as long as the gradation profile 1 (r) is given. Assuming both the host fluid and dielectric surface layer to be linear for (3) convenience, the effective third-order nonlinear susceptibility e for the graded colloidal crystal is given by (3) e =p
2 |Elin |2 Elin
|E0 |2 E02
(3)
¯ 1
(50) (3)
(3)
under a decoupling approximation, where the equivalent third-order nonlinear susceptibility ¯ 1 ≡ ¯ 1 (r = a) for the graded metallic core can be obtained [20] by solving d¯ 1 (r)/dr = ¯ 1 (r){(3 d¯1 (r)/dr)/(22 + ¯ 1 (r)) + [(d¯1 (r)/dr)/(22 + ¯ 1 (r))]∗ } (3)
(3)
+ ¯ 1 (r)(6Y + 2Y ∗ − 3)/r + 31 (r)/(5r) (3)
(3)
× |(¯1 (r) + 21 (r))/(31 (r))|2 [(¯1 (r) + 21 (r))/(31 (r))]2 × (5 + 18X 2 + 18|X|2 + 4X 3 + 12X|X|2 + 24|X|2 X 2 ),
(51)
with X=
¯ 1 (r) − 1 (r) , ¯ 1 (r) + 21 (r)
(52)
Y=
[1 (r) − 2 ][¯1 (r) − 1 (r)] , 1 (r)[¯1 (r) + 22 ]
(53)
(3)
(3)
as long as the gradation profile 1 (r) is also given. Here the intrinsic weak third-order nonlinear susceptibility 1 (r) satisfies the local constitutive relation between the displacement D1 (r) and the electric field E1 (r) inside the graded metallic core, (3)
D1 (r) = 1 (r)E1 (r) + 1 (r)|E1 (r)|2 E1 (r).
(54)
In Eq. (50), · · · denotes the volume average over the metallic region, and Elin the equivalent linear local electric field in the graded metallic core with the same gradation profile but with a vanishing nonlinear response at the frequency 2 can be expressed in the spectral representation as [57] concerned. Both |Elin |2 and Elin |Elin |2 = (E02 /p) dx |s|2 (x )/|s − x |2 , (55) 2 Elin = (E02 /p)
dx s 2 (x )/(s − x )2 ,
(56)
with material parameter s = 2 /(2 − ¯ 1 ), where (x ) is the spectral density. Since the e in Eq. (48) can be expressed as p e = 2 1 − ≡ 2 [1 − F (s)], (57) s − (1 − p)/3 according to 1 (x ) = − Im[F (x + i0+ )]
(58)
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we obtain
1 − p . (x ) = p x − 3
(59)
Here Im[· · ·] denotes the imaginary part of · · · . The point of achieving the resonant plasma band shown in Figs. 9 and 10 is that one needs a sufficiently large gradient rather than a crucially particular form of the dielectric function or gradation profiles. To show the effects of gradation, we have adopted the Drude form 1 (r) = 1 −
2p (r) ( + i )
(60)
with a model plasma-frequency gradation profile
r . (61) p (r) = p (0) 1 − C a One possible way to achieve such gradation is to fabricate a graded metallic core by using different noble metals as different layers inside the core. For focusing on the nonlinearity enhancement, we consider a model system where (3) (3) 1 (r) = 1 is a real and positive frequency-independent constant and does not have a gradation profile. In this case, (3) the equivalent nonlinear susceptibility ¯ 1 (r) should still depend on r because of the radial function 1 (r) [20]. For a given thickness of shell, when q varies from bct to bcc, to fcc lattices, the volume fraction p first decreases from bct, reaches a minimum at bcc, then increases again towards fcc (Table 1). At the same time, the longitudinal local field factor decreases from 1.09 at bct lattices almost monotonically to 1 at bcc and fcc lattices. Thus, for the bct case, the large p and large should give rise to a large red shift (namely, the plasma resonant peak and band are caused to be located at lower frequencies) from the single particle case with p ≈ 0 where the lattice effect disappears and only the gradation effect exists. For bcc lattices, the red shift should be the smallest due to small p and = 1, and hence for fcc lattices, the red shift should lie between those of bct and bct lattices. This is because does not change much while p changes significantly (Table 1). From Fig. 9, it is evident that for a given lattice, the effective linear and nonlinear responses depend strongly on the thickness parameter t. Both the redshift and strength of the plasma resonant peak or band is largest at smallest t [Fig. 9(a)–(b)]. This is a combination of the local field effect and the volume fraction effect in the colloidal crystal. However, for a given thickness parameter t, the dependence of these responses on the crystal structure is not prominent (no figures shown here). In fact, the plasma resonant band in Fig. 9 is caused to appear by the gradation, as discussed in Ref. [21] in which the results for the case of various C (and hence various degree of gradation) have also been reported. Similar results are displayed in Fig. 10 where the transverse field cases are investigated. In comparison with the longitudinal field cases (Fig. 9), the framework of the responses in transverse field cases (Fig. 10) is slightly blue-shifted (i.e., located at higher frequency). Owing to the volume fraction effect, this behavior is more evident for the smallest t, but small enough to be neglected at the largest t. The difference between the results predicted in Figs. 9 and10 is generally small because = 1.09298 is so close to ⊥ = 0.95351. However, the small shift can actually be detected in experiments. We believe dielectrophoresis can offer a convenient way of preparing a colloidal crystal [58]. It is not unusual that one fabricates graded colloidal crystals by dielectrophoresis. In this case, the particles in a medium may have different dielectric properties but they must be of the same size (so as to form colloidal crystals). In a nonuniform applied field, the different particles experiences different dielectrophoretic forces according to their strength of polarization. Regarding the fabrication of graded metallic spheres, a practical choice in experiments might possibly be to fabricate multilayered particles with or without dielectric anisotropy [59,60]. In addition, metallic alloying can also be a promising means. In the latter, the dielectric function of the particles no longer obeys the Drude form, and the theoretical calculations are formidable because it involves a Green function formalism for band structure, and a linear response theory (i.e., Kubo formula) for transport properties [61]. In Section 3.1, the objective of considering tetragonal lattices is to achieve anisotropy by changing the lattice parameters, in order to produce a larger optical nonlinearity than in an isotropic system such as a cubic structure. In summary, we theoretically investigate a class of nonlinear optical materials based on colloidal crystals of graded metallodielectric nanoparticles. Such materials can have both an enhancement and a red shift of optical nonlinearity, due to the gradation inside the metallic core as well as the lattice effects arising from the periodic structure.
J.P. Huang, K.W. Yu / Physics Reports 431 (2006) 87 – 172
103
2 t =1.2
L
t = 2.0
log10 [Im (εe) ]
t =3.0 0
-2
(a) 4
log10 [ |χe(3)| / χ1(3) ]
L
0
-4
-8 (b)
L
log10 ( FOM)
2
0
-2
(c) 0
0.2
0.4
0.6
0.8
1
ω /ωp (0) (3)
(3)
Fig. 9. For the bct lattice, (a) the linear optical absorption Im(e ), (b) the enhancement of the third-order optical nonlinearity |e |/1 , and (c)
(3) (3) the FOM ≡ |e |/[1 Im(e )] versus the normalized incident angular frequency Parameters: 2 = (3/2)2 , C = 0.3, and = 0.02p (0).
/p (0) for longitudinal field cases (L), for different t [49].
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2 T
t =1.2 t = 2.0 t =3.0
log10 [ Im (εe)]
0
-2
(a) 4
log10 [ |χe(3)| / χ1(3) ]
T
0
-4
-8 (b)
T
log10 (FOM)
2
0
(c) -2
0
0.2
0.4
0.6 ω /ωp (0)
0.8
1
Fig. 10. Same as Fig. 9, but for transverse field cases (T ).
3.2. With a graded-index host We shall theoretically investigate a colloidal crystal immersed in a graded-index host and demonstrate a giant enhanced optical nonlinearity band, which is controllable by the gradient and by the easily-tunable colloid structure as
J.P. Huang, K.W. Yu / Physics Reports 431 (2006) 87 – 172
105
Table 1 The volume fraction p of the metallic component for bct (i.e., q = 0.87358 or = 1.09298), bcc (i.e., q = 1.0 or = 1.0) and fcc (i.e., q = 21/3 or = 1.0) lattices at various thickness parameter t
bct bcc fcc
t = 1.2
2.0
3.0
p = 0.40401 p = 0.39362 p = 0.42852
0.08727 0.08502 0.09256
0.02586 0.02519 0.02743
well. We basically used the quasistatic point-dipole approximation, which suffices in terms of characterizing both the gradient effects and the lattice effects, otherwise the solution is formidable, either from a Green’s function formalism or from first-principles [62]. The theoretical calculations are deployed on a model tetragonal lattice with uniaxial anisotropy [see Fig. 11] [52]. Without loss of generality, we will only discuss the bct, bcc and the fcc lattices, respectively. The bcc has the lowest packing density while the fcc has the highest one among the three cases [52]. Extensions to other colloidal structures such as the simple tetragonal lattice are straightforward and similar results are expected. Taking advantages of the interlayer interaction tensor T (i.e., Tij denotes the interaction strength between two in-plane dipole arrays) given by the Lekner summation method [64–66], we solved the self-consistent equations Ei =
N 1 (0) N Tij · (j Ej ) + Ei , a3
(62)
j =1
where a is the lattice constant as shown in Fig. 11, j is the layer-dependent linear bare polarizability, here i, j label (0) the crystal layer and N denotes the total layer number. Ei in the self-consistent equation is not simply the externally applied electric field E0 due to the presence of gradient. However, it is the field inside the graded host medium, which is thus determined by virtue of the continuity of the normal component of the electric displacement D in the longitudinal case, i.e., E0 parallel to the uniaxial axis. It is the z-axis as shown in Fig. 11 in our case. Nevertheless in the transverse case (E0 perpendicular to the uniaxial axis), we exactly used the applied field E0 because the boundary condition now becomes the continuity of the tangential component of electric field. We compared the effective linear and nonlinear optical responses of colloidal crystals with the different lattice structures (i.e., bct, bcc and fcc), made of metallic nanoparticles of linear dielectric constant 1 and third-order nonlinear susceptibility 1 , suspended in a host fluid of m [see Fig. 11]. Both the longitudinal (L) and transverse (T) results will be discussed. The self-consistent equations over i = 1, 2 . . . , N are then combined together to take into account the lattice effect and is being able to be transformed into a matrix form as E = TAE + E(0) .
(63) (L,T )
} is simply N -dimensional vector and A is More precisely, in the longitudinal and the transverse cases, E = {Ei N × N diagonal matrix of the polarizability, which relates the induced dipole moment of the particle in the layer i and the local field Ei , and indeed consists of the linear and nonlinear contributions. That is pi = i Ei + i |Ei |2 Ei /3,
(64)
where i = m r 3 (1 − m )/(1 + 2m ). Here r is the radii of the metallic nanoparticles. In view of weak nonlinearity in the colloidal particles with the nonlinear relationship D = 1 E + 1 |E|2 E, by using the perturbation expansion method [67], we obtain
2
3m 2 3 3m
r 1 . (65) i = 1 + 2m 1 + 2m It is noteworthy that the linear local field Ei around the particles in the layer i are actually obtained by assuming no intrinsic nonlinear response, i.e., we set 1 = 0 for solving the self-consistent equations, which is appropriate provided
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a
b
c
Fig. 11. Schematic of the colloidal crystals immersed in a graded host medium. a = b is assumed in the numerical calculations, which forms square dipolar lattice in xy-plane [63].
that the nonlinear responses are much less than the linear ones. Next we use the resultant linear local fields Ei to extract the enhancement factor of the effective nonlinear susceptibility [67,68]
=
|Ei |2 Ei2 i 1 = , 1 3E04 1
(66)
where = [(q 3 + 2)/q]3/2 /24 is the total volume fraction of the metallic colloidal nanoparticles. 1 − q quantifies the degree of anisotropy of the periodic lattice [52], which also determines the interlayer interaction T, thus results in structure-controllable optical properties. Note that the averaging . . . in Eq. (66) is taken over the layers i (i=1, . . . , N ) instead of over the nanoparticles spatial volume, because in this approximation the local fields inside each of the particles are homogeneous. We also assume no nonlinear response in the host, which is in fact relatively neglectable comparing to that in the metal. Additionally, a gradient of the dielectric constant of the host fluid is introduced along the uniaxial direction of the colloidal crystallines, i.e, m = m (zi ) in our case. In this regard, we treat the host as a continuously(0) layered film, thus one explicitly has E0 = {Ei = E0 /m (zi )} in the longitudinal case. The formation of the gradient in the host might be achieved by dispersing different polymers in it, by selectively filling with microfluidic materials [69], or induced by the presence of a temperature gradient, etc. One can also simply coat the nanoparticles with different coverage shells. But it still remains a challenge because the novel properties from our prediction require a reasonably large gradient in the dielectric constant of the host. Fig. 12 shows in logarithmic scale the longitudinal optical absorption and the modulus of the nonlinearity enhancement factor defined in Eq. (66), as functions of the reduced frequency [see the figure caption for more details]. We specifically compared the results of bct (q =0.87358), bcc (q =1.0) and fcc (q =1.25992) as shown in the three columns. The presence of the inhomogeneity in the host fluid obviously leads to a broadened and giant enhanced resonant band in the low-frequency region. This is interesting for potential telecommunication applications. The results of the same colloid suspended in homogeneous host medium with m = 1 (dotted-lines) and m = 2.25 (dashed-lines) are also presented, in an attempt to demonstrate that the broadened resonant band in some sense stems from the hybridization of the non-graded structures. From the absorption spectrum and the enhancement in the third-order nonlinear susceptibility, we would expect an attractive figure of merit [19–23]. That is, the materials effectively exhibit large nonlinearity and relatively small absorption. This is certainly superior to pure metal because it generally has large nonlinearity and unwanted absorption concomitantly. In fact, the optical absorption arises from the surface plasmon resonance, which is obtained from the imaginary part of the effective dielectric constant that is extracted from the generalized Clausius–Mossotti formula [52,62]. Note that the plasmon resonant peaks in the cases of homogeneous host fluid are redshifted with respect to the corresponding ones [not shown] predicted form the Maxwell–Garnett theory. We actually set a gap (e.g., the coated layer thickness in experiments) of 2r/5 between the nearest lattice particles in order to avoid the severe complications arising from the multiple image interactions. It is well known that the multipolar interactions play crucial roles when two particles approach [62]. The introduction of the gap indeed makes the nanoparticles size, and thus the dipole factor relatively small and somewhat unfavorably suppresses the effect arising from variation of the lattice structure, as seen in Fig. 12.
J.P. Huang, K.W. Yu / Physics Reports 431 (2006) 87 – 172
bct
Absorption (arb.units)
102
|γ |
(a)
bcc
102
100
100
10-2
10-2
10-2
0
0.2
0.4
0.6
0.8
1
(b)
10-4
0
0.2
0.4
0.6
0.8
1
(c)
10-4
106
106
106
103
103
103
100
100
100
10-3
10-3
10-3
10-6 0
0.2
(d)
0.4 0.6 ω/ωp
0.8
10-6
1 (e)
0
0.2
0.4 0.6 ω/ωp
fcc
102
100
10-4
107
0.8
10-6
1 (f)
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6 ω/ωp
0.8
1
Fig. 12. The effective linear absorption and third-order nonlinearity enhancement factor of the periodic colloid nanoparticles (diameter a = 1) immersed in a graded host fluid whose dielectric constant varies as a function of the layer index i as 2 (i) = 1 + 1.25i/N , and in homogeneous host fluids with dielectric constant 2 = 1.52 (dashed) and 2 = 1.0 (dotted), respectively. The dielectric function of the metallic colloidal nanoparticles are simply denoted by the Drude form 1 = 1 − 2p /(2 + i) [63]. Parameters: = 0.02p , N = 25, r = 1, and E0 = 1.
Furthermore, due to the fact that we treated the continuous variations of dielectric function in the host as layered ones in obtaining i and i , and the fact that the dipoles are actually distributed in discrete lattice nodes, a series of sharp peaks are also observable in Fig. 12. The peaks are merged in the broadened band and they become more notable for a increased gradient in the host dielectric constant [not shown], whereas they tend to disappear as the crystal layer N is increased. This is also understandable in the generalized Bergman–Milton spectral representation in graded composites [70]. In detail, the merging is explicitly shown in Fig. 13, where we increased the layer to N = 50. The fcc is taken as an example and we present both the longitudinal and the transverse results. The peaks in Fig. 12(c) and (f) are distinctly smeared out in Fig. 13(a) and (b), respectively. The transverse results [see Fig. 13(c) and (d)] in the presence of the gradient is slightly different to that in the longitudinal case, but still retain the broadened bands. We ascribe this to the fact that the layer-to-layer interactions fall off exponentially due to the screening effect from the lattice [64–66], therefore give no much layer-structure-dependent interactions for both the two cases. Note that the longitudinal and transverse results of crystals in homogeneous host [dotted and dashed lines in Fig. 13] do not differ much as well. In conclusion, we have theoretically investigated the optical resonant enhancement due to lattice effect and gradient effect in colloidal crystals, which are made out of suspended metallic nanoparticles in a graded-index host. The gradient in the fluid and the colloid structure are easily subjected to tunability, for example, the structure transformation might be induced by electrorheological effects or by self-assembly of two kinds of particles with biochemically different surface properties, etc. 4. Metallic films Thin films can possess different optical properties (see, e.g., Ref. [71]) when comparing with their bulk counterparts. Recently, some authors found experimentally that the graded thin films may have better dielectric properties than a single-layer film [72]. Graded materials [39] are the materials whose material properties can vary continuously in space. These materials have attracted much interest in various engineering applications [73].
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102
102
Absorption (arb.units)
(a)
(c)
L
100
100
10-2
10-2
106
106 (d)
(b)
|γ |
T
103
103
100
100
10-3
10-3 0
0.2
0.4 0.6 ω/ωp
0.8
1
0
0.2
0.4
0.6 ω/ωp
0.8
1
Fig. 13. The same as Fig. 12, but with totally N = 50 layers in the fcc lattice: (a) and (b) the longitudinal case; (c) and (d) the transverse case [63].
The problem becomes more complicated by the presence of nonlinearity in realistic composites. Besides gradation (inhomogeneity), nonlinearity plays also an important role in the effective material properties of composite media [3,6,17,36,68,74–79]. A large nonlinearity enhancement was found indeed when the authors studied a sub-wavelength multilayer of titanium dioxide and conjugated polymer [4]. The surface plasmon resonant nonlinearity enhancement is often accompanied by a strong absorption, and this behavior renders the figure of merit (FOM) of the resonant enhancement peak to be too small to be useful. To circumvent this problem, we proposed to exploit new materials, namely, inhomogeneous (graded) metallic films, in order to achieve a large nonlinearity enhancement and an optimal FOM [21]. As a matter of fact, in practice it is more convenient to fabricate multilayer metallic films than graded films as multilayer metallic films can be readily prepared in a filtered arc deposition system. Therefore, it is necessary to discuss the multilayer effect as the number of layers inside the films increases. In this regard, this should be expected to have practical relevance. As the number of layers N increases, we shall show a gradual transition from sharp peaks to an emerging broad continuous band and the graded film results recover in the limit N → ∞. 4.1. Inhomogeneous metallic films Let us consider an inhomogeneous (graded) metallic film with width L, and the gradation under consideration is in the direction perpendicular to the film. The local constitutive relation between the displacement (D) and the electric field (E) inside the graded layered geometry is given by D(z, ) = (z, )E(z, ) + (z, )|E(z, )|2 E(z, ),
(67)
where (z, ) and (z, ) are respectively the linear dielectric constant and third-order nonlinear susceptibility. Note that both (z, ) and (z, ) are gradation profiles as a function of position r. Here we assume that the weak nonlinearity condition is satisfied, that is, the contribution of the second term (nonlinear part (z, )|E(z, )|2 ) in the right-hand side of Eq. (67) is much less than that of the first term (linear part (z, )) [74]. We further restrict our discussion to the quasi-static approximation, under which the whole layered geometry can be regarded as an effective homogeneous one with effective (overall) linear dielectric constant ¯ () and effective (overall) third-order nonlinear
J.P. Huang, K.W. Yu / Physics Reports 431 (2006) 87 – 172
109
susceptibility ¯ (). To show the definitions of ¯ () and ¯ (), we have [74] D = ¯ ()E0 + ¯ ()|E0 |2 E0 ,
(68)
where · · · denotes the spatial average, and E0 = E0 eˆz is the applied field along the z-axis. We adopt the graded Drude dielectric profile (z, ) = 1 −
2p (z) ( + i (z))
,
0 z L.
(69)
In Eq. (69), we adopted various plasma-frequency gradation profile p (z) = p (0)(1 − C · z/L),
(70)
and relaxation-rate gradation profile [80]
(z) = (∞) +
C , z/L
(71)
where C is a dimensionless constant (gradient). Here (∞) denotes the damping coefficient in the corresponding bulk material. C is a constant (gradient) which is related to the Fermi velocity. A z-dependent profile for the plasma frequency and the relaxation time can be achieved experimentally. One possible way may be to impose a temperature profile, because it has been observed that surface-enhanced Raman scattering is sensitive to temperature [81]. Thus one can tune the surface plasmon frequency by imposing an appropriate temperature gradient [82]. A temperature gradient can also be used in materials with a small bandgap or with a profile on dopant concentrations. In this case one can impose a charge carrier concentration profile to a certain extent. This effect, when coupled with materials with a significant intrinsic nonlinear susceptibility, will give us a way to control the effective nonlinear response. For less-conducting materials one can replace the Drude form of the dielectric constants by a Lorentz oscillator form. It may also be possible to fabricate dirty metal films in which the degree of disorder varies in the z direction and hence leads to a relaxation-rate gradation profile. Due to the simple layered geometry, we can use the equivalent capacitance of series combination to calculate the linear response, i.e., the optical absorption for the metallic film: 1 1 L dz = . (72) ¯ () L 0 (z, ) The calculation of nonlinear optical response can proceed as follows. We first calculate local electric field E(z, ) by the identity (z, )E(z, ) = ¯ ()E0 by virtue of the continuity of electric displacement, where E0 is the applied field. In view of the existence of nonlinearity inside the graded film, the effective nonlinear response ¯ () can be written as [74] ¯ ()E04 = (z, )|Elin (z)|2 Elin (z)2 ,
(73)
where Elin is the linear local electric field. Next, the effective nonlinear response can be written as an integral over the layer such as
¯ () 2 ¯ () 2 1 L
¯ () = dz (z, )
. (74) L 0 (z, ) (z, ) For numerical calculations, we set (z, ) to be constant (1 ), in an attempt to emphasize the enhancement of the optical nonlinearity. Without loss of generality, the layer width L is taken to be 1. Fig. 14 displays the optical absorption ∼ Im[¯()], the modulus of the effective third-order optical nonlinearity enhancement |¯()|/1 , as well as the FOM (figure of merit) |¯()|/{1 Im[¯()]} as a function of the incident angular
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J.P. Huang, K.W. Yu / Physics Reports 431 (2006) 87 – 172
4 Cω= 0.0
log10 {Im [ε(ω)]}
Cω= 0.2 Cω= 0.4
2
Cω= 0.6 Cω= 0.8
0
-2
0
0.2
0.4 0.6 ω/ωp (0)
0.8
1
0
0.2
0.4 0.6 ω /ωp (0)
0.8
1
0
0.2
0.4 0.6 ω /ωp (0)
0.8
1
(a) 4
log10 [|χ (ω)|/χ1]
3
2
1
0
-1 (b) 4
log10 (FOM)
2
0
-2
-4 (c)
Fig. 14. (a) The linear optical absorption Im[¯()], (b) the enhancement of the third-order optical nonlinearity |¯()|/1 , and (c) the FOM (figure of merit) ≡ |¯()|/{1 Im[¯()]} versus the normalized incident angular frequency /p (0) for dielectric function gradation profile (z, ) = 1 − 2p (z)/[( + i (z))] with various plasma-frequency gradation profile p (z) = p (0)(1 − C · z/L) and relaxation-rate gradation profile (z) = (∞) + C /(z/L) [21]. Parameters: (∞) = 0.02p (0) and C = 0.0.
J.P. Huang, K.W. Yu / Physics Reports 431 (2006) 87 – 172
111
frequency . Here Im[· · ·] means the imaginary part of · · · . To one’s interest, when the positional dependence of p (z) is taken into account (namely, C = 0), a broad resonant plasmon band is observed. As expected, the broad band is caused to appear by the effect of the positional dependence of the plasma frequency of the graded metallic film. In particular, this band can be observed within almost the whole range of frequency, as the gradient C is large enough. In other words, as long as the film under consideration is strongly inhomogeneous, a resonant plasmon band is expected to appear over the whole range of frequency. In addition, it is also shown that increasing C causes the resonant bands to be red-shifted (namely, located at a lower frequency region). In a word, although the enhancement of the effective third-order optical nonlinearity is often accompanied with the appearance of the optical absorption, the FOM is still possible to be quite attractive due to the presence of the gradation of the metallic film. Similarly, in Fig. 15, we investigate the effect of the inhomogeneity of the relaxation rates [ (z)], which comes from the graded metallic film. It is evident to show that, in the low-frequency region, the positional dependence of relaxation rate (z) enhances not only the third-order optical nonlinearity but also the FOM of such kind of graded metallic films. Consequently, graded metallic films can be a suitable candidate material for obtaining the optimal FOM. Thus, corresponding experiments are expected to be done to check our theoretical predictions since graded films can be fabricated easily. We have discussed a graded metallic film (layered geometry), in an attempt to investigate the effect of gradation on the nonlinear enhancement and FOM of such materials. In the conventional theory of surface plasmon resonant nonlinearity enhancement, there is often a dielectric component in the system of interest. In this regard, it turns out that it is not difficult to add a homogeneous dielectric layer on the metallic film. The same theory still works but a prominent surface plasmon resonant peak appears at somewhat lower frequencies in addition to the surface plasmon band. Due to the concomitantly strong absorption, the FOM of the resonant enhancement peak is too small to be useful. In the limit of vanishing volume fraction of the dielectric component, however, the present results recover. Nevertheless, the present results do not depend crucially on the particular form of the dielectric function. The only requirement is that we must have a sufficiently large gradient, either in p (z) or in (z) to yield a broad plasmon band. To sum up, we have investigated the effective linear and third-order nonlinear susceptibility of graded metallic films with weak nonlinearity. We calculated the effective linear dielectric constant and third-order nonlinear susceptibility. It has been found that the presence of gradation in metallic films yields a broad resonant plasmon band in the optical region, resulting in a large nonlinearity enhancement and hence an optimal FOM. 4.2. Multilayer metallic films To discuss the multilayer effect on the effective nonlinear optical response, let us first start from a general case, i.e., graded metallic film. In detail, we consider a graded metallic film with width L, and its gradation is in the direction perpendicular to the film. As a matter of fact, for graded films, the formalism has been detailedly derived in Ref. [21]. So, below we shall do a brief review, and further add some related backgrounds accordingly. Inside the graded film, the local constitutive relation between the displacement D and the electric field E is given by D(z, ) = (z, )E(z, ) + (z, )|E(z, )|2 E(z, ),
(75)
where (z, ) and (z, ) stand for the linear dielectric constant and third-order nonlinear susceptibility, respectively, and both are gradation profiles as a function of position z. The weak nonlinearity condition is assumed to be satisfied. That is, the contribution of the second term (i.e., nonlinear part (z, )|E(z, )|2 ) in the right-hand side of Eq. (75) is much less than that of the first term (namely, linear part (z, )) [74]. In the quasi-static approximation, the whole graded film can be regarded as an effective homogeneous one with effective linear dielectric constant ¯ () and effective third-order nonlinear susceptibility ¯ (). Both ¯ () and ¯ () are defined as [74] D = ¯ ()E0 + ¯ ()|E0 |2 E0 ,
(76)
where · · · denotes the spatial average, and E0 = E0 eˆz is the applied field along z axis. Then, we adopt the following graded Drude dielectric profile (z, ) = 1 −
2p (z) ( + i )
,
(77)
J.P. Huang, K.W. Yu / Physics Reports 431 (2006) 87 – 172
4 Cγ = 0.0
log10 {Im [ε(ω)]}
Cγ = 5.0X10-3ωp (0) Cγ = 1.0X10-2ωp (0)
2
Cγ =1.5X10-2ωp (0) Cγ = 2.0X10-2ωp (0)
0
-2
0
0.2
0.4 0.6 ω/ωp (0)
0.8
1
0
0.2
0.4 0.6 ω /ωp (0)
0.8
1
0
0.2
0.4 0.6 ω /ωp (0)
0.8
1
(a) 4
log10 [|χ (ω)|/χ1]
3
2
1
0
-1 (b) 4
2 log10 (FOM)
112
0
-2
-4 (c)
Fig. 15. Same as Fig. 14 [21]. Parameters: (∞) = 0.02p (0) and C = 0.6.
J.P. Huang, K.W. Yu / Physics Reports 431 (2006) 87 – 172
113
Fig. 16. Schematic graph showing multilayer metallic films.
where 0 z L, and stands for the damping coefficient in the corresponding bulk material. The general form in Eq. (77) allows for the possibility of a gradation profile in the plasma frequency p (z) [e.g., Eq. (70)]. In view of the z-dependent profile, let us use the equivalent capacitance for capacitors in series to evaluate the effective perpendicular linear response for a given frequency ¯ () [21], 1 1 L dz = . (78) ¯ () L 0 (z, ) Next, we take one step forward to write the effective nonlinear response ¯ () as an integral over the film [21],
¯ () 2 ¯ () 2 1 L
dz (z, )
, ¯ () = L 0 (z, ) (z, )
(79)
where (z, ) denotes the local third-order nonlinear susceptibility for a given frequency. It is worth noting that the real ¯ () should involve an integral over x, y, and z of the local (x, y, z, ) multiplied by terms involving (x, y, z, ). Thus, Eq. (79) offers an approximate ¯ (), as expected. To investigate the multilayer effect, we shall use some finite difference approximation of the graded Drude profile [Eq. (77)] for a finite number of layers. To mimic a multilayer system (Fig. 16), we divide the interval [0, L] into N equally spaced sub-intervals, [0, z1 ], (z1 , z2 ], . . . , (zN−1 , zN ]. Then we adopt the midpoint value of p (z) for each sub-interval as the plasma frequency of that sublayer. In this way, we calculate the effective dielectric constant, the effective third-order nonlinear susceptibility, as well as the figure of merit for each N. It is worth noting that for N → ∞ (e.g., N = 1024) the graded film results [21] recover in this limit. In what follows, we shall do some numerical calculations. We assume that the metal layers within the film have the same real and positive frequency-independent third-order nonlinear susceptibility (z, ) = 1 , and do not have a gradation profile. In doing so, we could focus on the enhancement of the optical nonlinearity. Without loss of generality, the film width L is taken to be unity. For numerical calculations, we take as a model plasma-frequency gradation profile p (z) = p (0)(1 − C · z),
(80)
where C is a constant (gradient) tuning the profile. Figs. 17–19, respectively, display the optical absorption ∼ Im[¯()], the modulus of the effective third-order optical nonlinearity enhancement |¯()|/1 , as well as the FOM |¯()|/{1 Im[¯()]} as a function of frequency /p (0). Here Im[· · ·] means the imaginary part of · · ·. In each panel of Figs. 17–19, the corresponding graded film results are shown as well. It is evident from Figs. 17–19 that for a few layers, say N = 2, 4, 8 [(a)–(c)], the optical absorption spectrum and the enhancement of optical nonlinearity consist mainly of sharp peaks. However, the strong optical absorption and the large fluctuation of the nonlinear optical enhancement near these sharp peaks render the FOM too small to be useful.
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3 N=2 graded
log10 {Im [ε (ω)]}
2
N=4 graded
N=8 graded
1 0 -1 (a)
(b)
(c)
-2 3 N=16 graded
log10 {Im [ε (ω)]}
2
N=64 graded
N=32 graded
1 0 -1 (d) -2
0
0.2
(e) 0.4 0.6 ω /ωp (0)
0.8
1
0
0.2
(f ) 0.4 0.6 ω /ωp (0)
0.8
1
0
0.2
0.4 0.6 ω /ωp (0)
0.8
1
Fig. 17. The linear optical absorption Im[¯()] versus the normalized incident angular frequency /p (0) for dielectric function gradation profile (z, ) = 1 − 2p (z)/[( + i )] with various plasma-frequency gradation profile p (z) = p (0)(1 − C · z) [83]. Parameters: = 0.02p (0), C = 0.8, L = 1, and 1 = 1.
When the number of layers becomes large [(d)–(f)], the sharp peaks accumulate to a broad band while the fluctuation has been reduced significantly. In this limit, the broad continuous absorption band emerges, and a large FOM persists. In a word, Figs. 17–19 show a gradual transition from sharp peaks to a broad continuous band as the number of layers increases. This also gives an explanation of the intriguing findings reported in our recent work [21]. To sum up, we have investigated the effective nonlinear optical response of metallic films as the number of layers inside the film increases until the graded film results recover. This is of practical value since in practice it is more convenient to fabricate multilayer metallic films than graded films. 5. Graded composites 5.1. Compositionally graded metal–dielectric composite films of anisotropic particles Chemical deposition techniques [Fig. 20(a)] as well as diffusing techniques [Fig. 20(b)] can be used to produce inhomogeneous composite films or interfaces. Metal–dielectric composites have received much attention due to the potential application of their linear and nonlinear optical properties [3,4,6,11,36,68,74–76,79,84–88]. Crucial elements for control of these properties are the micro-structure of the composite, particle shape, and material dispersity. For anisotropically shaped metallic nanoparticles, the resonant plasmon bands split up for orientations along major and minor axes. Furthermore, in case of a large size aspect ratio, the plasmon bands may shift into the infrared, thus enabling the use of metal nanostructures in telecommunication applications in this wavelength range. Compared with spherically shaped particles, anisotropically shaped metallic particles can show reduced plasmon relaxation times [89] as well as enhanced nonlinear responses [90], and may thus be used as building blocks in a variety of optical devices. Some techniques have been developed to fabricate rod-shaped metallic nanoparticles by using lithographic means [91] or
J.P. Huang, K.W. Yu / Physics Reports 431 (2006) 87 – 172
115
5 N=2 graded
log10 [| χ (ω) | /χ1]
4 3 2 1
(a)
N=4 graded
(b)
N=8 graded
(c)
0
log10 [| χ (ω) | /χ1]
5
4
3 N=16 graded
2
N=32 graded
(d) 1
0
0.2
N=64 graded
(e) 0.4 0.6 ω /ωp (0)
0.8
1
0
0.2
(f) 0.4 0.6 0.8 ω /ωp (0)
1
0
0.2
0.4 0.6 ω /ωp (0)
0.8
1
Fig. 18. Same as Fig. 17, but for the enhancement of the third-order optical nonlinearity |¯()|/1 [83].
log10 [FOM]
4
2
N=2 graded
0
N=4 graded
(a)
-2
N=8 graded
(b)
(c)
log10 [FOM]
4
2 N=16 graded
0
-2
N=32 graded
(d) 0
0.2
N=64 graded
(e) 0.4
0.6
ω /ωp (0)
0.8
1
0
0.2
(f) 0.4
0.6
ω /ωp (0)
0.8
1
0
0.2
0.4
0.6
ω /ωp (0)
Fig. 19. Same as Fig. 17, but for the FOM ≡ |¯()|/{1 Im[¯()]} [83].
0.8
1
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Fig. 20. Schematic graph showing (a) the chemical deposition technique to form a graded composite film, and (b) the diffusing technique to form a graded interface (composite film) between two bulk composites [94].
Fig. 21. Schematic graph to show the geometry of a metal–dielectric composite film with a variation of volume fraction of anisotropic particles along z axis perpendicular to the film [99]. The electric field E is parallel to the gradient (z axis), thus being perpendicular to the film.
anisotropic growth. Recently, one has demonstrated that mega-electron-volt ion irradiation can also be used to modify the shape of nanoparticles [92]. This ion-beam-induced anisotropic deformation effect is known to occur not only for a broad range of amorphous materials [93], but also for crystalline materials including metals [88]. That is, prolate spheroidal metallic particles in a dielectric host can readily be formed by irradiation of mega-electron-volt ions. The optical property of thin films [21,23,71,95,96] is often different from that of the corresponding bulk material. Recently, a metal–dielectric composite (Au:BaTiO3 ) film was investigated, and large figure of merit (FOM) was observed [97]. In general, graded materials [39,98] have quite different physical properties from the homogeneous materials. In particular, the graded thin films were found to show better dielectric properties than the single-layer films [72]. A large nonlinearity enhancement was recently found for a sub-wavelength multilayer film of titanium dioxide and conjugated polymer [4]. However, the surface-plasmon resonant nonlinearity enhancement often occurs concomitantly with a strong absorption, and unfortunately this behavior renders the FOM of the resonant enhancement peak to be too small to be useful. To circumvent this problem, we shall consider a kind of graded metal–dielectric composite film, in which a dielectric (or metallic) component is introduced as anisotropically shaped particles embedded in a metallic (or dielectric) component with a compositional and/or shape-dependent gradation profile. Let us consider a metal–dielectric composite film with a variation of volume fraction of anisotropic particles along z axis perpendicular to the film (Fig. 21). In this case, the local constitutive relation between the displacement field D(r, ) and electric field E(r, ) is given by D(r, ) = (r, )E(r, ) + (r, )|E(r, )|2 E(r, ),
(81)
where (r, ) and (r, ) are the linear dielectric constant and third-order nonlinear susceptibility of a layer inside the graded film, respectively. Here both (r, ) and (r, ) are the gradation profiles as a function of position r and field frequency . The weak nonlinearity condition is assumed to be satisfied. That is, the contribution of the second term [nonlinear part (r, )|E(r, )|2 ] in the right-hand side of Eq. (81) is much less than that of the first term [linear part (r, )] [74]. Next, we turn to the quasi-static approximation, under which the whole graded structure can be regarded as an effective homogeneous one with effective linear dielectric constant ¯ () and effective third-order nonlinear susceptibility ¯ (). In this connection, ¯ () and ¯ () are defined as [74] D0 = ¯ ()E0 + ¯ ()|E0 |2 E0 ,
(82)
J.P. Huang, K.W. Yu / Physics Reports 431 (2006) 87 – 172
117
where D0 and E0 (=E0 eˆ z ) are, respectively, the volume-averaged displacement field and electric field within the whole graded composite film. For calculating the nonlinear optical response, we shall apply a two-step solution. In Step A, we first derive the responses of a layer inside the graded film, ¯ (z, ) and ¯ (z, ). In Step B, the overall responses of the graded film, ¯ () and ¯ (), are derived. In the two-step solution, it should be remarked that the local field inside the spheroidal particles is uniform, and the effective nonlinear response of a layer is therefore exact within the Maxwell–Garnett theory. When we have a nonlinear host, we shall have to invoke the decoupling approximation [100]. It is worth noting that Step B is exact, see Eqs. (96)–(98) below. In a word, in Step A, we homogenize the composite film along xy plane while in Step B, we further homogenize the graded film along z axis. 5.1.1. Responses of a layer inside the graded film: ¯ (z, ) and ¯ (z, ) It is not possible to calculate ¯ (z, ) exactly in terms of the compositional and/or shape-dependent gradation profile. Nevertheless, to obtain an estimate of ¯ (z, ), we can take a small volume element inside the layer, at a position z. Further, this small volume element can be seen as a composite where the locations of the inclusion particles are random in the host medium. This, however, is a highly directional distribution since the long or short axis of prolate or oblate particles is parallel to the gradient along z axis. Accordingly, the volume fraction of the inclusion is p(z) for the dielectric or 1 − p(z) for the metal. In this regard, the well-known Maxwell–Garnett approximation holds very well for computing ¯ (z, ) [as shown in Eqs. (83) and (84) below]. In detail, for the dielectric particles embedded in the metallic component in a layer, ¯ (z, ) can be given by the first-kind of Maxwell–Garnett approximation (MGA1) [27,28,101] ¯ (z, ) − 1 () (2) (2) Lz ¯ (z, ) + (1 − Lz )1 ()
= p(z)
2 − 1 () (2) Lz 2
(2)
+ (1 − Lz )1 ()
,
(2)
(83) (2)
(2)
where Lz is the depolarization factor of the dielectric particles along z-axis, and satisfies a sum rule Lz + 2Lx = 1. (2) Here Lx is the depolarization factor of the dielectric particles along x(y)-axis, 2 (or 1 ()) stands for the dielectric constant of the dielectric (or metallic) particles, and p(z) denotes the volume fraction of the dielectric particles in each layer which is thus a compositional gradation profile as a function of position z. Alternatively, for the metallic particles embedded in the dielectric host, the second kind of Maxwell–Garnett approximation (MGA2) can be used to determine ¯ (z, ), such that [27,28,101] ¯ (z, ) − 2 (1) (1) Lz ¯ (z, ) + (1 − Lz )2
= (1 − p(z))
1 () − 2 , (1) (1) Lz 1 () + (1 − Lz )2
(1)
(84) (1)
(1)
where Lz is the depolarization factor of the metallic particles along z-axis. Similarly, there exists Lz + 2Lx = 1 (1) where Lx is the depolarization factor of the metallic particles along x(y)-axis. It is worth noting that Lz < 1/3, =1/3 and > 1/3 indicates the fact that the metallic (or dielectric) particles exist in the form of prolate spheroid, sphere and oblate spheroid, respectively. In Eqs. (83) and (84), the dielectric constant of the metal 1 () is given by the known Drude expression 1 () = 1 −
2p ( + i )
,
(85)
where p denotes the bulk plasmon frequency, and the collision frequency. In case of 1 () > 2 , the MGA1 always gives an upper bound while the MGA2 a lower bound, and vice versa. The exact result must lie between the two bounds [102]. For both the MGA1 and MGA2, the particles under discussion are randomly embedded but their orientations are all along z axis (i.e., perpendicular to the film). The reason is that in experiments the prolate spheroidal metallic particles can be highly oriented along the direction of irradiated ions [88]. For completeness, we shall also numerically calculate the case of oblate spheroids. Then, we calculate the effective nonlinear response for a layer at position z, ¯ (z, ) [74], ¯ (z, )E(z)4 = (1 − p(z))1 |E1 (z)|2 E1 (z)2 + p(z)2 |E2 (z)|2 E2 (z)2 ,
(86)
where 1 and 2 are, respectively, the (intrinsic) third-order nonlinear susceptibility of the metallic and dielectric components, E1 (z) (or E2 (z)) represents the local electric field inside the metallic (or dielectric) component within
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a layer at position z, E(z) denotes the volume-averaged electric field within the layer, and · · · stands for the volume average of · · · within the layer. In order to estimate ¯ (z, ), due to the existence of nonlinear host we have to invoke the decoupling approximation [100] |Ei (z)|2 Ei (z)2 = |Ei (z)|2 Ei (z)2 ,
i = 1, 2.
(87)
For the sake of consistency, the local field averages |Ei (z)|2 and Ei (z)2 should be determined by using the Maxwell–Garnett technique [103]. For the MGA1, we obtain the local field averages such that (Lz )−2 1 ()2 (2)
E2 (z)2 =
E(z)2 , (2) [(1 − p(z))2 + ((Lz )−1 − (1 − p(z)))1 ()]2 (2) (2) p(z)(Lz )−1 [(Lz )−1 1 ()2 − (1 − p(z))(2 − 1 ())2 ] 2 E1 (z) = 1 − E(z)2 , (2) [(1 − p(z))2 + ((Lz )−1 − (1 − p(z)))1 ()]2 (Lz )−2 |1 ()|2
(88)
(89)
(2)
|E2 (z)|2 =
E(z)2 , (2) |(1 − p(z))2 + ((Lz )−1 − (1 − p(z)))1 ()|2 (2) (2) p(z)(Lz )−1 [(Lz )−1 |1 ()|2 − (1 − p(z))|2 − 1 ()|2 ] 2 |E1 (z)| = 1 − E(z)2 , (2) −1 2 |(1 − p(z))2 + ((Lz ) − (1 − p(z)))1 ()|
(90)
(91)
with = 1/(1 − p(z)). Similarly, for the MGA2, the local field averages are given by (Lz )−2 22 (1)
E1 (z)2 =
(1) [p(z)1 () + ((Lz )−1
E2 (z)2 = 1 −
− p(z))2
]2
E(z)2 ,
(1 − p(z))(Lz )−1 [(Lz )−1 22 − p(z)(1 () − 2 )2 ] (1)
(1)
(92)
[p(z)1 () + ((Lz )−1 − p(z))2 ]2 (1)
(Lz )−2 |2 |2
E(z)2 ,
(93)
(1)
|E1 (z)|2 =
(1) |p(z)1 () + ((Lz )−1
|E2 (z)| =
2
E(z)2 ,
(1 − p(z))(Lz )−1 [(Lz )−1 |2 |2 − p(z)|1 () − 2 |2 ] (1)
1−
− p(z))2
|2
(1)
|p(z)1 () + ((Lz )−1 − p(z))2 |2 (1)
(94) E(z)2 ,
(95)
with = 1/p(z). 5.1.2. Overall responses of the graded film: ¯ () and ¯ () Owing to the simple graded structure (Fig. 21), we can use the equivalent capacitance of series combination to calculate the linear response (i.e., optical absorption for the graded film), ¯ (), W 1 dz 1 = , (96) ¯ () W 0 ¯ (z, ) where W is the thickness of the film. By virtue of the continuity of electric displacement, there is a relation ¯ (z, )E(z) = ¯ ()E0 .
(97)
Then, we take one step forward to obtain the effective third-order nonlinear susceptibility ¯ () as an integral over the graded film,
W
¯ () 2 ¯ () 2 1
¯ () = dz ¯ (z, ) . (98) W 0 ¯ (z, ) ¯ (z, )
J.P. Huang, K.W. Yu / Physics Reports 431 (2006) 87 – 172
119
In what follows, we shall do some numerical calculations. Set both 1 and 2 to be a real and positive frequencyindependent constant 0 , so that we could emphasize the enhancement of the optical nonlinearity. Without loss of generality, the film thickness W is taken to be unity. For the model calculation, we shall use the gradation profile p(z) = azm ,
(99)
where a and m are constants tuning the profile. Fig. 22 shows the results obtained from the MGA1 [Eq. (83)]. In this figure, we display: (a) the optical absorption ∼ Im[¯()]; (b) the modulus of the effective third-order optical nonlinearity enhancement |¯()|/0 ; and (c) the FOM (2) |¯()|/{0 Im[¯()]} as a function of the incident angular frequency , for different Lz . Here Im[· · ·] means the imaginary part of · · ·. When the layer gradation profile p(z) = azm is taken into account, a broad resonant plasmon (2) band is observed for any Lz of interest. In other words, the broad band is caused to appear by the effect of the positional dependence of the dielectric or metallic component. This conclusion may be made by comparing the curves in Fig. 22 with those of n = 0 (corresponding to the case where the effects of gradation as well as non-spherical shape (2) are excluded) in Fig. 27. Moreover, we find that decreasing Lz makes the resonant bands in both optical nonlinearity and optical absorption broader. Although the enhancement of the effective third-order optical nonlinearity is often accompanied with the appearance of the optical absorption, the FOM is still possible to be very attractive [see Fig. 22(c)] due to the positional dependence of the dielectric or metallic components. In particular, the particle shape can also be used to enhance the FOM significantly. It is worth noting that there is a prominent surface plasmon resonant peak which appears at somewhat higher frequencies in addition to the surface plasmon band at lower frequencies. Fig. 23 displays the results which were obtained from the MGA2 [Eq. (84)]. For this figure, we see the parallel shaped metallic particles are randomly embedded in the dielectric host in each layer. In contrast, the surface plasmon resonant peak is found to locate at lower frequencies in addition to the surface plasmon band which locates at higher frequencies. Also, it is shown that the broad plasmon bands in optical nonlinearity and absorption is caused to appear by the effect of gradation, when comparing the curves in Fig. 23 with those of n = 0 in Fig. 28. Note the latter corresponds (1) to the case without the effects of gradation and non-spherical shape. Decreasing Lz causes the plasmon bands to be broadened. This effect makes the FOM can be very attractive. The MGA1 was applied to plot Fig. 24, in an attempt to discuss the effect of the gradation of the volume fraction of the dielectric by means of the gradation profile p(z) = azm for different (a–c) a and (d–f) m. In other words, we investigate a compositional gradation profile in the film, in which the dielectric particles possess a positional-dependent volume fraction. In detail, increasing a causes the resonant plasmon bands in optical nonlinearity and absorption to be broadened, see Fig. 24(a–b). Accordingly, in case of gradation, the FOM can be more attractive, see Fig. 24(c). Similarly, Fig. 24(d–f) displays the influence of m. It is apparent to see that the broad resonant plasmon bands in optical nonlinearity and absorption can be enhanced by increasing m [see Fig. 24(d–e)]. Finally, the FOM can become more attractive in the frequency range 0.3p < < 0.7p , as m increases, see Fig. 24(f). Similar to Fig. 24, Fig. 25 is plotted via the MGA2. First, we discuss the effect of a. In both optical nonlinearity and absorption, the resonant plasmon bands are caused to be both enhanced and broadened by increasing a, as yielding a very attractive FOM in Fig. 25(c). On the other hand, the m effect on the optical nonlinearity and FOM plays a role too [see Fig. 25(e–f)], and accordingly both the optical nonlinearity and FOM can be enhanced accordingly. In addition, as a or m varies, the plasmon resonant peaks in Fig. 25 have the same red-shift (located at lower frequency) or blue-shift (located at higher frequency) behavior as those shown in Fig. 24 where the MGA1 was used instead. During ion irradiation, the ion energy can be much larger at the top of the film than that at the bottom. Therefore, (1) the particles can be much prolate at the top, but they are relatively spherical at the bottom. In other words, both Lz (2) and Lz can be small at the top of the film, while increases to roughly 1/3 at the bottom of the film. In this regard, we could introduce a gradation in the depolarization factor (Fig. 26) rather than in the volume fraction. Namely, in this case, Lz (z) is a function of z (Fig. 26). For convenience, we keep the volume fraction to be constant [e.g., p(z) = 0.85] for each layer throughout the film, and take a physical profile Lz (z) = (1/3)zn . In particular, as n = 0, we have Lz (z) = 1/3, i.e., the gradation in the depolarization factor and the non-spherical shape effect disappear. For different n, the corresponding results are shown in Figs. 27 and 28 for the MGA1 and MGA2, respectively. It is shown that the Lz (z) profile does have a significant impact on the optical response, as expected. In Fig. 27(a), the plasmon peak shows a reduction as well as a blue-shift as n changes from zero (without gradation) to nonzero (with gradation), and accordingly the optical nonlinearity and hence the FOM is reduced. The difference between the results for different
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4 Lz(2) = 0.1
log10 {Im [ ε (ω)]}
Lz(2) =0.2 2
Lz(2) =1/3 Lz(2) = 0.4 Lz(2) = 0.5
0
-2 (a) 8
log10 [| χ (ω)|/χ 0 ]
6
4
2
(b)
0 6
log10 (FOM)
4
2
0
0 (c)
0.5
ω /ωp
1
1.5
Fig. 22. Results for the MGA1 [Eq. (83)] [99]: (a) Linear optical absorption Im[¯()]; (b) enhancement of the third-order optical nonlinearity |¯()|/0 ; and (c) FOM ≡ |¯()|/{0 Im[¯()]} versus the normalized incident angular frequency /p for layer dielectric profile p(z) = azm , (2)
for different Lz . Parameters: a = 0.8, m = 1, /p = 0.01, and 2 = (3/2)2 .
J.P. Huang, K.W. Yu / Physics Reports 431 (2006) 87 – 172
121
Lz(1)= 0.1
log10 {Im [⎯ε (ω)]}
2
Lz(1)= 0.2 Lz(1)=1/3 Lz(1) = 0.4
0 Lz(1) = 0.5
-2 (a) 8
log10 [|⎯χ (ω) |/χ0]
6
4
2
(b) 0 5
log10 (FOM)
4
3
2
1 (c)
0
0.5
ω /ωp
1
1.5
Fig. 23. Results for the MGA2 [Eq. (84)] [99]: (a) Linear optical absorption Im[¯()]; (b) enhancement of the third-order optical nonlinearity |¯()|/0 ; and (c) FOM ≡ |¯()|/{0 Im[¯()]} versus the normalized incident angular frequency /p for layer dielectric profile p(z) = azm , (1)
for different Lz . Parameters: a = 0.8, m = 1, /p = 0.01, and 2 = (3/2)2 .
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J.P. Huang, K.W. Yu / Physics Reports 431 (2006) 87 – 172
log10 {Im[⎯ε (ω)]}
4 a = 0.2 a = 0.4 a = 0.6 a = 0.8
2
m = 0.6 m =1.0 m =1.4 m =1.8
0
-2 (d)
(a)
log10 [|⎯χ (ω)| /χ0]
6
4
2
(e)
(b) 0 6
log10 (FOM)
4 2 0 -2 0 (c)
0.5
ω /ωp
1
1.5
0 (f)
0.5
ω /ωp
1
1.5
Fig. 24. Results for the MGA1 [Eq. (83)] [99]: (a) and (d) Im[¯()]; (b) and (e) |¯()|/0 , and (c) and (f) FOM ≡ |¯()|/{0 Im[¯()]} versus /p (a)–(c) for different a at m = 1.0, and (d)–(f) for different m at a = 0.8. Parameters: Lz = 0.1, /p = 0.01, and 2 = (3/2)2 .
nonzero n (i.e., n = 4, 8, 12) is not very distinct (Fig. 27). Interestingly, the Lz gradation gives rise to an additional peak which appears at a lower frequency. For the MGA2 (Fig. 28), the surface plasmon resonant bands in optical absorption and nonlinearity are clearly visible for various Lz (z) profiles, see Fig. 28(a–b). In the presence of gradation, i.e., n becomes nonzero, the prominent plasmon absorption peak at n = 0 has been broadened into a plasmon band, and an additional peak is induced to appear at lower frequency. Concomitantly, a plasmon band and a peak in optical nonlinearity are also caused to appear [Fig. 28(b)] and hence the FOM can be enhanced accordingly, see Fig. 28(c). On the other hand, we also find that the plasmon bands in optical absorption and nonlinearity can be further broadened (and enhanced) by adopting a wider gradation profile such as Lz (z) = 0.5zn (no figures shown here). For this sort of profile, there are prolate particles at the top, but oblate particles at the bottom of the film. It is possible to realize such oblate particles near the bottom of the film due to the reaction stress from the substance. Finally, in Figs. 23(a), 25(a) and 28(a), there are always a plasmon band plus an absorption peak as long as the gradation profile exists. Recently, an absorption peak plus a slim plasmon absorption band was indeed observed [88], when one investigated the optical extinction spectra for ensembles of core-shell colloids with Au cores and shells embedded in an index-matching fluid. But, after irradiation with 30 MeV (mega electron volt) Cu ions, a broadening of the plasmon absorption band was also observed, which was thought to attribute to the formation of Au nanorods. To account for this behavior, we believe the particle shape, and gradation in the depolarization factor of metals and in the volume fraction of the metallic (or dielectric) component should be expected to play an important role.
J.P. Huang, K.W. Yu / Physics Reports 431 (2006) 87 – 172
123
log10 {Im[⎯ε (ω)]}
4 a = 0.2 a = 0.4 a = 0.6 a = 0.8
2
m = 0.6 m =1.0 m =1.4 m =1.8
0
-2 (a)
(d)
(b)
(e)
(c)
(f)
log10 [|⎯χ (ω)| /χ 0 ]
8 6 4 2 0
log10 (FOM)
6
4
2
0
0
1
0.5 ω /ωp
1.5 0
1
0.5
1.5
ω /ωp
Fig. 25. Results for the MGA2 [Eq. (84)] [99]: (a) and (d) Im[¯()]; (b) and (e) |¯()|/0 ; and (c) and (f) FOM ≡ |¯()|/{0 Im[¯()]} versus /p (a)–(c) for different a at m = 1.0, and (d)–(f) for different m at a = 0.8. Parameters: Lz = 0.1, /p = 0.01, and 2 = (3/2)2 .
Fig. 26. Schematic graph to show the geometry of a metal–dielectric composite film with a variation of depolarization factor of particles along z axis perpendicular to the film [99]. The electric field E is parallel to the gradient (z axis), thus being perpendicular to the film.
In a word, the sharp plasmon peak comes naturally from the existence of metal–dielectric interfaces. In the case of graded metallic films, there should be a broad band only, but no sharp peak. So, for the graded metal–dielectric composite film under present consideration, both the plasmon peak and the broad plasmon band should appear as predicted above.
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2
n=0
log10 {Im [⎯ε (ω)]}
n=4 n=8 0
n=12
-2
(a) -4
log10 [|⎯χ( ω)| /χ 0 ]
6
4
2
(b) 0 6
log10 (FOM)
4
2
0
-2 0 (c)
1
0.5
1.5
ω /ωp
Fig. 27. Results for the MGA1 [Eq. (83)] [99]: (a) Linear optical absorption Im[¯()]; (b) enhancement of the third-order optical nonlinearity |¯()|/0 ; and (c) FOM ≡ |¯()|/{0 Im[¯()]} versus the normalized incident angular frequency /p for the gradation profile of the depolar(2)
ization factor of dielectric particles Lz (z) = (1/3)zn , for different n. Parameters: p(z) = 0.85, /p = 0.01, and 2 = (3/2)2 .
J.P. Huang, K.W. Yu / Physics Reports 431 (2006) 87 – 172
log10 {Im [⎯ε (ω)]}
2
125
n= 0 n =4 n =8
0
n =12
-2
(a) -4
log10 [|⎯χ (ω)| /χ0]
6
4
2
(b) 0
4
log10 (FOM)
3
2
1
0 (c)
0.5
ω /ωp
1
1.5
Fig. 28. Results for the MGA2 [Eq. (84)] [99]: (a) Linear optical absorption Im[¯()]; (b) enhancement of the third-order optical nonlinearity |¯()|/0 ; and (c) FOM ≡ |¯()|/{0 Im[¯()]} versus the normalized incident angular frequency /p for the gradation profile of the depolar(1)
ization factor of metallic particles Lz (z) = (1/3)zn , for different n. Parameters: p(z) = 0.85, /p = 0.01, and 2 = (3/2)2 .
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We have studied the effective nonlinear optical response of a graded metal–dielectric composite film of anisotropic particles. Based on the MGA1 and MGA2 [Eqs. (83) and (84)], we derived the local electric field inside the film, and hence obtained the effective linear dielectric constant [Eq. (96)] and third-order nonlinear susceptibility [Eq. (98)] of the graded composite film. In comparison with textbook formulae, the formulae [Eqs. (83) and (84)] only differ from the z-dependent volume fraction p(z), in the sense that we could discuss the gradation which is perpendicular to the film and then leads to nonlinearity enhancement. As a matter of fact, the present results do not depend crucially on the particular form of the dielectric gradation profile p(z) or the depolarization-factor gradation profile Lz (z). The only requirement is that we must have a compositional or shape-dependent gradation to yield a broad plasmon band for the composite film. It should be remarked that the optical response of the graded structure depends on polarization of the incident light, because the incident optical field can always be resolved into two polarizations. However, a large nonlinearity enhancement occurs only when the electric field is parallel to the direction of the gradient [4], and the other polarization does not produce nonlinearity enhancement at all [4]. The nonlinear susceptibilities of both the parallel and perpendicular polarizations are related to the nonlinear phase shift which can be measured by using a z-scan method [4]. Following Roorda et al. [88], one could fabricate the film under the present discussion by using mega-electron-volt ion irradiation. Its third-order nonlinear susceptibility could also be measured by a degenerated four-wave-mixing method, which has been used for the Au/SiO2 composite film [12]. It is of interest to extend the present theory to composites in which graded spherical particles are embedded in a host medium [20] to account for mutual interactions among graded particles. To sum up, we have studied the effective linear dielectric constant and third-order nonlinear susceptibility of a graded metal–dielectric composite film of anisotropic particles with weak nonlinearity by invoking the local field effects exactly within the Maxwell–Garnett theory.We have numerically demonstrated that this kind of film can serve as a novel optical material for producing a broad structure in both the linear and nonlinear response and an enhancement in the nonlinear response. 5.2. Compositionally graded metal–dielectric films: effects of microstructure One of the crucial elements for control of the linear and/or nonlinear optical responses is the microstructure of composites (e.g., see Refs. [3,57] and references therein). For discussing the effect of microstructure, the usual two methods are the well-known Maxwell–Garnett approximation (MGA) [27,28] and Bruggeman effective medium approximation (EMA) [29]. It is worth noting that the MGA is an asymmetrical theory (Section 2.1) whereas the EMA is a symmetrical theory (Section 2.2). As mentioned above, the nonlinearity enhancement often occurs concomitantly with a strong absorption, and unfortunately this behavior renders the corresponding FOM to be too small to be useful. To circumvent this problem, we shall consider a kind of compositionally graded metal–dielectric film in which a dielectric component is introduced as particulates embedded in the metallic component. To our interest, most recently compositionally graded ferroelectric films were investigated, and new phenomenon was shown indeed [104]. In this regard, we expect the compositionally graded metal–dielectric film under our investigation can show some new interesting phenomenon concerning the nonlinear and/or linear optical responses, too. Let us consider a nonlinear graded film with width L, in which the dielectric particles of dielectric constant 2 is embedded in the host metal of 1 (), and the volume fraction of the embedded particles p2 (z) varies along z-axis. Here the gradient of gradation is in the direction perpendicular to the film, i.e., in z-axis. In this connection, the local constitutive relation between the displacement D and electric field E is given by D(z, ) = (z, )E(z, ) + (z, )|E(z, )|2 E(z, ),
(100)
where (z, ) and (z, ) are, respectively, the linear dielectric constant and third-order nonlinear susceptibility of a layer inside the graded film. It is worth mentioning that both (z, ) and (z, ) are gradation profiles as a function of position z. Let us further assume that the weak nonlinearity condition is satisfied. That is, the contribution of the second term (nonlinear part (z, )|E(z, )|2 ) in the right-hand side of Eq. (100) is much less than that of the first term (linear part (z, )) [74]. Next, we restrict our discussion to the quasi-static approximation, i.e., d/ 1, where d is the characteristic size of the particle and is the wavelength of the incident light. In the quasi-static approximation, the whole graded
J.P. Huang, K.W. Yu / Physics Reports 431 (2006) 87 – 172
127
structure can be regarded as an effective homogeneous one with effective (overall) linear dielectric constant ¯ () and effective (overall) third-order nonlinear susceptibility ¯ (). That is, ¯ () and ¯ () is defined as [74] D = ¯ ()E0 + ¯ ()|E0 |2 E0 ,
(101)
where · · · stands for the spatial average of · · ·, and E0 = E0 eˆz the applied field along z-axis. Now let us take into account the detailed microstructures in order to obtain ¯ () and ¯ (). First, we consider an asymmetrical microstructure [27,28] in which one component constitutes inclusion particles while the other serves as a host. For such a microstructure, the MGA (Maxwell–Garnett approximation) is valid. In detail, for each layer of the graded metal–dielectric film, the effective dielectric constant (z, ) is the solution of the MGA equation, (z, ) − 1 () 2 − 1 () = p2 (z) , (z, ) + 21 () 2 + 21 ()
(102)
On the other hand, we focus on a symmetrical microstructure in which the two sorts of particles can be exchanged, yielding, however, no effect on the effective dielectric constant (z, ) of each layer. For treating such a symmetric microstructure, the EMA (effective medium approximation or Bruggeman approximation) [29] works for calculating (z, ) by solving the self-consistent equation such that (1 − p2 (z))
1 () − (z, ) 2 − (z, ) + p2 (z) = 0. 1 () + 2(z, ) 2 + 2(z, )
(103)
Regarding Eqs. (102) and (103), we should remark more. In fact, it is not possible to calculate (z, ) exactly in terms of the layer dielectric profile p2 (z). Nevertheless, to obtain an estimate of (z, ), we can take a small volume element inside the layer, at a position z. Further, this small volume element can be seen as a composite where the dielectric particles are randomly embedded in the metallic component. Accordingly, the volume fraction of the dielectric particles is p2 (z). In this regard, the above-mentioned MGA and EMA should be expected to hold well for computing (z, ). Owing to the simple graded structure, we can use the equivalent capacitance of series combination to calculate the linear response (i.e., optical absorption), 1 1 L dz = . (104) ¯ () L 0 (z, ) To investigate the nonlinear optical response, we first calculate local electric field E(z, ) by means of the identity (z, )E(z, ) = ¯ ()E0
(105)
due to the virtue of the continuity of the electric displacement. In view of the existence of nonlinearity inside the graded film, the effective nonlinear response ¯ () can be given by [74] ¯ ()E04 = (z, )|Elin (z)|2 Elin (z)2 ,
(106)
where Elin denotes the linear local electric field. Next, we take one step forward to express the effective nonlinear response as an integral over the film,
¯ () 2 ¯ () 2 1 L
dz (z, ) . (107) ¯ () = L 0 (z, ) (z, ) For the following numerical calculations, we adopt a Drude-type dielectric function for metallic particles, namely, 1 () = 1 −
2p ( + i )
,
(108)
where p denotes the bulk plasmon frequency, and the damping constant. In addition, we set = 0.01p (typical value for noble bulk metals) and 2 = (3/2)2 (dielectric constant of glass). Now we are in a position to do some numerical calculations in an attempt to discuss the effect of the above-mentioned microstructures. Set (z, ) to be a constant 1 , so that we could emphasize the enhancement of the optical nonlinearity.
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Regarding the layer dielectric profile, take a power form p2 (z) = azm . Without loss of generality, the layer width L is set to be unity. In Figs. 29–32, we plot (a) the optical absorption ∼ Im[¯()], (b) the modulus of the effective third-order optical nonlinearity enhancement |¯()|/1 , and (c) the FOM (figure of merit) |¯()|/{1 Im[¯()]} as a function of the incident angular frequency , respectively. Here Im[· · ·] means the imaginary part of · · · . It is well known that the MGA correctly predicts the surface plasmon resonance of bulk metal–dielectric composite, and this properties is shown as well in our model for a metal–dielectric film. Figs. 29 and 30 show the optical properties based on the MGA. In Fig. 29, we display the effect of the coefficient a. When the layer dielectric profile p2 (z) is taken into account, a broad resonant plasmon band is observed always. In other words, the broad band is caused to appear by the effect of the positional dependence of the dielectric or metal. Also, we find that increasing a causes the resonant band not only to be enhanced, but also red-shifted (namely, located at a lower frequency region). In a word, although the enhancement of the effective third-order optical nonlinearity is often accompanied with the appearance of the optical absorption, the FOM is still possible to be very attractive due to the presence of the positional dependence of the dielectric or metallic components. Moreover, it is worth noting that a prominent surface plasmon resonant peak appears at somewhat higher frequencies in addition to the surface plasmon band. As a increases, this peak is blue-shifted (i.e., locates at a higher frequency region) accordingly. Similarly, Fig. 30 displays the influence of m. It is apparent to see that the broad resonant plasmon band can be enhanced significantly by adjusting m. However, no distinct red-shift occurs for the plasmon band as m varies. In contrast, we notice that increasing m can make the surface plasmon resonant peak red-shifted. For the symmetrical microstructure (EMA model), we also display the effects of a (Fig. 31) and m (Fig. 32), respectively. For this kind of microstructure, a plasmon band exists as well. However, the surface plasmon resonance becomes broad and weak, and the resonance peak disappears at a large volume fraction of metallic particles (e.g., a=0.2, m=1.0). This is different from the MGA prediction that whatever the volume fraction of the metal component is there always exists a sharp resonance. That is, such a difference is caused by introducing the two different (asymmetrical and symmetrical) microstructures. Moreover, it is well known that there is a percolation threshold predicted by the EMA, at which the properties of the metal–dielectric composite change significantly. For metal–dielectric composites the percolation threshold is p2 (z) = 2/3 [i.e., 1 − p2 (z) = 1/3] at which the conductivity of the composite becomes nonzero. In other words, as 1 − p2 (z) > 1/3 the composite behaves as a metal rather that a dielectric. Thus, in Figs. 31 and 32, there is no apparent resonance peak especially when the volume fraction of the metal is large. Also, the nonlinearity enhancement is unimpressive and the FOM is generally small. Nevertheless, the plasmon band always exists, too. For the Maxwell–Garnett model, the microstructure of interest should be asymmetrical. That is, the dielectric particles are surrounded by the metallic component. In other words, the dielectric particles are randomly dispersed in a metallic host so that the dielectric particles cannot touch each other. On the other hand, for the Bruggeman model, the microstructure should be symmetrical. That is, both the dielectric particles and metallic particles are mixed randomly. It is not possible to distinguish the embedded component with the host medium. It is known that the Maxwell–Garnett model (without gradation) can predict a sharp plasmon resonance peak whereas the Bruggeman model will give a broad plasmon band. For the Maxwell–Garnett model with gradation, a broad resonant plasmon band is observed, see Figs. 29–30. It is because the inhomogeneity due to gradation leads to a further broadening of the plasmon resonance peak. In more detail, as a or m increases, see Fig. 29 or 30, the resonant frequency takes on values within a broader range across the film, and hence leads to a broad plasmon band. The further broadening in the plasmon band also occurs in the Bruggeman model. We have discussed the effective linear and nonlinear optical responses of a compositionally graded metal–dielectric film in an attempt to study the effect of microstructure. For two asymmetrical and symmetrical microstructures, we used the MGA and EMA to calculate the effective responses, respectively. The appearance of plasmon bands is interesting, and comes about from the gradual changes in the volume fraction of the metallic component in one direction, i.e. like a one-dimensional tight binding band of surface plasmon modes at each layer. As a matter of fact, the present results do not depend crucially on the particular form of the layer dielectric profile p(z). However, the microstructure can significantly affect the linear and nonlinear optical response, as showed above. As the volume fraction of the metal component increases, the MGA (or EMA) predicts a sharp (or broad and weak) resonance. However, a plasmon band was observed always, regardless of the detailed microstructure. To obtain such results, the only requirement is that one must have a composition-dependent layer inside the graded film.
J.P. Huang, K.W. Yu / Physics Reports 431 (2006) 87 – 172
129
log10 {Im [⎯ε (ω)] }
4 a =0.2, m=1.0 a =0.4, m =1.0 a=0.6, m=1.0 a =0.8, m =1.0
2
0
-2 (a)
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
6
log10 [|⎯χ (ω)| / χ1]
4
2
0
-2 (b) 4
log10 (FOM)
2
0
-2
-4 (c)
ω /ωp
Fig. 29. Results of the MGA: (a) Linear optical absorption Im[¯()]; (b) enhancement of the third-order optical nonlinearity |¯()|/1 ; and (c) FOM (figure of merit) ≡ |¯()|/{1 Im[¯()]} versus the normalized incident angular frequency /p for layer dielectric profile p2 (z) = azm . Parameters: /p = 0.01 and 2 = (3/2)2 .
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log10 {Im [⎯ε (ω)] }
4
(a)
a = 0.8, m = 0.2 a = 0.8, m = 0.6 a = 0.8, m =1.0 a = 0.8, m =1.4 a = 0.8, m =1.8
2
0
-2
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
log10 [|⎯χ (ω)|/ χ1]
5
3
1
(b)
-1
log10 (FOM)
4
2
0 (c)
ω /ωp
Fig. 30. Results of the MGA. Same as Fig. 29, but for different m at a = 0.8.
J.P. Huang, K.W. Yu / Physics Reports 431 (2006) 87 – 172
a =0.2, m =1.0 a=0.4, m =1.0 a =0.6, m =1.0 a =0.8, m =1.0
log10 {Im [⎯ε (ω)] }
2
(a)
131
0
-2
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
4
log10 [|⎯χ (ω)| /χ1]
3
2
1
0
(b)
-1
log10 (FOM)
2
0
-2 (c)
ω /ωp Fig. 31. Results of the EMA. Others are the same as Fig. 29.
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3
a =0.8, m =0.2 a =0.8, m = 0.6 a = 0.8, m =1.0 a = 0.8, m =1.4 a =0.8, m =1.8
log10 {Im [⎯ε (ω)]}
2
1
0
(a)
-1
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
5
log10 [|⎯χ (ω)| /χ1]
4
3
2
1
0
(b)
-1 3
log10 (FOM)
2
1
0
-1 (c)
ω /ωp Fig. 32. Results of the EMA. Others are the same as Fig. 30.
J.P. Huang, K.W. Yu / Physics Reports 431 (2006) 87 – 172
133
In fact, a direct numerical evaluation of the linear problem giving Elin can give an independent check for the validity of Eqs. (104) and (105). In so doing, one needs to invoke a random resistor network for the genuine random composite. It is also interesting to extend the present consideration to composites in which graded spherical particles are embedded in a host medium [20]. To calculate (z, ) for an asymmetrical microstructure, we used the MGA [Eq. (102)] for which the metal (or dielectric) component serves as a host (or inclusion). Inversely, we could see the metal (or dielectric) component as an inclusion (or host), and use the same form as Eq. (102) by exchanging 1 () and 2 , and p2 (z) and 1 − p2 (z). In so doing, as 1 () > 2 the former MGA [Eq. (102)] always gives an upper bound while the latter offers a lower bound, and vice versa. But the exact result must lie between the two bounds, as already pointed out by Hashin and Shtrikman [102]. To sum up, we have studied a compositionally graded metal–dielectric film by investigating two asymmetrical and symmetrical microstructures, respectively, and found that the effective linear and nonlinear optical responses are sensitive to the microstructure. Thus, it is possible to gain a large nonlinearity enhancement and optimal FOM by choosing an appropriate microstructure in compositionally graded metal–dielectric films. 5.3. Composite media of graded spherical particles Graded materials [39,73], whose properties vary gradually as a function of position, have received much attention as one of the advanced heterogeneous composites in various engineering applications by using the gradients in thermal [105], electric [106] and mechanical properties [107]. This gradation may occur naturally or may be a product of manufacturing processes. It was reported [4,5,21,23] that graded (inhomogeneous) materials can show stronger nonlinear optical responses than the corresponding homogeneous ones. Also, it is desirable and of interest to use dielectric–coated metallic nanoparticles with varying shell thickness to form a dielectric constant gradient [56]. In nature, there are also many graded materials, such as liquid crystal droplets [108] and biological cells [24], because of the inhomogeneous compartment inside them. Physically, graded materials are quite different from the homogeneous ones and other conventional composites. Therefore, the composite media consisting of graded particles can be more useful and interesting than those of homogeneous inclusions. The traditional theories used to deal with the homogeneous materials [109], however fail to deal with composites of graded inclusions directly. To treat these composites, we have recently developed a first-principles approach [19,40,110] and a differential effective dipole theory (Section 2.4) [40,111]. The problem becomes more complicated by the presence of nonlinearity in realistic composites. Besides inhomogeneity, such nonlinearity plays also an important role in the effective material properties of composite media [3,6,17,68,74,75]. It is thus necessary to establish a new theory to study the effective nonlinear properties of graded composite media. In fact, the introduction of dielectric gradation profiles in nonlinear composites is able to provide an alternative way to control the local field fluctuation, and hence let us obtain the desired effective nonlinear response. In fact, the previous one-shell model [78] and multi-shell model [76], which were used to study the effective nonlinear optical property, can be seen as an initial model of graded inclusions. In Section 5.3, we shall put forth a nonlinear differential effective dipole approximation (NDEDA) to investigate the effective linear and nonlinear dielectric properties of composite media containing a very small volume fraction of nonlinear graded spherical particles (inclusions). For such particles, the linear and nonlinear physical properties will continuously vary along their radius. 5.3.1. Model and definition of effective linear and nonlinear responses Let us consider a nonlinear composite system, in which identical graded spherical inclusions with radius a, are randomly embedded in a linear host medium of dielectric constant 2 . The local constitutive relation between the displacement (D) and the electric field (E) inside the graded particle is given by D = (r)E + (r)|E|2 E,
(109)
where (r) and (r) are, respectively, the linear dielectric constant and third-order nonlinear susceptibility. Note both (r) and (r) are radial functions. Here we assume that the weak nonlinearity condition is satisfied [74]. In other words, the contribution of the second (nonlinear) part [s (r)|E|2 ] in the right-hand side of Eq. (109) is much less than that of the first (linear) part (r). We restrict further our discussion to the quasi-static approximation, under which the whole
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composite medium can be regarded as an effective homogeneous one with effective linear dielectric constant e and effective third-order nonlinear susceptibility e . To show the definitions of e and e , we have [74] D = e E0 + e |E0 |2 E0 ,
(110)
where · · · represents the spatial average, and E0 = E0 ez is the external applied field along z axis. The effective linear dielectric constant e is given by 1 i Elin,i dV = f (r)Elin,1 + (1 − f )2 Elin,2 , e E0 = V V
(111)
where f is the volume fraction of the graded particles and the subscript stands for the linear local field [i.e., obtained for the same system but with (r) = 0]. In view of the existence of nonlinearity inside the graded particles, e can then be written as [74,100] 1 1 2 2 2 e E04 = i |E|2lin,i Elin,i dV = (r)|E|2lin,1 Elin,1 dV = f (r)|E|2lin,1 Elin,1 . (112) V V V i In the next section, we will develop a NDEDA (nonlinear differential effective dipole approximation), in an attempt to derive the equivalent linear dielectric constant ¯ (a) and third-order nonlinear susceptibility ¯ (a) of the nonlinear graded inclusions. Then, the effective linear dielectric constant and third-order nonlinear susceptibility of the composite media of nonlinear graded inclusions will be derived accordingly in the dilute limit. 5.3.2. Nonlinear differential effective dipole approximation To establish the NDEDA, we first mimic the gradation profile by a multi-shell construction. That is, we build up the dielectric profile by adding shells gradually [112]. We start with an infinitesimal spherical core with linear dielectric constant (0) and third-order nonlinear susceptibility (0), and keep on adding spherical shells with linear dielectric constant (r) and third-order nonlinear susceptibility (r) at radius r, until r = a is reached. At radius r, the inhomogeneous spherical particle with space-dependent dielectric gradation profiles (r) and (r) can be replaced by a homogenous sphere with the equivalent dielectric properties ¯ (r) and ¯ (r). Here the homogeneous sphere should induce the same dipole moment as the original inhomogeneous sphere. Next, we add to the sphere a spherical shell of infinitesimal thickness dr, with dielectric constant (r) and nonlinear susceptibility (r). In this sense, the coated inclusions is composed of a spherical core with radius r, linear dielectric constant ¯ (r) and nonlinear susceptibility ¯ (r), and a shell with outermost radius r + dr, linear dielectric constant (r) and nonlinear susceptibility (r). Since these coated inclusions with a very small volume fraction are randomly embedded in a linear host medium, under the quasi-static approximation, we can readily obtain the linear electric potentials in the core, shell and host medium by solving the Laplace equation [76] c = − E0 AR cos , R < r, Cr 3 s = − E0 BR − 2 cos , r < R < r + dr, R D(r + dr)3 h = − E0 R − cos , R > r + dr, R2 where A=
92 (r) , Q
D=
[(r) − 2 ][¯(r) + 2(r)] + [2 + 2(r)][¯(r) − (r)] , Q
B=
32 [¯(r) + 2(r)] , Q
C=
32 [¯(r) − (r)] , Q
with interfacial parameter ≡ [r/(r + dr)]3 , and Q = [(r) + 22 ][¯(r) + 2(r)] + 2[(r) − 2 ][¯(r) − (r)].
(113)
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135
The effective (overall) linear dielectric constant of the system is determined by the dilute-limit expression [36] e = 2 + 3p2 D,
(114)
where p is the volume fraction of graded particles with radius r. The equivalent dielectric constant ¯ (r + dr) for the graded particles with radius r + dr can be obtained self-consistently by the vanishing of the dipole factor D by replacing 2 with ¯ (r + dr). Taking the limit dr → 0 and keeping to the first order in dr, we obtain ¯ (r) − (r) ¯ (r)(1 − ) + (r)(2 + ) ¯ (r) − (r) ¯ (r) − (r) = ¯ (r) − · 3+ dr. r (r)
¯ (r + dr) = (r) + 3(r) ·
(115)
Thus, we have the differential equation for the equivalent dielectric constant ¯ (r) as [112] d¯(r) [(r) − ¯ (r)] · [¯(r) + 2(r)] = . dr r(r)
(116)
Note that Eq. (116) is just the Tartar formula, derived for assemblages of spheres with varying radial and tangential conductivity [39]. Next, we speculate on how to derive the equivalent nonlinear susceptibility ¯ (r). After applying Eq. (112) to the coated particles with radius r + dr, we have ¯ (r + dr)
|E|2 E2 R r+dr |E0 |2 E02
= ¯(r)
|E|2 E2 R r |E0 |2 E02
+ (1 − )
(r)|E|2 E2 r
.
(117)
As dr → 0, the left-hand side of the above equation admits
2
2
|E|2 E2 R r+dr 32 32
¯ (r + dr) = ¯ (r + dr)
¯ (r + dr) + 2 ¯ (r + dr) + 2 |E0 |2 E02 2 2 d¯(r)/dr ∗ 2 2 2 2 3d¯(r)/dr = ¯ (r)|K| K − dr ¯ (r)|K| K + 22 + ¯ (r) 22 + ¯ (r) + |K|2 K 2
d¯ (r) dr, dr
with K = (32 )/[¯(r) + 22 ]. The first part of the right-hand side of Eq. (117) is written as ¯ (r)|E|2 E2 R r dr 2 2 ∗ 1 + (6y + 2y , = ¯ (r)|K| K − 3) r |E0 |2 E02
(118)
(119)
where y=
[(r) − 2 ][¯(r) − (r)] . (r)[¯(r) + 22 ]
The second part of the right-hand side of Eq. (117) has the form [36] (1 − )
(r)|E|2 E2 r
=
3(r) dr|z|2 z2 (5 + 18x 2 + 18|x|2 + 4x 3 + 12x|x|2 + 24|x|2 x 2 ), 5r
where x=
¯ (r) − (r) ¯ (r) + 2(r)
and
z=
2 [¯(r) + 2(r)] . (r)[¯(r) + 22 ]
(120)
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Substituting Eqs. (118), (119) and (120) into Eq. (117), we have a differential equation for the equivalent nonlinear susceptibility ¯ (r), namely,
6y + 2y ∗ − 3 3(r)
¯ (r) + 2(r)
2 3d¯(r)/dr d¯(r)/dr ∗ d¯ (r) + ¯ (r) · = ¯ (r) + + ·
dr 22 + ¯ (r) 22 + ¯ (r) r 5r 3(r) ¯ (r) + 2(r) 2 × (5 + 18x 2 + 18|x|2 + 4x 3 + 12x|x|2 + 24|x|2 x 2 ). (121) 3(r) So far, the equivalent ¯ (r) and ¯ (r) of graded spherical particles of radius r can be calculated, at least numerically, by solving the differential equations Eqs. (116) and (121), as long as (r) (dielectric-constant gradation profile) and (r) (nonlinear-susceptibility gradation profile) are given. Here we would like to mention that, even though (r) is independent of r, the equivalent ¯ (r) should still be dependent on r because of (r) as a function of r. Moreover, for both (r) = 1 and (r) = 1 (i.e., they are both constant and independent of r), Eqs. (116) and (121) will naturally reduce to the solutions ¯ (r) = 1 and ¯ (r) = 1 . To obtain ¯ (r = a) and ¯ (r = a), we integrate Eqs. (116) and (121) numerically at given initial conditions ¯ (r → 0) and ¯ (r → 0). Once ¯ (r = a) and ¯ (r = a) are calculated, we can take one step forward to work out the effective linear and nonlinear responses e and e of the whole composite in the dilute limit, i.e. [74], e = 2 + 32 f and
¯ (r = a) − 2 , ¯ (r = a) + 22
e = f ¯ (r = a)
2 2
32 32
. ¯ (r = a) + 22 ¯ (r = a) + 22
(122)
(123)
5.3.3. Exact solution for power-law gradation profiles Based on the first-principles approach, we have found that, for a power-law dielectric gradation profile, i.e., (r) = A(r/a)n , the potential in the graded inclusions and the host medium can be exactly given by [113] i (r) = − 1 E0 r s cos ,
r < a,
2 E0 cos , r > a, r2 where the coefficients 1 and 2 have the form h (r) = − E0 r cos +
1 =
3a 1−s 2 sA + 22
and
2 =
(124)
sA − 2 3 a , sA + 22
and s is given by 1 s= 9 + 2n + n2 − (1 + n) . 2 The local electric field inside the graded inclusions can be derived from the potential E = −∇, Ei = 1 E0 r s−1 (s cos er − sin e ) = 1 E0 r s−1 {(s − 1) cos sin cos ex + (s − 1) cos sin sin ey + [(s − 1)cos2 + 1]ez },
(125)
where er , e , and ex , ey and ez are unix vectors in spherical coordinates and in Cartesian coordinates. In the dilute limit, from Eq. (111), we can obtain the effective linear dielectric constant as follows 1 e = 2 + [A(r/a)n − 2 ]ez · Ei dV V E 0 i A 2 2+s = 2 + 32 f − . (126) sA + 22 2 + n + s 2+s
J.P. Huang, K.W. Yu / Physics Reports 431 (2006) 87 – 172
137
1.10
εe
1.05
n = 0.1 exact n =1.0 exact n = 3.0 exact n = 0.1 NDEDA n = 1.0 NDEDA n = 3.0 NDEDA
1.00
0.95
0.90 0
1
2 A
3
4
Fig. 33. The effective linear dielectric constant e versus A for the power-law dielectric gradation profile (r) = A(r/a)n in the dilute limit f = 0.05 [20]. Lines: numerical results from the NDEDA [Eq. (116)]; Symbols: exact results [Eq. (126)].
On the other hand, the substitution of Eq. (125) into Eq. (112) yields 1 (r)|1 |2 21 (s 2 cos2 + sin2 )2 r 4s−2 sin dr d d e = V i a f (r)r 4s−2 dr. = 3 |1 |2 21 (8 + 4s + 3s 4 ) · 5a 0 For example, for a linear profile of (r), i.e., (r) = k1 + k2 · r/a, Eq. (127) leads to
2 32 f 32
2 k2 4k1 2 4 (8 + 4s + 3s ) + . e =
20 sA + 22 sA + 22 s 4s − 1 In addition, for a power-law profile of (r), namely, (r) = k1 (r/a)k2 , Eq. (127) produces
2 32 8 + 4s 2 + 3s 4 f
32
2 e = . k1 5 sA + 22 sA + 22 k2 − 1 + 4s
(127)
(128)
(129)
We are now in a position to evaluate the NDEDA. For the comparison between the first-principles approach and the NDEDA, we first perform numerical calculations for the case where the dielectric constant exhibits power-law gradation profiles (r) = A(r/a)n , while the third-order nonlinear susceptibility shows two model gradation profiles: (a) linear profile (r) = k1 + k2 · r/a, and (b) power-law profile (r) = k1 (r/a)k2 . Without loss of generality, we take 2 = 1 and a = 1 for numerical calculations. The fourth-order Runge–Kutta algorithm is adopted to integrate the differential equations [Eqs. (116) and (121)] with step size 0.01. Meanwhile, the initial core radius is set to be 0.001. It was verified that this step size guarantees accurate numerics. In Fig. 33, the effective linear dielectric constant (e ) is plotted as a function of A for various indices n. It is shown that e exhibits a monotonic increase for increasing A (and decreasing n). This can be understood by using the equivalent dielectric constant ¯ (r = a) which increases as A increases (n decreases). Moreover, the excellent agreement between the NDEDA [Eq. (116)] and the first-principles approach [Eq. (126)] is shown as well. Next, the effective third-order nonlinear susceptibility (e ) is plotted as a function of A for the linear gradation profile (r) = k1 + k2 · r/a (Fig. 34), and for the power-law profile (r) = k1 (r/a)k2 (Fig. 35). We find that the effective nonlinear susceptibility decreases for increasing A. The reason is that, as mentioned above, for larger A, the graded inclusions possess larger equivalent dielectric constant, and the local field inside the nonlinear inclusions will become
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J.P. Huang, K.W. Yu / Physics Reports 431 (2006) 87 – 172
16 n =0.1 exact n =1.0 exact n =3.0 exact
12
n = 0.1 NDEDA n =1.0 NDEDA χe/ k1
n =3.0 NDEDA 8
4
0 0
1
(a)
2
3
4
A 20
χe/ k1
15
10
5
0 0 (b)
1
2 A
3
4
Fig. 34. The effective third-order nonlinear susceptibility e versus A for power-law dielectric-constant gradation profile (r) = A(r/a)n and linear nonlinear-susceptibility gradation profile (r) = k1 + k2 · r/a with (a) k1 = 1 and k2 = 1, and (b) k1 = 2 and k2 = 3 [20]. Lines: numerical results from the NDEDA [Eqs. (116) and (121)]; Symbols: exact results [Eq. (127)].
more weak, which results in a weaker effective nonlinear susceptibility (e ). In addition, increasing n leads generally to increasing e , and such a trend is clearly observed at large A. Again, we obtain the excellent agreement between the first-principles approach [Eqs. (128) and (129)] and the NDEDA [Eqs. (116) and (121)]. In what follows, we investigate the surface plasmon resonance effect on the metal–dielectric composite. We adopt the Drude-like dielectric constant for graded metal particles, namely, (r) = 1 −
2p (r) [ + i (r)]
,
(130)
where p (r) and (r) are the radius-dependent plasma frequency and damping coefficient, respectively. For the sake of simplicity, set (r) = 1 to be independent of r, in an attempt to emphasize the enhancement of the effective optical
J.P. Huang, K.W. Yu / Physics Reports 431 (2006) 87 – 172
139
4 n = 0.1 exact n =1.0 exact n = 3.0 exact
χ e / k1
n = 0.1 NDEDA n =1.0 NDEDA n =3.0 NDEDA
2
0 0
1
2
3
4
A
(a) 3
χ e / k1
2
1
0 0 (b)
1
2 A
3
4
Fig. 35. Same as Fig. 34, but for power-law nonlinear-susceptibility gradation profile (r) = k1 (r/a)k2 [20].
nonlinearity, and 2 = 1.77 (the dielectric constant of water). We assume further p (r) to be
r , r < a. p (r) = p 1 − k · a
(131)
This form is quite physical for k > 0, since the center of grains can be better metallic so that p (r) is larger, while the boundary of the grain may be poorer metallic so that p (r) is much smaller. Such the variation can also appear because of the temperature effect [114]. For small particles, we have the radius-dependent (r) as [80]
(r) = (∞) +
k , r/a
r < a,
(132)
where (∞) stands for the damping coefficient in the bulk material. Here k is a constant which is related to the Fermi velocity vF . In this case, the exact solution being predicted by a first-principles approach is absent. Fortunately, we can resort to the NDEDA instead.
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2
log10 [Im (εe)]
1 0 -1 -2 -3 -4
0.2
0.4
0.6 ω /ωp
0.8
1.0
6
6
4 2 kω = 0.0 kω = 0.1 kω = 0.2 kω = 0.3
0 -2 -4 0.2
0.4
0.6 ω /ωp
0.8
1.0
0.2
0.4
0.6 ω /ωp
0.8
2
log10 (FOM)
log10 (|χe|/χ1)
4
0 1.0
Fig. 36. (a) The linear optical absorption Im(e ), (b) the enhancement of the third-order optical nonlinearity |e |/1 , and (c) the figure of merit ≡ |e |/Im(e ) versus the incident angular frequency /p for dielectric-constant gradation profile (r) = 1 − 2p (r)/[( + i (r))] with p (r) = p (1 − k · r/a) and (r) = 0.01p [20]. Parameters: 2 = 1.77 and f = 0.05.
In Fig. 36, we plot the optical absorption [∼ Im(e )], the modulus of the effective third-order optical nonlinearity enhancement (|e |/1 ) and the figure of merit (|e |/Im(e )) versus the incident angular frequency . For the case of the homogeneous particles, i.e., k = 0, there is a single sharp peak at ≈ 0.5p , corresponding to the surface plasmon resonance, as expected. However, for the case of the graded particles, i.e. k = 0, besides a sharp peak, a broad continuous resonant band in the high-frequency region is apparently observed. The position of the sharp peak can be estimated from the resonant condition Re[¯(r = a)] + 22 = 0, while the broad continuous spectrum is indeed a salient result of the gradation profile. More exactly, the broad spectrum results from the effect of the radius-dependent plasma frequency. In Ref. [78], we found that, when the shell model is taken into account, a broad continuous spectrum should be expected to occur around the large pole in the spectral density function. In fact, the graded particles under consideration can be regarded as a certain limit of multi shells, which thus should yield the broader spectra in Im(e ), |e |/1 as well as |e |/Im(e ). In addition, we note that increasing k makes both the surface plasmon frequency and the center of the resonant bands red-shifted. In particular, the resonant bands can become more broad due to strong inhomogeneity of the particles. From the figure, we conclude that, although the third-order optical nonlinearity is always accompanied with the optical absorption, the figure of merit in the high frequency region is still attractive due to the presence of weak optical absorption. Thus, we believe that graded particles have potential applications in obtaining the optimal figure of merit, and make the composite media more realistic for practical applications. Finally, we focus on the effect of (r) on the nonlinear optical property in Fig. 37. As evident from the results, the variation of k plays an important role in the magnitude of the effective optical properties, particularly at the surface plasmon resonance frequency. We have developed an NDEDA (nonlinear differential effective dipole approximation) to calculate the effective linear and nonlinear dielectric responses of composite media containing nonlinear graded inclusions. The results obtained from the NDEDA are compared with the exact solutions derived from a first-principles approach for the power-law
J.P. Huang, K.W. Yu / Physics Reports 431 (2006) 87 – 172
141
log10 [Im (εe)]
1 0 -1 -2 -3 0.2
0.4
0.6 ω /ωp
0.8
1.0
log10(|χe|/χ1)
6
4
k γ =1*10-3ωp
4
2
log10 (FOM)
6 k γ =0 k γ = 2*10-3ωp k γ = 5*10-3ωp
2 0 0.2
0.4
0.6 ω /ωp
0.8
1.0
0.2
0.4
0.6 ω /ωp
0.8
1.0
Fig. 37. Same as Fig. 36, but with p (r) = p and (r) = (∞) + k /(r/a) for (∞) = 0.01p [20].
dielectric gradation profiles, and the excellent agreement between them has been shown. We should remark that the exact solutions are also obtainable for the linear dielectric gradation profiles with small slopes (the derivation not shown here). In this case, the excellent agreement between the two methods can be shown as well since the NDEDA is valid indeed for arbitrary gradation profiles. In general, the exact solution is quite few in realistic composite research, and thus our NDEDA can be used as a benchmark. The NDEDA was derived for the composite containing the nonlinear graded inclusions in a linear host. Interestingly, it can be readily generalized to the composite system where the graded inclusions and the host are both nonlinear [4]. In this situation, the effective third-order nonlinear response can be written as [11,79],
2
2
32 32
e = f ¯ (r = a)
+ 2 (1 − f ) ¯ (r = a) + 22 ¯ (r = a) + 22 18 2 18 2 6 2 2 3 8 2 2 ∗ + 2 f 3 + + + || + || + + || , (133) 5 5 5 5 5 where ≡ [¯(r = a) − 2 ]/[¯(r = a) + 22 ] and 2 is the third-order nonlinear susceptibility of the host medium. As a matter of fact, for this purpose, the perturbation method can also be adopted [115]. The NDEDA is strictly valid in the dilute limit. To achieve the strong optical nonlinearity enhancement, we need possibly nonlinear inclusions with high volume fractions. In this connection, the effect of the volume fraction is expected to cause a further broadening of the resonant peak, and possibly, a desired separation of the optical absorption peak from the nonlinearity enhancement due to mutual interactions [7]. Therefore, it is of particular interest to generalize the NDEDA for treating the case of high volume fractions. It is also instructive to develop the first-principles approach to weakly nonlinear graded composites. The perturbation approach [67] in weakly nonlinear composites is just suitable for this problem. Moreover, with the aim of the variational approach [116], the NDEDA may be applied to the cases of strong nonlinearity, where the linear part [(r)] in Eq. (109) vanishes. On the other hand, based on the self-consistent mean-field approximation [117], the applicability of NDEDA to more general cases, where linear [(r)] and nonlinear [(r)|E|2 ] parts can be comparable, may be also possible.
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To sum up, we put forth an NDEDA and a first-principles approach for investigating the optical responses of nonlinear graded spherical particles. The excellent agreement between the two methods has been shown. As an application, we applied the NDEDA to discuss the surface plasmon resonance effect on the effective linear and nonlinear optical properties like the optical absorption, the optical nonlinearity enhancement, and the figure of merit. It is found that the dielectric gradation profile can be used to control the surface plasmon resonance and achieve the large figure of merit in the high-frequency region, where the optical absorption is quite small. 5.4. Composites of graded particles with dielectric anisotropy The above gradation models were built up under the assumption that the graded inclusions exhibit isotropic dielectric response. However, dielectric anisotropy occurs naturally due to the presence of gradation inside the particles. Moreover, there are many inhomogeneous materials with spatial anisotropy, like polycrystal aggregates of a single anisotropic component [118], liquid crystal droplets [119], and cell membranes containing mobile charges [120]. In these situations, the local dielectric coefficient should be tensorial. Thus, for a better understanding of the dielectric–anisotropy effect, it is necessary to generalize our previous isotropic gradation models [19,110–112] accordingly. The nonlinearity is a common phenomena in realistic graded materials. And the spatial anisotropy effect has not yet been investigated in the traditional theories. Below we shall develop a new theory, in an attempt to study the effective linear and nonlinear optical properties of composite media, by taking into account the dielectric anisotropy of the nonlinear graded particles. For the dielectric tensor of these graded inclusions, the components of the tensorial dielectric constant of interest will be assumed to vary along the radius of the particles continuously. 5.4.1. Model and definition of effective linear dielectric constant and third-order nonlinear susceptibility Let us consider a dilute composite material, where the identical spherical inclusions having a dielectric constant ↔ tensor ε 1 , with radius a, are randomly embedded in a linear isotropic host with (scalar) dielectric constant 2 . Inside the anisotropic inclusions, the local constitutive relation between the displacement (D) and the electric field (E) is given by [121] Di =
ij Ej +
j
ij kl Ej Ek El∗ .
(134)
j kl
Here Di and Ei are, respectively, the ith Cartesian components of D and E. It is worth remarking that ij and ij kl are the second-rank and fourth-rank Cartesian tensors, respectively. Our analysis will be limited to the case of weak nonlinearity. In other words, the nonlinear part in Eq. (134) will be assumed to be small when compared with the linear part. In what follows, the dielectric tensor for the the anisotropic spherical inclusions is assumed to be diagonal in spherical coordinates, with a value 1t (r) in the tangential directions and 1r (r) in the radial direction [25,120,122]. Here, both dielectric gradation profiles 1r (r) and 1t (r) will be mathematically represented as radial functions [25]. In view of ↔ the spherical symmetry, we can express the dielectric constant tensor ε 1 (r) of graded particles in the form ⎛ ↔ ⎜ ε 1 (r) = ⎝
ε1r (r)
0
0
⎞
0
1t (r)
0
⎟ ⎠.
0
0
ε1t (r)
(135)
Note the above form is in spherical coordinates, rather than in Cartesian coordinates. Nevertheless, it can also be represented in Cartesian coordinates by a transformation using appropriate rotation matrices. As the graded inclusions with dielectric anisotropy are randomly oriented, the whole sample should be macroscopically isotropic. Thus, we can define the effective linear dielectric constant e and the third-order nonlinear susceptibility e of the whole composite as [74,121] D = e E0 + e |E0 |2 E0 ,
(136)
J.P. Huang, K.W. Yu / Physics Reports 431 (2006) 87 – 172
143
where · · · stands for the spatial average, and E0 = E0 ez denotes the external applied field along z-axis. In Eq. (136), the effective linear dielectric constant e is given by 1 e E0 = V
↔
↔
ε Elin dV = f ε 1 · E1,lin + (1 − f )2 E2,lin ,
(137)
V
where f is the volume fraction of the graded inclusions. Here, the subscript ‘lin’ denotes the linear local field inside the graded inclusions or the host. In view of the existence of nonlinearity inside the anisotropic graded particles, e is given by [74,123] e E04 =
1 ∗ ij kl Elin,i Elin,j Elin,k Elin,l dV = f ij kl Elin,i Elin,j Elin,k Elin,l . V V ij kl
(138)
ij kl
Here Elin,i denotes the Cartesian component of the linear local electric field. Then, just as in isotropic composites [74], both e and e in nonlinear composite media with local dielectric anisotropy can be expressed (to the lowest order in the nonlinearity) in terms of the electric field in the related linear medium as well. In the next section, we will develop an NADEDA (nonlinear anisotropic differential effective dipole approximation), so as to derive the equivalent linear dielectric constant ¯ (a) and third-order nonlinear susceptibility ¯ (a) of the nonlinear graded inclusions. In this connection, e and e of this anisotropic graded composite media can further be derived in the dilute limit. 5.4.2. Nonlinear anisotropic differential effective dipole approximation To put forth an NADEDA (nonlinear anisotropic differential effective dipole approximation) for graded particles with dielectric anisotropy, we regard the gradation profiles as a multi-shell construction. In detail, we build up the dielectric profile by adding shells gradually [112]. Let us start with an infinitesimal spherical core with linear dielectric constants 1r (r = 0) = 1t (r = 0) = (0) and nonlinear susceptibility ij kl , and keep on adding shells with the tangential and radial dielectric constant 1t (r) and 1t (r), and the Cartesian fourth-rank tensorial nonlinear susceptibility ij kl (to show the optical nonlinearity enhancement, we always assume ij kl to be independent of r), at radius r, until r = a is reached. At radius r, we have an inhomogeneous spherical particle with spatially varying dielectric constant, which are characterized by the gradation profiles 1r (r) and 1t (r), and with tensorial nonlinear susceptibility ij kl . Then, we can regard such an inhomogeneous particle as an effective homogeneous one with the equivalent isotropic dielectric properties ¯ (r) and ¯ (r). Here the homogeneous sphere should induce the same dipole moment as the original inhomogeneous sphere. Then, we add to the homogeneous particle a spherical shell of infinitesimal thickness dr, with linear dielectric constants, 1r (r) and 1t (r), and nonlinear susceptibility, ij kl . In this situation, the coated inclusions are composed of a spherical core with radius r, linear dielectric constant ¯ (r) as well as nonlinear susceptibility ¯ (r), and a shell with outermost radius r + dr, linear dielectric constants 1r (r) and 1t (r), as well as nonlinear susceptibility ij kl . For the graded particles with dielectric anisotropy describe Eq. (135), the displacement vector is related to the field, ↔ D = ε 1 (r) · E. In view of E = −∇, we have the following electrostatic equation: ↔
∇ · ( ε 1 (r) · ∇) = 0.
(139)
In spherical coordinates, Eq. (139) can be cast into 1 j r 2 jr
1 j 1 j j j j r 2 1r (r) + 2 sin 1t (r) + 2 2 1t (r) = 0. jr r sin j
jr j r sin j
(140)
Let us consider the composite where the coated inclusions are randomly embedded in the linear host medium. Under the quasi-static approximation, we can readily obtain the linear electric potentials inside the core, shell and host medium
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by solving Eq. (140), c = − E0 AR cos ,
R < r,
Cr (1+2 ) (r + dr)1− s = − E0 B(r + dr) R − R +1 D(r + dr)3 h = − E0 R − cos , R > r + dr. R2 1−
cos ,
r < R < r + dr,
(141)
Here the four unknown parameters A, B, C and D can be determined by applying the appropriate boundary conditions on the interfaces. As a result, we obtain A=
3(1 + 2 )2 1r (r)( −1)/3 , Q
D=
[ · 1r (r) − 2 ][¯(r) + (1 + )1r (r)] + (1+2 )/3 [2 + (1 + )1r (r)][¯(r) − · 1r (r)] , Q
B=
32 [¯(r) + (1 + )1r (r)] , Q
with interfacial parameter ≡ [r/(r + dr)]3 and ≡ −1/2 +
√
C=
32 [¯(r) − · 1r (r)] , Q
1/4 + 21t (r)/1r (r), and
Q = [ · 1r (r) + 22 ][¯(r) + (1 + )1r (r)] + (1+2 )/3 [(1 + )1r (r) − 22 ][¯(r) − · 1r (r)]. If 1t (r) = 1r (r), the physical parameter = 1, and then Eq. (8) degenerates to the isotropic form. The effective (overall) linear dielectric constant of the system is determined by the dilute-limit expression [36] e = 2 + 3p2 D,
(142)
where p is the volume fraction of the graded particles with radius r. The equivalent dielectric constant ¯ (r + dr) for the graded particles with radius r + dr can be self-consistently obtained by the vanishing of the dipole factor D by replacing 2 with ¯ (r + dr). Taking the limit dr → 0 and keeping to the first order in dr, we obtain ¯ (r + dr) = 1r (r) 1 + = ¯ (r) +
¯ (r)[( − 1) + ( + 2)(1+2 )/3 ] + 1r (r)[( 2 − 1) − ( + 2) (1+2 )/3 ]
¯ (r)(1 − (1+2 )/3 ) + 1r (r)[ + 1 + · (1+2 )/3 ]
[ 1r (r) − ¯ (r)][( + 1)1r (r) + ¯ (r)] dr. r1r (r)
(143)
Thus, we have the differential equation for the equivalent dielectric constant ¯ (r), d¯(r) ( + 1)[1r (r)]2 − ¯ (r)1r (r) − [¯(r)]2 = . dr r1r (r)
(144)
Note that Eq. (144) is just the Tartar formula, which was derived for assemblages of spheres with varying radial and tangential conductivity [39]. If 1r is independent of r, namely 1r = 1 , we have = 1 due to isotropic property at r = 0, and then Eq. (144) predicts ¯ (r) = 1 , as expected. Next, we speculate on how to derive the equivalent nonlinear susceptibility ¯ (r). After applying Eq. (138) to the coated particles with radius r + dr, we have ij kl Ei Ej Ek El∗ r
J.P. Huang, K.W. Yu / Physics Reports 431 (2006) 87 – 172
145
As dr → 0, the left-hand side of the above equation admits ¯ (r + dr)
|E|2 E 2 R r+dr |E0 |2 E02
2 2
32 32
¯ (r + dr) + 22 ¯ (r + dr) + 22 d¯(r)/dr ∗ 2 2 2 2 3d¯(r)/dr + = ¯ (r)|K| K − dr ¯ (r)|K| K 22 + ¯ (r) 22 + ¯ (r)
= ¯ (r + dr)
+ |K|2 K 2
d¯ (r) · dr, dr
(146)
with K = (32 )/[¯(r) + 22 ]. The first part of the right-hand side of Eq. (145) is written as
¯ (r)|E|2 E 2 R r |E0 |2 E02
dr = ¯ (r)|K|2 K 2 1 + (3y + y ∗ − 3) , r
(147)
where y=
[(1 + )1r (r) − 22 ][¯(r) − · 1r (r)] − + 1. 1r (r)[¯(r) + 22 ]
The term U ≡ ( ij kl ij kl (r)Ei Ej Ek El∗ r
|K|2 K 2 , 315
where Up1 = [B2 ( − 1) + C2 (2 + )]3 · [B2∗ ( − 1) + C2∗ (2 + )], Up2 = [B2 ( − 1) + C2 (2 + )]2 · [|C2 |2 (5 + 2 + 5 2 ) + (C2 B2∗ + B2 C2∗ )(−4 + 5 + 5 2 ) + |B2 |2 (8 + 8 + 5 2 )], Up3 = [B2∗ ( − 1) + C2∗ (2 + )] · [B23 (−8 + 3 2 + 5 3 ) + 3B22 C2 (8 + 2 + 6 2 + 5 3 ) + 3B2 C22 (−7 + 5 + 9 2 + 5 3 ) + C23 (10 + 9 + 12 2 + 5 3 )], Up4 = [B22 |B2 |2 (128 + 64 + 48 2 + 40 3 + 35 4 ) + B22 (3C2 B2∗ + B2 C2∗ )(−112 − 8 + 30 2 + 55 3 + 35 4 ) + 3B2 C2 (C2 B2∗ + B2 C2∗ )(104 + 4 + 39 2 + 70 3 + 35 4 ) + C22 (C2 B2∗ + 3B2 C2∗ )(−94 + 43 + 75 2 + 85 3 + 35 4 ) + C22 |C2 |2 (107 + 52 + 138 2 + 100 3 + 35 4 )],
(148)
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with ¯ (r) + (1 + )1r (r) (1 + 2 )1r (r)
B2 =
and
¯ (r) − · 1r (r) . (1 + 2 )1r (r)
C2 =
Substituting Eqs. (146)–(148) into Eq. (145), we have a differential equation for the equivalent nonlinear susceptibility ¯ (r), namely, 3y + y ∗ − 3 3 3d¯(r)/dr d¯(r)/dr ∗ U d¯ (r) + ¯ (r) · = ¯ (r) + + · . dr 22 + ¯ (r) 22 + ¯ (r) r r |K|2 K 2
(149)
From Eqs. (148) and (149), it is evident that ij kl does not contribute to the equivalent nonlinear susceptibility, except for the cases with equal pair indices. So far, the equivalent ¯ (r) and ¯ (r) of the anisotropic graded spherical particles with radius r can be calculated, at least numerically, by solving the differential equations Eqs. (144) and (149), as long as 1r (r), 1t (r) (dielectric-constant gradation profiles) and ij kl are given. Here we would like to mention that, even though ij kl is independent of r, the equivalent ¯ (r) should still be dependent on r. This is because both 1r (r) and 1t (r) are r-dependent. Moreover, if ij kl = 0, Eq. (145) admits ¯ (r) = 0, as expected as well. To obtain ¯ (r = a) and ¯ (r = a), we integrate Eqs. (144) and (149) numerically, for given initial conditions ¯ (r → 0) and ¯ (r → 0). Once ¯ (r = a) and ¯ (r = a) are calculated, we can take one step forward to work out the effective linear and nonlinear responses of the whole composite e and e [121], e = 2 + 32 f
¯ (r = a) − 2 , ¯ (r = a) + 22
(150)
and
e = f ¯ (r = a)
2 2
32 32
. ¯ (r = a) + 22 ¯ (r = a) + 22
(151)
5.4.3. Exact solution for linear gradation profiles Based on the first-principles approach, the potentials within the graded inclusions and the host medium can be exactly obtained, when the dielectric gradation profiles are the linear radial functions with small slopes, i.e., 1r (r) = (0) + g(r/a) and 1t (r) = (0) + h(r/a). Here g [< a(0)] and h are two constants, and (0) denotes the linear dielectric constant at radius r = 0. The potentials within the graded spheres and the host medium are respectively given by c (r, ) = − E0 A1
∞
Ck
k=0
m (r, ) = − E0 r cos +
gr a(0)
k+1 cos ,
D1 E0 cos , r2
r < a,
(152)
r > a,
where the coefficients A1 and D1 have the following forms: A1 =
32 a ((0) + g)v2 + 22 v1
and
D1 =
((0) + g)v2 − 2 v1 3 a . ((0) + g)v2 + 22 v1
Here v1 and v2 are given by v1 =
∞ k=0
Ck
g (0)
k+1 and
v2 =
∞ k=0
Ck (k + 1)
g (0)
k+1 ,
J.P. Huang, K.W. Yu / Physics Reports 431 (2006) 87 – 172
147
with Ck satisfying the following recurrent relation: Ck+1 = −
(k + 1)(k + 3) − 2h/g Ck . (k + 2)(k + 3) − 2
The local electric field inside the anisotropic graded inclusions can be derived from the relation E = −∇, and we have Ec = A1 E0
∞
Ck
k=0
= A1 E0
∞
Ck
k=0
g a(0) g a(0)
k+1 r k [(k + 1) cos er − sin e ] k+1 r k {k cos sin cos ex + k cos sin sin ey
× [(k + 1)cos2 + sin2 ]ez }.
(153)
Then, the corresponding displacement admits ↔
Dc = ε 1 (r) · Ec = A1 E0
∞
Ck
k=0
= A1 E0
∞ k=0
Ck
g a(0)
g a(0)
k+1 r k [1r (r)(k + 1) cos er − 1t (r) sin e ]
k+1 r k {[1r (r)(k + 1) − 1t (r)] cos sin cos ex
+ [1r (k + 1) − 1t (r)] cos sin sin ey + [1r (r)(k + 1)cos2 + 1t (r)sin2 ]ez },
(154)
where er and e (ex , ey and ez ) are the unix vectors in spherical coordinates (Cartesian coordinates). In the dilute limit, from Eq. (137), we can obtain the effective linear dielectric constant as e = 2 +
1 ↔ ( ε 1 (r) · E − 2 E) · ez dV V E 0 i
= 2 + 3f 2
[(0) − 2 ] v1 + gv 3 + 2hv 4 , [(0) + g] v2 + 22 v1
(155)
where v3 =
∞ k=0
Ck
1+k g k+1 4 + k (0)
and
v4 =
∞ k=0
Ck
1 g k+1 . 4 + k (0)
On the other hand, the substitution of Eq. (153) into Eq. (138) yields e =
1 ij kl Ei Ej Ek El∗ dV V ij kl i
= f [(xxyy + yxxy + xyxy + xyyx + yxyx + yyxx + 3xxxx + 3yyyy )Uq1 × (xxzz + xzxz + zxxz + yyzz + yzyz + zyyz )Uq2 + (zzxx + zxzx + xzzx + zzyy + yzzy + zyzy )Uq3 + zzzz Uq4 ],
(156)
148
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where ∞ ∞ ∞ ∞ 1 |A2 |2 A22 Uq1 = 105
Ck1 Ck2 Ck3 Ck4
k1 =0 k2 =0 k3 =0 k4 =0
g (0)
k1 +k2 +k3 +3
k 1 k 2 k3 k4 , × 3 + k 1 + k2 + k3 + k4 ∞ ∞ ∞ ∞ g k1 +k2 +k3 +3 2 2 Uq2 = 3|A2 | A2 Ck1 Ck2 Ck3 Ck4 (0)
g (0)
k4 +1 ∗
k1 =0 k2 =0 k3 =0 k4 =0
k4 +1 ∗
1 1 1 k1 k2 + (k3 + k4 ) + k3 k4 , × 3 + k1 + k2 + k3 + k4 15 35 63 ∞ ∞ ∞ ∞ g k1 +k2 +k3 +3 Uq3 = 3|A2 |2 A22 Ck1 Ck2 Ck3 Ck4 (0)
g (0)
k1 =0 k2 =0 k3 =0 k4 =0
k4 +1 ∗
k3 k 4 1 1 1 + (k1 + k2 ) + k1 k2 , × 3 + k1 + k2 + k3 + k4 15 35 63 ⎧ ∞ ∞ ∞ ⎨ ∞ g k1 +k2 +k3 +3 2 2 Uq4 = 3|A2 | A2 Ck1 Ck2 Ck3 Ck4 ⎩ (0)
g (0)
k1 =0 k2 =0 k3 =0 k4 =0
×
g (0)
k4 +1 ∗
⎡ 4 3 4 1 1 ⎣1 + 1 ki + ki kj 3 + k 1 + k2 + k3 + k4 3 5
⎤⎫ 3 4 2 ⎬ 1 1 × ki kj kl + k 1 k 2 k3 k 4 ⎦ , ⎭ 7 9
i=1
i=1 j =i+1
i=1 j =i+1 l=j +1
with A2 = (32 )/{[(0) + g]v2 + 22 v1 }. To illustrate the NADEDA, we first perform numerical calculations for the linear dielectric gradation profiles, that is, 1r (r) = (0) + gr/a (radial dielectric constant), and 1t (r) = (0) + hr/a (tangential dielectric constant). In this situation, the exact results for e and e exist, and thus it allows us to show the correctness of the NADEDA. For model calculations, we set h > g (Note that our formulae can still be used for hg). For the NADEDA, we numerically integrate Eqs. (144) and (149) by using Mathematica with the initial radius r = 0.001. In Fig. 38, the effective linear dielectric constant (e ) is plotted as a function of the dielectric constant of anisotropic graded particles at radius r = 0 [(0)], for various gradients h and g. It is shown that e increases monotonically with the increase of (0). Moreover, increasing the gradient g causes e to increase as well. This can be understood by the fact that the increases of both (0) and g lead to the increase of the equivalent dielectric constant ¯ (a) of the graded particles, thus increasing the effective response of the whole system. For e , the NADEDA shows good agreement with the first-principles approach. Next, we investigate the effective third-order nonlinear susceptibility. Let us set the tensorial dielectric susceptibility of the particles to be independent of r, in an attempt to focus on the nonlinearity enhancement. As a result, it is shown that the nonlinearity enhancement decreases with the increase of (0) and g. As mentioned above, for larger e and g, the graded inclusions possess a larger equivalent dielectric constant, and hence the ith Cartesian component of the local field should become more weak accordingly. Then, the weaker effective nonlinear susceptibility is obtained. As displayed in Fig. 39, we show three typical cases of nonlinearity enhancement. Here, all the physical parameters in use are real, and thus the nonlinearity enhancement for zzxx (the only nonzero component) is the same as that for xxzz .
J.P. Huang, K.W. Yu / Physics Reports 431 (2006) 87 – 172
149
1.11
1.10 εe
g =1 exact g =2 exact g =3 exact 1.09 g =1 NADEDA g=2 NADEDA g =3 NADEDA 1.08 1
2
3 ε (0)
4
5
Fig. 38. The effective linear dielectric constant (e ) versus the dielectric constant of the spherical core with radius r = 0+ [(0)], for the linear dielectric gradation profiles with various radial gradients g [60]. Parameters: volume fraction f =0.05, the tangential gradient h=8. Lines: numerical results obtained from the NADEDA [Eqs. (144) and (150)]; Symbols: exact results predicted by the first-principles approach [Eq. (155)]. Note that the exact results are available for (0) > g.
Moreover, for other nonzero components of the tensorial nonlinear susceptibility, the nonlinearity enhancement will be the same as one of these shown in Fig. 39. For example, Fig. 39(a) can also show the nonlinearity enhancement for 3xxxx , 3yyyy , yxxy , etc. Again, the excellent agreement is numerically demonstrated between the first-principles approach and the NADEDA [Eqs. (144), (149) and (151)]. In what follows, we shall investigate the surface plasma resonance effect on the nonlinear metal–dielectric composite. As a model calculation, we assume the radial and tangential dielectric constants for the graded metal particles to be Drude-like, namely, 1r (r) = 1 −
2pr (r) ( + i )
and
1t (r) = 1 −
2pt (r) ( + i )
,
(157)
where pr (r) and pt (r) are the radius-dependent radial and tangential plasma frequencies, respectively, and is the damping coefficient. For the linear dielectric host, we choose 2 = 1.77 (a typical dielectric constant of water). We further assume pr (r) and pt (r) to be
r r and pt (r) = p 1 − kt · , r < a. (158) pr (r) = p 1 − kr · a a The above form is quite physical for 0 < kr (kt ) < 1, since the center of grains can be better metallic so that p (r) is larger, while the boundary of the grain may be poorer metallic so that p (r) is much smaller. In fact, such a variation can also appear owing to the temperature effect [114]. Moreover, we choose kt kr , in view of the strong metallic behavior in the tangential direction. Fig. 40 displays the real and imaginary parts of effective dielectric constant e as a function of the incident angular frequency /p . For kr = 0, there exists a frequency region, where the real part of the effective dielectric constant is negative. With increasing kr , this region becomes narrow generally, in accompanied with less negative Re(e ) [see Fig. 40(a)]. This is due to the fact that increasing kr decreases the influence of the metallic behavior [owing to the decrease of pr (r)]. In the mean time, the sharp peak for Im(e ) turn weak with kr [see Fig. 40(b)]. Furthermore, for kr = 0, the continuous resonant bands in the high frequency region appear always, and this region becomes more
150
J.P. Huang, K.W. Yu / Physics Reports 431 (2006) 87 – 172
-4
χe /χ xxyy
-5 -6 -7 (a) -8 g=1 exact -1.2 g=2 exact g=3 exact g=1 NADEDA g=2 NADEDA g=3 NADEDA -1.6
-3
-4
-2.0
-5
-2.4 1
(b)
χe /χzzzz
χe /χ xxzz
-2
2
3 ε (0)
4
5
1 (c)
2
3 ε (0)
4
5
Fig. 39. (a) e /xxyy versus (0), for the linear dielectric gradation profiles with various g, at h = 8 [60]. Here xxyy is the only nonzero component of the tensorial nonlinear susceptibility. Lines: numerical results obtained from the NADEDA [Eqs. (144), (149) and (151)]; Symbols: exact results predicted by the first-principles approach [Eq. (156)]. (b) Same as (a), but e /xxzz versus (0), with xxzz being the only nonzero component. (c) Same as (a), but e /zzzz versus (0), with zzzz being the only nonzero component.
broad as kr increases. In this case, the appearance of the resonant bands results from the radius-dependent plasma frequency p (r). This phenomenon has already been observed, when a shell model [78] or nonspherical model [124] was taken into account. In our previous works [78,124], a broad continuous spectrum is shown to be around the larger pole in the corresponding spectral density function. Here, the graded particles under consideration can be regarded as a construction of multi shells, which hence should be expected to yield the broader spectra for the optical absorption [Im(e )]. In addition, we note that, as kr increases, both the surface plasma frequency and the center of resonant bands are red-shifted. In particular, for larger kr , the resonant bands can become broader, owing to strong inhomogeneity inside the particles. Then, we speculate on how gradation and anisotropy affect the optical nonlinearity enhancement in metal–dielectric composites. As shown in Fig. 41, no matter which component of the nonlinear susceptibility tensor is nonzero, e can be substantially enhanced within a certain frequency region. In particular, this enhancement becomes quite strong for zzzz (the only nonzero component). In fact, the physical origin of this huge enhancement is the large increase in the local field component Ez . In addition, the nonlinearity enhancement will become more strong, for the system with a larger kr which is related to a higher contrast between 1t and 1r . For example, |e /zzzz | > 104 in the frequency region 0.2 /p 1.0 for kr = 1. From Fig. 41, we also find that the optical nonlinearity enhancement obtained for four nonzero components, respectively, displays the similar qualitative behaviors. This should be in contrast to those observed in a polycrystalline quasi-one-dimensional conductor [121,123,125], where the effective optical nonlinearity for four elements of the nonlinear susceptibility tensor exhibit quite different behaviors [121] (the differences become more distinct by using spectral representation approximation [123]). Actually, the differences result from two different kinds of dielectric anisotropy (and hence two different kinds of tensorial dielectric constants) under consideration. We focus on the particles with spatially varying, but spherically symmetric, dielectric anisotropy, whereas, in the previous works [121,123,125], the authors studied uniaxial anisotropy in the Cartesian coordinate system.
J.P. Huang, K.W. Yu / Physics Reports 431 (2006) 87 – 172
151
8
6
Re (εe )
4
2
0
-2
(a) -4 2 kr =0.0 1
kr = 0.2
log10 [Im (εe)]
kr = 0.4 kr =0.6
0
kr =0.8
-1
-2
-3 0.0 (b)
0.2
0.4
ω /ωp
0.6
0.8
1.0
Fig. 40. The real and imaginary parts of e versus frequency /p , for various kr [60]. Parameters: p / = 0.01 and kt = 0.
Although the optical nonlinearity enhancements for four typical nonzero components of the nonlinear susceptibility (ij kl ) take on quite similar behaviors, their contributions to the magnitude of the effective optical nonlinearity are different (see Fig. 42). As shown in Fig. 42, the strongest (weakest) nonlinearity enhancement occurs for the case with zzzz (xxyy ) being only nonzero component. Moreover, the differences between the two cases of xxzz = 0 only and zzxx = 0 only are clearly shown for Re(e /1 ) and Im(e /1 ). For practical applications, a most useful parameter is the figure of merit (FOM), which is defined as the ratio of |e | to Im(e ). In Fig. 43, we investigate the figure of merit. Here the only nonzero component is assumed to be zzzz . We find that the increase of kr (namely, the rapid decrease of the radial metallic behavior) results in a large enhancement of the FOM, especially in the high frequency range [see Fig. 43(a)]. However, the increase of kt (i.e., the rapid decrease of the tangential metallic behavior) causes the FOM in the high-frequency region to decrease [see Fig. 43(b)].
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J.P. Huang, K.W. Yu / Physics Reports 431 (2006) 87 – 172
5
0
0 kr=0.0 kr=0.2 kr=0.4 kr=0.6 kr=0.8 kr=1.0
-5
-5
log10 (|χe/χxxzz|)
log10 (|χe/χxxyy|)
5
-10
-10 (b)
(a) 5
0
-5
-10 0.0 (c)
4
2
0.2
0.4 0.6 ω /ωp
0.8
1.0
0.0 (d)
0.2
0.4 0.6 ω /ωp
0.8
log10 (|χe/χzzzz|)
log10 (|χe/χzzxx|)
6
0 1.0
Fig. 41. Same as Fig. 40, but (a) |e /xxyy | versus /p , with xxyy being the only nonzero component; (b) |e /xxzz | versus /p , with xxzz being the only nonzero component; (c) |e /zzxx | versus /p , with zzxx being the only nonzero component; (d) |e /zzzz | versus /p , with zzzz being the only nonzero component [60].
For instance, we attain FOM> 105 (which is quite large) in the frequency region 0.3 /p 1.0 for kr = 0.6 and kt = 0. Therefore, it is possible to achieve a large figure of merit by introducing the radial gradation and keeping the tangential dielectric constant unchanged. We have developed an NADEDA to investigate the effective linear dielectric constant and third-order nonlinear susceptibility of composite media consisting of nonlinear inclusions with spatially varying dielectric anisotropy. Alternatively, based on the first-principles approach, we have derived the exact expressions for e and e , for the linear dielectric-constant profiles with small slopes. To our interest, excellent agreement is found between the approximation results (NADEDA) and the exact results (first-principles approach). It is worth noting that exact solutions are very few in composite research, and thus our NADEDA provides an effective way to estimate the effective nonlinear properties in composite media consisting of anisotropic graded inclusions. An application, we apply the NADEDA to study the surface plasma resonance effect on the effective linear dielectric constant, the optical nonlinearity enhancement and the figure of merit in metal–dielectric composites, in which the metal particles possess the tensorial dielectric constants with dielectric gradation profiles. It is found that the gradation profiles in radial dielectric constants are a useful way to control the local-field effects, thus being able to enhance the figure of merit hugely. The present methods are strictly valid in the dilute limit. The presence of both gradation and dielectric anisotropy is shown to be helpful to achieve the large figure of merit, but unable to realize the separation of the absorption peak from the nonlinearity enhancement peak. In this regard, we may intentionally manipulate composite microstructures, e.g., by using the shape distribution of graded inclusions [7], and by using fractal [6] and anisotropic microstructures [90] with large volume fractions. When the volume fraction of graded inclusions is large, percolation behaviors can occur. To this end, the further broadening of the enhancement peak as well as the desired separation of the optical absorption from the nonlinearity peak is expected to be realized.
J.P. Huang, K.W. Yu / Physics Reports 431 (2006) 87 – 172
153
0.2 0.0
0.0 χ1 = χxxyy χ1 = χxxzz χ1 = χzzxx χ1 = χzzzz
-0.2
-0.1
Im(χe/χ1)(106)
Re (χe/χ1) (106)
kr=1.0, kt=0.0
-0.2
-0.4
log10 (|χe/χ1|)
6
4
2
0 0.2
0.4
0.6 ω /ωp
0.8
1.0
Fig. 42. The real and imaginary parts, and the modulus of the optical nonlinearity enhancement for kr = 1 and kt = 0 [60].
To sum up, we put forth an NADEDA, in an attempt to discuss the effects of gradation as well as anisotropy on the optical properties of composite media. For the linear dielectric-constant profiles, the NADEDA has been numerically demonstrated in good agreement with the first-principles approach. Interestingly, both the huge nonlinearity enhancement and the large figure of merit are shown to be achievable by the presence of gradation as well as local anisotropy inside the inclusions. 5.5. Spectral representation for understanding the effective dielectric constant of graded and multilayer composites The idea of using a spectral representation for understanding the effective dielectric constant of graded and multilayer composites has been motivated by a recent study of the optical absorption spectrum of a graded metallic film [21]. In that work, a broad surface plasmon absorption band was observed in addition to a strong Drude absorption peak at zero frequency. Such a broad absorption band has been shown to be responsible for the enhanced nonlinear optical response as well as an attractive FOM. Yuen et al. [90] pointed out that such an absorption spectrum, being related to the imaginary part of the effective dielectric constant, should equally well be reflected in the Bergman–Milton spectral representation of the effective dielectric constant [3,126,127]. Bergman–Milton spectral representation was originally developed for calculating the effective dielectric constant and other response functions of two-component composites [3,126,127]. However, the two concerned components are all homogeneous. Therefore, it is worth extending the spectral representation to graded composite materials. The work on graded films is just a simple example of a more general graded composite in three dimensions. One of the main purpose of this idea help to identify the physical origin of the broad absorption band. It turns out that, unlike in the case of homogeneous materials, the characteristic function of a graded composite is a continuous function because of the continuous variation of the dielectric function within the constituent component. We shall apply our theory to a special case of graded composites, i.e., multilayer material, which is more convenient to fabricate in practice than graded material [128], and many algorithms are now available for designing of multilayer coatings [129,130]. Thus, the present idea is necessary in the sense that we shall discuss the multilayer effect as the number of layers inside the material increases. In this regard, this idea should be expected to have practical relevance.
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7 kr =0.2, k t =0.2 kr = 0.4, k t =0.2 k r =0.6, k t =0.2 kr = 0.8, k t =0.2
log10 (FOM)
6
5
4
3 (a)
0.2
0.4
0.6
0.8
1.0
7
log10 (FOM)
6
5
kr = 0.6, k t =0.0 kr = 0.6, k t = 0.2 kr = 0.6, k t = 0.4 kr = 0.6, k t = 0.6
4
3 0.2 (b)
0.4
0.6
0.8
1.0
ω /ωp
Fig. 43. The figure of merit ≡ |e /zzzz |/Im(e ) versus /p , with zzzz being the only nonzero component [60].
As the number of layers N increases, we shall show a gradual transition from sharp peaks to a broad continuous band until the graded composite results are recovered by the limit of N → ∞. We consider a two-component composite in which graded inclusions of dielectric constant 1 (r) are embedded in a homogeneous host medium of dielectric constant 2 . It is noted that the dielectric constant 1 (r) is a gradation profile as a function of the position r. And we will restrict our discussion and calculation to the quasi-static approximation, i.e., dc/ 1, where d is the characteristic size of the inclusion, c is the speed of light in vacuum and is the frequency of the applied field. In the quasi-static approximation, the whole graded structure can be regarded as an effective
J.P. Huang, K.W. Yu / Physics Reports 431 (2006) 87 – 172
homogeneous one with effective (overall) linear dielectric constant defined as [74] 1 E·D dV , e = V E02
155
(159)
where E0 is the applied electric field along z direction, E and D are the local electric field and local displacement, respectively. The object of the present section is to solve the Laplace’s equation ∇ · ((r)∇(r)) = 0
(160)
subject to the boundary condition 0 = −E0 z. The dielectric function (r) varies from component to component but has a fixed mathematical expression for a given component. It can be expressed as [3] 1 (161) (r) = 2 1 − (r) , s where s=[1−ref /2 ]−1 is the material parameter and ref is some reference dielectric constant in the graded component. We are considering a microstructure gradient (r) in Eq. (161). Thus, the following spectral representation is valid for arbitrary gradation (microstructure) profiles. The characteristic function (r) may be written in terms of a real function f (r) as $ 1 + f (r) in inclusion, (r) = 0 in host, which accords for the microstructure of graded composites. The function f (r) depends on the specific variation of the dielectric constant in the inclusion component. For homogeneous constituent component, i.e., f (r) = 0, (r) = 1 in the inclusion component, while (r) = 0 in the host medium. For graded systems, (r) can be a continuous function in the inclusion component because of the continuous variation of the dielectric function within the inclusion component. Thus, Eq. (159) can be solved 1 (r) = −E0 z + (162) dV (r )∇ G(r − r ) · ∇ (r ), s where G(r − r ) is a Green’s function satisfying: ∇ 2 G(r − r ) = − 3 (r − r ) G=0
for r in V,
for r on the boundary.
ˆ which satisfies In order to obtain a solution for Eq. (160), we introduce an integral-differential Hermitian operator , ˆ ≡ dV (r )∇ G(r − r ) · ∇ , and define an inner product as | = dV (r)∇∗ · ∇.
(163)
With the above definitions, Eq. (162) can be simplified to 1ˆ (r) = −E0 z + (r). s ˆ Then, the generalized eigenvalue problem Let sn and |n be the nth eigenvalue and eigenfunction of operator . becomes ∇ · ((r)∇n ) = sn ∇ 2 n .
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J.P. Huang, K.W. Yu / Physics Reports 431 (2006) 87 – 172
The potential | can be expanded in series of eigenfunctions, s |n n |z | ≡ , sn − s n |n n
(164)
where we choose E0 = 1 for convenience. Since (r) is a real function, the eigenvalues sn will be real. Also, for graded component, (r) is a continuous function, which will cover the full region, i.e., −∞ (r) ∞. Therefore, the eigenvalues sn , which depend on the continuously graded microstructure (r), do not lie within the interval [0, 1] but extend to −∞sn ∞ as first pointed by Gu and Gong [33] for three-component composites case. However, eigenvalues sn still lie in [0, 1] for 0 (r)1. We are now in the position to find an analytical representation for the effective dielectric constant e according to Eq. (159). We take advantage of Green’s theorem, the boundary condition 0 = −z, and the Maxwell equation ∇ · D = 0 to obtain the effective dielectric constant e 1 = (−∇) · D dV 2 2 V −1 1 = zˆ · 1 − (r) ∇ dV V s =1+
1 z|. sV
(165)
If we now introduce the reduced response [3] F (s) = 1 −
e , 2
(166)
and substitute Eq. (164) into Eq. (165) we find
2 1 z|n 1 F (s) = . V n n |n s − sn We can now express the effective dielectric constant as fn e = 2 1 − , s − sn n
(167)
where fn is given by
2
1 z|n fn = . V n |n Using the above equations, we obtain the following sum rule: n
1 z|z V 1 = dV (r)∇z · ∇z V 1 = dV (r). V
fn =
(168)
It is worth noting that the sum rule will not equal to the volume fraction of inclusion. This is different from the Bergman–Milton spectral representation for two homogeneous systems, in which the sum rule equals to the volume fraction of the inclusion.
J.P. Huang, K.W. Yu / Physics Reports 431 (2006) 87 – 172
ˆ has a continuous spectrum, Eq. (167) should be replaced with the integral form When the operator m(s ) , e = 2 1 − ds s − s where m(s ) is the spectral density function. Then, the reduced response becomes m(s ) . F (s) = ds s − s
157
(169)
(170)
If we write s as s + i0+ , the right side of Eq. (170) becomes m(s ) P ds − im(s), s − s and thus, m(s ) is given through the limiting process 1 m(s ) = − Im[F (s + i0+ )].
(171)
This final result is identical in form to Bergman’s expression for the analogous function in scalar composite materials. However, there are differences in the derivation, namely, the definition of the inner product Eq. (163), the continuous graded microstructure (r), the sum rule, as well as the range of eigenvalues sn . From Eq. (169) it is evident that if the spectral density function m(s ) is known, the effective dielectric constant can be obtained accurately, and vice versa. The spectral representation has been used to analyze the effective dielectric properties of composites. Recently, Levy and Bergman [131] also used it in their study of nonlinear optical susceptibility. In this regard, Sheng and coworkers [132] developed a practical algorithm for calculating the effective dielectric constants based on the spectral representation. In what follows, we restrict ourselves to a graded composite both in one dimension and three dimensions, and corresponding multilayer composites. 5.5.1. Spectral density function of a graded film We consider a graded dielectric film of width L, in which two media meet at a planar interface as shown in Fig. 44 (a). The first medium 1 (z) varies along z-axis, while the second medium 2 is homogeneous. We define the graded microstructure as $ 1 + az, 0 < z h, (z) = (172) 0, h < z < L, where a and h are real constants. They can be varied to describe different graded films. Thus, according to Eq. (161), the dielectric function of graded film can be expressed as (z) . (173) (z) = 2 1 − s Owing to the simple geometry of a graded film, we can use the equivalent capacitance of a series combination to calculate the effective dielectric constant as 1 L 1 1 = dz. (174) e L 0 (z) Substituting Eqs. (172) and (173) into Eq. (174), we obtain 1 1 − h s[ln(1 − ((0)/s)) − ln(1 − ((h)/s))] = + , e 2 a2 with the assumption L = 1.
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1.6 a =-0.4 a =-0.8
ε (z)/ε0
1.4
1.2
1 0
0.2
0.4
0.6
0.8
1
z
(a) 1.6
ε (r)/ε0
1.4
1.2
1 0 (b)
0.5
1 r
1.5
2
Fig. 44. (a) Dielectric profile of various graded films at h = 0.6; (b) Dielectric proifle of various graded sphere with unit radius [70]. Parameters used: 2 = 1 and s = −2.
We are now in a position to extend the Bergman–Milton spectral representation of the effective dielectric constant [3,126,127] to a graded film. For a graded system, (z) can be a continuous function in the inclusion medium. Using Eqs. (169)–(171), we obtain the spectral density function for a graded film as m(s ) = −
as arg((s − 1)/(s − ah − 1)) [(s arg((s − 1)/(s − ah − 1)))2 + (a(h − 1) − s ln((s − 1)/(s − ah − 1)))2 ]
where s = Re[s] and arg[· · ·] denote the arguments of complex functions.
,
(175)
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159
5.5.2. Spectral density function of a graded sphere The above theory can be generalized to graded composites in three dimensions. We consider a graded sphere with dielectric constant 1 (r) embedded into a homogeneous host medium with dielectric constant 2 . The dielectric constant of the graded sphere 1 (r) varies along the radius r. We can obtain the effective dielectric constant of a graded sphere using the spectral representation. We consider the graded microstructure as $ (r) =
1 + ar, 0 < r R, 0,
r > R,
where R is the radius of the graded sphere. Thus, from Eq. (161) the dielectric constant in the graded sphere is given by (r) = 2
(r) 1− . s
(176)
In the dilute limit the effective dielectric constant of a small volume fraction p of graded spheres embedded in a host medium is given by [36,133] e = 2 + 32 pb,
(177)
where b is the dipole factor of graded spheres embedded in a host as given in Ref. [19]. Using Eqs. (166) and (177), the reduced response can be obtained as F (s) = −32 pb.
(178)
Thus, the spectral density function of a graded sphere can be given through a numerical evaluation of Eq. (171). 5.5.3. Spectral density function of multilayer composites A multilayer composite is a special case of graded composites. The gradation becomes continuous as the number of layers approaches infinity. To investigate the multilayer effect, we shall use a finite difference approximation for the graded profile (Eqs. (173) and (176)) for a finite number of layers. To mimic a multilayer system, we divide the interval [0, 1] into N equally spaced sub-intervals, [0, z1 ], (z1 , z2 ], . . . , (zN − 1, 1]. Then we adopt the midpoint value of (z) for each sub-interval as the dielectric constant of that sublayer. In this way, we calculate the effective dielectric constant, eigenvalues, as well spectral density function for each N. It is worth noting that the results of N → ∞ (e.g., N = 1024 recovers the results of graded composites). In addition to multilayer films, we can use the above approach to study the much simpler problem of a two-layer film. In this system, we have two layers of dielectric constants 1 , 2 , and host 0 . Thickness are hy, h(1 − y), and 1 − h, respectively, where y is the length ratio between component 1 and component 2 . We also define two microstructure parameters, 1 and 2 . If we let s = 1/(1 − 1 /0 ), then 1 = 1, and 2 = (0 − 2 )/(0 − 1 ). According to Eq. (173), the effective dielectric constant of the two-layer film is now given by 1 hy h(1 − y) 1 − h = + + . e 1 2 0 According to Eq. (166), the reduced response can be given by F (s) =
F1 F2 + , s − s1 s − s2
(179)
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where F1 =
h(s1 (y − y + ) − ) , s1 − s 2
h(s2 (y − y + ) − ) , s1 − s 2 1 s1 = 1 − h(y − y + ) + 2 % 2 − 4(−1 + h) + (1 − h(y − y + ) + ) ,
F2 = −
1 s2 = 1 − h(y − y + ) + 2 % + 4(−1 + h) + (1 − h(y − y + ) + )2 .
(180) (181) (182) (183) (184) (185)
From the sums of F1 and F2 and the integral of graded microstructure (z) given by Eq. (172), we can check that the sum rule expressed by Eq. (168) is obeyed. It should also be noted that there are two poles in the expression for the reduced response corresponding to two peaks in the spectral density function. If h = 1, then s1 = 0, that is, one peak is located at zero, which is explicitly shown in Fig. 45(a). Similarly, we can also apply our graded spectral representation to a single-shell sphere of core dielectric constant 1 , covered by a shell of 2 , and suspended in a host of 0 . In this example, we can also define two microstructure parameters 1 and 2 . If we let s = 1/(1 − 1 /0 ), then 1 = 1, and 2 = (0 − 2 )/(0 − 1 ). The dipole factor of single-shell sphere is given as [36,133] b=
2 − 0 + (0 + 22 ) 1 f 3 , 2 + 20 + 2(2 − 0 ) 1 f 3
where f is the ratio between radius core and radius shell, and 1 is given by 1 =
1 − 2 . 1 + 22
Then, we can also write Eq. (178) similarly to Eq. (179), where the residues and eigenvalues are given by F1 =
−3ps 1 [(−1 + )y 3 − ] − p[1 − 2(−1 + )y 3 + 2] , 3(s1 − s2 )
3ps 2 [(−1 + )y 3 − ] + p[1 − 2(−1 + )y 3 + 2] , 3(s1 − s2 ) % 1 3 3 2 s1 = 1 + 3 − 1 + (2 − 8y ) + (1 + 8y ) , 6 % 1 3 3 2 s2 = 1 + 3 + 1 + (2 − 8y ) + (1 + 8y ) . 6
F2 =
(186) (187) (188) (189)
Analysis shows that the spectral representation for N = 2 contains two simple poles corresponding to two peaks in the spectral density function. Therefore, we draw the conclusion that, N peaks are a result of N layers. Moreover, N − 1 peaks will accumulate into a continuous broad absorption spectrum when N tends toward infinity, which can be seen from Figs. 47(f) and 48(f). We are now in a position to do some numerical calculations of the spectral density function from Eqs. (169) and (171). A small but finite imaginary part in the complex parameter has been used in the calculations. Without any loss of generality, we choose L = 1 and R = 1 for convenience. We show the effect of different graded profiles, as well
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161
4 a=-0.2 a=-0.4 a=-0.6
2
log10 [m(s)]
a=-0.8
0
-2
(a)
-4
0
0.2
0.4
0
0.2
0.4
0.6
0.8
1
0.6
0.8
1
4
log10 [m(s)]
2
0
-2
-4 (b)
s
Fig. 45. (a) Spectral density function of a graded film without an interface, i.e., h = 1.0; (b) Spectral density function of a graded film meeting a homogeneous medium at an interface h = 0.5, and 2 = 1 [70].
as the effect of the thickness of the inclusion. It should be noted, that in all figures the range of s is limited to [0, 1], because we chose −1 < a < 0 which limits the value of into [0, 1]. Fig. 44 displays the dielectric profile of a graded film (Fig. 44(a)) and a graded sphere (Fig. 44(b)). This figure obviously shows that the dielectric constant varies with the position in inclusion while a constant in host medium. Also, different values of a accord with different graded materials.
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2 a = -0.2 a = -0.4 a = -0.6
0
log10 [m(s)]
a = -0.8
-2
-4
-6
0
0.2
0.4
0.6
0.8
1
s Fig. 46. Spectral density function of a graded sphere with volume fraction p = 0.1 [70].
In Fig. 45(a), we plot the spectral density function m(s) of a graded film without an interface against the spectral parameter for various graded microstructures (z). It is evident that there is always a broad continuous band in the spectral density function. Both the strength as well as the width of the continuous part of m(s) increase with the gradient of the dielectric profile. Thus, the previous results of the broad surface-plasmon band can be expected. Note that there is a sharp peak at s = 0, which is also present in a homogeneous film. In Fig. 45(b), we plot the spectral density function of a graded film meeting a homogeneous medium at an interface for various graded microstructure (z). Again, there is always a broad continuous band in the spectral density function. However, the sharp peak has now shifted to a finite value of s, which is also present in a homogeneous film. In Fig. 46, the spectral density function of graded sphere is displayed for a volume fraction p = 0.1. In this case, the interface always exists. It is clear that a broad continuous function in the spectral density function is always observed, as well as the shift of the sharp peak. However, the decrease of the broad continuous function is more abrupt for graded sphere than for graded film with increasing s. Figs. 47 and 48 display the spectral density function for a multilayer film and a sphere, respectively. It is clear that there are always N sharp peaks for N layers. Moreover, it is worth noting that there occurs a transition from sharp peaks to a broad continuous band with increasing N (see Figs. 47(f) and 48(f)), that is, the graded results are recovered by the limit results of N → ∞. In particular, we had obtained the analytical expression of spectral density function for N = 2. There are two resonances corresponding to the two peaks in Figs. 47(a) and 48(a). We have investigated a graded composite film and a sphere by means of the Bergman–Milton spectral representation. It has been shown that the spectral density function can be obtained analytically for a graded system. However, unlike in the case of homogeneous constituent components, the characteristic function is a continuous function due to the presence of gradation. Moreover, the derivation as well as some salient properties, namely, the sum rule, the definition of inner product, the definition of the integral-differential operators, and the range of spectral parameters, do change because of the continuous variation of the dielectric profile within the constituent components. It should be noted that in graded composite, the eigenvalues are not limited to [0, 1], and they can be extended to −∞ sn ∞ for the full region , i.e., −∞ ∞. However, for simplicity, we investigated the spectral density function in 0 s 1 by choosing −1 < a < 0 to limit the value of into [0, 1]. We also study multilayer composites and calculated the spectral density function versus the number of layers, to explicitly demonstrate that the broad continuous spectrum arises from the accumulation of poles when the number
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163
2 graded N=22
graded N=21
graded N=23
log10 [m(s)]
0
-2
(a)
(b)
(c)
-4 2 graded N=24
graded N =25
graded N = 210
log10 [m(s)]
0
-2
(d) -4
0
(f)
(e) 0.2
0.4
0.6
0.8
1
0
0.2
s
0.4
0.6 s
0.8
1
0
0.2
0.4
0.6
0.8
1
s
Fig. 47. Spectral density function of various multilayer film with 0 = 1 and a = −0.8 [70].
of layers tends to infinity. This finding coincides with the broad surface-plasmon absorption band associated with the optical properties of graded composites. To sum up, we have investigated the spectral density function of graded film and graded sphere, as well as multilayer cases. There is always a broad continuous function in the spectral density function in graded composite, but simple poles in multilayer composite, the number of pole depends on the number of layers. Moreover, there is a gradual transition from sharp peaks to a broad continuous band until the graded composite results recover in the limit of N → ∞. 6. Magneto-controlled nonlinear optical materials The most common way to obtain a nonlinear optical material is to search for materials in which the components possess an inherently large nonlinear response [2]. In contrast, in this section we shall exploit theoretically a nonlinear optical material whose nonlinear optical properties and nonlinearity enhancement can be tuned by applying an external magnetic field—thus called magneto-controlled nonlinear optical materials. Devices that could benefit from these materials include optical switches, optical limiters, etc. Ferromagnetic nanoparticles, typically consisting of magnetite or cobalt, have a typical diameter of 10 nm, and carry a permanent magnetic moment (e.g., of the strength ∼ 2.4×104 B for magnetite nanoparticles, where B denotes the Bohr magneton) [134]. As the ferromagnetic nanoparticles are suspended in a host fluid like water, they can easily form particle chains under the application of external magnetic fields [134–136], thus yielding a magnetic-field-induced anisotropical structure. Recently, a nonmagnetic golden shell was used to enhance the stability of the ferromagnetic nanoparticle against air and moisture [137]. Below we shall show that the effective nonlinear optical response of the suspension which contains ferromagnetic nanoparticles with
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2 graded N =22
graded N= 21
graded N = 23
log10 [m(s)]
0
-2
-4 (a)
(b)
(c)
-6 2 graded N = 24
graded N =25
graded N =210
log10 [m(s)]
0
-2
-4 (e)
(d) -6
0
0.2
0.4
0.6 s
0.8
1
0
(f) 0.2
0.4
0.6 s
0.8
1
0
0.2
0.4
0.6
0.8
1
s
Fig. 48. Spectral density function of various multi-shell sphere with 0 = 1, a = −0.8 and p = 0.1 [70].
Fig. 49. (Color online) Schematic graph showing the design of magneto-controlled ferrofluid-based nonlinear optical materials [138].
metallic nonlinear shells (see Fig. 49) can be enhanced significantly due to the effect of the magnetic-field-induced anisotropy. For the research on nonlinear optical responses, the introduction of a controllable element (e.g., external magnetic field) should be expected to open a fascinating field of new phenomena. The third-order nonlinear susceptibility s of metallic (say typically, noble metals like gold and silver) shells is very large when compared to that of the magnetite or cobalt core and the host fluid like water. Let us start by considering ferromagnetic linear nanoparticles of linear dielectric constant 1 coated with a nonmagnetic metallic nonlinear shell
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165
of 1 and s which are suspended in a linear host fluid of 2 . That is, in the shells, there is a nonlinear relation between the displacement Ds and the electric field Es , Ds = 1 Es + s |Es |2 Es , where 1 is given by the Drude form, 1 = 1 − 2p /[( + i)], where p and stand for the plasmon frequency and the relaxation rate, respectively, and denotes the frequency of the incident light. In what follows, the thickness of the shell and the radius of the core are respectively denoted as d and R. Next, we restrict our discussion to the quasi-static approximation, under which the structured particle or the whole suspension can be regarded as an effective homogeneous one. It is known that the effective third-order nonlinear susceptibility ¯ of an area [Note the area represents the structured particle and the whole suspension, respectively, see Eqs. (196) and (199) below] is defined as [36,74] 1 ¯ = (r)|∇0 (r)|2 [∇0 (r)]2 dr, (190) V |E0 |2 E02 V which is in terms of zeroth-order potential 0 (r) only, see Eqs. (192)–(194) below. In Eq. (190) E0 denotes the external applied electric field, V the volume of the area under consideration, r the local position inside the medium (r the distance from the particle center to the point of interest), and (r) an r-dependent third-order nonlinear susceptibility. To obtain the effective nonlinear susceptibility of the structured particle which contains a linear core with a nonlinear shell, we should obtain the zeroth-order potentials which are actually obtained for the system in which the nonlinear characteristic of shells disappears, s = 0. Under the quasi-static approximation, the Maxwell equations read ∇ × E = 0 and ∇ · D = 0,
(191)
and hence E = −∇, where is an electric potential. Solving Eqs. (191) [or the corresponding Laplace equation ∇ 2 = 0], we obtain the zeroth-order potentials for the core c0 , the shell s0 , and the host h0 c0 = − c1 E0 r cos ,
(192)
r < R,
s0 = − E0 (c2 r − c3 r −2 ) cos , h0 = − E0 (r − c4 r −2 ) cos ,
R < r < R + d,
(193)
r > R + d,
(194)
where is the angle between the external field and the line joining the particle center and the point under investigation, and the coefficients c1 , c2 , c3 , and c4 are determined by the appropriate boundary conditions. Owing to Eq. (190), the effective third-order nonlinear susceptibility of the structured particle 1 can be given by 1
|∇0 (r)|2 [∇0 (r)]2 r R+d |E0 |2 E02
= f s
|∇0 (r)|2 [∇0 (r)]2 R
,
(195)
where f is the volume ratio of the shell to the core. Thus, we obtain 1 = s
,
where = (3/5)[1/(1 − f )1/3 − 1]|z|2 z2 (5 + 18x 2 + 18|x|2 + 4x 3 + 12x|x|2 + 24|x|2 x 2 ) and
2 2
2 2
=
2 + (/3)(1 − 2 ) 2 + (/3)(1 − 2 )
(196)
(197)
with x = (1 − 1 )/(1 + 21 ) and z = (1/3)[2 (1 + 21 )]/{1 [2 + (/3)(1 − 2 )]}. In Eq. (197), the effective linear dielectric constant 1 of each structured particle can be determined by the well-known asymmetrical Maxwell–Garnett approximation 1 − 1 − 1 = (1 − f ) 1 . 1 + 21 1 + 21
(198)
It is worth noting that for the above derivation a local field factor has already been added, see Eq. (197). As a matter of fact, should be 1 for the direct evaluation of Eqs. (190)–(194). The aim of the intended introduction of into Eq. (197) is to include the field-induced anisotropy in the system, at least qualitatively. In detail, denotes the local
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field factors L and T for longitudinal and transverse field cases, respectively. Here the longitudinal (or transverse) field case corresponds to the fact that the E-field of the light is parallel (or perpendicular) to the particle chain. Similar factors in electrorheological fluids were measured by using computer simulations [139,140], and obtained theoretically [52,53] according to the Ewald–Kornfeld formulation.There is a sum rule for L and T , L + 2T = 3 [55]. The parameter measures the degree of anisotropy, which is induced by the applied magnetic field H. More precisely, the degree of the field-induced anisotropy is measured by how much deviates from unity, 1 < T < 3 for transverse field cases and 0 < L < 1 for longitudinal field cases. As H increases T and L should tend to 3 and 0, respectively, which is indicative of the formation of more and more particle chains as evident in experiments [134]. So, a crude estimate of can be obtained from the contribution of chains [141], namely, = [4(d + R)3 /p] ∞ n=1 n n (H )gn , where p denotes the volume fraction of the structured particles in the suspension, gn the depolarization factor for a chain with n structured particles, and n (H ) the density of the chain which is a function of H. It is noteworthy that for given p, n (H ) also depends on the dipolar coupling constant which relates the dipole–dipole interaction energy of two contacting particles to the thermal energy. Now, the system of interest can be equivalent to the one in which all the particles with linear dielectric constant 1 [Eq. (198)] and nonlinear susceptibility 1 [Eq. (196)] are embedded in a host fluid with 2 . For the equivalent system, it is easy to solve the corresponding Maxwell equations [Eqs. (191)], in order to get the zeroth-order potentials in the particles and the host.According to Eq. (190), we obtain the effective third-order nonlinear susceptibility of the whole suspension e as e = p1 , which can be rewritten as e = ps ,
(199)
The substitution of = 1.0 (i.e., the isotropic limit) into Eq. (199) yields the same expression as derived in Ref. [36] in which the dielectric constants of the core and shell of structured particles were, however, assumed to be real rather than complex. On the other hand, the effective linear dielectric constant of the whole suspension under present consideration e can be given by the developed Maxwell–Garnett approximation which works for suspensions with field-induced anisotropic structures [52] e − 2 1 − 2 =p . e + (3 − )2 1 + 22
(200)
For numerical calculations, we take f = 0.65, p = 0.2, 1 = −25 + 4i, 2 = 1.77 (dielectric constant of water), and = 0.01p . We further see s to be a real and positive frequency-independent constant, in order to focus on the nonlinearity enhancement. Figs. 50 and 51 display the linear optical absorption Im(e ), the enhancement of the third-order optical nonlinearity |e |/s , and the figure of merit (FOM) |e |/[s Im(e )] (see Section 1), as a function of normalized frequency /p , for (Fig. 50) longitudinal and (Fig. 51) transverse field cases. Here the frequency is normalized by p (rather than a specific value of p ), so that the result could be valid for general cases. As mentioned before, = 1.0 corresponds to the isotropic limit. In this case, there is no external magnetic field, and hence all the structured particles are randomly suspended. The figures show that the existence of nonlinear shells causes an enhancement of nonlinearity to appear, see Figs. 50(b) and 51(b), thus yielding a large FOM, see Fig. 50(c) and Fig. 51(c). Such a nonlinearity enhancement induced by shell effects was already reported [36]. The main feature of Figs. 50 and 51 is the effects of external magnetic fields. As L changes from 1.0, to 0.6, and to 0.2, (namely, as T varies from 1.0, to 1.2, and to 1.4) the external magnetic field is adjusted from zero, to low strength, and to high strength. Due to the interaction between the ferromagnetic nanoparticles and the magnetic field, more and more particle chains are caused to appear naturally, thus yielding a magnetic-field-induced anisotropic structure in the suspension. It is evident to observe that the plasmon peak is caused to be blue-shifted for longitudinal field cases as the magnetic field increases. However, for transverse field cases, the plasmon peak displays a red-shift for the increasing magnetic field. In other words, the optical absorption is induced to be anisotropic due to the application of the external magnetic field which produces an anisotropic structure. In fact, the optical absorption arises from the surface plasmon resonance, which is obtained from the imaginary part of the effective dielectric constant. For single metallic particles in the dilute limit, it is well known that there is a large absorption when the resonant condition 1 + 22 = 0 is fulfilled. When there is a larger volume fraction p of structured particles and an anisotropy of the suspension, the effective dielectric constant should be obtained from Eq. (200), thus yielding a modified resonant condition (1 − p)1 + (2 + p)2 = 0. So, the resonant frequency becomes larger (smaller) than the isotropic limit ( = 1) when becomes smaller (larger) than 1. In other words, there is a blue (red) shift for the longitudinal (transversal) field cases. More interestingly, for longitudinal field cases, a giant enhancement of nonlinearity is shown as the magnetic field increases, see Fig. 50(b).
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167
2 T
α L =1.0
1
log10 [Im (εe )]
α L = 0.6 α L = 0.2
0
-1
-2
(a) -3 15 T
log10 [|χ e|/χ s]
10
5
0
(b) -5 40 T
log10 [FOM}
30
20
10
0
-10 (c)
0
1
0.5
1.5
ω /ωp
Fig. 50. (a) The linear optical absorption Im(e ), (b) the enhancement of the third-order optical nonlinearity |e |/s , and (c) the FOM |e |/[s Im(e )] versus the normalized incident angular frequency /p , for various strengths of the external magnetic field which are represented by local-field factors L , for longitudinal field cases (L) [9].
J.P. Huang, K.W. Yu / Physics Reports 431 (2006) 87 – 172
2 T
α L =1.0
1
log10 [Im (εe )]
α L = 0.6 α L = 0.2
0
-1
-2
(a) -3 15 T
log10 [|χ e|/χ s]
10
5
0
(b) -5 40 T 30
log10 [FOM}
168
20
10
0
-10 (c)
0
1
0.5
1.5
ω /ωp
Fig. 51. Same as Fig. 50, but for transverse field cases (T ) [9].
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169
In detail, the nonlinearity enhancement of a high-field case (say, = 0.2) can be of five orders of magnitude larger than that of the zero-field case ( = 1.0). Inversely, a reduction of nonlinearity is found for transverse field cases, see Fig. 51(b). The magnitude of the nonlinearity reduction is very small in the transverse field case, when compared to that of the nonlinearity enhancement in the longitudinal field case. Owing to the giant enhancement of nonlinearity [see Fig. 50(b)], the FOM becomes much more attractive for longitudinal field cases [see Fig. 50(c)]. The FOM of a high-field case (say, = 0.2) can even be ten-order-of-magnitude enhanced in the longitudinal field case. However, the effect of the magnetic field on the FOM for transverse field cases seems to be uninteresting since the FOM is caused to be decreased slightly due to the nonlinearity reduction shown in Fig. 51(b).Since the permanent magnetic moment of the magnetite nanoparticles m is approximately 2.4 × 104 B [134], we can estimate the threshold magnetic field Hc = 14.3 kA/m (or threshold magnetic induction Bc = 0.018 T) above which the corresponding magnetic energy can overcome the thermal energy 1/40 eV so as to obtain appreciable anisotropy. Besides the magnetic energy, we should also compare the interaction energy. For instance, for two touching magnetite nanoparticles, the interaction between them is proportional to m2 /[2(d + R)]3 , assuming the two structured particles to be in a head-to-tail alignment. Since the magnetic moment m goes as (2R)3 , the interaction energy could vary as [2R 2 /(d + R)]3 . In order to break up the two touching nanoparticles, the thermal energy should be larger than the interaction energy. So, threshold field Hc = 14.3 kA/m serves as an upper estimate. Nevertheless, for cobalt nanoparticles, the threshold field Hc should be lower due to larger permanent magnetic moments. To sum up, by including a metallic nonlinear shell in the system, one can tune the nonlinear optical properties by applying a magnetic field. Such a proposed magneto-controlled nonlinear optical material can serve as optical materials which have anisotropic nonlinear optical properties and a giant enhancement of nonlinearity, as well as an attractive FOM. 7. Summary Throughout the article, theoretical analysis has been conducted. In looking for experimental support, we find that the separation between absorption peak and nonlinearity enhancement peak proposed by one of us [90] has recently been confirmed experimentally by Guan et al. [142]. In the same article by Guan et al. [142], they also experimentally confirmed our theoretical prediction [54] that the peak of third order nonlinear susceptibilities shifts to longer wavelengths with an increasing applied electric field (which, during fabrication, causes metallic particles to have the form of prolate spheroid). Composite effects are always expected to open a fascinating field of new phenomena in nonlinear optics. Owing to composite effects, two or more materials can be combined in such a manner that the effective nonlinear optical susceptibility of the composite exceeds those of the constituent materials, which is thus called enhanced nonlinear optical responses. We have investigated the physical processes of the composite effects on the enhanced third-order nonlinear optical responses of the recently-proposed nonlinear optical materials: 1. Colloidal nanocrystals with inhomogeneous metallodielectric particles or a graded-index host Such materials can have both an enhancement and a red shift of optical nonlinearity, due to the gradation inside the metallic core or host as well as the lattice effects arising from the periodic structure. 2. Metallic films with inhomogeneous microstructures adjusted by ion doping or temperature gradient It has been found that the presence of gradation (or multilayer) in metallic films yields a broad resonant plasmon band in the optical region, resulting in a large nonlinearity enhancement and hence an optimal figure of merit. 3. Composites with compositional gradation or graded particles We found enhanced nonlinear optical responses can be achieved due to the presence of gradation inside compositionally graded metal–dielectric films and/or dielectrically-graded particles with/without dielectric anisotropy. A spectral representation was developed to understand the enhanced responses. 4. Magneto-controlled ferrofluid-based nonlinear optical materials By including a metallic nonlinear shell in field-responsive ferrofluids, one can tune the enhanced nonlinear optical properties by applying an external magnetic field. Such a proposed magneto-controlled nonlinear optical material can serve as optical materials which have anisotropic nonlinear optical properties and a giant enhancement of nonlinearity, as well as an attractive figure of merit. The composite effects, which result from strong enhancement or fluctuations of the local fields in the different microstructures, yield enhanced nonlinearity enhancement of the proposed nonlinear optical materials. These materials
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have some features: Firstly, they can offer higher (or anisotropic) effective nonlinear optical responses and hence desired faster response times, due to the many-body (local-field) effect and the long-range lattice effect in the particle chains/columns or clusters in the system subjected to external fields. Secondly, the nonlinear optical responses or the response times can be real-time-adjusted by choosing appropriate external electric or magnetic fields, due to the fieldinduced change of the microstructure in the suspensions. Finally, cost can be saved much since the suspension-based nonlinear optical materials have fluidity and are difficult to be abraded. Except for enhanced third-order nonlinear optical susceptibilities discussed in the review, there are a number of other optical processes [10,143], e.g., four-wave mixing, second or third harmonic generation, etc. We may also consider composite effects on such nonlinear optical processes, which are also interesting and useful in the sense that they are accessible to experimentalists working in the field. Following the approaches mentioned in the review, new types of nonlinear dielectric materials can also be designed [54,144] for use in electronic and microwave components, sensor windows, and so on. Acknowledgments For completing the review we have profited from valuable and stimulating collaborations and discussions with Prof. Z. Y. Li, Prof. G.Q. Gu, Prof. L. Gao, Dr. M. Karttunen, Prof. K. Yakubo, Prof. T. Nakayama, Prof. C. Holm, Prof. P. M. Hui, Mr. J. J. Xiao, and Ms. L. Dong. J.P.H. would also like to express his gratitude to Ms. C. Z. Fan, Mr. G. Wang, Mr. W. J. Tian, and Ms. Y. J. Zhao, for their helpful assistance. The original research was supported by the Research Grants Council of the Hong Kong SAR Government, by the Alexander von Humboldt Foundation in Germany, by the German Research Foundation under Grant No. HO 1108/8-4, by the Department of Physics, Fudan University, China, and by the Grant-in-Aid for Scientific Research organized by Japan Society for the Promotion of Science. J.P.H. also acknowledges the financial support by the Shanghai Education Committee and the Shanghai Education Development Foundation (“Shu Guang” project) under Grant No. KBH1512203, by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry, China, by the National Natural Science Foundation of China under Grant No. 10321003, and by Jiangsu Key Laboratory of Thin Films, Suzhou University. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]
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Physics Reports 431 (2006) 173 – 230 www.elsevier.com/locate/physrep
The periodic Anderson model: Symmetry-based results and some exact solutions Canio Nocea, b,∗ a Laboratorio Regionale SuperMat, CNR-INFM-Salerno, Italy b Dipartimento di Fisica “E. R. Caianiello”, Università di Salerno, I-84081 Baronissi (Salerno), Italy
Accepted 4 May 2006 editor: A.A. Maradudin
Abstract The topic “magnetic impurities in metals” is certainly one of the most studied problems of the solid-state physics in the last years. The interest toward this argument relies on the fact that the interaction between the magnetic moment of the impurities and the conduction electrons of the host metal, is responsible for a large variety of physical phenomena. The simplest model that captures the essential physics of the systems previously mentioned is certainly the periodic Anderson model. This model appeared in the literature for the first time in 1961 in a paper by P.W. Anderson as an attempt to describe in a simplified way the effects of correlations for d-electrons in transition metals. The Hamiltonian of this model cannot be exactly solved in general. Nevertheless, exact results are known in some special cases. The argument of this review is the discussion of some of these exact solutions and the symmetry properties exhibited by the microscopic model Hamiltonian. The review has been organized in such a way that an introductory material is presented to make the main points intelligible to a non-specialist reader even though very recent developments on this topic are also presented. In particular, we will discuss special solutions of the model, holding in any dimension, when one of the interacting couplings of the model vanishes. We want to mention that, in spite of the crudeness of the models so derived, some physical insights can be derived from these simplified versions of the Anderson Hamiltonian. The impossibility of ordering, magnetic or superconducting, will be also discussed. These results hold for any temperature, electron filling and any strength of the parameters of the model, but are confined to low-dimensional cases and are based on the application of the Bogoliubov’s inequality. It is also discussed the T = 0 version of the Bogoliubov’s inequality and it is shown that quantum effects disorder the system, at least in one dimension. Recent studies of the Anderson model showing exact solutions holding for specific values of the microscopic parameters and/or for special filling will be also analyzed. These results are based on the application of spin reflection positivity and on symmetry properties exhibited by Anderson Hamiltonian. Some results in the U = ∞ limit are also presented; namely, we discuss the conditions under which a ferromagnetic ground state is established in one dimension when the number of electrons exceeds by one the number of sites and then, for decorated lattices, we derive the ground-state energy and we construct the corresponding eigenstate. Finally, a simple theorem on the total momentum of the ground state of the symmetric version of the Hamiltonian is presented. © 2006 Published by Elsevier B.V. PACS: 71.27.+a; 75.20.Hr; 71.28.+d; 71.10.Fd; 75.10.Lp; 75.30.Mb
∗ Corresponding author at: Università di Salerno, INFM-CNR Salerno, Via S. Allende, I-84081 Baronissi (SA), Italy.
E-mail address:
[email protected]. 0370-1573/$ - see front matter © 2006 Published by Elsevier B.V. doi:10.1016/j.physrep.2006.05.003
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C. Noce / Physics Reports 431 (2006) 173 – 230
Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Derivation of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Symmetries properties of the PAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Simple exact solutions holding in any dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Atomic limit tij = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1. NTOT = 0 and S = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2. NTOT = 1 and S = 21 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3. NTOT = 2 and S = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.4. NTOT = 2 and S = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.5. NTOT = 3 and S = 21 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.6. NTOT = 4 and S = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Uncorrelated limit U = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Non-mixing limit V = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. Unconstrained-range hopping(s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Absence of long-range orders in low dimensions: finite and zero temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Proof of the Bogoliubov’s inequality at finite temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Application of Bogoliubov’s inequality to HPAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1. Magnetic long-range orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2. Superconducting pairing of s-wave type or generalized pairing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3. Superconducting pairing of dx 2 .y 2 -wave type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4. T = 0 Bogoliubov’s inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1. Magnetic long-range orders at T = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2. Superconducting pairing of generalized type pairing at T = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Spin-reflection positivity method applied to PAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Supersolid in the PAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Spin and charge gaps in PAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. Antiferromagnetism in PAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. U = ∞ limit: special solutions of PAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1. Ferromagnetism at quarter filling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3. Exact solution of the PAM for decorated lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Total momentum of the one-dimensional PAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12. Conclusions and perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1. Introduction The study of metallic systems containing localized magnetic moments has assumed a remarkable and growing relevance in the solid-state physics. When localized moments, supplied by atoms of rare earth (mainly cerium) or actinides (mainly uranium) are introduced in a host metal, the whole system exhibits, at low temperatures, unusual behaviors in transport properties. This effect has to be certainly ascribed to the existence of special kinds of interactions between the localized magnetic moments and the electrons of the host metal. It is well known, for instance, that localized impurities may act as scattering centers for conduction electrons and, below a characteristic temperature, they can produce a partial or total compensation of the local magnetic moment giving
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rise to a minimum in the resistivity (the Kondo effect) below a special temperature, called the Kondo temperature. In concentrated systems, the conduction electrons can mediate the exchange interaction between the localized moments, producing the so-called Ruderman, Kittel, Kasuya and Yosida interaction (RKKY) that, at low temperatures, may originate various magnetic orders, mainly antiferromagnetic. The realization of metallic systems with magnetic moments cannot be possible with any magnetic ion. Indeed, an atom having a magnetic moment can lose its magnetic properties, partially or totally, when it is embedded in a metallic matrix. This often happens when transition metal ions are considered. We notice that in such systems, the magnetic properties are produced by partially filled d-shells. In dense systems, electrons belonging to these shells being not strongly localized, give rise to d-band rather than keep localized character. On the other hand, the magnetic moment of rare earth or actinide ions is associated with partially filled f-shells that are inner than d-shells and are described by wave functions generally confined to distances less than the interatomic spacing. This implies that ions of these metals, when embedded in the host metal, tend to keep the same magnetic moment of the isolated case. Particular relevance have some intermetallic compounds containing rare earth ions such Ce, Yb, Sm, Tm and Eu because the atoms of these ions seem to have a non-integer valence. In normal conditions, these ions have a chemical valence equal to +2 or +3. This property is experimentally checked measuring the lattice constant. Indeed, the lattice constant is determined by the atomic radius that, in turn, is related to the number of electrons of the core. In some compounds such as SmS, CePd3 , YbCuAl, the experimental value of the lattice constant is in between two values corresponding to two states whose valence differs by one. For this reason, these compounds are called mixed or intermediated valence systems. The appearance of a mixed valence state is always accompanied by anomalies, both in equilibrium properties and transport quantities. A qualitative explanation of this phenomenon can be done as follows. First of all, a mixed valence system has to be considered as homogeneous in the sense that all the ions present in the compound have the same non-integer value of the valence. In this respect, the intermediate valence systems are different from systems such as Fe3 O4 , where some iron ions have valence +2, or +3 depending on their positions in the crystal lattice. On the contrary, we can assume that, in each rare earth ion, the electronic configuration with n electrons in the f-shell and that produced from the promotion of one of these electrons into the conduction band, are nearly degenerate. Following this qualitative interpretation, we assume, for each ion, time fluctuations between these configurations, one having magnetic nature and the other non-magnetic. For instance, when a cerium ion is introduced in Pd host, the Fermi level of the f-electron is very close to the Fermi level of the system so that the configurations (a) Ce+3 (4f 1 5d 1 6s 2 , magnetic configuration) and (b) Ce+4 (4f 0 5d 1 6s 2 , non-magnetic configuration) become almost degenerate and thus strong fluctuations can appear. In this case, of course the valence of cerium ion assumes a value between 3 and 4 and this fact justifies the name given to this effect. This kind of description is confirmed by the experiments because, measurements based on a small time rate compared to that of the valence fluctuations show evidence of both configurations while slow experiments, involving low energy, show a state having intermediate property as a consequence of some sort of average between the two configurations. In the first case, one uses high-energy source such as X-ray while in the latter case, anelastic neutron scattering is usually used. Finally, it is worth mentioning that the mechanism of valence fluctuations yields an instability in the magnetic state of the system, in the sense that no magnetic ordering appears in intermediate valence systems. There is another class of systems with rare earth and actinide metals having (quasi) integer valence and thus very different from mixed valence compounds. These compounds are called Kondo lattice. In these systems such as CeCu2 Si2 , CeAl3 , etc. the degree of the degeneration of the two above-mentioned configurations is reduced with respect mixed valence systems, but not enough to define a magnetic regime. This effect sometimes produces a RKKY interaction, tending to align the magnetic moments, that overcomes the Kondo effect, whose role is to compensate the magnetic moments, so that magnetically ordered ground states can appear. This is for example the case of CeB6 and U2 Zn17 . When the Kondo effect is more effective, it is possible to define a special temperature T , playing the role of the Kondo temperature defined for dilute systems, below which these systems behave like Fermi liquids with high value of the density of states at the Fermi energy. This assumption has been done looking at the linear term for the specific heat at low temperatures. In an equivalent way, one can say that electrons near the Fermi surface possess a large effective mass. Typical values of this mass are roughly thousands larger than the electron mass. Various experimental techniques have shown that the heavy electrons at Fermi energy come from partially filled f-shells of the magnetic ions. These carriers acquire a certain degree of itineracy, reduced compared to the usual mobility of conduction electrons, due to the formation, by means of a hybridization mechanism between conduction electrons and localized ones of f-bands and, as consequence, the electrons moving within this band have a large effective mass.
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Above T this behavior changes in a radical way: the systems can be considered as normal metals, with normal electrons, and the localized magnetic moments contribute to the magnetic susceptibility producing a Curie-type law. The ground state of heavy fermion systems can be also of superconductive type besides of magnetic, when RKKY interaction is more effective, or non-magnetic, in the Kondo regime. Systems like CeCu2 Si2 , UPt3 , UBe13 , etc. become superconducting at low temperatures with Cooper pairs formed by f-electrons trough a non-phononic coupling. It is worth noticing that mixed valence systems and heavy fermion compounds exhibit rather similar transport properties. This fact suggests that a theoretical description of both classes of systems may be done within the same type of model in different regime for the microscopic parameters. The simplest model that captures the essential physics of the phenomena previously described is certainly the periodic Anderson model (PAM) [1]. The model Hamiltonian is built up by different contributions that reproduce the complex dynamics due to the interplay between band electrons and correlated electrons. Namely, the Hamiltonian contains the following terms: an uncorrelated conduction band term which describes electrons, supposed independent, in a conduction band with a specified energy spectrum; a lattice of localized magnetic impurities with on-site energy whose electrons are experiencing a Coulomb repulsion when are located at the same lattice site with opposite spin; and finally, the term responsible for the hybridization of conduction electrons and localized correlated electrons when both are present on the same lattice site. This Hamiltonian cannot be exactly solved in general. Referring to the microscopic approximate methods so far introduced, we mention the approaches based on perturbative expansion in one of the parameters of the Hamiltonian; mean-field methods applied to the Hamiltonians deduced from Anderson Hamiltonian when special transformations eliminate the Coulomb repulsion term; variational approaches based on suitable choices for the ground states. Nevertheless, exact results are known in some special cases: this review is devoted to the discussion of these exact solutions with special emphasis given to the description of the techniques used to achieve these results. The review is organized as follows: in Section 2, the PAM is derived starting from a first quantized description of impurities in a metal; then, in Section 3 the symmetries exhibited by the model Hamiltonian are presented and discussed. Section 4 is devoted to special solutions of the model, holding in any dimension, when one of the interacting couplings of the model vanishes. We want to mention that, in spite of the crudeness of the models so derived, some physical insights can be derived from these simplified versions of the Anderson Hamiltonian. In Section 5, the impossibility of ordering, magnetic or superconducting, is discussed. These results hold for any finite temperature, electron filling and any strength of the parameters of the models, but are confined in one and two dimensions. It is also discussed the T = 0 version of the Bogoliubov’s inequality looking at the long-range magnetic and superconducting orders and it is shown that, if an excitation gap opens, in the low-dimensional cases, these long-range orders will not appear at zero temperature. More precisely, we show that the Anderson Hamiltonian exhibits a disordered phase at T = 0 in one dimension, while numerical results indicate a magnetically ordered phase in two dimensions. Some exact solutions holding for specific values of the parameters and/or for special filling are discussed in Section 6, where the spinreflection positivity condition is applied to the Anderson model. In particular, it is shown that the ground state of the model, at half-filling, is a spin and pseudospin singlet. Moreover, we show in Section 7 that the lower-energy state is rotationally invariant in the pseudospin space, thus supporting the simultaneous presence of superconductivity and charge–density wave (CDW) order. This amount to conclude that this state for negative value of the Coulomb repulsion can represent a supersolid. A discussion on the excitation gaps is presented in Section 8, while a study on the antiferromagnetism of the model is illustrated in Section 9. Section 10 is devoted to the U = ∞ limit of the model. Making use of the Perron–Frobenius theorem, an interesting result related to the magnetic nature of the ground state at quarter filling is discussed. Then, a special exact solution of the model for decorated lattices is analyzed and presented within the Brandt–Giesekus approach. In Section 11, a simple theorem on the total momentum of the ground state of the symmetric PAM on a one-dimensional bipartite lattice with periodic boundary conditions is stated and proven, and finally, Section 12 contains a summary of the review and the perspectives on the theoretical studies on the PAM. For completeness, we mention that via Bethe ansatz [2–14] or renormalization group method [15,16] an exact solution of the Anderson model can be achieved. There are already an enormous number of treatises on these methods. In view of this situation, we avoid repetition of such treatments in this review; for a simple and concise description of these approaches see for instance [17]. Besides, no mention will be done to approximate solutions of the model being the review focused on the exact treatments of the Anderson Hamiltonian. However, for sake of completeness here we cite some of the methods usually applied to the Anderson model, and to other strongly correlated electron models too. We mention the Gutzwiller projection technique [18–21]; the slave boson method [22,23]; the large-N approximation [24]; perturbative expansion
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with respect to the on-site Coulomb repulsion [25–28]; non-standard diagrammatic expansions [29–31] or non-crossing approximation (NCA) [32,33]. Because of the rather technical aspects of these approaches, we omit here a discussion and refer the reader to the some reviews which cover several of the above theories in detail [34–39]. As final remark, we would like to notice that there are a few interesting reviews devoted to some of the problems and topics afforded in this paper. In this respect, we cite the excellent reviews on the spin reflection positivity in strongly correlated systems [40,41], on the connection between spin reflection positivity and Perron–Frobenius theorem [42] and on the application of Bogoliubov’ inequality to low-dimensional many-body models [43,44]. The readers interested into these specific arguments can give a look at these papers. 2. Derivation of the model There is experimental evidence that different kinds of impurities embedded in different host metals, exhibit a wide class of physical behaviors. Indeed, the magnetic properties of the impurities, being related to the 3d, 4d, 4f or 5f partially filled shells, affect in a different manner the properties of the host matrix and then of the whole system. The behavior of the impurities is produced by the following mechanisms: the atomic Coulomb interaction, the exchange interaction, and the admixture of their wave functions with the conduction electron band of the host [45]. Moreover, any theoretical model describing this kind of systems has to handle the circumstances that produce a localized moment in the metal and the effect of the interaction between the localized moments and the conduction electrons. A satisfactory answer to these questions is certainly given by Anderson model. The Hamiltonian for the PAM we refer to throughout this review is the following: HPAM = Hc + Hf + Hf c .
(1)
In this Hamiltonian, Hc describes a set of uncorrelated conduction electrons: Hc = tij ci† cj , ij
where tij is the hopping term whose form depends on the crystal structure as well as on the hopping processes allowed; Hf , the part related to the correlated electrons, is given by † H f = εf fi fi + U ni↑ ni↓ , i
i
where the correlated electrons are assumed localized on the a level with energy εf and strongly interacting, with a strength U, when two of them are localized on the same level; finally Hf c corresponds to the hybridization between correlated and uncorrelated electrons: Hf c = V (ci† fi, + h.c.). i
Here V is the hybridization amplitude, assumed constant. In the above terms, we have indicated with ci† (fi† ) the creation operator for an uncorrelated (correlated) electron with spin at the i site. Moreover, n = fi† fi is the number operator for f-electron with spin at i site. The starting point to derive this Hamiltonian is to consider at beginning an impurity embedded into a metallic host [5]. The more general first quantized form of the Hamiltonian describing the interaction between a metal and an impurity is the following H = H0 + Himp + Hint , where H0 is the Hamiltonian for electrons moving in a periodic lattice; Himp describes one of the N impurity potential Himp =
N i=1
Vimp (ri ) +
1 e2 , 2 rij i=j
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Vimp being the potential of the impurity when all the electrons of the outer shell are removed, while the second term is the Coulomb repulsion. Finally, Hint is the spin–orbit coupling produced by the impurity potential, i.e. Hint =
N i=1
1 dVimp Li · i . mcri dri
Here we have indicated with m, L, , the mass, the orbital angular momentum, the spin of the impurity, respectively. To derive the Anderson model, we consider that the energy levels of the Himp are very close to the Fermi surface. This assumption implies the formation of narrow resonances, whose reduced width is a consequence of the difference between the interatomic distance for the conduction electrons, given by the inverse of the Fermi wave vector kF−1 , and the Bohr radius of the d or f shells. This consideration and the fact that the wave function of electrons near the Fermi level contains both the localized and the itinerant states turns out very helpful in writing down the Hamiltonian H. Indeed, we can express the eigenfunctions of H using as the basis a linear combination of orbital and band wave functions. In the second quantized notation we can write † † † (r) = l,m (k, r)ck,l,m, + d (r)fm, , m,
l,m,
† † where ck,l,m, is the creation operator for the spherical wave function at the site r; fm, corresponds to the creation operator for a localized electron. The first summation is restricted to wave vectors less than the inverse of the Bohr radius. Besides, the coefficients are given by
l,m (k, r) = Rl (kr)Ylm (ϑ, ) and
d (r) = Rl0
r rB
Yl0 m (ϑ, ).
These functions are not orthogonal, but the corresponding overlap integral is small and then, we neglect their nonorthogonality. Of course this consideration is a consequence of the different scales given by the Bohr radius rB and the Fermi wave vector kF . Now, we can rewrite the Hamiltonian H in the second quantized form in terms of the field operator , noticing that it contains a large number of terms H = † (r)[H0 + Himp + Hint ](r) d3 r. Let us discuss them one by one looking at their meaning and strength. We start with the terms containing only c† and c operators. These terms, put together, represent the Hamiltonian of the host metal. Neglecting the many particle corrections and assuming that the spectrum of the electrons is spherically symmetric near the Fermi level, we have † ε(k)ck,l,m, Hc = ck,l,m, . k,l,m,
Now, we consider the terms containing only f † and f operators. These terms, containing only the field operators for the localized electrons, correspond to the Hamiltonian for a d or f-ions in the crystal field of the surrounding conduction electrons. The more general form of this Hamiltonian is † εmn fm, W (m1 1 , m2 2 ; m3 3 , m4 4 )fm† 1 1 fm† 2 2 fm3 3 fm4 4 − KLf · Sf , Hf = fn, + m,n,
mi ,i
where εmn denotes the single-electron atomic energy, measured from the Fermi level: 2 r 1 r j r l0 (l0+1 ) Vimp (r) + − rR l0 r 2 dr Rl0 Rl0 εmn = mn 2 2 rB 2mr rB jr rB 2mr
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plus the crystal field term; W (m1 1 , m2 2 ; m3 3 , m4 4 ) is the non-relativistic part of the interaction between d or f-electrons and obeys the constraints: m1 + m2 = m3 + m4 and 1 + 2 = 3 + 4 [46]; and the last term is the spin–orbit coupling for the localized electrons. Then, we discuss the terms containing one-particle mixing between f and c operators. The single-particle mixing interaction is the so-called hybridization interaction: † Hf c = V (ln, l0 m)(ck,l,n, fm, + h.c.), l,m,n,
where the hybridization coupling is r ∗ Yl0 m (ϑ, ) d3 r. (ϑ, )[H0 + Vimp (r)]Rl0 V (ln, l0 m) = Rl (kr)Yln rB We have neglected in the evaluation of the hybridization coupling the effect of the crystal field and of the spin–orbit interaction. This means that we limit, in the above expression for V (ln, l0 m), to terms that are invariant under rotation in the coordinate space. Besides, the spherical symmetry of the problem implies that only conduction electrons with angular momentum l0 contribute to V so that V (ln, l0 m) = V (k)ll 0 mn [47]. In conclusion, considering only the conduction electrons in the partial wave with l = l0 , the Anderson Hamiltonian for a single impurity reads as † † ε(k)ck,m, V (k)(ck,m, H= ck,m, + Hf + fm, + h.c.). k,m,
k,m,
2.1. Discussion We notice that the terms of the Hamiltonian equation (1) are slightly different from those now derived. Hence, some simplifications and approximations have to be made to recover the Hamiltonian equation (1). Concerning the term describing the conduction electrons only, it is further simplified assuming a s-wave form, i.e. isotropic k-dependence, and the energy spectrum is related to the crystal lattice structure through the well-known relationship ε(k) = tij e−i(Ri −Rj )k , ij
where we assumed the translational invariance of the lattice. The actual geometry of the lattice is contained in the hopping matrix tij connecting two lattice sites Ri and Rj . For the local f part of the Hamiltonian the following assumptions are considered. The local energy εmn in Hf is written simply as εf neglecting the crystal field contribution; the spin–orbit coupling is also neglected and finally the interacting term is approximated as U n↑ n↓ . When a lattice of impurities is considered, the local Hamiltonian assumes the following expression: † fi fi + U ni↑ ni↓ . Hf = εf i
i
Finally, referring to the hybridization interaction V (k), we assume a contact expression so that only s-wave are scattered by the impurity and thus V (k) = V . Besides, in presence of a lattice of f ions, the hybridization term is written as V (ci† fi, + h.c.). Hf c = i
We would like to mention that, within renormalization group approach [15], it has been rigorously shown that the corrections induced by an energy-dependent hybridization are unimportant, so that it is quite unnecessary to worry about the energy dependence of coupling constant V and thus the Hf c takes the form previously introduced.
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For sake of completeness, we mention that in the above derivation we have disregarded some terms containing a few f and c operators. Namely, there are five non-single-particle terms with f and c operators: (1) c† f † cf ; (2) c† c† ff ; f † f † cc; (3) c† c† cf ; f † f † f c; corresponding to the contact exchange coupling (term 1), renormalized Coulomb interaction (terms 2) and renormalized virtual mixing (terms 3) [48]. All these terms have been neglected in the Anderson derivation of the model Hamiltonian equation (1). 3. Symmetries properties of the PAM The Anderson Hamiltonian equation (1) exhibits several symmetries, and this property, as it will be shown later, turns out to be very useful since it makes the analysis of the model easier providing many exact relationships between relevant many-body quantities. In this section we give a summary of these symmetries. The PAM exhibits the usual SU(2) rotational symmetry in spin space. The generators of the rotations are the spin operators given by 1 † † † † Sz = (ci↑ ci↑ − ci↓ ci↓ + fi↑ fi↑ − fi↓ fi↓ − 2), (2) 2 i
S + = (S − )† =
† † (ci↑ ci↓ + fi↑ fi↓ ).
(3)
i
For the Hubbard model it is known that there is an additional SU(2) symmetry in charge space, provided that the underlying lattice is bipartite [49,50]. Since the PAM may be considered as a generalized Hubbard model with a special connectivity, if each c and f orbital is assigned to an independent site, the PAM on a bipartite lattice has the additional SU(2) symmetry too [51,52]. A more detailed discussion on this point will be done in Section 6. To understand the meaning of this symmetry, let us first define a bipartite lattice. A lattice of size L is bipartite if it can be divided in two sublattices A and B so that the hopping processes are effective only between sites belonging to different sublattices. According to Shen [40], this condition is realized by the structure function of the lattice i 1 if i ∈ A, i = −1 if i ∈ B. We notice that the definition of this structure function of the bipartite lattice is based on the symmetry of the model Hamiltonian. Now, we introduce two unitary transformations which apply to the field operators c and f and, consequently modify the PAM Hamiltonian into a new Hamiltonian. Let us introduce the following unitary transformation W: W ci W −1 = i ci† ,
(4)
Wf i W −1 = −i fi† .
(5)
We notice that, under this transformation, the PAM Hamiltonian is invariant, providing that the symmetric condition 2εf + U = 0 is realized. A mapping of positive U PAM to the attractive U model is obtained through the following unitary transformation R acting only on one species of spin indices of the field operators c and f, namely down spin: Rci↑ R −1 = ci↑ ,
(6)
† Rci↓ R −1 = i ci↓ ,
(7)
Rf i↑ R −1 = fi↑ ,
(8)
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Table 1 Mapping realized by the electron–hole transformation R defined in the text H (U, εf )
H (−U, −εf )
Sz S+ S−
Jz J+ J−
† † i i (ci ci − fi fi ) † † i i (ci↑ ci↓ − fi↑ fi↓ ) † † i i (ci↓ ci↑ − fi↓ fi↑ )
† † i i (ci ci − fi fi ) † † † † i (ci↑ ci↓ + fi↑ fi↓ ) i (ci↓ ci↑ + fi↓ fi↑ )
We report how the Hamiltonian, the spin, and some relevant operators (left column) change when the unitary transformation R is applied (right column). † Rf i↓ R −1 = −i fi↓ .
(9)
Thus, performing this particle–hole transformation on one of the spin species, one can easily prove that the U > 0 model is mapped into a negative U one. This amounts to say that the original model has another SU(2) hidden symmetry represented by the so-called pseudospin J operator. This operator can be derived applying the transformation R to the spin operators Eqs. (2)–(3); its components are Jz =
1 † † † † (ci↑ ci↑ + ci↓ ci↓ + fi↑ fi↑ + fi↓ fi↓ − 2), 2
(10)
i
J + = (J − )† =
† † † † i (ci↑ ci↓ − fi↑ fi↓ ).
(11)
i
We want to point out that the z component of J is equal to Jz = NTOT /2 − L, where NTOT is the total number operator, implying that Jz is nothing but the total number operator. By means of this transformation the PAM with positive on-site Coulomb repulsion U for f-electrons is mapped to the one with negative U. However, the symmetric condition is needed, otherwise it appears into the transformed Hamiltonian a spin-dependent chemical potential corresponding to the application of a magnetic field. In this case, the symmetry in spin space is reduced from spherical one to axial corresponding to the U (1) symmetry. Obviously, S and J have the same parity, and Sz and Jz do as well. When the total number of electrons is even, Sz and Jz both have an integer eigenvalue, not necessarily identical to each other, while they have a half-integer eigenvalue when the electron number is odd. The real spin and pseudospin operators mutually commute, constituting an SO(4) symmetry. Therefore, if the symmetric condition is satisfied and if the lattice is bipartite, all the eigenstates of the symmetric PAM can be classified by the set of quantum numbers (S, Sz , J, Jz ). As previously mentioned, the z component of the pseudospin operator is nothing but the total number operator, while its transverse components are the staggered pairing operators [49]. Similarly, the z component of the staggered magnetic moment for the positive U PAM is mapped to the CDW operator for the negative U model, and the transverse components of the staggered magnetic moment for U > 0 to the uniform pairing operators for U < 0. These relations are summarized in Table 1. We notice that, under this transformation the longitudinal part of the spin density wave operator i i (ci† ci − fi† fi ) is transformed in the charged density wave, or equivalently orbital, operator i i (ci† ci − fi† fi ), while the transverse parts of the † † † † spin density wave operator i i (ci↑ ci↓ − fi↑ fi↓ ) and i i (ci↓ ci↑ − fi↓ fi↑ ) are mapped into the on site pairing † † † † creation operator i (ci↑ ci↓ + fi↑ fi↓ ) and on site pairing annihilation operator i (ci↓ ci↑ + fi↓ fi↑ ), respectively. As far as the PAM Hamiltonian is considered, we notice that W H (U, εf )W −1 = H (−U, −εf ) + constant,
(12)
which implies that H (U, εf ) and H (−U, −εf ) are related to each other, when the symmetric condition 2εf + U is assumed. Moreover, this relationship implies that all the spectrum of H (−U, −εf ) can be deduced from that of H (U, εf ).
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When the R transformation is concerned, we notice that the mapping between repulsive and attractive PAM is valid too and a relationship similar to that obtained for W holds. Indeed, we have in the symmetric limit: RH (U, εf )R −1 = H (−U, −εf ).
(13)
Finally, let us discuss the properties of the following unitary transformation G: † , Gci↑ G−1 = i ci↑
(14)
† , Gci↓ G−1 = ci↓
(15)
Gf i G−1 = −i fi† ,
(16)
Gf †i G−1 = −i fi .
(17)
Applying G to the PAM Hamiltonian in the symmetric limit, we obtain GH (U, εf )G−1 = H (U, −εf ) + constant,
(18)
which implies that H (U, εf ) and H (U, −εf ) are related to each other. Also for this case, Eq. (18) implies that all the spectrum of H (U, −εf ) can be derived from H (U, εf ). 4. Simple exact solutions holding in any dimensions The PAM admits some exact solutions, holding in any dimensions, when one of the parameters of the Hamiltonian is set equal to zero and also when the hopping term is unconstrained. Here we discuss these different limits separately, and then some considerations on their physical relevance are presented. For simplicity, in this section we add to the Hamiltonian of Eq. (1) the following, irrelevant, on-site term for the uncorrelated electrons: † c i ci . Hc0 = εc i
4.1. Atomic limit tij = 0 If we assume that the hopping term vanishes for any lattice sites i and j, i.e. tij = 0, the Anderson Hamiltonian can be exactly solved because the different lattice sites are decoupled. In this case the PAM Hamiltonian can be rewritten as i HPAM , HPAM = i i where HPAM
is written in terms of field operators acting only on fermions belonging to i site. Since the total Hamiltonian is the sum of terms which mutually commute, the solution of the problem can be obtained solving a single-site problem. Considering that the Fock space for c-electrons, as well as for f-electrons, is four dimensional, one may obtain the i solution of the problem diagonalizing a 16×16 matrix. Moreover, since HPAM commutes with the total number operator NTOT , the total spin operator S and the total third spin component Sz , the 16 × 16 matrix can be reduced into a block form containing: (i) two one-dimensional blocks, corresponding to the vacuum and to the double c- and f-occupation; (ii) two bi-dimensional blocks for single and triple occupation and total third component Sz equal to 21 and similarly for Sz = − 21 ; (iii) when the occupation number is equal to two, one has a three-dimensional diagonal block corresponding to S equal to 1 and a three-dimensional block for total spin equal zero. For completeness, below are reported the block matrices and their eigenvalues with the corresponding eigenvectors. We notice that, matrices in the different NTOT and S channels are build-up considering the following rule: the more generic state vector is of the form |f↑ c↑ f↓ c↓ = f↑† c↑† f↓† c↓† |0, i.e. we have defined the ordering of c and f operators entering a generic vector state.
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4.1.1. NTOT = 0 and S = 0 The matrix representation of this subspace is (0) and the eigenstate is |0 and it has zero energy. 4.1.2. NTOT = 1 and S = 21 i Using the representation above introduced for basis vectors, the matrix representation of HPAM in this subspace is εc V . V εf Diagonalizing this Hamiltonian, we obtain the following eigenvalues: εf + ε c ± 1 , 2 where 1 = (εf − εc )2 + 4V 2 , while the corresponding eigenvectors are E1,2 =
|1 = cos ϑf † |0 + sin ϑc† |0, |2 = − sin ϑf † |0 + cos ϑc† |0. Here we have defined
1 ε f − εc . − cos ϑ = 2 21 We notice that due to Sz symmetry of the model, the matrix, the eigenvalues and the eigenvectors above reported refer to Sz = ± 21 . 4.1.3. NTOT = 2 and S = 1 i The matrix HPAM in this subspace is given by ⎞ ⎛ 0 0 εc + εf ⎟ ⎜ εc + εf 0 ⎠. ⎝ 0 ε c + εf √ The eigenstates are f↑† c↑† |0, 1/ 2 (f↑† c↓† |0 − c↑† f↓† |0) and f↓† c↓† |0, they are degenerate and their energy is εc + εf . Of course, they correspond to Sz = 1, 0, −1, respectively. 0
0
4.1.4. NTOT = 2 and S = 0 i In this case the matrix representation of HPAM is √ ⎛ 2ε ⎞ 2V 0 c √ ⎜ √2V ε + ε 2V ⎟ c f ⎝ ⎠. √ 2V 2εf + U 0 The eigenvalues can be calculated diagonalizing the above reported 3 × 3 matrix while the eigenvectors can be written in the following way: |Ii = xi f↑† f↓† |0 − yi (f↑† c↓† − f↓† c↑† )|0 + zi c↑† c↓† |0,
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where i = 1, 2, 3 and the nine coefficients {xi , yi , zi } are expressed in the following form: −1/2 2 2 2V 2V + +2 , yi = Ei − 2εf − U Ei − 2εc xi =
2V yi , Ei − 2εf − U
zi =
2V yi , Ei − 2εc
Ei being the eigenvalues of the above matrix. 4.1.5. NTOT = 3 and S = 21 i As for NTOT = 1, the matrix representation of HPAM in this subspace is 2 × 2 and it has the following: 2εc + εf V . V εc + 2εf + U Diagonalizing this matrix Hamiltonian we obtain the following eigenvalues: 3εf + 3εc + U ± 2 , 2 where 2 = (εf − εc + U )2 + 4V 2 , while the corresponding eigenvectors are E1,2 =
† |1 = cos f† c† f−† |0 + sin f† c† c− |0, † |2 = − sin f† c† f−† |0 + cos f† c† c− |0.
Here we have defined
1 εf + U − ε c cos = . − 2 22 As consequence of Sz symmetry, the matrix, the eigenvalues and the eigenvectors above reported refer to Sz = ± 21 . 4.1.6. NTOT = 4 and S = 0 i In this subspace the matrix representation of HPAM is (2εc + 2εf + U ). The eigenstate is f↑† c↑† f↓† c↓† |0 and it has energy equal to 2εc + 2εf + U . 4.2. Uncorrelated limit U = 0 In this case, the total Hamiltonian can be cast in the form of two equal terms that differ only for the spin HPAM = HPAM↓ + HPAM↑ , i.e. there is no correlation between the electrons in the different spin channels and one has simply N independent one-particle two-band systems. The one-particle models can be diagonalized yielding, for the periodic model, the new eigenenergies 1 E±,k = εk + εf ± 4V 2 + (εk − εf )2 , 2
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2
ε (k)
1
0
-1
-2 −π
0 k
π
Fig. 1. Energy bands for uncorrelated case (U = 0) for εf = −1, V = 0.5 and εc = 0. We have assumed for the uncorrelated c-electrons the following energy spectrum ε(k) = −2 cos k in unit of t.
where εk is the Fourier transform of the following quantity: ε(k) = (tij − εc ij )e−i(Ri −Rj )k . ij
Then, we have an excitation spectrum consisting of two bands separated by an hybridization gap. A typical spectrum is shown in Fig. 1 where we have assumed for conduction electrons a nearest-neighbor hopping on a square lattice. The flat regions of the two bands give rise to high peaks in the f-electron density of states, thus indicating that the quasi-particle excitations near the hybridization gap have mainly f character. The Green functions are correspondingly given by Ga =
R1a (k) R2a (k) + , − E+ (k) + − E− (k) +
a = f, c,
where the residues Ria (k) have the form ⎛ f
c R1,2 (k) = R2,1 (k) =
1⎜ ⎝1 ± 2
⎞ ε f − εk
(εf − εk )
2
+ 4V 2
⎟ ⎠.
4.3. Non-mixing limit V = 0 When V =0 HPAM can be exactly solved because there is no coupling between correlated f-electrons and uncorrelated c-electrons. The Hamiltonian for localized f-electrons can be expressed as the sum of identical Hamiltonians for singlesite as previously described for tij = 0 and trivially the energy spectrum can be obtained. There are three configurations for correlated electrons corresponding to different occupation of f-level: (1) zero occupation with an energy E0 = 0; (2) single occupation by a spin with energy E1, = εf ; (3) double occupation with a spin-up and a spin-down electrons with energy E2 = 2εf + U . If the ground state corresponds to single occupation then the state has two-fold degeneracy corresponding to spin one-half. It will have an associated magnetic moment which will give a Curie law contribution to the susceptibility. The other two configurations are non-degenerate and consequently have no magnetic moment. In this atomic limit the condition for a local momentum to exist is that the singly occupied configuration lies lowest, which
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requires that the f-level energy εf has to be lower than the Fermi level, so that it is favourable to add one electron, and εf + U has to be higher than the Fermi level so that it is unfavourable to add a second electron. Obviously, referring to Hc , this Hamiltonian describes a free-electron Hamiltonian and the energy spectrum can be easily obtained when the hopping term tij has been fixed. 4.4. Unconstrained-range hopping(s) We assume now that the itineracy of conduction electrons and their hybridization with localized f-like states may be both realized through unconstrained inter-site processes. In other words, we will assume that the hopping processes of particles have infinite range and occur with equal probability. We notice that this limit was firstly adopted by Van Dongen and Vollhardt [53] for the Hubbard model and subsequently applied to other correlated electron models such as the t − J model [54], the simple [55] and the multichannel Kondo model [56] and the Falicov and Kimball model [57]. Here we discuss the solution of the PAM presented in [58]. We begin considering that the hopping processes for conduction electrons are allowed to take place with equal amplitude, regardless of the pair of lattice sites involved. Then, we write the Hamiltonian in this way HPAM = H1 + H2 , where H1 refer to the hopping term for conduction electrons while H2 describes the local part of the model containing only on-site terms. Assuming a constant value for the hopping matrix term for every pair of lattice sites, we have ε(k) = −tLk,0 , where L denotes the number of lattice sites and ε(k) is the Fourier transform of tij . The key point which leads to the exact solution of the model is the fact that in the thermodynamic limit the non-local part H1 commutes with the local part H2 of the total Hamiltonian. Indeed, if in H2 the contribution from k = 0 is neglected, one introduces an error of the order of 1/L which vanishes when the thermodynamic limit is concerned. In this case H1 and H2 commute and, consequently, the total partition function can be factorized as Z(H ) = Z(H1 )Z(H2 ). Here Z(H1 ) is proportional to exp(2N t), while Z(H2 ) has the form Z(H2 ) =
16
L exp[−(Ei − Ni )]
,
i=1
where is related to the inverse of the temperature through the Boltzmann constant kB =1/kB T , Ei denote the atomic eigenenergies and Ni the corresponding eigenvalue of the total number operator. We notice that the form of Z(H2 ) reflects the fact that H2 is the sum of L identical site Hamiltonian, while the eigenenergies can be readily calculated using the matrices and their eigenvalues introduced in the previous section. From the knowledge of Z one immediately gets the whole thermodynamics of the model through the grand-canonical potential =−
1 1 ln Z = − (ln Z1 + ln Z2 ).
Now, we are in the position to discuss some physical properties of this unconstrained PAM. Defining the double-occupancy average as 1 † † D= fi↑ fi↑ fi↓ fi↓ , L i
we find that, when the density n is lower than 2, the density of double f-occupation D = 0; for n2, D is non-vanishing and U-dependent at zero as well as at finite temperatures, providing that the f-level energy is deep enough. More precisely, in the range 2 n3, we find the transition from D = 1 to 0 occurring around U = −εf , which becomes sharper and sharper as the value of the hybridization is reduced. In the limit of vanishing hybridization, the correlation
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Fig. 2. Double occupancy density of f levels as function of U/|εf | for the total density of electrons equal to 3.5 for different values of V, namely V = 0.0012 (solid line), V = 0.012 (dashed line), V = 0.6 (dotted line), and V = 1.2 (dashed-dotted line).
function D behaves as the step function, i.e. D(U ) = (−εf − U ). A similar situation is found when 3 n4; the only difference is that in the region U −εf the transition occurs from D = 1 to n − 3. The dependence on U of D we find here is strictly related to the fact that the effect of infinite-range hopping on the double-occupancy is indirect; indeed, the f-electrons get their mobility only through the hybridization with c-electrons. As an example in Fig. 2 it is reported the double occupancy of f -level as function of the Coulomb interaction U, normalized to the εf , for different values of the hybridization strength V. In particular, we choose V = 0.0012 (solid line), V = 0.012 (dashed line), V = 0.6 (dotted line), and V = 1.2 (dashed-dotted line). We can observe that for U εf a transition between D = 1 and 0.5 occurs and this transition becomes sharper and sharper as the value of the hybridization V is reduced. We focus now on the conduction properties of the model, looking at the Mattis criterion. Introducing the quantities
+ = EG (N + 1) − EG (N ),
− = EG (N ) − EG (N − 1), where EG (N ) is the ground-state energy for N-particle states, we can say that the system is in a metallic state when ≡ + . − is equal to zero, otherwise it is in an insulating state. In the case here treated, we find that at exactly half-filling, i.e. for n = 2, there is always a kink in the plot of chemical potential versus n, that is is positive and the system is an insulator. On the other hand, away from half-filling a metallic behavior is found. It is also worth pointing out that another limit in which the PAM can be exactly solved is when the hopping and the hybridization are both characterized by constant infinite-range amplitudes, under the assumption that the local hybridization is not allowed to occur. In this case, the eigenvalues of H2 , the local part of the total Hamiltonian, assume the very simple structure f
Ei = Ni ε − Ni + UTi , f
where Ni is the number of f-electrons in the eigenstate |i, Ni is the total number eigenvalue and Ti is non-vanishing and equal to one if |i contains two f-electrons.
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Moreover, the non-local Hamiltonian H1 takes the form † [tc0 c0 + V (c0† f0 + f0† c0 )]. H1 = −L
From its diagonalization one gets a two-level energy spectrum given by 1,2 = − 21 L t ± t 2 + 4V 2 . 4.5. Discussion The exact solutions presented above refer to non-interesting physical situations, even if they contain the qualitative features of the complete Anderson model. The relevance of these solutions mainly resides on the fact that they can be considered as the zeroth-order of suitable perturbative scheme. Indeed, starting from the exact solutions 4.1–4.3, one may introduce perturbation expansions in terms of tij , V and U, respectively. For instance, the perturbative expansion that considers as small parameter the hopping amplitude may result useful to analyze systems having reduced conduction band: the zeroth-order corresponding to the limit of vanishing conduction band [59–62]. When the correlated and uncorrelated electrons weekly hybridize among them, one can start from a perturbation in terms of V. Usually, the value of the hybridization strength is of the order of 10−2 eV when, apart from cerium, rare-earth impurities are considered. For completeness, we notice that usually U is about 10 eV. In both these cases, the perturbative approaches one has to resort, are non-standard. Indeed, due to the presence in the “free” Hamiltonian H0 of the correlated four-field term proportional to U, the Wick conventional theorem does not hold. The interested reader can give a look to the many interesting reviews on this subject [34,35]. The standard perturbative approach, i.e. the Wick theorem and Feynman graphs, can be applied when the U term is treated as a perturbation [25–28,63,64]. The only problem related to this approach is that relevant physical systems are usually characterized by non-small Coulomb repulsion U. As final consideration, it is worth pointing out that, despite the absence of itinerancy effects, the atomic limit 4.1 retains some essential features of the complete model, especially in the description of the thermodynamical properties related to the behavior of the quasi-localized f-electrons. Indeed, this limit captures some specific features of the PAM coming out from a simultaneous non-perturbative treatment of the hybridization coupling and the on-site repulsion between f-electrons. This is, for instance, made evident by the temperature dependence of the specific heat and the static magnetic susceptibility, which in the atomic limit are easily evaluated from the expressions CV = kB 2
j2 j2
ln Z,
2B j2 ln Z, lim h→0 jh2 where Z = i exp(−Ei ) is the partition function and B is the Bohr magneton. We notice that the evaluation of the magnetic susceptibility requires the introduction of an external magnetic field h which renormalizes the f-electron site energy only, i.e. εf −→εf + B h, where = ± according to the sign of the Sz . Indeed, the field h is assumed to be effective on f-electrons, since there is experimental evidence that, in the compounds usually described in terms of the PAM, the conduction electrons exhibit a paramagnetic behavior [65]. As concrete example, in Fig. 3 it is reported the temperature behavior of the specific heat for different values of the hybridization V, for U = 3 and εf = −1. The curves plotted refer to V = 0.15 (solid line), V = 0.1 (dashed line) and V = 0.05 (dotted line). For the same choice of the parameters, in Fig. 4 it is plotted the temperature behavior of the magnetic susceptibility. From these curves, we infer that the specific heat exhibits a Schottky-like peak whose height is not affected by variations of the hybridization coupling V and the value of the temperature where the peak appears increases with increasing V. It is worth noticing that the temperature of the Schottky peak in the specific heat and the temperature where the susceptibility has the maximum are the same for any value of V. Therefore, we may consider this temperature as the “Kondo temperature” of the model. Nevertheless, this identification is made essentially on a qualitative ground being absent in this approximation a conduction band of finite width. As consequence, this =
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Fig. 3. Temperature behavior of the specific heat for the Anderson model in the atomic limit. The model parameters are U = 3 and εf = −1. The curves plotted refer to V = 0.15 (solid line), V = 0.1 (dashed line) and V = 0.05 (dotted line).
Fig. 4. Temperature behavior of the magnetic susceptibility for the Anderson model in the atomic limit, in 2B units. The model parameters are the same as in Fig. 3.
“Kondo temperature” increases with increasing V with some square root dependence rather than with the more realistic exponential behavior [36]. As a final consideration, referring to the case of unconstrained hopping cases, we notice that the assumption of infinite-range hopping processes may, of course, look rather unrealistic, specially if one consider that in a tight-binding picture, electrons are usually allowed to hop and hybridize with constant hopping matrix elements mainly between nearest-neighbor sites, being the other hopping processes negligible. Nevertheless, the exact solution of the model presented above in the thermodynamic limit and at finite temperatures, leads to some physically interesting results concerning the conduction property of the model. We have indeed shown significative results about the behavior of the double f-occupation density and we have explicitly demonstrated the existence, at any temperature and at exact half-filling, of a metal–insulator phase transition.
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5. Absence of long-range orders in low dimensions: finite and zero temperature In this section, we discuss the conditions that favor the possibility to have long-range-ordered states for the PAM. To this end, we firstly introduce the framework where this problem may be treated. We refer to the statistical mechanics and specifically phase transitions within this frame. We would like to point out that by means of the procedure below described and applied to the PAM, some exact results are obtained on the existence, or non-existence, of long-range orders. It is well-known, and is one of the main issues in the field of statistical mechanics, that the search for criteria for the existence of phase transitions in physical systems is a crucial question. This issue can usually be handled by the identification of the order parameter which vanishes on one side of the transition, but takes on a finite value on the other side. In a continuous phase transition, this quantity may gradually evolve from zero, at the critical point, to a finite value on one side of the transition. Usually this happens on the low-temperature side of the transition. Of course, for different kinds of phases, different order parameters has to be chosen and studied. The occurrence of a phase transition is often related to the failure of one of the phases to exhibit a certain symmetry property of the underlying Hamiltonians. In this respect, Bogoliubov introduced a method based on some inequalities formulated to describe the occurrence of spontaneous symmetry-breaking in terms of quasi-averages [66–69]. We want to stress that, the Bogoliubov inequality is a rigorous relation between two arbitrary operators and a Hamiltonian of a physical system. Hohenberg [70] was the first to notice that the Bogoliubov inequality can be used to rule out the possibility of a phase transition, specifically showing that, there is no finite-temperature phase transition in one- and two-dimensional superfluid systems. Then, many other interesting results have been obtained using the Bogoliubov inequality in several contests, such as spin models or strongly correlated electron Hamiltonians. So, referring to spin systems only, Mermin and Wagner [71] considered the case of spontaneous magnetization in the Heisenberg model excluding magnetic ordering at finite temperatures in dimensions less than three. Using the same approach, Walker ruled out the possibility of phase transition, in one and two dimensions in an excitonic insulating state [72]. Referring to models for strongly correlated electron systems, this procedure has been applied by Ghosh to the Hubbard model [73] and by Van den Bergh and Vertogen [74] and Robaszkiewicz and Micnas [75] to the s–d interaction models. The Bogoliubov inequality procedure has been applied by Uhrig and Su to generalized multiband Hubbard models [76]. A large class of correlated electron models has been discussed by Proetto and Lopez who confirmed the Mermin–Wagner result for the Heisenberg model for the Anderson lattice model and Kondo Hamiltonian [77]. The Anderson model has been also studied by Noce and Cuoco who rederived the Bogoliubov inequality for the magnetic phase as well as for different types of the superconducting order parameters including the d-wave symmetry [78]. This proof follows a similar one previously obtained by Su and Suzuki for the Hubbard model to exclude in two dimensions a dx 2 −y 2 wave pairing [79]. Finally, we would like to mention a recent paper by Gelfert and Nolting which includes a description of existing proofs of the absence of finite-temperature phase transitions in low-dimensional systems [43]. They also present new proofs for the absence of long-range order in Hubbard films, and of magnetic long-range order in the PAM films. Here we report the results available in one and in two dimensions for the PAM. In particular, we rigorously show that, the Bogoliubov inequality prevents, within the PAM, spontaneous ferromagnetic or antiferromagnetic ordering, as well as, long-range order for different kinds of pairing, s-wave and d-wave Cooper pairing and generalized -pairing. These results hold at arbitrary non-zero temperatures and for both repulsive and attractive Coulomb interaction between localized electrons. For completeness, we repeat here the proof to obtain the Bogoliubov inequality in the form given by Gelfert [43,44]. 5.1. Proof of the Bogoliubov’s inequality at finite temperature The Bogoliubov inequality is a rigorous relationship between two arbitrary operators A and B and a specific Hamiltonian H of a relevant physical system. In its original form proposed in [66], this inequality is given by | [B, A]|2 {A, A+ } [B + , [H, B]], 2
(19)
where X denotes the quasi-average of the operator X or the ensemble average in a restricted ensemble appropriate for the system under study having a broken symmetry. We would like to mention that, the operators A and B do not
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necessarily have a physical interpretation from the very beginning, so that the physical significance of Eq. (19) will depend on the choice for the operators involved. Indeed, the relevant quantity is the order parameter we would like to study, and this quantity has to appear in Eq. (19). This condition is the real constraint implying a suitable choice for A and B operators. In the above expression, [X, Y ] and {X, Y } denote the commutator and the anticommutator of the operators X and Y, respectively. The inequality can be proven by introducing a suitable “scalar” product which is based on the energy eigenvalues En and the corresponding set of normalized, and mutually orthogonal energy eigenstates |n of the Hamiltonian H verifying the completeness relation: I=
|n n|.
(20)
n
Introducing the statistical weight wn =
exp(−En ) , tr(exp(−H ))
(21)
we can define the Bogoliubov scalar product A, B of two arbitrary operators A and B as
A, B ≡
n|A+ |m m|B|n
n,mE n =Em
w m − wn , En − E m
(22)
where the sum is restricted to states |n having En = Em . In order to verify that A, B has indeed the properties one expects from the mathematical definition of a scalar product, we have to check if the following axioms are fulfilled: • A, B is a complex number with A, B = ( B, A)∗ ; • A, B is linear with respect to its arguments, i.e. A, 1 B1 + 2 B2 = 1 A, B1 + 2 A, B2 ; • A, B induces a positive semi-definite norm in the space of operators, i.e.
A, A0;
(23)
• A, A = 0 does not, in general, imply A = 0. The first axiom follows from the definition of the Bogoliubov scalar product by noticing that (wm − wn )/(En − Em ) is a real number and ( n|B + |m m|A|n)∗ = n|A+ |m m|B|n. The second axiom follows from the meaning of the matrix elements m|B|n; by using the first axiom, one derive an analogous relation for the first argument of A, B. The third axiom is a consequence of the relationship: (wm − wn )/(En − Em ) 0. Finally, referring to the last axiom we notice that for the Hamiltonian, we have H, H > 0, while H = 0 has no physically meaning. Since A, B is positive semi-definite, the Schwarz inequality holds: | A, B|2 A, A B, B. Now, if we choose the operator B in the following: B = [C + , H ],
(24)
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i.e. B is the commutator of the Hermitean conjugate of C operator with the model Hamiltonian H, the mixed product
A, B on the left-handed side of Eq. (24) is wm − wn
n|A+ |m m|[C + , H ]− |n
A, B = En − E m n,mE n =Em
=
n|A+ |m m|C + |n(wm − wn )
n,m
=
wm m|C + A+ |m −
m
wn n|A+ C + |n
n
= [C + , A+ ],
(25)
where use of the completeness relation and the definition of wn has been made. The norm B, B = B2 0 can be calculated from Eq. (25) by keeping in mind that B = [C + , H ]; the final result is
B, B = [C + , [H, C]]0.
(26)
For the remaining norm A, A = A2 , we start from Eq. (25) and note that the ratio (wm − wn )/(En − Em ) > 0 (Em = En ) can be bounded 1 (e−Em + e−En ) (e−Em − e−En ) wm − wn = En − E m tr[exp(−H )] (En − Em ) (e−Em + e−En ) (wm + wn ) (En − Em ) < (wm + wn ). tanh = En − E m 2 2
(27)
Here, the condition Em = En has again been used. It is also straightforward to show that the norm A2 is bounded
A, A <
2
n|A+ |m m|A|n(wm + wm )
n,mE n =Em
n|A+ |m m|A|n(wm + wm ) 2 n,m
=
wm m|{A, A+ }|m = {A, A+ }. 2 m 2
(28)
Using these results, we may easily derive the Bogoliubov inequality: making use of Eqs. (25)–(28), the Schwarz inequality equation (24) assumes the form of Eq. (19). Moreover, if the quantities entering Eq. (19) depend on a wave vector k, one can rewrite the Bogoliubov inequality in the following [43]: k
| [C, A]|2
{A, A+ }(k), +
[[C, H ], C ](k) 2
(29)
k
where k is within the first Brillouin zone in the reciprocal lattice. In the following, we will use this last form Eq. (29) of the Bogoliubov inequality. Before to apply the Bogoliubov inequality equation (29), we point out that, within this approach, when one is investigating a degenerate state, one should first remove the degeneracy and then study the so-called quasi-averages involved. We want to stress that degeneracy may be removed by introducing a symmetry-breaking term into the Hamiltonian under study.
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Therefore, in order to write down the Bogoliubov inequality for a specified many-body Hamiltonian and to prove that in the thermodynamic limit the order parameter vanishes, at least in the low-dimensional situations, one has to properly choose the A and B operators in such a way that they can reproduce the order parameter related to the long-range order under study. In the next subsections, we apply the Bogoliubov inequality to the PAM to rule out the possibility of long-range order of magnetic type, ferromagnetic or antiferromagnetic, and of superconducting nature. In this last case, we will also show that the model does not exhibit s-wave and d-wave pairing, as well as generalized -pairing. 5.2. Application of Bogoliubov’s inequality to HPAM Let us first introduce the different symmetry-breaking terms relevant for the long-range orders we will investigate. If we want to analyze the existence, or non-existence, of magnetic order we add to the Hamiltonian HPAM the following symmetry-breaking term: Hm = −h Slz e−iqRl l
for superconducting long-range order solutions of s-wave type or generalized -pairing, we add the U (1) symmetry breaking term Hs = − l e−iqRl , l
while, to investigate the dx 2 .y 2 -wave pairing long-range order, in two dimensions, we add the term Hd = −
− g()(+ i + i ). 2 l,
Here h, and are the amplitude of the symmetry-breaking fields, Siz is the total third component of the local † † † † spin operator, i is the total local on-site pairing raising operator, whose components are + i = ci↑ ci↓ + fi↑ fi↓ and † † † † + † + − + − i = (i ) ; i (i ) is the total non-local on-site pairing raising (lowering) operator, i = ci↑ ci+↓ + fi↑ fi+↓ ,
† ((+ i ) ). We notice that these pairing operators i and i satisfy the usual SU(2) Lie algebra. Finally, g() = 1(−1) ˆ ( = ±jˆa), where a is the lattice constant, and iˆ and jˆare the unit vectors along the x and y direction, for = ±ia respectively. For completeness, we give attention to the fact that the generalized pseudospin operator has been introduced by Su and Suzuki [79] in order to recover the usual SU(2) algebra for dx 2 .y 2 -wave pairing operators. Indeed, considering that g() = 0, ±1, the following quantity i, g()+ i can be decomposed, for any site i, in four terms each of which obeys the usual SU(2) symmetry. Before going into details, we summarize some useful commutation relations in Fourier space. To this end, we define the Fourier transform of a vectorial operator Ai in the usual way, i.e. A(k) = Ai exp(ikRi ), i
Ai =
1 A(k) exp(−ikRi ). L k
We easily verify that the Fourier transform of the spin operators satisfy the commutation relations [S ± (k), S z (k )] = ∓S ± (k + k ), [S + (k), S − (k )] = 2S z (k + k ) and one also has (S + (k))† = S − (−k), with L denoting the number of lattice sites.
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Of course, the local on-site pairing operators are defined as [± (k), z (k )] = ∓± (k + k ), [+ (k), − (k )] = 2z (k + k ) and similarly for non-local pairing operators [± (k), z (k )] = ∓± (k + k ), [(k), − (k )] = 2z (k + k ). Let us discuss separately the three cases above introduced. 5.2.1. Magnetic long-range orders The Hamiltonian related to this situation is H = HPAM + Hm . For our purpose we choose A = S − (−k − p) and B = S + (k). With this choice of the operators A and B, we are able to reproduce the magnetic order parameter, i.e. the magnetic moment S z . Moreover, after some tedious but straightforward algebra, we deduce the following inequality for the commutator containing the Hamiltonian | [[B, H ], B † ]| = t (Ri − Rj )(1 − exp(ip(Ri − Rj ))) exp(ik(Ri − Rj ))(nck↑ − nck↓ ) + 2hS z (−p) . i,j k Keeping in mind the following fact: (a) | k (nck↑ − nck↓ )| Nc , Nc being the total number of uncorrelated electrons, (b) |1 − cos(x)| < x 2 /2, (c) the translational invariance of the model, and (d) using the triangle inequality, we can write | [[B, H ], B † ]|
|t (Ri )|
i
Ri2 p 2 Nc + 2h| S z (−p)|. 2
Evaluating the other quantities appearing in the Bogoliubov’s inequality, we finally have −1 1 Nc m(p) , 8 Dk 2 + (2hN /Nc )m(p) 2
k
where m(p) = | S z (−p)|/L. Replacing the sum over k by the integral over the first Brillouin zone, we obtain for small h and in the thermodynamic limit m(p)
const |h| T
in one dimension, and 1 const , m(p) √ √ ln(|h|) T in two dimensions. From these expressions, it immediately follows that when h → 0, m(p) → 0 implying that several types of magnetic order are excluded. Moreover, for suitable choices of the wave vector p, we can show that ferromagnetic or antiferromagnetic orderings are excluded. Indeed, when p=0 the ferromagnetic ordering is forbidden; if we choose p in such a way that exp(ipRi ) = ±1 when Ri connects sites belonging to the same sublattice or different ones, respectively, we can conclude that the antiferromagnetic order is impossible either.
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5.2.2. Superconducting pairing of s-wave type or generalized pairing The Hamiltonian related to this situation is H = HPAM + Hs . In this case, in order to reproduce the superconducting order parameter, we apply the Bogoliubov’s inequality choosing A = + (−k − p) and B = z (k). Using the same method previously used, we deduce the following: | [[B, H ], B † ]| c c + − = t (Ri − Rj )(1 − exp(ipRi )) exp(ikRi )(nk↑ − nk↓ ) + ( (−p) + (p)) . k
i
Recalling the assumption (a)–(d) made for magnetic long-range orders one can write | [[B, H ], B † ]|
i
|t (Ri )|
Ri2 p 2 Nc + | + (−p) + − (p)|. 2
Since (+ (k))† = − (−k) and introducing the following quantity F (p) = + (−p)/L, corresponding to the per site pairing order parameter, we have −1 1 Nc |F(p)| . 2 Dk 2 + (2N/Nc )|F(p)| 2
k
Therefore, we have derived an inequality having the same form as the one deduced for the magnetization and thus, similar considerations can be made. In particular, for p = 0, F(0) defines the s-wave pairing order parameter, for p = ± , F(±) corresponds to the -pairing order parameter, and finally for p = 0,±, F(p) stands for generalized -pairing with momentum equal to p. In conclusion, in one- and two-dimensional PAM generalized pairing long-range orders will not appear at any temperature. 5.2.3. Superconducting pairing of dx 2 .y 2 -wave type Considering the remark made at the end of previous section, we can apply the procedure previously outlined for magnetic and superconducting long-range orders also to this case. Of course we has to refer to the following Hamiltonian: H = HPAM + Hd . Then, it is straightforward to deduce an inequality similar to the one previously derived, i.e. −1 1 Nc |G(p)| , 2 Dk 2 + (2N/Nc )|G(p)| 2
k
1
where G(p) = 2 i, g()+ i /L and k = (kx , ky ) is a two-dimensional wave vector belonging to the first Brillouin zone. Applying the same considerations as in the previous subsection, we conclude that the two-dimensional PAM does not exhibits dx 2 −y 2 pairing at any non-zero temperature. We notice that using the same argument, we also exclude the ˆ possibility of extended s-wave with pairing operator proportional to l, w()+ i , where w() = 1 for = ±ia and = ±jˆa and also this result holds in one and two dimensions. 5.3. Discussion In this section, we discuss some properties of the inequalities above derived as well as some limits of the Bogoliubov procedure. First of all, we want to stress that the quantity D is a well-defined function because the overlap integral between Wannier functions decreases rapidly with distance. Indeed, when t (Ri − Rj ) = −t for i and j nearest-neighbors, D = tz, z being the coordination number while in the case of next nearest-neighbors hopping integral t , D = tz + lt z , where l = 4 (2) in one (two) dimension and z is the number of next nearest-neighbor sites.
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Moreover, we point out that the pairing order parameter F is the sum of order parameters for pairing of uncorrelated electrons and of localized ones. As a consequence of the hybridization term that couples the two subsystems of electrons and forces the two types of electrons to have the same critical temperature [80], when F=0, we can state that, separately, the two order parameters vanish. In other words, we can rule out the existence of superconductivity both for conduction and correlated electrons. For completeness, we notice that Koma and Tasaki [81], using a different procedure from the Bogoliubov technique here outlined and applied, ruled out the possibility of superconducting electron pairing and of magnetic ordering, in one and two dimensions, for a large class of correlated electron models. In their approach, one calculates some two-point correlation functions and proves upper bounds for these functions. Of course, if these bounds decay exponentially or with power law, one rigorously rules out the possibility of a transition in an ordered state. The proof is based on a method previously developed for classical spin systems by McBryan and Spencer [82], and extended to quantum spin systems by Ito [83]. In these papers, the global continuous symmetry of the spin space plays an essential role. The class of correlated models Koma and Tasaki refer to corresponds to tight-binding models on one-dimensional lattice or planar lattice with: (1) hopping matrix terms txy vanishing when |x − y| exceeds a finite constant R, and less of another constant t when R is greater than |x − y|; (2) interaction terms being arbitrary functions of the number operators. Therefore, the class of Hamiltonians may include for instance the Hubbard model, the PAM, the t − J model and also models containing long-range, random or spin-dependent interactions. Nevertheless, we think that the presentation of the impossibility of long-range orders based on the Bogoliubov inequality is simpler than Koma–Tasaki method, even if, within the Bogoliubov procedure one has to choose suitable A and B operators related to the long-range order analyzed as well as to the model Hamiltonian under study. Finally, it must be stressed that other than two-point, long-range correlations are not excluded by the method here used. For instance, in the isotropic Heisenberg model the four-site spin–spin correlation decays exponentially with increasing distance but a more complicate correlation displays long-range behavior [84]. Another example is the topological order as it occurs in the XY model, due to Kosterlitz–Thouless transition. This kind of order is not ruled out by an extension of the arguments presented here to the XY model. It is possible that, below a finite temperature transition, the susceptibility diverges without the occurrence of a spontaneous magnetization because the correlations decay according to a power law [85]. 5.4. T = 0 Bogoliubov’s inequality Let us discuss now how one can extend to zero temperature the Bogoliubov’s inequality. As previously shown, the breaking of continuous symmetries at finite temperatures is ruled out for one- and twodimensional systems by the Mermin–Wagner theorem, under the assumption that the interaction falls off sufficiently rapidly with increasing the inter-particle distance [71]. We notice that the Mermin–Wagner theorem follows from the fact that in low-dimensional cases, a diverging number of infinitesimally low-lying excitations is created at any finite temperature, and thus the assumption of a non-vanishing order parameter is not self-consistent. Conversely, we can state that, in low dimensions, thermal and/or quantum fluctuations prevent long-range order. This consideration does not apply at T =0, implying that the ground state may be ordered. As an example we mention the two-dimensional ferro(antiferro)magnetism which can be realized also at zero temperature. In this case, quantum fluctuations oppose but do not prevent the appearance of a two-dimensional magnetically ordered phase. In contrast, for one-dimensional systems quantum fluctuations tend to become so strong that they usually prevent even ground-state ordering. Moreover, it is known that the ground state of the one-dimensional Hubbard model is a non-magnetic singlet at any band filling and for any value of Coulomb interaction U. This includes the large U limit at half-filling where the model becomes equivalent to the isotropic spin one-half-antiferromagnetic Heisenberg chain, which in turn cannot order either. However, we would like to emphasize that this consideration holds only if the order parameter is fluctuating, i.e. it does not commute with the Hamiltonian. The order parameter for ferromagnetism commutes with the Hubbard or Heisenberg Hamiltonian, and thus there is nothing to prevent a one-dimensional ferromagnet from becoming ordered at T = 0. As further example we refer to the fully polarized state that is an eigenstate of the Heisenberg model. If the coupling exchange is ferromagnetic, this state is the ground state of the model. Summarizing, when the T = 0 case is considered, great care and attention has to be used. To this end, it is worth noticing that one can extend the Mermin–Wagner theorem to zero temperature and then derive some exact results in one and two dimensions for correlated electron models. In other words, it is possible to deduce
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a T = 0 Bogoliubov-like inequality. For completeness, we notice that this kind of approach has been previously applied by Auerbach to exclude magnetic order in the Heisenberg model in one dimension [86] and by Su et al. to the Hubbard model [87]. To write down the T = 0-type Bogoliubov’s inequality we follow the following procedure. Let us assume that at T = 0 in the presence of an infinitesimal ordering field there is a unique ground state |0, i.e. we assume that the ground state is not degenerate, we comment on this assumption later. The expectation of a generic operator A is written in the usual way:
A = 0|A|0. Now, the T = 0 scalar product between any two operators A and B is defined as follows: (A, B) =
0|A† |m m|B|0 + 0|B|m m|A† |0 , Em − E 0
(30)
m=0
where |m is an eigenstate of the Hamiltonian with energy Em . It is straightforward to check that the quantity (A, B) verifies the axioms of a true scalar product, so that the Schwarz inequality holds: |(A, B)|2 (A, A)(B, B).
(31)
Now we choose the operator B as the commutator between a generic operator C and the model Hamiltonian H in the following: B = [C † , H ]. Substituting this expression for B in the scalar product we have (A, B) =
0|A† |m m|[C † , H ]|0 + 0|[C † , H ]|m m|A† |0 . Em − E 0
(32)
m=0
Keeping in mind that {|m} are eigenstate of H we have (A, B) =
(E0 − Em ) 0|A† |m m|C † |0 + (Em − E0 ) 0|C † |m m|A† |0 , Em − E 0
(33)
m=0
so that, after a simple simplification, we deduce (A, B) =
0|C † |m m|A† |0 − 0|A† |m m|C † |0.
(34)
m=0
Now, if we extend the summation over all the states, i.e. we also include the vacuum state |0 and subtract this contribution, we get (A, B) = 0|[C † , A† ]|0,
(35)
where we have used the spectral decomposition of the unity operator I= |m m|. m
Moreover, the “norm” of B is (B, B) = [C † , [H, C]].
(36)
Therefore, the T = 0 Bogoliubov’s inequality reads as | [C † , A† ]|2 (A, A) [C † , [H, C]].
(37)
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This inequality can be cast in a slight different way, more appropriate for the discussion of absence of long-range orders at zero temperature. Namely, if there is an excitation gap in the energy spectrum {Em }, denoting with the difference between the ground-state energy and the lowest excited energy we have 1 1 ( {A† , A} − 2| A† |2 ) {A† , A}. Therefore the inequality in Eq. (37) assumes the final form: (A, A)
(38)
1 (39)
{A† , A} [C † , [H, C]]. To look at spontaneous ordering at T = 0, following the procedure adopted at finite temperature, one should first remove the degeneracy of the model and then study the expectation values involved, the degeneracy being removed by introducing symmetry-breaking terms into the Hamiltonian. | [C † , A† ]|2
5.4.1. Magnetic long-range orders at T = 0 To look at magnetic long-range order, we apply Eq. (39) choosing A = S − (−k + p) and C = S + (k). After some straightforward algebra, we deduce the following inequality for the commutator containing the Hamiltonian: | [C † , [H, C]]| † t (Ri − Rj ){1 − exp[ip(Ri − Rj )]}(ci↑ cj ↓ − cj†↓ ci↓ ) + 2hS z (−p) . ij
(40)
Using the conditions used in previous subsections, we can write
| [C † , [H, C]]|
|t (Ri )|
i
Ri2 k 2 Nc + 2h| S z (−p)|. 2
(41)
Again, the quantity ≡ i |t (Ri )|Ri2 /2 is well defined since the matrix elements t (Ri ) become rapidly negligible with distance. Summing both sides of Eq. (41) over k and considering that
{S − (−k + p), S + (k + p)} = L
{Si− , Si+ } L2 , k
i
we finally have m(p)2 L
k
1 Nc k 2 + 2hLm(p)
−1 ,
(42)
where we have introduced the following quantity m(p) = | S z (−p)|/N . When we replace the sum by the integral over the first Brillouin zone in the above inequality and take the thermodynamic limit, we obtain for small h m(p)const |h|, in one dimension, and √ 1 m(p) const √ , ln |h|
(43)
(44)
in two dimensions. Both in one and two dimensions, from Eqs. (43)–(44) it follows that the magnetization m(p) goes to zero when the applied magnetic field h vanishes. Therefore, like for T = 0 case, ferro- and antiferromagnetic long-range orders are excluded. We want to stress that this conclusion holds assuming that is not vanishing, i.e. that there is a gap in the excitation energy spectrum [88].
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5.4.2. Superconducting pairing of generalized type pairing at T = 0 Looking at superconducting pairing, as previously done for magnetic orders, we properly choose A and C operators. In particular, A = + (−k − p) and C = z (k); we obtain the following: −1 1 L . (45) | (p)|2 2 Nc k 2 + 2L| (p)| k
Analogously, looking for a d-wave superconducting solution of dx 2 .y 2 type, we derive the inequality: −1 1 L |d (p)| . 2 Nc k 2 + 2L|d (p)| 2
(46)
k
In previous equations, (p) = + (−p)/N and d (p) = i, g()+ i (−p)/2N . We notice that in Eq. (46), k and p are two-dimensional wave vectors. The inequalities above reported have the same form as the one derived for the magnetic case and therefore, similar considerations can be made. In conclusion, in one and two dimensions pairing long-range orders do not appear at T = 0, as well as the two-dimensional d-wave pairing for the PAM [88]. 5.5. Discussion We have discussed the T = 0 version of the Bogoliubov’s inequality looking at the long-range magnetic and superconducting orders for the PAM. We have shown that, if an excitation gap opens in low-dimensional PAM, then long-range orders of magnetic or superconducting type will not appear at zero temperature. Interestingly, this energy gap in Eqs. (43)–(46) plays the role of temperature in conventional Bogoliubov’s inequality. We want to discuss now if there is a gap in the PAM, in such a way that previous inequalities prevent the existence of long-range magnetic and superconducting orders. It is widely accepted that low-dimensional Anderson model at half-filling might have an energy gap [89,90]. This assumption is based on the consideration that, at half-filling, the energy band of the conduction electrons is symmetric for bipartite lattices. With the inclusion of the mixing term V, a hybridization gap opens at zero energy. Therefore, the ground state for U = 0 is the state where the lower hybridized band is fully occupied, leaving the upper hybridized band empty and this is a unique ground state. Hence, there is a finite energy gap to the excited states; then, from the continuity with respect to U, the unique ground state for U > 0 has still no level crossing. However, we point out that this conclusion is based on a naively ground and indeed, we now show that it is not completely correct. It is well-known that the PAM at half-filling has been used as a theoretical model for the Kondo insulator at least in one dimension being, in the strong coupling limit, the PAM mapped into the Kondo lattice model. In one dimension this last model, with a half-filled conduction band, has always a finite spin excitation gap and the charge gap is larger than the spin gap; therefore, the ground state is an incompressible spin liquid for any non-vanishing exchange coupling [91] and this conclusion implies that this state is a quantum-disordered phase. As a consequence of this limit procedure, it is natural to expect that the PAM at half-filling also shows a ground state which is a spin liquid state for any interaction U. Summarizing, the results presented above and the speculations here made suggest that, at least in one dimension, the PAM at zero temperature is in a quantum-disordered state. This conclusion is further corroborated by numerical results obtained through exact diagonalization method and density matrix formulation. Indeed, they show that an excitation gap exists for any finite value of U [52]. The situation is different in two dimensions. Numerical data on the two-dimensional PAM, at half-filling obtained from quantum Monte Carlo simulations, show that there is a regime change from a spin liquid phase to an antiferromagnetically ordered phase [92] and this result agrees with the one obtained in two dimensions quantum Monte Carlo simulations on the related Kondo lattice model [93]. Thus, we can infer that the spin liquid phase becomes unstable against antiferromagnetic long-range order. Therefore, although proven numerically and not rigorously, the two-dimensional PAM exhibits an ordered phase at zero temperature and this in turn suggests that there are gapless magnetic excitations. Then, we can summarize this discussion saying that the PAM exhibits a disorder phase at T = 0 in one dimension while, as indicated by numerical results, a magnetically ordered phase is sit in two-dimensional case [88].
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As final consideration, we come back to the assumption of non-degenerate ground state. If the ground state is degenerate, according to a method devised by Su et al. [87], it is possible to define a new two-point function (A, B). It can be shown that this new correlation function verifies all the properties of a true scalar product. Thus, the inequality equation (39) still holds and one can state that, when there is an excitation gap in the spectrum of energy for the model Hamiltonian under study, at T = 0 there is no long-range order. 6. Spin-reflection positivity method applied to PAM The purpose of the method of reflection positivity in the spin space is to derive an inequality for the ground-state energy by means of the Schwarz inequality. This method was introduced firstly in quantum field theory by Osterwalder and Schrader [94], and then was applied to relevant models of statistical physics by Fröhlich and coworkers [95–98]. In a seminal paper, Lieb applied the reflection positivity in spin space to investigate the Hubbard model showing that the ground state is non-degenerate and calculating its total spin when the lattice is bipartite and at half-filling [99]. Subsequently, a generalized formalism was developed and applied to a series of many theoretical models for strongly correlated electron systems. We refer to the Heisenberg model, the PAM [89], the single-channel Kondo model [90], generalized Hubbard models [100] and other strongly correlated electron models [101–103]. Before going into details, let us summarize how this method works following the discussion reported in [41]. Let us start with a many-body Hamiltonian defined on a lattice and a set of all relevant operator of the system. Let us assume that this set is an algebra with the operations of addition, complex scalar multiplication and operator product. In several cases, this algebra can be written as the direct product of two subalgebras and an isomorphism between these subalgebras can be also introduced. For instance, the relevant algebra for strongly correlated electron models is the family of polynomials in all the fermion operators. Moreover, the two subalgebras can be chosen as the subset of polynomials in up operators and down operators, respectively. The isomorphism between these two sets can be introduced assuming that this one-to-one mapping changes spin-up operators in spin-down operators and viceversa. We call this map spin reflection mapping. Now, we consider an arbitrary Hamiltonian H, i.e. an Hermitean polynomial, and we define the following linear functional: H = Z −1 Tr(O exp(−H )),
O is a generic operator belonging to the algebra and Z the partition function. where O Referring to this functional, we call H reflection positive if and only if O)0
O( is the complex conjugate of X. for all the operators in the algebra. Here X We notice that a sufficient condition for an Hamiltonian to be reflection positive is contained in the following theorem: Theorem 1. If H belonging to the algebra can be cast in the form ⊗ I1 + I2 ⊗ (A) − H =A
N
i ⊗ (B i ), B
i=1
and B i belonging to one of the two subalgebras, then H is reflection positive. with A In the previous expression H is the Hamiltonian operator of the model and I1 and I2 denote the identity operator acting in the two subalgebras. A companion theorem has been demonstrated by Dyson–Lieb–Simon and it affirms that [96]: Theorem 2. If H assumes the form ⊗ I1 + I2 ⊗ C − H =A
N i=1
i ⊗ (B i ), B
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and C are elements of subalgebra 1 and subalgebra 2, respectively, then the ground-state energy E0 (A, C) where A satisfies the inequality C) 1 E0 (A, (A)) + 1 E0 ((C), C). E0 (A, 2 2 In the following, we will prove the validity of the Dyson–Lieb–Simon Theorem 2 considering the concrete example of the PAM. To this end, we introduce in a lattice a basis set; we denote by || the finite number of lattice sites. We consider the subspace Sz = 0 for an even number of electrons and we denote a set of basis states as X ≡ { }∈I which is an orthonormal basis set composed solely of spin- electrons. Here denotes a set of indices. The basis vectors can be ↑ ↓ written as a superposition of spin-up and spin-down basis states: { ≡ ⊗ }. Therefore, a generic wave function in this subspace can be written as a linear combination of : ↑ ↓ = C ⊗ . (47)
The coefficients C define a matrix that can be considered as a map from X ↑ to X ↓ , and these vector spaces have the same finite dimension. acting on X space. We will assume that O can be expressed as the Let us introduce now a generic operator O acts product of an even number of field operators. With these definitions it is straightforward to calculate how O on : ↑ = ↑ = ↑ | O C O C | |O
=
=
=
↑
↑
↑ | C |O ↑ ) C (O
↑ C) , (O
(48)
↑ ) = ↑ |O ↑ |↑ denote the matrix elements of O ↑ . where (O In a similar way we obtain ↓ = ↓t ) . O (C O
(49)
↓t O
↓ . Here indicates the transposed matrix of O Therefore, it is easy to show that ↑ O ↓ = ↑ ) C (O ↓t ) O (O
=
↑ C O ↓t ) . (O
(50)
For sake of simplicity, we will assume in the following that state vectors are expressed in terms of real coefficients are real; this in turn implies that O † = O t . so that O Let us assume now that the model Hamiltonian is given by ↑ ↓ H = K↑ + K↓ + V V , (51)
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where K and V are field operators for spin only. The term represents the interaction part of the Hamiltonian and define the strength of the interaction between electrons with different spins. According to previous results, the Schrödinger equation H = E gives ↑ ↓† K ↑ C + CK ↓† + V CV = EC, (52)
where we have assumed that K and V operators are real symmetric. The Hermitean conjugate of this equation is ↓ ↑† C † K ↑† + K ↓ C † + V C † V = EC † .
(53)
Thus, if up–down spin symmetry holds for basis vectors, C and C † verify the same Schrödinger equation and so C ±C † . In this way we can assume that C is Hermitean. We want to stress that, to assume that C is Hermitean two conditions must be verified: the reality of K and V operators and the up–down spin symmetry. Assuming that the trace of CC † is equal to one and that the state is normalized, multiplying Eq. (52) by C † and taking the trace, we have ↑ ↓† E(C) = Tr(C † K ↑ C + CK ↓ C † ) + Tr C † V CV . (54)
The matrix C, being Hermitean, can be diagonalized by a unitary matrix W in such a way that C|i = xi |i, where xi are real eigenvalues, associated with the eigenstates |i. Therefore, Tr C † V CV † =
i|C † V C|j j |V † |i ij
=
xi xj i|V |j j |V † |i
ij
=
xi xj | i|V |j |2
ij
|xi ||xj | i|V |j j |V † |i = Tr QVQV† ,
(55)
ij
where the matrix Q is given by (C † C)1/2 and its eigenvalues are |xi |. For negative values of , we have the following: (56)
E(C)E(Q), while if we assume that = E(C) = E(Q).
C
is the ground-state wave function, we have (57)
The last equation is the starting point to show that the ground state is unique; indeed in this case we should show that C = P or C = −P is a unique solution of the equation E(C) = E(Q). Following Lieb, let us introduce a positive semi-definite matrix P = Q − C; we will show that P = 0. Since P can be considered as a linear map from X ↓ to X ↑ , the condition we have to prove means that, for any vector w belonging to X↓ , P w = 0. In other words, the kernel of P is the whole X↓ space. If all the eigenvalues of C have the same sign, then C = ±P and the equality is proven. Then,
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we assume that C = −P . In this case, one of the eigenvalues of Q is zero implying that one of the eigenvectors of P is such that P w = 0. We also notice that Q satisfies the Schödinger equation and therefore: ↑ ↓† V P V = EP . (58) K ↑ P + P K ↓† +
If P w = 0, it follows that ↑ ↓†
w|V P V |w = 0,
(59)
whereas the up–down spin symmetry implies that ↑ ↓† ↓ ↑† ↓ r | w|V ||2 0.
w|V P V |w = w|V P V |w =
(60)
Here r = |x | − x 0. ↑ ↓† These equations imply that w|V P V |w = 0 for any . ↓† Since the matrix P is positive semi-definite, we have w |w = 0 when w = P 1/2 V |w. With this definition we obtain w = 0 which gives ↓†
↑
P V |w = w|V P = 0
(61)
and analogously ↓†
P K |w = 0.
(62)
To be specific, let us consider how the method works when the Hubbard model is concerned. In this case K corresponds to the kinetic term for electrons having spin and V is part of Coulomb interaction term expressed in term of number operator n . Assuming the connectivity condition for the lattice, we can produce any vector belonging ↑↓ to X↓ by iterated applications of K ↑↓ and Vi , for i ∈ , starting from a generic vector in X ↓ . Therefore, we have K = {x ∈ X↓ |P x = 0} = X ↓ and thus P = 0, that is Q = C. We can prove the uniqueness of the ground state in the following way. Assuming that there are two normalized ground state C1 and C2 with C1 = C2 , we have that the state C1 + C2 , where is real constant, is a ground state. Choosing the constant in an appropriate way, we can produce a state C1 +C2 which is never positive semi-definite contradicting the property of positive semi-definite of the ground state. The proof to show the uniqueness of the ground state has been done in two steps: 1. E(C) = E(Q), 2. C = Q (or C = −Q). The negative value for the constant implies that we are treating the attractive Hubbard model. Thus, the method presented holds for models having attractive interaction or that can be transformed into attractive ones by some unitary transformations which preserve the up–down spin symmetry. Now let us specialize the application of this method to the PAM. In particular, we will show that the ground state for the symmetric Anderson model at half-filling is unique and is a spin singlet, i.e. it has S = 0, at least if the lattice is bipartite. To prove this formal statement, we rewrite the Anderson Hamiltonian in the way that it looks like an Hubbard-like Hamiltonian [89]. To this end, we start considering two identical copies of the original lattice, namely L1 and L2 , and we double the lattice by connecting the corresponding lattice points of L1 and L2 with bonds of length 1. We denote ˆ the new layered lattice. Moreover, each point of ˆ has coordinates r = (i, m) with i being the lattice site position with ˆ has 2L lattice points, if L is the number of the lattice points of the real lattice and m = 1, 2. In this manner, the lattice . Then, we define a new operator dr in the following way: ci if m = 1, dr = fi if m = 2.
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ˆ and dr , the Hamiltonian HPAM can be considered as the Hamiltonian of a generalized With these definitions of ˆ when V is taken to be the hopping energy of d electrons between layer 1 and Hubbard model on the bipartite lattice , layer 2 [89]. Finally, HPAM can be rewritten as trs (dr† dh + h.c.) + U nr↑ nr↓ , (63) Hˆ =
rs,
r∈L2
where the hopping matrix trs is defined as ⎧ −t if r and s are n.n. in L1 , ⎪ ⎨ trs = V if r and s are in different layers connected by a bond, ⎪ ⎩ 0 otherwise. ˆ is bipartite with respect H ˆ and the electrons in the first layer are itinerant and We notice that the fictitious lattice uncorrelated while the electrons in the second layer are correlated and do not hop. With this identification, we follow the Lieb proof for the Hubbard model [99]. This proof consists of three parts: • If Ux 0 for every x, among the ground states there is one with spin S = 0. • If Ux < 0 for every x, the ground state is unique and hence has S = 0. • If Ux = U > 0 for all x, the ground state is unique and has S = 0, providing that the lattice is bipartite and the two sublattices have the same number of sites. The first part of Lieb proof can be certainly applied to the negative U PAM, and thus we can state that among the ground states there is at least one with S = 0. This result holds for any εf and not only for the symmetric case. Before going into the proof of the uniqueness, we comment on the repulsive PAM, which is our primary concern. In this case one can map the repulsive Hamiltonian to an attractive one, if the symmetric condition 2εf + U = 0 holds at all the sites, the half-filling situation NTOT = 2L is considered, and if the lattice is bipartite. This is done by using the electron–hole transformation for down-spin electrons. Namely, we consider the electron–hole transformation with respect to one species of spin, say down spin, realized by means of the unitary transformation W previously introduced, † ci↓ −→ i ci↓ , † , fi↓ −→ −i fi↓
where i is +1 for one sublattice and −1 for the other (see Eqs. (4)–(5)). It is immediate to observe that, providing the lattice is bipartite, the hopping terms remain unchanged as well as the hybridization term, due to the phase factors depending on the sublattice index. Referring to the problem of the uniqueness, Lieb’s proof is not applicable to the present case, since it requires a non-zero on-site interaction at every site. This difficulty is avoided by using the special connectivity of the conduction and localized orbitals of the model [89]. In particular, we will first prove the uniqueness for the attractive case, then the electron–hole symmetry will lead to show the uniqueness for the repulsive case. Because of rotation symmetry in spin space, it is enough to study the problem in the Sz = 0 subspace. To show the reflection positivity in spin space, it is convenient to write the ground-state wave function as follows: M . (64) =
We recall that the functions are a complete orthonormal set of real bases for L electrons for each spin including both c- and f-electrons. Besides, as before the time-reversal symmetry guarantees that the coefficient M can be chosen real symmetric. In order to prove the uniqueness of the ground state, following the procedure previously outlined, it needs to show that the real symmetric matrix M is semi-definite positive (or negative).
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Let us consider the Hermitean positive semi-definite matrix R and its kernel Q, R ≡ |M| − M,
Q ≡ {v ∈ X|Rv = 0}.
(65)
Now, we introduce the matrices for the one-particle part of the Anderson Hamiltonian K and the local densities N c and N f by ⎞⎤ ⎡⎛ † † ⎠ ⎦ ⎣ ⎝ + H.c. , (66) −t K ≡ c i cj + V c i fi
i,j i (Nic ) ≡ |ci† ci | ,
(67)
f
(Ni ) ≡ |fi† fi | .
(68) f
Then, we can show that K is a linear map from Q to Q as well as Ni . This last condition is proven using the fact that the f orbitals have a finite on-site Coulomb interaction; then, from the Schrödinger equation we easily show that for a vector v in the kernel Q f f f v KR + RK + U Ni RN i v = U R 1/2 Ni v2 = 0; (69) i
i
f
f
f
so that R 1/2 Ni v = 0 for every lattice site i. Multiplying from the left by R 1/2 , we obtain RN i v = 0, i.e. Ni v ∈ Q. We want to stress that up to this step, the procedure of demonstration follows the proof for the Hubbard model. For the PAM, we can also show that also Nic is a linear map from Q to Q although the c orbitals have no on-site Coulomb interaction. In order to show this, we notice that f
Nic = Ni +
1 f f f f [(1 − Ni )KN i , Ni K(1 − Ni )]. V2 f
(70)
f
Due to the fact that Ni (1 − Ni ) and K are linear map from Q to Q, the last equation implies that also the operator Nic maps Q to Q. Two considerations are important at this stage: (1) no on-site interactions at c orbitals are necessary for this property of Nic operator; (2) the above property holds because the hybridization term V is restricted to the same site; for longranged mixing terms, it is no longer true that Nic maps Q to Q. In conclusion, this is a consequence of the special form of the hybridization term in the PAM. The proof for uniqueness is analogous to that for the Hubbard model previously outlined. The operators K, Nix (x = c, f ) are defined in the configuration space X. Since R is a linear operator, its kernel Q is a subspace in the configuration space X = QQ. Here Q denotes the complement, with respect to X, of Q. The uniqueness condition is proven if we show that Q = X or Q = X. Due to their Hermiticity, we notice that Nix and K also map Q = X. f f To prove the uniqueness, we define a specific configuration of the c- and f-electrons by m={nc1 , . . . , ncL ; n1 , . . . , nL }, x with ni = 0 or 1 and a family of the projectors onto the m vector P
m
=
L $ $
[nxi Nix + (1 − nxi )(1 − Nix )].
(71)
x=c,f i=1
These projection operators map Q into Q and Q into Q. f Let us consider now the configuration with all the f orbitals occupied, ni = 1 for all i. Then the projection operator 0 P to this configuration is f
f
f
P 0 = N1 N2 . . . NL .
(72)
The image of the projection operator P 0 is then one dimensional and this imply that this configuration belongs to Q or Q. Besides, it is easily shown that starting from this configuration, we can construct all basis states of X by iterated
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applications of P m and K. Therefore, it is proven that Q = X or otherwise it is an empty set, meaning that M = ∓|M|. This result implies that the ground state is unique. As a final step, we have to show that the ground state has S = 0 for the repulsive case too. To this end, we note that the energy band of the conduction electrons εk is symmetric for the bipartite lattice. The mixing term V, when is different from zero, opens a hybridization gap at zero energy. Therefore, when the uncorrelated U = 0 case is concerned, the ground state is the state with the lower hybridized band fully occupied and the upper hybridized band empty. This state is the unique ground state and obviously has S = 0, and there is a finite energy gap to the excited states. Invoking a continuity argument with respect to U, the unique ground state for U > 0 has also S = 0 since there is no level crossing [89]. 6.1. Discussion From the result above derived, we now show that the ground state of the PAM at half-filling is also a pseudospin singlet [104]. Indeed, we can prove that: Lemma. If U is positive, the ground state of H (U, εf ) is unique and has pseudospin equal to zero. Let | be an eigenstate of H (U, εf ), J 2 , Jz , S 2 and Sz having eigenvalues E , j 2 , jz , s 2 and sz , respectively, i.e., | ≡ |U, εf , s, sz , j, jz . Using Eq. (12), we have H (−U, −εf )W | = (E + εf )W |.
(73)
Therefore W | is proportional to | ≡ | − U, −εf , j, −jz , s, sz . In particular if | is the ground state of H (U, εf ) in the subspace {J 2 , Jz , S 2 , Sz }, | is the ground state of H (−U, −εf ) in the subspace {S 2 , Sz , J 2 , −Jz }, and its energy is E = E + εf . Since the total pseudospin in the ground state of H (U, εf ) is related to J, we may refer to the real spin S in the ground state of H (−U, −εf ). This consideration allows us to state the ground state is indeed a pseudospin singlet. Therefore, we can conclude that ground state of the half-filled repulsive PAM is a spin and a pseudospin singlet. Hence, we would like comment on this pseudospin symmetry. For the Hubbard model at half-filling, Yang and Zhang thought that this symmetry is due to a suitable combination of the kinetic and the potential energy, which gives rise to a coherent propagation of the so called pairs; the pairs have zero kinetic energy and cannot be scattered by a third electron on the same site due to the Pauli principle. In the Anderson model a pair of conduction c and correlated f-electrons can coherently propagate only if the symmetric condition is assumed. In the pseudospin language the local f energy term and the Coulomb term can be considered like two opposite fields along the z direction so that one can freely rotate a pseudospin only if the total field is zero. 7. Supersolid in the PAM Starting from the results of previous section, we discuss now some physical properties of the ground state of the PAM on a bipartite lattice. We show first that the lower-energy state is rotationally invariant in the pseudospin space, and supporting the simultaneous presence of superconductivity and CDW order. We then conclude that this state, for negative value of the Coulomb repulsion, may represent a supersolid. Before going into details, it is worth defining the concept of supersolid. We say that the supersolid is a state of matter which exhibits simultaneously both solid and superfluid properties. In other words, it displays both long-range positional order (modulation of the charge density with characteristic wavelength) as well as finite superfluid density (naively, off-diagonal-long-range order (ODLRO)). Concerning the ODLRO, this order was first proposed to relate Bose–Einstein condensation in a system of bosons by Penrose [105]. The same order was also introduced in a system of fermions, and Yang proposed ODLRO to describe the superconductivity of a system of electrons [106]. Namely, he showed that the existence of ODLRO implies the phenomenon of quantized magnetic flux. Moreover, it has also been shown that ODLRO and gauge covariance lead to the Meissner effect, which is a typical characteristic of superconductivity [107]. We would like to emphasize that ODLRO may provides a possible tool to study superconductivity on a finite system, and this fact gives the possibility of doing numerical computations. In summary, to get proof of some rigorous examples of models for strongly correlated
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systems, with non-trivial interactions, possessing ODLRO may be highly relevant. In a pioneering work [49], Yang for the first time found that a class of eigenstates of the Hubbard model possess ODLRO by using the -pairing operators, although these states do not refer to the ground state. Soon later, using the pseudospin symmetry, Yang and Zhang proposed a criterion for -pairing superconductivity [49]. Then, Singh and Scalettar showed that this model, with infinite attractive U interaction, has ODLRO in the ground state [50]. Shen and Qiu proved that the ground state of the attractive Hubbard model on some bipartite lattices possesses ODLRO within a finite range of electron density, regardless of the lattice dimension [108]. Extended Hubbard models were also shown to have ODLRO [87]. Coming back to the supersolid concept, there is no generally accepted opinion regarding the experimental detection of the supersolid property. There are promising experimental data indicating supersolidity in 4 He, although the interpretation of data is controversial [109]. There is also a large class of fermionic systems, such as NbSe2 and A15 compounds, that at high temperatures yield a CDW and that upon cooling exhibit a second transition into a superconducting phase [110], and this evidence may support a supersolid state at low temperatures. From the theoretical point of view, there are many results supporting the existence of this state [111]. Moreover, it is found that several strongly correlated electron models may support the existence of states representing a supersolid. Indeed, Pu and Shen showed that the ground state of attractive Hubbard model, at half-filling, can support the coexistence of ODLRO and CDW ordering [112]; the same result holds for Hubbard model with bond–charge interaction [113]. Finally, we mention that it has been shown recently that a mixed boson–fermion model also exhibits this supersolidity property [103]. We discuss now the physical consequences of the pseudospin symmetry in the PAM and its connection to the supersolid. As shown in the previous section, in the symmetric limit the ground state is a pseudospin singlet and it is invariant for rotation in the pseudospin space implying that ODLRO and CDW co-exist [114]. As previously mentioned, the Anderson model includes two antagonist processes: the first one is the formation of local magnetic moments caused by the Coulomb repulsion, the second is the screening of this repulsion by means of conduction electrons, that may produce the Kondo effect. These phenomena are a features of the magnetic properties of the model. Nevertheless, the model possesses another symmetry, whose generators are given by J operator, which makes more interesting also the investigation of charge properties. The components of this operators verify the following: [H, J ± ] = ±EJ ±
(74)
with E = 2εf + U , implying that, apart from the symmetric limit, i.e. when 2εf + U = 0, only J 2 and Jz commute with the Hamiltonian. This amount to say that the eigenstates of J 2 , with different Jz , have different energy according to the relation (74). It is worth pointing out, and it will be useful later for visualizing the physical meaning of the result, that J x and J y operators correspond to pairing operators (see Eq. (11)) thus, ordering of the pseudospins in the x–y plane represents superconductivity in the PAM. We also notice that their magnitude is connected to the order parameter, and their angle in the x–y plane can be associated with the superconductng phase. In the following, starting from the consideration that the ground state of the PAM, in the symmetric limit, is a pseudospin singlet (rotationally invariant in the pseudospin space) we will show that it exhibits also the coexistence of CDW and ODLRO. To look at long-range order, we study the following equal time correlation function: − J +− (q) = |Jq+ J−q |,
(75)
where Jq± is the Fourier transform of Ji± Jq± =
e∓q·Ri Ji± .
i
If for q → q0 , q0 being a generic vector of the first Brillouin zone, the J +− (q0 ) correlation function is of order L (number of the lattice sites), according to Yang’s definition [106], the state | possesses ODLRO property. Let us look now at the correlation function for the z component of the pseudospin operator: z J zz (q) = |Jqz J−q |.
If also J zz (q), for a particular momentum is of order L, then it stands for a CDW.
(76)
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To prove that the ground state | of the PAM exhibits both the properties above mentioned, we use the property that | is a pseudospin singlet, that is J ± | = J z | = 0. Starting from this equation, we can easily obtain the following: y y
(77)
y
(78)
z x
|Jqx J−q | = |Jq J−q | = |Jqz J−q |
and y
z x
|Jqx J−q | = |Jq J−q | = |Jqz J−q | = 0.
Substituting Eqs. (77)–(78) into the definition of J +− (q) (Eq. (75)), one obtains x J +− (q) = |(Jqx + iJq )(J−q − iJ−q )| y
y
y y
x | + |Jq J−q | = 2J zz (q). = |Jqx J−q
(79)
The meaning of the last equation is that, if superconductivity characterized by ODLRO appears, then so does the CDW, and viceversa. It should be pointed out [112] the possibility to have this relationship between ODLRO and CDW comes in only for a pseudospin singlet state; when the system is not in the symmetric limit, the result does not hold anymore. 7.1. Discussion Let us comment on the physical interpretation of the above results, particularly on the relationship between the ground state with supersolid properties and the exact results [115,116]. In the case of positive U it has been shown [115–117] that the ground state has short-range antiferromagnetic correlations thus ruling out the possibility of an pairing type of superconductivity and consequently of a supersolid. Anyway, it is worth stressing that antiferromagnetic correlations in the ground state does not exclude the possibility of superconductivity with non-local pairing. Details on this topic will be given in Section 9. In the attractive U PAM, we may argue that the superconducting (or CDW) instability comes in, and this supports the idea that the ground state for negative U can represent a supersolid [114]. 8. Spin and charge gaps in PAM It is well-known that from the knowledge of excitation gaps, one can get deep insight into quantum transport and magnetic properties of strongly correlated electron systems. A powerful tool to better understand these properties is the investigation of charged and spin excitation gaps. We want to point out that this investigation is usually confined to some specific filling, namely at half-filling. In order to show how these gaps affect the mentioned transport properties, let us introduce the general definitions of the charged gaps and the spin excitation gaps at half-filling. The spin excitation gap is defined by S ≡ E0 (J = 0, S = 1) − E0 (J = 0, S = 0), while the charge excitation gap is defined by C ≡ E0 (J = 1, S = 0) − E0 (J = 0, S = 0). Here E0 (J = j, S = s) is the lowest eigenvalue of the corresponding Hamiltonian in the subspace with quantum numbers J = j and S = s, where J is the pseudospin quantum number and S the usual spin quantum number [52,118].
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209
Finally, we introduce the quasi-particle gap as qp ≡ E0 (NL + 1) + E0 (NL − 1) − 2E0 (NL ), where E0 (N ) corresponds to the ground state of the model for N particle, NL indicating the half-filled case (NL = 2L) for the PAM, L being the number of lattice sites [118,119]. Numerical calculations on one-dimensional periodic Anderson chains indicate that the charged gap is always larger than the spin excitation at half-filling [52]. Indeed, in this paper it is shown that the ratio between C and s increases monotonically, as function of the Coulomb repulsion, from the unity, diverging in the strong coupling limit. We now give a direct proof this inequality and we prove that the quasi-particle gap is larger than the spin excitation gap too [120–122]. We want to stress that the procedure below applied to the PAM has been successfully used, in a more general ground, for relevant strongly correlated electron models such as the Hubbard model and the Kondo lattice model [121,122]. The proof of the inequalities above claimed is based on the following theorems: Theorem 3. For the PAM defined on a d-dimensional simple cubic lattice, when the system is half-filled, the charged gap and the corresponding quasi-particle gap satisfy the following inequality: qp S . Theorem 4. For the PAM defined on a d-dimensional simple cubic lattice, when the system is half-filled, the charged gaps and the corresponding spin excitation gaps satisfy the following inequality: C S . To prove these theorems, we will apply a generalized version of Lieb spin-reflection positivity method described in Section 6. Let us sketch the main steps of the proof [121,122]. • Introduction of a unitary partial particle–hole transformation, and mapping of the original Hamiltonian to an equivalent Hamiltonian with negative coupling constants; • Writing of the transformed Hamiltonian into a form of the direct product of up-spin fermion operators and down-spin fermion operators; • Use of the spin-reflection positivity method to the transformed Hamiltonian and getting proof of an inequality for the lowest eigenvalues of this transformed Hamiltonian in the different subspaces; • Application of the inverse of the partial particle–hole transformation to the inequality to come back to the initial Hamiltonian. We proceed to the proof of the Theorems 3 and 4. It is a well-known that, for the symmetric PAM there exists a unitary transformation under which the Hamiltonian with positive interaction coupling constant is mapped into a corresponding Hamiltonian with negative interaction. To be more precise, we use the unitary transformation R defined in Section 3 (Eqs. (6)–(9)). We notice that the sign of U is changed but the sign of V is unchanged under the application of R to HPAM . Since the hybridization term can be mathematically treated as a generalized hopping term (see Section 6) the sign of V plays no role in the following proof. Besides, under R the half-filled subspace for the Hamiltonian is mapped into itself. In particular, since R is %PAM , unitary, the ground state of HPAM in the half-filled subspace is mapped onto its counterpart for transformed H in the same subspace. Moreover, other subspaces are not invariant under R. We also notice that the spin operator S is mapped into the corresponding pseudospin operator J, and vice versa (see Table 1). Consequently, under the action of R, an eigenstate | = |J = j, S = s of the original Hamiltonian is mapped into an eigenstate |% = |J = s, S = j of the transformed Hamiltonian. In particular, the ground states of the original Hamiltonian in the sector with quantum numbers J = j, S = s are mapped into the ground states of the transformed Hamiltonian in the sector with quantum numbers J = s, S = j . This properties will be applied in the following.
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Next, we write the HPAM Hamiltonian as a direct product of up-spin fermion operators with down-spin fermion operators. Following the results quoted in Refs. [123–125], we introduce the pseudofermion operators ei↑ = ai↑ ,
ei↓ = (−1)N↑ ai↓ ,
where ai stands for fi and ci and N↑ is the total number of fermions with up spin in the system. We would like to stress that the new fermion operators ei↑ commute with ei↓ and consequently, the R transformed PAM Hamiltonian in the symmetric limit can be rewritten as %PAM = ↑ ⊗ I + I ⊗ ↓ − U H
(ni↑ − 21 ) ⊗ (ni↓ − 21 ),
(80)
i
being some Hermitean polynomial of fermion operators with spin and I corresponds to the identity operator. %PAM Hamiltonian is written in a polynomial form of ei and e† with real Finally, we notice that the U term of the H i coefficients. Indeed, one can write U
(ni↑ − 21 ) ⊗ (ni↓ − 21 ) =
i
† † (Zˆ i↑ ⊗ Zˆ i↓ + Zˆ i↑ ⊗ Zˆ i↓ ),
(81)
i∈
where is a positive constant and all the operators {Zˆ i } and {Zˆ i† } in Eq. (81) are real operators. As final result, in this way we have written the Hamiltonian into a form of the direct product of up-spin and down-spin fermion operators. %PAM in the subspace corresponding to NL + 1 particles. Now, let us consider the ground-state vector |NL + 1 of H + − %PAM , we apply these operators to the initial Keeping in mind that the spin operators S and S commute with H ground state many times in such a way that we transform the initial state |NL + 1 into a new one characterized by the condition N↑ -N↓ = 1. This state |(NL + 1) has total third component quantum number Sz = 1/2 and its wave function (NL + 1), which has NL /2+1 up-spin fermions and NL /2 down-spin fermions, can be written as (NL + 1) =
↑
↓
Aij i ⊗ i ,
(82)
i,j
where i is a state vector defined as i ≡ ei†1 ei†2 . . . ei†M |0, with M = NL /2 + 1 for = ↑ and M = NL /2 for = ↓. The set {i } provides a natural basis for the subspace of M fermions with spin . Nevertheless, we notice that, if one chooses F↑ ≡ W↑ ((NL /2) + 1) and F↓ ≡ W↑ (NL /2) as the subspaces for up-spin and down-spin fermions, then the coefficient matrix A = (Aij ) will not be a square matrix. Therefore, we will define G↑ and G↓ by G = W (NL /2)W ((NL /2) + 1). Moreover, G↑ and G↓ have now the same dimension and consequently, the matrix A N /2 N /2+1 may be written as a square matrix P × P , where P = CNLL + CNLL , where Cnm denotes the dimension of the Fock space for m sites and n particles. The form of the matrix A is A=
O
C
O
O
,
N /2+1
(83) N /2
where C is an CNLL × CNLL non-zero matrix and O is a matrix with all the elements equal to zero. Since for arbitrary n × n matrix X, and hence not necessarily Hermitean, it is possible to find two n × n unitary matrices Q1 and Q2 and an n × n diagonal semi-positive definite matrix D (Dij = di ij and di 0 for i = 1, 2, . . . , n) such that [120] X = Q1 DQ2 ,
(84)
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211
we can find two unitary matrices V1 , V2 and a diagonal positive semi-definite matrix K, such that A = V1 KV 2 . Therefore, (NL + 1) assumes the following: (NL + 1) =
P P
↑
↓
Aij i ⊗ j
i=1 j =1
=
P P
↑
↓
(V1 KV 2 )ij i ⊗ j
i=1 j =1
=
P
↑
↓
h l l ⊗ l ,
(85)
l=1
where ↑
l =
P
↑
↓
(V1 )il i ,
l =
P
↓
(V2 )lj j .
(86)
j =1
i=1
↑
↓
Moreover, the matrices Vi (i =1, 2) are unitary matrices implying that {l } and {l } are orthonormal bases in subspaces G↑ and G↓ , respectively. Besides, (NL + 1) is an eigenvector of N↑ N↑ |(NL + 1) =
NL + 1 |(NL + 1) 2
(87)
and when the explicit form of (NL + 1) is considered, we may write P
↑
↓
hl [N↑ l ] ⊗ l =
l=1
P
hl
l=1
NL ↑ ↓ + 1 l ⊗ l . 2
(88)
↓
Taking the scalar product of last equation with l ↑ hl [N↑ l ] = hl
NL ↑ + 1 l , 2
(89)
↑
we find that the state l is an eigenvector of N↑ with eigenvalue NL /2 + 1, if hl = 0. The same conclusion also holds ↓ for the operator N↓ and states {l }. %PAM in the Using the representation of the ground state above reported (see Eq. (85)) the ground-state energy of H subspace W (NL + 1) is given by E0 (NL + 1) =
D
↑
l=1
−
⎛ ⎝
i
P
l1 ,l2 =1
i
−
↑
↓
↓
h2l [ l |↑ |l + l |↓ |l ]
⎛ ⎝
P l1 ,l2 =1
⎞ ↑
↑
↓
↓
hl1 hl2 l2 |Zi↑ |l1 l2 |Zi↓ |l1 ⎠ ⎞ ↑
↑
↓
↓
† † hl1 hl2 l2 |Zi↑ |l1 l2 |Zi↓ |l1 ⎠ .
(90)
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Applying the triangle inequality to each terms above calculated, we obtain E0 (NL + 1)
P P ' 1 ' & 1 2& hl l ||l + l ||l + h2l l ||l + l ||l 2 2 l=1
l=1
⎛
⎞ P ⎝ hl1 hl2 l2 |Zi |l1 l2 |Zi |l1 ⎠ − 2 l1 ,l2 =1
i
⎛
⎞ P ⎝ − hl1 hl2 l2 |Zi |l1 l2 |Zi |l1 ⎠ 2 l1 ,l2 =1
i
⎛
⎞ P ⎝ † † − hl1 hl2 l2 |Zi |l1 l2 |Zi |l1 ⎠ 2 l1 ,l2 =1
i
⎛
⎞ P ⎝ − hl1 hl2 l2 |Zi† |l1 l2 |Zi† |l1 ⎠ , 2 i
(91)
l1 ,l2 =1
where for simplicity, we have dropped the spin indices. Then, we introduce a couple of new wave functions f1 and f2 by f1 =
P
↑ ¯ ↓, h l l ⊗ l
f2 =
l=1
P
↑ ¯ ↓, hl l ⊗ l
(92)
l=1
¯ and ¯ . We notice that f1 and f2 are wave functions written in terms of the complex conjugates of l and l , that is l l in subspace W (NL /2 + 1, NL /2 + 1) and W (NL /2, NL /2), respectively. It is straightforward to see that
f1 |f1 = f2 |f2 =
P
h2l = (NL + 1)|(NL + 1) = 1.
(93)
l=1
The inequality for the ground-state energy E0 (NL + 1) may be rewritten in a different way in terms of f1 and f2 , considering that operators are Hermitean and {Zi } ({Zi† }) are real; we have %PAM |f1 + 1 f2 |H %PAM |f2 . E0 (NL + 1) 21 f1 |H 2
(94)
Then, by using the fact that
(L + 1)|N↑ |(L + 1) =
NL + 1, 2
we have, for instance,
f1 |N↑ |f1 = f1 |N↓ |f1 =
P
h2l l |N |l =
l=1
NL + 1. 2
(95)
This implies that E0 (NL + 1) 21 E0 (NL + 2) +
1 2
E0 (NL ).
(96)
%PAM , into the original PAM Finally, we apply the inverse transformation R −1 to map the attractive Hamiltonian H Hamiltonian. Since we are interested in the change of inequality Eq. (96) under this transformation, we will consider how each of E0 (NL ), E0 (NL + 1) and E0 (NL + 2) changes under the inverse of the unitary transformation R.
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213
As mentioned above, E0 (NL ) is unchanged under the partial particle–hole transformation; then it is the lowest eigenvalue of the original Hamiltonian in the half-filled subspace W (NTOT = NL ). Besides, E0 (NL + 1) is also invariant due to the fact that the partial particle–hole transformation only change the particle number of down-spin fermions from NL /2 to NL − NL /2 = NL /2 and keep the particle number of up-spin fermions, N↑ = NL /2 + 1 unchanged. This consideration implies that the subspace W (N↑ = NL /2 + 1, N↓ = NL /2) is mapped into itself. The situation is different for the ground-state energy E0 (NL + 2). As previously proven, in the subspace W (NL + 2), the ground states of HPAM have quantum numbers S = 0 and J = 1. Therefore, the inverse transformation R −1 map these states into the ground states of the original PAM Hamiltonian in the subspace with quantum numbers J = 0 and S = 1. In conclusion, these states are the ground states of the original Hamiltonian in the half-filled sector with S = 1. For these reasons, we obtain %PAM ) = E0 (NL , S = 1; HPAM ). E0 (NL + 2; H
(97)
Therefore, for the Hamiltonian HPAM , the inequality equation (96) may assume the following equivalent expression: E0 (NL + 1) 21 E0 (NL , S = 1) +
1 2
E0 (NL , S = 0),
(98)
E0 (NL + 1) − E0 (NL ) 21 E0 (NL , S = 1) − 21 E0 (NL , S = 0).
(99)
Since the PAM in the symmetric limit verifies the particle–hole symmetry, at half-filling E0 (NL + 1) = E0 (NL − 1), and it is easily seen that this last inequality reproduces the inequality claimed in the Theorem 3, when it is multiplied by 2. In order to prove the Theorem 4, we start with a different wave function, changing the wave function in Eq. (82) with the following: (N↑ = N/2, N↓ = NL − N/2) =
↑
↓
Bij i ⊗ i ,
(100)
i,j
where i is a state vector defined as i ≡ er†1 er†2 . . . er†M |0, with M = N/2 for = ↑ and M = NL − N/2 for = ↓. Here, N is an integer less that NL . We notice that the dimension of Fock spaces W↑ (N/2) and W↓ (NL − N/2) is the same [122]; thus B is a D × D square matrix. Therefore, we can apply the factorization lemma previously used so that the ground-state wave function may be rewritten as (N↑ = N/2, N↓ = NL − N/2) =
D D
↑
↓
Bij i ⊗ j
i=1 j =1
=
D D
↑
↓
(U1 H U 2 )ij i ⊗ j
i=1 j =1
=
D
↑
↓
hl l ⊗ l ,
(101)
l=1
where ↑ l
=
D i=1
↑ (U1 )il i ,
↓ l
=
D
↓
(U2 )lj j .
(102)
j =1 ↑
↓
Since Ui (i = 1, 2) are unitary matrices, {l } and {l } are also orthonormal bases in subspaces W↑ (N/2) and W↓ (NL − N/2), respectively.
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Using this expression for the wave function of the ground state, we can easily estimate the ground-state energy in the subspace W (N↑ = (N/2), N↓ = (NL − N/2)). E0 (N↑ = (N/2), N↓ = (NL − N/2)) %PAM |(N↑ = N/2, N↓ = NL − N/2) = (N↑ = N/2, N↓ = NL − N/2)|H ⎛ ⎞ D P ↑ ↑ ↓ ↓ ↑ ↑ ↓ ↓ ⎝ = h2l [ l |↑ |l + l |↓ |l ] − hl1 hl2 l2 |Zi↑ |l1 l2 |Zi↓ |l1 ⎠
−
⎛
P
⎝
l1 ,l2 =1
i
l1 ,l2 =1
i
l=1
⎞ ↑
↑
↓
↓
† † hl1 hl2 l2 |Zi↑ |l1 l2 |Zi↓ |l1 ⎠
P P 1 2 1 2 hl [ l ||l + l ||l ] + hl [ l ||l + l ||l ] 2 2 l=1
l=1
⎛
⎞ P ⎝ hl1 hl2 l2 |Zi |l1 l2 |Zi |l1 ⎠ − 2 l1 ,l2 =1
i
⎛
⎞ P ⎝ − hl1 hl2 l2 |Zi |l1 l2 |Zi |l1 ⎠ 2 l1 ,l2 =1
i
⎛
⎞ P ⎝ − hl1 hl2 l2 |Zi† |l1 l2 |Zi† |l1 ⎠ 2 l1 ,l2 =1
i
⎛
⎞ P ⎝ − hl1 hl2 l2 |Zi† |l1 l2 |Zi† |l1 ⎠ , 2
(103)
l1 ,l2 =1
i
where we have dropped the spin indices and applied the triangle inequality. As before, we introduce two new normalized wave functions g1 and g2 by f1 =
D l=1
↑
↓
¯ , h l l ⊗ l
f2 =
D
↑
↓
¯ , h l l ⊗ l
(104)
l=1
¯ l are the complex conjugates of and l , respectively. where ¯ l and l In terms of these functions, Eq. (103) can be rewritten as %PAM |g1 + 1 g2 |H %PAM |g2 . E0 (N↑ = N/2, N↓ = NL − N/2) 21 g1 |H 2
(105)
Considering that g1 is a wave function in the subspace W (N↑ = N/2, N↓ = N/2) and g2 in the subspace W (N↑ = NL − N/2, N↓ = NL − N/2), applying a variational argument we get E0 (N↑ = N/2, N↓ = NL − N/2) 21 E0 (N↑ = N/2, N↓ = N/2) + 21 E0 (N↑ = NL − N/2, N↓ = NL − N/2) =
1 2
E0 (N ) +
1 2
E0 (2NL − N ).
(106)
The last equality is a consequence of the fact that E0 (N↑ = N/2, N↓ = N/2) and E0 (N↑ = NL − N/2, N↓ = NL − N/2) are the ground states in the subspaces W (N) and W (2NL − N ).
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Finally, using the particle–hole symmetry we deduce that E0 (N ) = E0 (2NL − N ), which implies, when substituted in Eq. (106) E0 (N↑ = N/2, N↓ = NL − N/2)E0 (N ). In order to successfully complete the proof of the Theorem 4, we notice that under the particle–hole transformation R the subspace W (N↑ = N/2, N↓ = N/2) is mapped into the subspace W (N↑ = N/2, N↓ = NL − N/2), so that the %PAM ground state of the PAM Hamiltonian in the subspace W (N↑ = N/2, N↓ = N/2) is equal to the ground state of H %PAM , can be reinterpreted as in the subspace W (N↑ = N/2, N↓ = NL − N/2). Thus, the last inequality, proven for H %PAM ), E0 (N, HPAM )E0 (N, H
(107)
where we have emphasized the Hamiltonians whose the ground state refers to. Since this inequality holds for any N, we also have %PAM ). E0 (NL + 2, HPAM ) E0 (NL + 2, H
(108)
Let us consider now the right-handed side of this inequality. The energy refers to a state having S = 0 and J = 1 as demonstrated in Ref. [89]. Applying the particle–hole transformation to the corresponding state, we get a state with S = 1 and J = 0. In other words, we have obtained the ground state at half-filling for spin S = 1. Summarizing: %PAM ) = E0 (NL , J = 0, S = 1, HPAM ). E0 (NL + 2, H Therefore, the two last inequalities read as E0 (NL + 2, HPAM ) E0 (NL , J = 0, S = 1, HPAM ). Since E0 (NL , J = 1, S = 0, HPAM ) E0 (NL + 2, HPAM ), we easily get, from the last inequality when E0 (NL , HPAM ) is subtracted to both sides, the inequality claimed in the Theorem 4. 8.1. Discussion The inequality equation (99) can be strengthened. Indeed, following the suggestions reported in Ref. [120], one can easily show that, for any finite lattice, this inequality may be substituted by the analogous one without the equality symbol. For this reason, the system has a Mott charged gap. However, we have to keep in mind that, to study the Mott metal–insulator transition of the PAM in the thermodynamic limit, one needs to resort to other approaches. In the above proof, we have assumed that the coupling constant is site-independent. This assumption has been made to make the proof of the theorems simpler, but it can be relaxed as it still holds with site-dependent coupling constants [120]. We notice that, the definition of the quasi-particle qp may have some shortcomings when the Zeeman interaction term is added to HPAM or when small size clusters are considered. Indeed, in this case finite size effects makes qp strongly depending on the number of sites; in other words, as this number changes, qp greatly fluctuates. A possible solution to this shortcut has been given by Nishino [126] who proposed the following alternative definition of quasi-particle gap: % qp = E0 (n↑ + 1, n↓ + 1) + E0 (n↑ , n↓ ) − E0 (n↑ + 1, n↓ ) − E0 (n↑ , n↓ + 1), where E0 (n1 , n2 ) is the ground-state energy for a state with n1 spin-up electrons and n2 spin-down electrons with the constraint NTOT = n1 + n2 + 1. It can be shown numerically that this quantity is less parity dependent than qp and more importantly it has been rigorously proven that, for any couple of n↑ and n↓ allowed integer values, the quasi-particle gap qp is a non-negative quantity [127]. As concluding remark, we want to stress that the method applied to derive the inequalities for excitation gaps represents a very powerful tools to investigate, and more importantly to exactly prove, relevant results for strongly
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correlated electron models [121,122,127]. We would like to mention that this procedure has been successfully applied to the Hubbard model and the Kondo lattice model, and more recently to a two-band Hubbard model [128]. 9. Antiferromagnetism in PAM By using the results of previous section and a theorem proven in [129], we will show here that in the half-filled ground state of the symmetric PAM short-range antiferromagnetic correlations exists. For simplicity, following the proof outlined in Ref. [115], let us consider the PAM defined on a specific lattice: a two-dimensional square lattice with the lattice constant being set to be equal to one, in suitable units. Moreover, it is worth using the PAM Hamiltonian in a Hubburd-generalized form. To this end we take advantage of the results reported in Section 6 where we introduced a new “spinor” operator dr , which contains both correlated f operator and uncorrelated c one, and doubled the real lattice. To prove the result above claimed, it is useful to write the PAM Hamiltonian, in the symmetric limit, in the following way: HPAM = trs (fr† fh + h.c.) + U (nr↑ − 21 )(nr↓ − 21 ), (109)
rs,
r∈L2
where an unimportant constant term has been dropped. In this form, due to the presence of correlation only for one species of electrons, the Hamiltonian is not symmetric. To avoid this asymmetry between the two types of electrons, we add to the HPAM an auxiliary interaction term dependent on a parameter that we will let to zero at the end of the calculations, i.e. % Haux = U (nr↑ − 21 )(nr↓ − 21 ). r∈L1
%) of the repulsive symmetric PAM at half-filling, there are short-range In order to show that, in the ground state 0 (U, U antiferromagnetic spin correlations for f-electrons, we make use of the particle–hole transformation T on the d operators: T d r↑ T −1 = dr↑ , † T d r↓ T −1 = (r)dr↓ ,
where (r) = 1 if r∈ L1 and (r) = −1 if r∈ L2 . We notice that this unitary transformation is nothing that the unitary transformation R introduced in Section 3. Then, the repulsive PAM Hamiltonian is mapped into the attractive one and, in this form, the Hamiltonian exhibits the reflection positivity symmetry in the spin space. The attractive PAM Hamiltonian has the same spectrum as the repulsive one at half-filling; thus, we may study the properties of the ground state of the T-transformed Hamiltonian %) and then we transform back to the ground state 0 (U, U %) by applying to the state 0 (−U, −U %) the 0 (−U, −U operator T −1 . Keeping in mind these considerations and considering that the T-transformed HPAM commutes with the total number operator N , we can classify the eigenstates of the transformed Hamiltonian in terms of n1 and n2 , the eigenvalues of the operators N↑ and N↓ , respectively, we can prove the following theorem: Theorem 5. For any two lattice points n and m we have %)|On† Om |0 (−U, −U %)0,
0 (−U, −U
(110)
when the half-filled case n1 = n2 = L is concerned, and Ox = dx↑ dx↓ . The proof of the inequality claimed in this theorem is easily obtained by using the result obtained in the Section 6. Indeed, we can write the ground state in the form j %) = 0 (−U, −U Wij i↑ ⊗ ↓ , ij
C. Noce / Physics Reports 431 (2006) 173 – 230
217
where W is a Hermitean positive definite matrix and i is an orthogonal real basis for species of L/2 fermions. Thus, the expectation value of O operator can be written as follows: %)|On† Om |0 (−U, −U %) = 0 (−U, −U %)|f † f † fm↑ fm↓ |0 (−U, −U %)
0 (−U, −U n↓ n↑ j † † %)|(f † fn↑ )(f † fm↓ )|0 (−U, −U %) = = 0 (−U, −U Wij Wlm i↑ |fn↑ fm↑ |l↑ ↓ |fn↓ fm↓ |m ↓ , n↑ m↓ ij
ij
lm
j
† † Wj†i Wlm i↑ |fn↑ fm↑ |l↑ ↓ |fn↓ fm↓ |m ↓
lm
=
ij
j
† † † † Wj i i↑ |fn↑ fn↑ |l↑ Wlm m ↓ |(fn↓ fm↓ ) |↓ = Tr(W MW M ),
(111)
lm
where we have defined M ≡ dn† dm , and used the fact that W is an Hermitean matrix and the reality of the basis i . Since W is a definite positive matrix, W 1/2 is well defined so that we have Tr(W MW M † ) = Tr[(W 1/2 MW 1/2 )(W 1/2 MW 1/2 )† ] 0
(112)
and this result implies Eq. (110). Now, applying the inverse particle–hole transformation we easily deduce the following theorem: Theorem 6. In the ground state of the repulsive symmetric PAM at half-filling we have 0 if n and m belong to the same sublattice, + − % %
0 (U, U )|Sn Sm |0 (U, U ) 0 if n and m not belong to the same sublattice,
(113)
† † † where the spin operator S refers to the operator d, i.e. Sx+ = dx↑ dx↓ , Sx− = (Sx+ )† and S z x = 21 (dx↑ dx↑ − dx↓ dx↓ ).
The proof of this inequality is easily obtained considering Eq. (110) and applying to it the inverse particle–hole transformation [115]. Therefore, the short-range transverse spin correlations of correlated electrons in the ground state of positive (ε, U ) symmetric PAM at half-filling are antiferromagnetic. The theorems above presented refer to a ground-state property, and then it is worth thinking about finite temperatures and discussing if the same conclusions still hold true. From the experimental point of view, we can imagine that, with increasing the temperature, the antiferromagnetic correlations will become eventually short ranged but they will be still dominant. Nevertheless, we do not know rigorously if the antiferromagnetic spin-correlation inequalities above deduced, should be valid at finite temperatures. Indeed, when the temperature grows, from zero to finite values, lowlying excited states become activated. Since they are orthogonal to the ground state, we cannot apply the positive definiteness of the ground-state wave function at finite temperatures, and hence a new procedure has to be introduced. In this respect, we will use a technique, introduced by Kubo and Kishi [130]. This method could be considered as the finite-temperature version of spin-reflection positivity method. Especially, they applied this technique to the Hubbard model, calculating upper bounds to the spin susceptibility for the attractive case and to the on-site pairing susceptibility for the half-filled repulsive model. For completeness, we notice that the approach used by Kubo and Kishi has been previously introduced in Ref. [78] to study the so-called Gaussian domination in a quantum model for magnetism. Here, we apply this method showing that, at any finite temperatures, the antiferromagnetic spin correlations in the symmetric PAM are always dominant [131]. Let us start introducing the canonical thermal average of a certain operator O:
ON =
1 Tr F (N) (e−HPAM O), ZN
where ZN = Tr F (N) (e−HPAM ) is the partition function. In this definition, F (N ) is a subspace of the Fock space for the PAM at fixed number of particle N. We notice that we can specify the occupation number, due to the fact that the PAM Hamiltonian commutes with N, the total particle
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number operator and therefore, the particle number is a good quantum number. Furthermore, the Hilbert space of the symmetric PAM can be divided into the subspaces F (N ), each of them is characterized by a specific particle number N. In particular, we will refer to the subspace F (NL ) as to the half-filled subspace. Since, HPAM also commutes with N↑ and N↓ , the number operators of particles with up-spin and down-spin, respectively, each subspace F (N ) can be further reduced to a direct sum of subspaces F (N1 , N2 ) of N1 up-spin and N2 down-spin particles with N1 +N2 = N . For instance, the half-filled subspace F (NL ) can be written as F (NL ) = F (0, NL )F (1, NL − 1) · · · F (NL − 1, 1)F (NL − 1, 0).
(114)
We prove now the following theorem: Theorem 7. At any T = 0, the transverse spin-correlation functions of the symmetric PAM, at half-filling, are antiferromagnetic. This theorem implies that, for any pair of lattice sites i and j, the following: 0 if n and m belong to the same sublattice, − NL
Sn+ Sm 0 if n and m not belong to the same sublattice.
(115)
We have proven above this inequality for the ground states of the PAM (T = 0); to verify that this result still holds at T = 0, we notice that ZNL , the partition function, is by definition a positive quantity. The inequality equation − ) verifies the same inequality. We also notice that the (115) is then satisfied if the numerator Tr F (N) (e−HPAM Sn+ Sm decomposition Eq. (114) of subspace F (NL ), indicates that the proof of the inequality equation (115) is reduced to − ). establish the same inequality in each partial-trace Tr F (NL −N,N) (e−HPAM Sn+ Sm As for the ground-state proof, due to the repulsive interaction in the PAM model, the finite-temperature spin-reflectionpositivity method does not work. Then, we introduce the partial particle–hole transformation used in the spin-reflection positivity section; this transformation maps the positive-U PAM Hamiltonian into a negative-U Hamiltonian of the same form. Moreover, each subspace F (NL − N1 , N1 ) is mapped into subspace F (N1 , N1 ), when the particle–hole transformation is concerned. Using these considerations, we now show that: Theorem 8. At any T = 0, the following relation holds: − Tr F (N,N) (e−HPAM (−U ) + n m ) 0
(116)
when any pair of lattice sites i and j are taken and the attractive PAM is considered. We notice that the previous Theorem 7 is nothing but a corollary of the present theorem. Indeed, assuming that this last Theorem 8 holds, by applying to the inequality equation (116) the particle–hole transformation we obtain − ) 0 (n)(m)Tr F (N,N) (e−HPAM (U ) Sn+ Sm
(117)
which is an antiferromagnetic quantity due to the presence of the form factors (i). Now, we simply outline the proof of the Theorem 8 since it strictly follows the proof of the similar inequality for negative-U Hubbard model [130]. Indeed, as extensively discussed, the PAM can be viewed as a generalized Hubbard model when a site index i is identified with a pair comprising the site index and a “flavor” of the orbitals [89]. Then, one firstly rewrites the trace to be evaluated as − Tr F (N,N) (e−HPAM (−U ) + n m ) (
− = Tr F (N,N) + n m exp − T↑ − 1/2
Uk (nk↑ − 1/2)2
k
+T↓ − 1/2
k
Uk (nk↓ − 1/2) − 1/2 2
) Uk (nk↑ − nk↓ )
2
,
k
where T represents the generalized hopping operator for fermions with spin .
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219
Then, we decouple the operators nk↑ from nk↓ in two steps. Firstly, by means of the well-known Suzuki–Trotter decomposition scheme [132], we decompose the quantity exp(−HPAM ) into a product of exponential operators, i.e. M Ui (ni↑ − ni↓ )2 , exp(−HPAM ) = lim exp(−H0 /M) × exp − M−→∞ 2M i
where we have indicated with H0 the single-particle part of the PAM Hamiltonian. As second step, we apply the well-known Hubbard–Stratonovich transformation to the exponential operators containing the interaction terms: +∞ 1 2 ) = √ exp(−O dp exp(−p 2 /4 + ip O). 2 −∞ Performing the above-described steps, we may rewrite the quantity under study in a form that the traces are complex conjugate to each other. Indeed, one has − Tr F (N,N) (e−HPAM (−U ) + n m )
= lim
M−→∞
1
√ (2 )MN L
× Tr F (N↑ =N)
× Tr F (N↓ =N)
$
⎛ dpl,m exp ⎝−
m,l
⎧ ⎨
† fn↑ fm↑
⎩
$
m,l
† fn↓ fm↓
⎩
$
4
⎞ ⎠
⎫ ⎬ exp − K↑ × exp i pm,l Ul /2Mnl↑ ⎭ M
l
m,l
⎧ ⎨
2 pm,l
exp −
m,l
K↓ × exp i M
l
pm,l Ul /2Mnl↓
⎫ ⎬ ⎭
,
where we have introduced the operator K = T − 1/2 l Ul (nl − 1/2)2 . Concluding, their product is a non-negative quantity, since the other factors in the integrand are also positive, and as a result, the whole expression is a positive quantity and the Theorem 8 is proven. 9.1. Discussion Since the PAM Hamiltonian has the SU(2) spin symmetry and its ground state is non-degenerate at half-filling, one can also show that the above reported Theorems 7 and 8 on spin correlations hold for longitudinal spin correlation functions. The proof of this claim is based on the fact that the ground state is a spin-singlet [89] so that S ± |0 = S z |0 = 0. Starting from this relation on the ground state, we can easily obtain the expected result applying the same procedure outlined in the previous Section 8 to show that the ground state is a supersolid. Of course, we have to replace the pseudospin operator with the spin operator. We want to stress that in the ground state of symmetric PAM, at half-filling, there is short-range antiferromagnetic order but we do not infer that the ground state exhibits long-range antiferromagnetic order. Indeed, we have shown that, in low-dimensional cases, the Anderson model for any filling, and thus also in the symmetric case, does not possesses any magnetic long-range order (see Section 5). Referring to the finite temperature case, we finally notice that in Ref. [131], apart from the proof for the PAM we had outlined above, it is also reported a similar result for the Kondo lattice model. 10. U = ∞ limit: special solutions of PAM In this section, we discuss some exact results holding in the limit of infinite Coulomb interaction strength. The first result refers to one-dimensional chains with open boundary conditions. We show that the PAM exhibits ferromagnetism
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when the filling is close to the quarter filling, providing that the position of the energy level for correlated electrons is sufficiently deep relative to the conduction band. Then, we present an exact solution of the PAM calculating the groundstate energy, and the corresponding eigenvector, when a D-dimensional decorated lattices with periodic boundary conditions for any dimension D 2 and arbitrary size is considered. 10.1. Ferromagnetism at quarter filling Since we will discuss on the ferromagnetic properties of the PAM, it may be useful to summarize some recent results on ferromagnetism in strongly correlated electron models [133]. Apart from the Heisenberg model [134] describing a scenario where the spin-independent Coulomb interaction and the Pauli exclusion principle generate the exchange interaction between localized electron spins, the one-band Hubbard model with nearest-neighbor hopping was one of the first models proposed to describe itinerant ferromagnetism [135]. Nevertheless, rigorous examples of ferromagnetism, or ferrimagnetism, in this model have been limited to singular situations such as the case of infinitely large Coulomb interaction (Nagaoka–Thouless ferromagnetism) [136] or when the magnetization is due to dispersionless band (Lieb ferromagnetism [99] and flat-band ferromagnetism [137–141]). Moreover, local stability of ferromagnetism in the Hubbard model with nearly flat bands was proven [142]. Apart from these rigorous theoretical examples, several numerical calculations confined to finite systems have shown that the Hubbard model, at special band filling and for suitable boundary conditions, exhibits fully polarized ferromagnetic ground states [143]. For real systems, the presence of orbital degeneracy is expected to be relevant. Indeed, it was firstly suggested by Slater [144], and then stressed by van Vleck [145] that the intra-atomic exchange, leading to the Hund rule, might be responsible for bulk ferromagnetism. Hnece, microscopic calculations stem from two-band models which are the natural extension of the single-band Hubbard model [146]. The first result of great relevance on the physics of the twoband models was the derivation of an effective strong-coupling Hamiltonian at quarter filling by Kugel’ and Khomskii [147], and by Cyrot and Lyon-Caen [148] who included the effect of the on-site pair hopping. In particular, it has been shown that this effective Hamiltonian leads to an insulating ferromagnetic ground state. This result has been confirmed by exact diagonalization studies of finite systems in one dimension, when the strong coupling limit is considered [149]. Exact results are still more rare for models describing electrons belonging to different orbital states, namely the Anderson model and its extensions such as for instance the Falicov–Kimball model [150–153]. Here, we will show that, for one-dimensional chains with open boundary conditions, the PAM exhibits ferromagnetism in the limit of strong Coulomb repulsion. This result holds close to the quarter filling, providing that the position of the local energy level for correlated electrons is deep enough compared the conduction band [154,155]. Theorem 9. Let us consider the one-dimensional case with open boundary conditions in the infinite U-limit. Denoting with NTOT the total number of electrons and with L the number of lattice sites, for a filling corresponding to NTOT =L+1 the restriction of the PAM to the subspace with basis vectors containing at most two conduction electrons exhibits a ground state which has total spin S = (NTOT − 1)/2 and is unique, apart from Sz degeneracy. The proof of this theorem is based on the fact that when the Hilbert space is truncated in such a way as to contain states with no more than two conduction electrons, the off-diagonal matrix elements of the Hamiltonian, restricted to this Hilbert space, are all non-positive, if the configuration space is connected by repeated applications of HPAM . Once the non-positive-definiteness of the off-diagonal matrix elements of HPAM is shown, one can apply the Perron–Frobenius theorem to the matrix representation of HPAM [156]. The Perron–Frobenius theorem concerns real matrices with non-positive, or equivalently non-negative elements. Specially, Perron proved a theorem on the spectra of positive matrices, then Frobenius generalized this theorem to irreducible non-negative matrices, considering that a positive matrix is a special case of an irreducible non-negative matrix. Before going into details, and to state the Perron–Frobenius theorem in a formal way, let us begin with some standard definitions [157]. A square matrix A is called reducible if under a permutations of its rows (columns) can be put into the form B 0 A= , (118) C D where B and D are square matrices; otherwise A is called irreducible.
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We write BC if and only if bij cij , for any i, j . Here, bij cij denote the elements of the B and C matrices, respectively. Of course, we will write B > C if the equality sign is omitted in the above inequalities. A matrix A is called non-negative, and indicated as A 0, or positive, and indicated as A > 0, if all the elements of A, aij are non-negative or positive, respectively. The maximum among the absolute value of the eigenvalues of A will be denoted a (A). Now, we are in the position to formally state the Perron–Frobenius theorem [156]: Theorem 10. Let A be an irreducible non-negative square matrix. Then • • • • •
(A) > 0 and is an eigenvalue of A; To this eigenvalue corresponds an eigenvector with positive coordinates; If A B 0, then for any eigenvalues i of B we have (A) |i |; If B has an eigenvalue |i | = (A), then A = B; (A) is a simple eigenvalue, i.e. a simple root of the secular equation.
The (A) eigenvalue is called the Frobenius eigenvalue of A. Before going in the proof of the ferromagnetism in the PAM, some comments need on the above-stated Theorem 10. If we multiply A by −1, we make all the matrix elements negative and we can speak on the lowest eigenvalue and we notice that the diagonal elements are not important. Finally, according to this theorem, the ground state is unique and is expressed as a linear combination of basis states with positive coefficients [42]. Coming back to our problem, it can be also shown that, under the assumptions above specified, in such ground state the total spin is maximized. We want to stress that the existence of a high-spin ground state can be proven only if the Hamiltonian is restricted to a reduced Hilbert space in which the role of the states with more than two conduction electrons can be neglected. This situation is expected to be realized when the f-electron site energy εf is deep with respect to the conduction bandwidth, and larger, in absolute value, than the hybridization coupling (|εf |?V ). This condition ensures that the system tends to maximize the number of f-electrons, with one electron per site compatibly with the constraint U = ∞. This implies that for NTOT + 1 electrons distributed on L sites, there is the tendency to have Nf = NTOT and Nc = 1. According to these considerations, we expect that a critical value εf c of εf exists, and below this value the spins of the f-electrons in the ground state are ferromagnetically ordered. Here, we give a proof of this result following the scheme reported in Refs. [154,155]. Under the assumptions of the Theorem 9, we restrict ourselves to a truncated basis set of states containing at most two conduction electrons. These state vectors are: |i; {n } = ci† f1†1 f2†2 . . . fN†TOT N
TOT
|0,
where {n } represents the set of NTOT (n = 1, 2, . . . , NTOT ) f-electrons. Applying to these states the Hamiltonian, we obtain † † † † † HPAM |i; {n } = − t(ci+1 + ci−1 )f11 f22 . . . fNTOT N
+ V ci†
j
TOT
|0 + NTOT εf |i; {n }
(−1)j −1 cj† f1†1 f2†2 . . . j . . . fN†TOT N |0, -./0 TOT
where we have introduced the symbol -./0 i to indicate that in the state, the creation f -operator of f-electron at i site is absent. Besides, we notice that the off-diagonal states proportional to t are always non-negative if we assume that t > 0. If we indicate with |i ↑↓; {n } the states † † † |i ↑, i ↓; {n } = |i ↓, i ↑; {n } = −(−)i ci↑ ci↓ f11 f2†2 . . . -./0 i . . . fN† N |0,
and with |ij ; {n } the states |ij ; {n } = (−)j ci† cj† f1†1 f2†2 . . . j . . . fN†TOT N |0, -./0 TOT
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we easily obtain
ij ; {n }|HPAM |i; {n } = −V ,
i = j ,
i ↑↓; {n }; |HPAM |i; {n } = −V −i . All these matrix elements are negative because the hybridization V is non-negative. In the subspace under study there are other states of the type |ij, k ; {n } = Ci j,k ci† ck† f1†1 f2†2 . . . j . . . fN†TOT N |0 -./0 TOT and if we arrange the phase factors Ci j,k in a suitable way, we can have negative values when expectation values between these states and the states |ij ; {n } are evaluated. Therefore, we can conclude that all the off-diagonal elements are negative and, according to the Perron–Frobenius theorem, the lowest eigenvalue state is unique and it can be written as a linear combination of all basis vectors with strictly positive coefficients. We can also prove that such state has total spin S given by S = (NTOT − 1)/2. Indeed, in the subspace Sz = (NTOT − 1)/2, we can find a state with S = (NTOT − 1)/2. This state is a linear combination of vector states in which one of the vectors contains a localized f-electron forming a singlet with a conduction electron vector. If we apply to this state the operator S − , we get a new state having the same S = (NTOT − 1)/2 but lower Sz . Since this state is a linear combination of basis states with non-negative coefficients, it has non-vanishing overlap with the ground state. This implies that the ground state has the total spin S = (NTOT − 1)/2. 10.2. Discussion The result here proven holds for one-dimensional chains with open boundary conditions; a rigorous proof of the theorem for closed chains being still lacking. However, from the application of the same procedure to a generalization of the Anderson model [158], one can see that at least for small clusters, a ferromagnetic configuration can be clearly identified even when periodic boundary conditions are considered, for suitable choice of the parameters of the model Hamiltonian. Moreover, from general arguments this result is expected to be valid also in the case of infinite lattices at arbitrary dimensions. Indeed, using the Schrieffer-Wolff transformation [159], the Anderson Hamiltonian can be mapped into a Kondo lattice model for which the ground-state configuration results from hopping processes involving pairs of sites occupied with localized equal-spin electrons. Thus, when εf is deep enough and in the case of NTOT =L+1, it is energetically favorable to accommodate NTOT f-electrons with the same spin while the remaining c-electron will partially compensate the momentum of f-electrons via a Kondo screening effect [160]. Finally, we mention the results presented in Ref. [161]. In this paper, the authors consider the strong hybridization and the strong coupling regime of the one dimensional PAM. They show rigorously that the low energy sector of the PAM maps into an effective Hamiltonian that exhibits a ferromagnetic ground state for any electron density between half and three-quarter filling. This wide region of concentrations clearly indicates that the microscopic mechanism is not related to an RKKY interaction, the stabilization of the ferromagnetic ground state being due to the coherent propagation of the Kondo singlet in a ferromagnetic background, similarly to the Nagaoka mechanism operating in the U-infinite Hubbard model. 10.3. Exact solution of the PAM for decorated lattices Let us discuss now the exact solution presented by Brandt and Giesekus in their seminal paper [162] on decorated PAM. In order to show the Brandt–Giesekus solution, we start with some definitions. A D-dimensional decorated cubic lattice is build-up in the following way. Let us denote the cubic lattice vectors by R, and the D basis vectors by ei . The electronic states are located at the decorated lattice sites ri = R + 1/2ei . In other words, they are located at the centers of cubic bonds. Obviously, the unit cell contains D sites. For the following, it is useful to introduce the set N (R) composed by 2D sites ri = R ± 1/2ei . For example, for D = 3 the set N (R) forms octahedra and the structure is similar to ReO3 . We note that the lattice structure of this compound is similar to the perovskite structure CaTiO3 without the calcium ion. Hence, the basis vectors are b1 = (0, 0, 0) for Re ion and b2 = (a/2, 0, 0), b3 = (0, a/2, 0), and b4 = (0, 0, a/2) for the O ions. Here, a is the lattice constant (see Fig. 5).
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Fig. 5. Crystal structure of ReO3 compound. The black balls denote the rhenium ions while the white ones the oxygen ions.
The hopping processes are allowed only between nearest-neighbor sites and those next-nearest-neighbor sites that are connected across the center of a D-dimensional octahedron. Both types of hopping are required to have the same matrix element −t, where t is positive. Finally, periodic boundary conditions are assumed. Therefore, a generic hopping term reads Ht = −t
R r=r ∈N(R)
cr† cr .
(119)
Analogously, the localized orbitals are described by the on-site term Hf = εf
R
fR† fR
(120)
while the effect of the hybridization between itinerant and localized electrons is included in the usual hybridization term Hf c = V (fR† cr + H.c.). (121) R r∈N(R)
Within Brandt–Giesekus approach, the Anderson model is studied in the limit of infinite Coulomb interaction strength. This limit simply excludes double occupancy of orbitals at the same localized lattice site. In this manner, the interaction reduces the Hilbert space of the Anderson Hamiltonian to those states having occupation number of localized electrons at any R site not exceeding one. The operator projecting the Anderson Hamiltonian onto this subspace takes the form: P=
$
† † (1 − fR↑ fR↑ fR↓ fR↓ ).
(122)
R
Then, the Anderson Hamiltonian excluding f-orbital double occupancy is %PAM = P (Ht + Hf + Hf c )P . H
(123)
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Now, to solve the model we introduce the following generalized operators ⎛ ⎞ 1 cr† ⎠ + bf †R , †R = a ⎝ √ 2D r∈N(R) being |a|2 + |b|2 = 1. The Hamiltonian then reads as √ 2Dt b 2Dt b2 % Hf − Hf c P HPAM = P Hc − εf a 2 V a ( 2 ) 2Dt 4DtN b † = 2 P. R P R − P − 2t Nc + (2 − )D Nf 2 a a a
(124)
(125)
R
The parameter above introduced allows to adjust the sum R fR† fR = /εf Hf + (1 − )Nf in a way it can be interpreted as the on site energy term or f-particle number operator Nf . This representation of the Hamiltonian is an operator identity and no approximations are made. If we are able to %PAM and a positive rewrite this operator expression in term of some c-numbers, some operators commuting with H semi-definite remainder, we can immediately estimate the ground-state energy from below. If the coefficients of Hf and Hf c become identical to −1 and the combination of occupation number operators in the right-hand side add up to produce the conserved quantity Nc + Nf , then we obtain the simple solution: V 2 = t2 −
tεf , 2
(126)
implying that εf must be smaller than 2t. Therefore, the ground-state energy E0 is larger than 2t (Nc + Nf ) − 4DtN /a 2 . To show that E0 is indeed equal to the upper bound, we consider the following projected state: (127) | = P |0 , 1 %PAM | vanishes. where |0 = R, †R |0. Since P †R P †R = 0 the only non-trivial in the eigenvalue equation H % As consequence, the state | is an eigenstate of HPAM and the following inequality holds: %PAM | = 2t (Nc + Nf ) − 4DtN/a 2 . E0 |H
(128)
Since the upper bound and the lower bound on the ground-state energy coincide, we have obtained the true ground-state energy. We notice finally, that the quantity a 2 into the expression of E0 is given by 2t − εf −1 2 . a = 1+ 4Dt Differently from the results obtained when this procedure is applied to the Hubbard model, the solution here presented is restricted to a special region of (εf , V ) space parameter. Nevertheless, when real systems are concerned, the above restriction is located in a physical relevant regime. 10.4. Discussion Although the solution above presented refers to lattice with special topology and for suitable choice of parameters of the PAM, it may provide a basis for future studies, in which one can extracts various physics out of the exact ground state of the Brandt–Giesekus type. It is worth stressing that Strack has generalized the approach here presented, considering the infinite U Anderson model with arbitrary spin degeneracy in one-dimensional case for a restricted, but physical reasonable, parameter regime at half-filling [163], while Tasaki proved the uniqueness of the ground state for a specified electron density
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[164]. This solution has been also extended to two and three dimensions [165,166]. The first exact ground states at finite value of the Coulomb repulsion have been recently published in one-dimensional [167,168] as well as in twodimensional case [169–171]. These solutions have been obtained by a decomposition of the Hamiltonian in terms of positive semi-definite operators and in this way the ground states emerge on continuous, although restricted, regions of the T = 0 phase diagram of the system. This phase diagram extend from the low U limit up to the high U limit. To decompose the Hamiltonian in positive semi-definite operators one uses the following procedure: firstly, the Coulomb repulsion term is exactly transformed into a positive semi-definite term, then the remaining part of the Hamiltonian is written again as positive semi-definite expression using cell operators. Namely, in one-dimensional one can make use of bond operators while in two-dimensional one can use the elementary plaquette operators. Nevertheless, we notice that the Anderson model one refers to in this case contains always the non-local hybridization matrix term while exact results can be achieved with or without the direct hopping term between f-electrons. Finally, we mention that very recently it has been shown that it is possible to construct, within the same procedure above outlined, exact ground-state wave functions describing metallic and insulating phase for the Anderson model at non-integer electron filling even in three dimensions. By means of this approach, it has been demonstrated the insulating and/or the conducting nature of the solutions; the presence of strong variations in the compressibility of the system when leaving the insulating phase and finally the non-Fermi liquid nature of metallic phase. We want to stress that here we do not devote more details to these papers since they refer to Anderson Hamiltonian with or without hopping term for correlated electrons and more importantly a non-local hybridization is always considered. 11. Total momentum of the one-dimensional PAM In this section, we state and prove a result that can be deduced only for the one-dimensional PAM with periodic boundary conditions. Namely, it is possible to show the following theorem [104]: Theorem 11. The ground state of the symmetric PAM on a one-dimensional bipartite lattice with periodic boundary conditions has total momentum equal to 0 for any even number of lattice sites L. The proof of this theorem is here reported: for U = 0 the periodic Anderson Hamiltonian reduces to the kinetic and hybridization terms only and from the assumption on the dimensionality of the lattice we have the following quasi-particle energy spectrum (see Section 4.2): k ± 2k + 4V 2 ± , (129) k = 2 where the wave vector k assumes the following values: k=n
2 L
with L being the number of lattice sites, n = 0, 1, 2 . . . L − 1 and k = −2t cos k (See Section 4). We can rearrange the wave vectors k as follows: 2 2 1 2 2 1 , 2 ; 2 1 − , {}, k = {0}, 2 ; 2 1 − L L L L
(130)
(131)
in such a way that for the k values enclosed between parenthesis we have the same energy ± k. At the half-filling, the ground state is unique, as shown by spin reflection positivity, and the lower band − is completely filled. The total momentum P of the ground state is then 2k = 2(L − 1) = 0 [mod 2]. (132) P= k
Therefore, at vanishing U the ground state is obtained filling completely the lower hybridized band and leaving the upper band empty. It is obvious that there is a finite energy gap to the excited states, so that from a continuity argument with respect to U the unique ground state for any positive or negative U also has P = 0 since there is no level crossing.
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11.1. Discussion It is worth noticing that, from the parity symmetry of the model, the unique ground state has to possess necessarily a total momentum equal to 0 or , in 2 unit. Nevertheless, one cannot determine what among 0 and is the value of the momentum both in the repulsive and attractive cases. The above proven Theorem 11 solves this ambiguity characterizing the total momentum of the ground state of the model. Furthermore, using the results of previous Section 6, the half-filled ground state may be fully characterized in terms of quantum numbers associated with S 2 , J 2 and the total momentum. Indeed, we can state that the ground state of the symmetric one-dimensional PAM on a bipartite lattice with periodic boundary conditions is a spin and pseudospin singlet and has total momentum equal to zero. We want to mention that a similar characterization of the ground state of the half-filled one dimensional Hubbard model with periodic boundary conditions has been given in [172]. In this case it has been proven that the ground state is a spin and a pseudospin singlet and has total momentum equal to 0 or for repulsive case if the number of sites is odd or even, respectively, and total momentum equal to zero when the attractive case is considered. As final consideration we notice that numerical calculations, in spite of many advantages, have severe memory limitations imposed by the size of clusters that can be studied. Indeed, the basis set of vectors employed in the diagonalization procedure grows largely with the increasing of the system size. This problem may be considerably alleviated using the symmetries of the Hamiltonian that reduce the matrix Hamiltonian to a block form with submatrices having reduced dimension. The most obvious symmetry is the U(1) symmetry that is usually conserved at least for fermionic problems. The third component of total spin operator S may also be a good quantum number and, for translational invariant Hamiltonians, the total momentum P of the system is conserved as well. Therefore, the full quantum characterization of the ground state as that here presented could turn out useful in numerical study of the PAM, at least in one-dimensional case with periodic boundary conditions. 12. Conclusions and perspectives As a model for strongly correlated electron systems, the periodic Anderson model has been widely used to discuss many questions concerning among others the nature of Fermi liquid state with heavy mass; the interplay between the Kondo effect and the RKKY interaction; the magnetic nature of the ground state, ferromagnetic or antiferromagnetic, the superconductivity and the metal–insulator transition. The model is largely applied in the study of real physical systems whose properties can be interpreted in a two-band picture such as heavy-fermion systems [173], mixedvalence compounds [174] or even high-temperature superconductors [175]. However, the physics described by the Anderson model is almost exclusively interpreted on the basis of approximations. This is a consequence of the fact that only few results are exactly known about the behavior of the PAM. Therefore, the possibility to establish some rigorous results for this model, can be the starting point to master some key features of realistic systems. Indeed, rigorous results, numerical or analytical, greatly aid the study of strong correlations in microscopic models. Considering the single impurity Anderson model, the renormalization group method and the Bethe ansatz solution are perhaps the best examples of solid approaches and exact solutions of a rather simple model that changed and solidified the thinking in what was a highly controversial and puzzling problem area. For dense systems, the natural extension being the PAM, the renormalization group procedure and Bethe ansatz solution are still lacking even in one dimension. What remains true, however, is the connection between the strong and weak coupling limits via a natural extension of the Schrieffer–Wolff transformation. Moreover, many of available exact results are based on symmetry properties exhibited by the microscopic Hamiltonians. We notice that, symmetry arguments not only may simplify the calculations of physical quantities, but also often give a detailed insight into the physical situation. Indeed, symmetry is central to our understanding and description of natural phenomena. The fundamental conservation laws of physics, such as the conservation of momentum and energy, are the consequences of symmetries in space and time; also, our understanding of the forces of nature is based on a local symmetry, i.e. the gauge symmetry. Therefore it is amenable to carry out a symmetry analysis before undertaking any calculations or experimental investigations on specific problem. This is the aim of the review here presented: we have described some rigorous results on the PAM and we have given a detailed description of symmetry properties exhibited by the model. Among the exact results, first of all, we have described some solutions holding for any filling and dimensions for special choices of interaction couplings. We refer to, for instance, to the solution of the model in the atomic limit.
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In this case, we can classify the eigenstates of the Hamiltonian in terms of conserved quantities such as the total occupation number and total spin. Although these solutions are oversimplified, they retain in some cases the properties of the general model. Then we have studied the possibility to have ordered states at finite temperatures using the Bogoliubov’s inequality and we have presented a very recent result on the T = 0 case. We have shown that, at least in one dimension, the quantum fluctuations disorder the system preventing the formation of ordered states of magnetic or superconducting nature. We have subsequently introduced the spin-reflection positivity method and we have discussed many exact results. Namely, making use of spin up-down symmetry we have chosen a suitable basis set of states and expressed a generic state as square matrix. We have proven the positive definiteness of the ground state and presented many results on correlation functions. Among the relevant results, we have shown that at half-filling the ground state is a spin and pseudospin singlet and we have deduced that the ground state can be a supersolid. Finally, we have discussed some rigorous results holding when the U = ∞ limit is considered. Nevertheless, several questions are still open. Among the more relevant, we mention the condition of half-filling extensively used in the application of spin reflection positivity procedure. This is a very strong restriction since, at the present whether the antiferromagnetic or ferromagnetic long-range order could survive when the deviation of half-filling case is concerned, is still a challenging problem. Referring to the application of Bogoliubov’s inequality, the results presented here, and for other correlated models, are valid in reduced dimensions. At the moment, results based on the application of this inequality are inconclusive when three-dimensional realistic situations are treated. Furthermore, an interesting theoretical issue to be investigated is whether the coexistence of ODLRO and CDW at the special point of pseudospin rotational invariance can be extended to a finite region of the parameters by introducing additional couplings. Moreover, we expect that numerical calculations can shed light on the physical properties of the PAM. Indeed, we think that they could suggest the search of new and powerful theoretical methods to inquire this model, as a specific example, and also other strongly correlated electron models. Finally, we would like to mention a recent paper on the Anderson model [176] whose title is “The Anderson model: why does it continue to be so fascinating?” Apart from the appealing title and thus from the interest towards this model, in that paper important questions are addressed. Specifically the relevance of the solution of the single impurity Anderson Hamiltonian; the relationship between the Anderson model and the Kondo model; the important role played by the Anderson model within many-body physics and finally the difficulty to manage the lattice problems. In light of the review here presented, we hope that modestly, at least for the two last questions, some positive answers have been provided. Acknowledgments I am indebted to Mario Cuoco who over the years has been a source of stimulation and encouragement, and has increased my depth of understanding of the subject of this review. I am also grateful to him for careful reading of early drafts of the manuscript. I also wish to thank Alfonso Romano for many useful discussions concerning the physics of the Anderson model we had during last years. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
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Physics Reports 431 (2006) 231 – 259 www.elsevier.com/locate/physrep
Inhomogeneous superconductivity and the “pseudogap” state of novel superconductors Vladimir Z. Kresina,∗ , Yurii N. Ovchinnikovb , Stuart A. Wolfc a Lawrence Berkeley Laboratory, University of California at Berkeley, CA 94720, USA b L.D. Landau Institute for Theoretical Physics, Russian Academy of Sciences, 117334 Moscow, Russia c Department of Materials Science and Engineering and Department of Physics, University of Virginia, Charlottesville, VA 22903, USA
Accepted 20 May 2006 editor: David D. Awschalom
Abstract Many novel superconducting compounds such as the high Tc oxides are intrinsically inhomogeneous systems by virtue of the superconductivity being closely related to the carrier density which is in turn provided in most cases by doping. An inhomogeneous structure is thus created by the statistical nature of the distribution of dopants. At the same time doping also leads to pair-breaking and, consequently, to a local depression of Tc . This is a major factor leading to inhomogeneity. As a result, the critical temperature is spatially dependent: Tc ≡ Tc (r). The “pseudogap” state is characterized by several energy scales: T ∗ , Tc∗ , and Tc . The highest energy scale (T ∗ ) corresponds to phase separation (at T < T ∗ ) into a mixed metallic-insulating structure. Especially interesting is the region Tc∗ > T > Tc where the compound contains superconducting “islands” embedded in a normal metallic matrix. As a result, the system is characterized by a normal conductance along with an energy gap structure, anomalous diamagnetism, unusual a.c. properties, an isotope effect, and a “giant” Josephson proximity effect. An energy gap may persist to temperatures above Tc∗ caused by the presence of a charge density wave (CDW) or spin density wave (SDW) in the region T > Tc∗ but less than T ∗ , whereas below Tc∗ superconducting pairing also makes a contribution to the energy gap (Tc∗ is an “intrinsic” critical temperature). The values of T ∗ , Tc∗ , Tc depend on the compound and the doping level. The transition at Tc into the dissipationless (R = 0) macroscopically coherent state is of a percolation nature. © 2006 Elsevier B.V. All rights reserved. PACS: 74.72.−h; 74.81.−g; 74.62.−c
Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 2. “Pseudogap” state: main properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 2.1. Conventional superconductors: fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 2.2. Anomalous diamagnetism above Tc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 2.3. Energy gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 2.4. Transport properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 2.5. “Giant” Josephson proximity effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 ∗ Corresponding author.
E-mail address:
[email protected] (V.Z. Kresin). 0370-1573/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physrep.2006.05.006
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2.6. Isotope effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 2.7. Various models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 3. Inhomogenous superconductivity and the “pseudogap” phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 3.1. Qualitative picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 3.2. Phase separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 3.3. Inhomogeneity: experimental data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 3.4. Energy scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 4. Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 4.1. General equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 4.2. Diamagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 4.3. Density of states: gap structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 4.4. ac transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 4.5. Isotope effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 5. “Giant” Josephson proximity effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 6. Other systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 6.1. Borocarbides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 6.2. W O 3 + N a compound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 6.3. Granular superconductors; Pb+Ag system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 7. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
1. Introduction Novel superconducting systems and, first of all, the high Tc oxides, display many properties in the normal state above Tc , which are drastically different from those for conventional materials. This unusual normal (it is called “normal” because of finite resistance) state was dubbed the “pseudogap” state. The first observation of this state was reported in 1989, that is, shortly after the discovery of high Tc superconductivity (Bednorz and Muller, 1986). NMR measurements demonstrated the presence of an energy gap for spin excitations (Alloul et al., 1989; Warren et al., 1989). Later, the presence of an energy gap was observed by various techniques, such as tunneling, infrared, photoemission, heat capacity, etc. One should stress that the presence of the gap is important, but, nevertheless, not the only special feature of the state above Tc . One can also observe peculiar magnetic transport and microwave properties as well as an isotope effect. An unusual “giant” Josephson proximity effect has also been observed. In the following review we will describe (Section 2) the main properties of the “pseudogap” state. The study of the “pseudogap” state has attracted a lot of attention. There are a number of experimental and theoretical papers describing interesting data and various theoretical models (see, e.g., reviews by Timusk and Statt, 1999; Orenstein and Millis, 2000; Kivelson et al., 2003; Norman et al., 2005). There is a general question about the definition of the “pseudogap” state. The consensus is that we are dealing with an unusual normal state, but such a definition is too vague. Some authors define the “pseudogap” state as a state above Tc with an energy gap; this even resonates with the notation of the state. A more general definition implies that above Tc the sample is in a peculiar state which is intermediate between fully superconducting and normal. Indeed, as in any normal metal, the sample displays a finite resistance. In addition, such a key feature as macroscopic phase coherence does not persists above Tc . However, one can observe some features typical for the superconducting state such as the energy gap, anomalous diamagnetism, and the isotope effect. It is interesting that the temperature scales (e.g., for the energy gap vs. diamagnetism) could be different. In addition, the manifestation of the “pseudogap” state depends on the doping level, being the strongest for the underdoped region. Although the major focus of recent experimental studies has been on the high Tc cuprates, the presence of the “pseudogap” state was reported also for other superconducting systems, such as borocarbides, bronzes, as well as for some more unusual conventional superconductors. We will discuss these properties in Section 6. During the last several years we have been greatly involved in the description of this peculiar state (Ovchinnikov et al., 1999, 2001, 2002; Kresin et al., 2000, 2003, 2004).
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The structure of the paper is as follows. Section 2 contains a cross-cut description of the major experimental data. The concept of intrinsic inhomogeneity and the model describing the “pseudogap” state will be analyzed qualitatively in Section 3. The different energy scales (T ∗ , Tc∗ , Tc ) will be introduced. Section 4 contains a detailed theoretical analysis and comparison with experimental data. The “giant” Josephson proximity effect is analyzed in Section 5. Section 6 focuses on other novel superconductors. And finally, Section 7 contains concluding remarks and some open questions. 2. “Pseudogap” state: main properties In general, novel superconducting systems and, in particular, cuprates are characterized by a finite resistance above Tc and also macroscopic phase coherence disappear above Tc . However, a number of other features typical of the superconducting state, such as anomalous diamagnetism, energy gap, peculiar a.c. properties, etc. are observed above Tc . Of course, the presence of each isolated feature (e.g., energy gap, see below) might have several alternative explanations, but the theory should provide an unified explanation describing all relevant data. Note also, that the manifestations of various anomalous properties depends on the level of doping. The strongest anomalies occur in the underdoped region. In this section we describe the main properties of the “pseudogap” state. As was mentioned above, the first experimental observation of this state was carried out with the use of the NMR technique. Later, many experimental methods have been employed; (some of them will be described below; see also review: Timusk and Statt, 1999). Further below in this section (not in chronological order) we will follow some sequence which allows us to analyze the data in a consistent way. 2.1. Conventional superconductors: fundamentals In order to contrast the behavior of novel superconductors above Tc with that for usual systems, we describe in this section several fundamental properties of conventional superconductors; see also Table 1 which directly contrasts the two classes. The most fundamental feature is the anomalous diamagnetism (Meissner effect) observed below the critical temperature. As for the region above Tc , the magnetic response of conventional metals is relatively small and almost temperature independent. The energy gap is a fundamental microscopic parameter. As we know, = 0 in the normal phase and opens up at Tc . One should note, however, that the energy gap is, indeed, an important parameter, but its presence is not a crucial factor for superconductivity. For example, one can observe “gapless superconductivity” (Abrikosov and Gor’kov, 1961), caused by the pair-breaking effect, e.g., by the presence of localized magnetic moments. As we know, superconductors have a finite resistance above Tc whereas below the critical temperature they are in the dissipationless state (R = 0). In addition, a.c. transport behaves in a peculiar way, namely, above Tc , with high accuracy (see, e.g., Landau and Lifshitz, 1960), Re Z = Im Z (Z is the surface impedance). Contrary to this, below Tc one can observe a strong inequality Re Z = Im Z. The superconducting state is also characterized by macroscopic phase coherence. For example, such a remarkable phenomenon as the Josephson effect is directly related to this feature. Table 1 “Pseudogap” state vs conventional superconducting and normal states
Resistance Energy gap Anomalous diamagnetism Macroscopic phase coherence Josephson effect Isotope effect Impedance Z
Superconducting state (T < Tc )
Normal state (T > Tc )
“Pseudogap” state (Tc∗ > T > Tc )
R=0 = 0 Yes Yes Yes Yes Re Z = Im Z
R = 0 =0 No No No No Re Z = Im Z
R = 0 = 0 Yes No “Giant” effect Yes Re Z = Im Z
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Of course, because of fluctuations, the superconducting state can persists above Tc , but such a contribution can be properly analyzed (see e.g., Larkin and Varlamov, 2005). One can continue the list of properties contrasting the superconducting and normal state, but here we restrict ourselves to the aforementioned features (see also Table 1), because it is sufficient for our purpose. 2.2. Anomalous diamagnetism above Tc Usual normal metals display relatively weak response to a small external magnetic field. Indeed, the electronic gas is characterized by small Pauli paramagnetism. The magnetic susceptibility of real metals consists of several contributions (see, e.g., Ashcroff and Merman, 1976) and the resulting response might be diamagnetic, but the total susceptibility is almost temperature independent. The situation above Tc in the cuprates appears to be drastically different. Unusual magnetic properties of the “pseudogap” state have been observed by several groups. The scanning SQUID microscopy was used by Iguchi et al. (2001) to study the underdoped La2−x Sr x CuO4 compound. This technique allows one to create a local magnetic image of the surface, its “magnetic” map. The critical temperature of the underdoped LSCO films was: Tc ≈ 18 K. A peculiar inhomogeneous picture has been observed (Fig. 1): the film contains diamagnetic domains and their presence persists up to 80 K (!). The total size of the diamagnetic regions is growing as the temperature is decreased (Fig. 2). The diamagnetic response appears to be strongly temperature dependent; this is a very unusual feature of the materials. A strong temperature dependent diamagnetic response above Tc ≈ 15 K has also been observed by Bergemann et al. (1998) by using the torque magnetometry technique for the overdoped Tl2 Ba2 CaO+ compound. Like LSCO, the diamagnetic moment was also strongly temperature dependent (Fig. 3). Torque magnetometry was also employed recently by Wang et al. (2005) to study the Bi2212 compound. Similarly, diamagnetic response was observed. It is essential that the analysis ruled out fluctuations as a key source of the observed diamagnetism. An interesting study of the YBCO compound has been carried out by Caretta et al. (2000) and by Lascialfari et al. (2002). The magnetization of the underdoped vs. optimally doped samples was measured by using special SQUID magnetometers. The observed diamagnetic response could be caused by superconducting fluctuations above Tc (see e.g., Larkin and Varlamov, 2005). An anisotropic Ginzburg–Landau functional was employed to analyze the data. It has been concluded that fluctuations play an exceptional role and the diamagnetism above Tc for the sample with optimum doping can be explained by their presence. However, and this is important for our analysis, their contributions are not sufficient to explain the data for the underdoped sample, and it is necessary to take into account the inhomogeneity of the structure (see below) to account for the observed diamagnetism. This effect is especially significant for the underdoped cuprates. The strong diamagnetic response above Tc has been reported not only for the cuprates, but also for other novel superconducting systems. We will describe these systems below (Section 6). 2.3. Energy gap The presence of the energy gap above Tc has been observed using various experimental methods. Even the title “pseudogap state” reflects the existence of the gap structure. In connection with this it is worth noting that this title is confusing, since we are dealing not with a “pseudogap” but with a real gap, that is, with real dip of the density states in the low energy region. Let us start with tunneling spectroscopy which allows one to perform the most detailed and reliable study of the gap spectrum. The data obtained by Renner et al. (1998) for an underdoped Bi2212 crystal are shown in Fig. 4. The scanning tunneling microscopy (STM) of the crystals cleaved in vacuum was employed. One can see directly the dip in the density of states (energy gap) which persists above Tc ≈ 83 K and stays up to ≈ 200 K (!). One should note several interesting features of the data plotted in Fig. 4. First of all, the gap structure changes continuously from the superconducting region T < Tc to the “pseudogap” state (T > Tc ); there is not any noticeable change at Tc . This can be considered as an indication that the gap structure above Tc is related to superconducting pairing. The second interesting feature is that an increase in temperature affects the depth of the dip, but not the value of the gap, which is determined by its width. Such an unusual independence of the gap magnitude on temperature will be
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Fig. 1. Development of magnetic “islands” with temperature (magnetic imaging of LSCO films).
discussed in more detail below (Section 4.2). Tunneling measurements with similar results have been also carried out by Ekino et al. (1999). The presence of an energy gap has been also determined using infrared spectroscopy. As a matter of fact, infrared spectroscopy was used in the pioneering works by Glover and Tinkham (1956), Ginsberg and Tinkham (1960), see Tinkham (1996); this was the first experimental observation of the gap in conventional superconductors. The measurements of the c-axis conductivity in the underdoped YBCO compound and the corresponding analysis (Homes et al., 1993; see also Orenstein et al., 1990; Opel et al., 1999; Kaindl et al., 2000) reveal the presence of a gap at temperatures above Tc (Fig. 5). Again, it is interesting to note that the value of the gap (≈ 400 cm−1 ) is not noticeably affected by the transition Tc to the dissipationless state at higher temperatures. The gap persists up to T ≈ 300 K. It is important that an increase in temperature leads to an decrease in depth of the gap, but the value of the gap (the width) remains almost constant and this is very similar to what is observed in tunneling (see above).
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Fig. 2. Parameters of magnetic domains vs temperature for LSCO films.
Fig. 3. Diamagnetic susceptibility for the Tl2 Ba2 CuO6+ (Tc = 15 K): experimental data; solid line-theory.
A special type of spectroscopy, the so-called “intrinsic tunneling spectroscopy” was developed by Suzuki et al. (1999) and by Krasnov et al. (2000). It is based on the fact that below Tc the transport in the c-direction represents an intrinsic Josephson current (Kleiner et al., 1992; Scheekga et al., 1998). This method can provide detailed information about the density of states. At the same time, intrinsic tunneling spectroscopy is sensitive to the stoichiometry of the sample,
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Fig. 4. Tunneling conductance of states for the underdoped Bi 2212 crystal.
Fig. 5. The optical conductivity of Yba2 Cu3 O6.7 along the c-axis.
so that the method can be reliably used in the region near optimum Tc . According to the study of the temperature dependence of the I–V characteristics, one can distinguish two energy gaps, one is the pairing gap at T < Tc and the second is a gap which differs from zero in a large temperature range below T ∗ which is associated with the pseudogap.
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The presence of an energy gap for spin excitations has been established using NMR. Actually, the “pseudogap” state was initially observed using this method. It has been observed in YBCO for different nuclei in both the Knight shift and spin-relaxation rate experiments: for 89Y (Alloul et al., 1989, for 63 Cu; Warren et al., 1989; Walstedt and Warren, 1990; Walstedt et al., 1990), for 17 O; Takigawa et al., 1989, and in HgBa2 Cu4+x (Bobroff et al., 1997). Photoemission spectroscopy has also revealed the presence of an energy gap at T > Tc (see the reviews by Shen and Dessau, 1995; Randeria and Campuzano, 1997). For example, it has been demonstrated that the energy gap persists in an underdoped sample of Bi2212 up to 95 K; this temperature is much higher than Tc = 79 K (Loeser et al., 1996), see also Ding et al. (1996). The data above presents direct spectroscopic observations of the gap structure above Tc . A gap in the spectrum can also be inferred from heat capacity data. One should note that measurements carried out by Loram et al. (1994), Wade et al. (1994) were one of the first observations of the “pseudogap” state. The measurements of the Sommerfeld constant (T ) for YBa2 Cu3 O6+x display a loss of entropy caused by the gap structure. The data for the energy gap (T ) are derived from values of the electronic entropy S(T , x) for 0.73 < x < 0.97. Again, it has been observed that the energy gap persists for T > Tc , and the effect is especially strong for the underdoped samples. The substitution of Zn impurities for Cu2+ on the CuO2 planes leads to depression of Tc and a strong impact on the value of (T ) and thermoelectric power (Tallon et al., 1995). The La2−x Sr x CuO4 compound was studied in detail by Momono et al. (1999). The data are similar to those of Loram et al. (1994). The authors also performed measurements for samples with Ni substitution. 2.4. Transport properties Let us now discuss other interesting features observed above the resistive Tc , in the “pseudogap” region. As we know, the resistivity in the normal state at optimum doping is described by a linear temperature dependence n ∝ T . However, a noticeable deviation from a simple linear law has been observed in the underdoped region (Takagi et al., 1992; Hopfengartner et al., 1994; Fruchter et al., 2004; Sfar et al., 2005). The microwave properties and a.c. transport have been studied by Kusco et al. (2002). It was shown that above Tc , that is, in the normal state Re Z = Im Z, where Z is the surface impedance. This is an unusual property, since in ordinary normal metals with a high degree of precision, the real and imaginary part of the impedance are equal (see e.g., Landau and Lifshitz, 1960), that is Re Zn Im Zn . The observed inequality Re Z = Im Z is typically observed in superconducting materials. In describing transport properties, one should also mention interesting data on the thermal Nernst effect (Xu et al., 2000; Wang et al., 2001; Ong and Wang, 2004). This effect is analogous to the Hall effect, but it is manifested by the appearance of an electric field created by an external magnetic field in the presence of a thermal gradient. Based on the data, the authors concluded that above Tc , in the “pseudogap” state, one can detect the presence of vortex-like excitations. Indeed, the value of the Nernst coefficient greatly exceeds that for any normal metal. On other hand, conventional superconductors of II type in the mixed state display a large value of this coefficient (Josephson, 1965). A careful analysis of the Nernst data (Wang et al., 2005) reveals that the sharp rise in Nernst coefficient occurs very close to Tc , that is in the region that can have large superconducting clusters which in principle can contain vortices (see below, Section 3.4). As was noted above (Section 2.2), Wang et al. recently performed torque measurements which reveal diamagnetism above Tc ; as a result, it was concluded, that the superconducting state persists above Tc . One should also note that previous torque measurements (Bergemann et al., 1998) on the overdoped T l-based cuprate, also display diamagnetism above Tc (see Sections 2.3 and 4.2) and the vortices were not detected even though this technique is quite sensitive to their presence. A direct probe of the fluctuations above Tc was described recently by Bergeal et al. (2006). The experiment was carried out with use of a Josephson-like junction, so that the presence of macroscopic phase coherent pairing state above Tc was checked. It has been demonstrated that fluctuations survive only in the region close to Tc (T ≈ 10–15 K) and cannot be responsible for the “ pseudogap’ behavior in all region. The measurements of normal resistivity (Darhmaoui and Jung, 1998;Yan et al., 2000; Jung et al., 2000) have revealed a strong inhomogeneity of YBCO and TBCCO samples. The presence of two different phases and inhomogeneous structure of the order parameter has been demonstrated.
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Interesting data on thermal conductivity for various high Tc compounds were described by Sun et al. (2006). The measurements performed for different doping levels reveal the absence of a universal dependence for the thermal flow and indicate the importance of strong inhomogeneity in the cuprates. 2.5. “Giant” Josephson proximity effect The so-called “Giant” Josephson proximity effect is another interesting phenomenon observed in the “pseudogap” region above Tc (Bozovic et al., 2004). The films of La0.85 Sr 0.15 CuO4 (Tc ≈ 45 K) were used as electrodes whereas the underdoped LaCuO compound (Tc ≈ 25 K) formed the barrier which was prepared in the c-geometry (the coherence length c ≈ 4 A). The measurements were performed at Tc < T < 35 K, so that the barrier was in the “pseudogap” state. Since T > Tc , we are dealing with an SNS junction. As is known, for such a junction, the thickness of the barrier should not exceed the coherence length N = hv ¯ F /2T which is of the order of c . However, the Josephson current was observed for thicknesses of the barrier up to 200 A(!). Such a “giant” effect cannot be explained using conventional theory. We will discuss this effect in detail in Section 5. 2.6. Isotope effect Another interesting property of the “pseudogap” state is the strong isotope effect. This effect has been observed by Lanzara et al. (1999) for La2−x Sr x CuO4 using X-ray absorption near-edge spectroscopy (XANES). The effect has been also observed by Temprano et al. (2000) for the HoBa2 Cu4 O8 compound. The slightly underdoped HoBa2 Cu4 O8 sample was studied by neutron spectroscopy. As we know (see, e.g., review by Mesot and Furrer, 1997), the opening of the gap which could be associated with the “pseudogap” affects the relaxation rate of crystal field excitations. The isotopic substitution 16 O→18 O leads to a drastic change in the value of the pseudogap temperature Tc∗ (Tc∗ ≈ 170 K → Tc∗ ≈ 220 K). Such a large isotope shift corresponds to a value of the isotope coefficient ∗ = −2.2 ± 0.6. Note that, contrary to the typical superconductor, its value is negative. 2.7. Various models As was noted in the Introduction, the unusual “pseudogap” state observed above Tc , has attracted a lot of attention. Correspondingly, several theoretical models have been proposed. These models are mainly related to the origin of the energy gap above Tc . Since there are multiple causes for the presence of a gap, it is not surprising that a number of different models have been developed. The proposed models can be separated into two major groups. The first group considers the gap above Tc as being totally unrelated to superconducting pairing. For example, one can assume that the gap is caused by a charge density wave (CDW) instability. Such a point of view looks very reasonable, especially for cuprates containing low dimensional structural units, such as planes and chains. Indeed, the presence of chains in YBCO leads to nesting near the Fermi surface and, correspondingly, to a possible CDW. Another similar scenario, which assumes a spin density wave (SDW) instability was proposed shortly after the discovery of the “pseudogap” state (see, e.g., Alloul et al., 1989; review by Norman et al., 2005); this state was observed first with the use of NMR spectroscopy (see above, Section 2.3). An energy gap can be also caused by Coulomb disorder (Canfield et al., 1998). Another possible scenario (Tallon and Loram, 2000) which also belongs to the first group introduces an entirely new order parameter, some hidden quantity which competes with the pairing. The value of this parameter depends on the carrier concentration and is different from zero below some critical value nc which lies slightly above that for optimum doping. In other words, the “pseudogap” state is absent for the overdoped state. According to the second group of models, there is some relation between the pairing gap at T < Tc and the “pseudogap” at T > Tc . Emery and Kivelson (1995) assumed at a temperature dubbed T ∗ , so-called precursor pairs are formed. However, only at the lowest temperature, Tc , do these pairs form a phase coherent state. All these models contain realistic explanations for the appearance of an energy gap above Tc , in the “ pseudogap” region. One should also realize that these explanations are not necessary mutually exclusive; indeed, it might be that several factors contribute to the energy spectrum. However, and this was emphasized in previous sections, the energy gap is an important, but not a unique feature of the “pseudogap” state. Any theory of this novel behavior should provide an unified explanation of various phenomena
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(energy gap above Tc , anomalous diamagnetism, isotope effect, “giant” Josephson effect, etc.). To the best of our knowledge, such a challenge has not been met in the frameworks of various models. The most straightforward method allowing one to bridge the “pseudogap” state with pairing below Tc is to assume that the “pseudogap” state is caused by fluctuations. Of course, the fluctuations contain all ingredients of the usual superconducting state. Moreover, in the region close to Tc , fluctuations are very essential. However, the “ pseudogap” region extends to very high temperatures (e.g., of order of 80 K for the LaSrCuO underdoped compound with Tc ≈ 18 K) and we are dealing with temperature region which is too large for fluctuations. This has been demonstrated by a special experiment (Bergeal et al., 2006) described above, Section 2.4. According to the data, fluctuating pairs could be present up to about 10 K above Tc . Furthermore, as was discussed in Section 2.3, fluctuations alone are not able to provide an explanation for the observed diamagnetism. We think that the unified explanation of the various phenomena observed (see above, Sections 2.1–2.6) requires the presence of an intrinsic inhomogeneity of the compound. This does not mean that other factors are not relevant. For example, the formation of the gap can be greatly affected by CDW and SDW instabilities. As will be discussed below (Section 3.4), the “pseudogap” region is characterized by two characteristic energy scales (Tc∗ and T ∗ ), and the contributions of CDW and SDW formations are especially important in the region Tc∗ < T < T ∗ . However, a complete description of the “pseudogap” state requires the presence of an intrinsically inhomogeneous structure, and below we describe this concept in detail. 3. Inhomogenous superconductivity and the “pseudogap” phenomenon As was described above, intensive experimental studies reveal a number of unusual features of the cuprates above the resistive transition Tc . We described anomalous diamagnetism which strongly depends on temperature, an energy gap structure, a strong inequality Re Z = Im Z (Z is the surface impedance), a “giant” Josephson proximity effect, and an isotope effect on Tc∗ . All these phenomena can be explained in an unified way (Ovchinnikov et al., 1999, 2001; Ovchinnikov and Kresin, 2002; Kresin et al., 2003, 2004; Kresin and Wolf, 1994). We focus in this section on a qualitative picture and discuss some relevant experimental data. A more detailed theory will be described below (Section 4). 3.1. Qualitative picture Consider an inhomogeneous superconductor, so that Tc =Tc (r). The system contains a set of superconducting regions “islands” embedded in a normal metallic matrix (Fig. 6). Properties of such a system correspond to the “pseudogap” state. Indeed, the normal metallic matrix provides finite resistance whereas the existence of the superconducting “islands” leads to an energy gap structure and the diamagnetic moment. As was mentioned above (Section 2.2), the presence of diamagnetic “islands” has been observed directly by Iguchi et al. (2001). The superconducting “islands” are embedded in the normal metallic matrix. As a result, we are dealing
Fig. 6. Inhomogeneous structure. “Islands” are characterized by values of Tc ’s higher than the matrix.
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with the superconductor–normal metal interface and the proximity effect plays a crucial role. The proximity effect determines a minimum length scale of the superconducting regions which is of order of the coherence length 0 . Indeed, if a superconducting “island” has a size smaller than 0 , its superconducting state would be totally depressed by the proximity effect between the superconducting region and the normal metallic phase. As temperature decreases towards Tc , the size of the superconducting regions increases as does the number of “islands”. The critical temperature Tc corresponds to the percolation transition, that is to the formation of a macroscopic superconducting region (“infinite cluster” in terms of the percolation theory see, e.g. Shklovskii and Efros, 1984; Stuffer and Aharony, 1992), and to phase coherence and dissipationless superconducting phenomena. In conventional superconductors the resistive and Meissner transitions occur at the same temperature, Tc . The picture in the “pseudogap” state is different. The resistive and Meissner transition are split. The Meissner transition (the appearance of the diamagnetism) occurs at Tc∗ , whereas the resistive transition, that is, the transition to the macroscopic dissipationless state takes place at Tc , and Tc < Tc∗ . Note that in our papers (Ovchinnikov et al., 1999, 2001) we used also the notations Tcres and TcMeis . In this article we use only the notations Tc and Tc∗ , so that Tc ≡ Tcres , Tc∗ ≡ TcMeis . It is important to note also that the “pseudogap” state in the region Tc < T < Tc∗ is not a phase coherent one; each superconducting “island” has its own phase. At T = Tc the macroscopic superconducting region is formed, and below Tc we are dealing with macroscopically phase coherent phenomenon. As was noted above, the presence of superconducting “islands” embedded in a normal metallic matrix implies an inhomogeneity of the compound. There are two possible scenarios for such an inhomogeneous structure: I. inhomogeneous distribution of pair-breakers, and II. inhomogeneous distribution of carriers leading to spatial dependence of the coupling constant. Both scenarios lead to an inhomogeneous superconductivity but for the cuprates the first of them is dominant. Let us discuss it in more detail. Pair breaking can be caused by localized magnetic moments (Abrikosov and Gor’kov, 1961; de Gennes, 1964, 1966). Qualitatively, the picture of pair-breaking can be visualized in the following way. A Cooper pair consists of two carriers with opposite spins (for singlet pairing; this is the case for both, s- or d-wave scenarios). A localized magnetic moment acts to align both spins in the same direction and this leads to pair-breaking. It is well known that a pair-breaking effect leads to a depression in Tc . It is also known, that for d-wave pairing nonmagnetic impurities are also pair-breakers. Therefore, a non-uniform distribution of pair-breakers makes the critical temperature spatially dependent: Tc ≡ Tc (r). Such a distribution is caused by the statistical nature of doping. The region which contains a larger number of pair-breakers is characterized by a smaller value of the local Tc . A detailed theoretical analysis of the inhomogeneous structure caused by the presence of isolated non-magnetic and magnetic defects has revealed unusual properties for the density of states in superconductors (Vechter et al., 2003; Shytov et al., 2003; see also review by Balatsky et al., 2006). They concluded that, strictly speaking, the density of states is finite everywhere in the superconducting gap. The density could be exponentially small, but different from zero even for an s-wave order parameter; the tail extends into the mean field gap. It was also shown (Zhu et al., 2005) that isolated defects create long-range elastic deformations which lead to a local depression of the order parameter. As was mentioned above, an inhomogeneity can be caused also by an inhomogeneous distribution of carriers (see Ovchinnikov et al., 2001). However, since the inhomogeneous distribution of pair-breakers (dopants) appears to be the dominant source of inhomogeneity, we focus below (see Section 4.3) on such a channel. The percolative nature of the transition at T = Tc is due to the statistical nature of doping. The picture is similar to that introduced in manganites which represents another family of doped oxides (see Gor’kov and Kresin, 1998, 2000; Dzero et al., 2000; see also review by Gor’kov and Kresin, 2004). Manganites (e.g., La0.7 Sr 0.3 MnO3 ) are characterized by the presence of ferromagnetic metallic regions embedded in the low conductivity paramagnetic matrix above T = TCurie , TCurie is the Curie temperature. At T = TCurie one can observe a percolative transition to the macroscopic ferromagnetic metallic state. The transition from the “pseudogap” to the macroscopic dissipationless state is also of percolative nature. 3.2. Phase separation The picture described above can be treated in the framework of a general concept of phase separation. This concept was introduced by Gor’kov and Sokol (1987) shortly after the discovery of the high Tc oxides (Bednorz and
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Muller, 1986) and then was studied in many papers (see, e.g., book by Sigmund and Mueller, 1994). This concept implies the coexistence of metallic and insulating phases, and such a coexistence is a very important ingredient of the physics of doped cuprates. Of course, this concept means that the nominal state of the compound is in itself inhomogeneous; this feature is manifested by the separation of metallic and insulating phases. An interesting analysis of NMR data on nuclear spin relaxation in cuprates (Gor’kov and Teitelbaum, 2004) has demonstrated that below T ∗ the temperature dependence of 63 T1−1 can be presented as a sum of two contribution: 63
T1−1 = 63 T1−1 (x) + 63 T˜1−1 (T ),
(3.1)
where T1 (x) is sample dependent and T˜1 (T ) is an universal function of temperature. The first term corresponds to the “stripe” like excitations and the second one to moving metallic and antiferromagnetic subphases. Our picture of the “pseudogap” state (Fig. 6) implies the next step in the picture of inhomogeneity. Namely, in addition to the mixture of metallic and insulating phases, the metallic phase is itself inhomogeneous; we are dealing with the coexistence of normal and superconducting phases within the metallic phase. One should again stress an important aspect of the inhomogeneity of the metallic phase (mixture of normal and superconducting regions, Fig. 7. Namely, one should take into account the proximity effect between the normal matrix and superconducting “islands”. For example, as was mentioned above, the proximity effect determines the minimum size of a superconducting “island” which is of the order of the coherence length 0 . If the size is smaller than 0 , the superconducting state of the “islands” would be completely destroyed by the normal metallic matrix. 3.3. Inhomogeneity: experimental data The picture of inhomogeneity described above has strong experimental support. The inhomogeneous structure of the cuprates has been observed using neutron diffraction (Egami and Billinge, 1996; Gutmann et al., 2000). The underdoped compound is very inhomogeneous but becomes more homogeneous if it is doped towards the optimum level (maximum Tc ), Bozin et al. (2000). Such a picture is totally consistent with our scenario for the “pseudogap”, because this phenomenon is very strongly manifested in the underdoped region. Scanning tunneling microscopy (STM) provides a wealth of information about local structure. An atomic scale study using the STM technique has been performed by the Davis group (Cornell University). It has been demonstrated (Pan et al., 2001) that the presence of oxygen dopants in the Bi2 Sr 2 CaCu2 O8+ compound leads to inhomogeneity which is manifested by a spatial dependence of the density of states, and, correspondingly, the order parameter is inhomogeneous. Recent STM measurements (McElroy et al., 2005b) have demonstrated that individual dopants are the cause of local disorder, and, therefore, indeed, their statistical distribution leads to observed inhomogeneity of the sample. A picture of the phase separation has been observed by Lang et al. (2002). A spatial Fourier transform method allows one to study atom-scale modulations and the doping dependence of the nanoscale electronic structure (McElroy et al., 2003, 2005a). STM measurements have been performed at different locations on the surface of a BiSrCaCuO sample (at T 4.2 K). The energy gap defined as the distance between the peaks of the density of states displays a strong spatial dependence (see also Truscott, 2000). These observations provide strong experimental support for the concept of the inhomogeneity of the metallic phase. It was stated above (Section 3.1) that one should distinguish the resistive (Tc ) and magnetic transitions (Tc∗ ). Experimentally this has been demonstrated by Dos Santos et al. (2003); the transition temperature onset, Tc (x), in the Bi2 Sr 2 Ca1−x Pr x Cu2 O8+ (Bi 2212+Pr) compound was analyzed in detail by resistivity and magnetization measurements. The behavior of TCR (x) and TCM (x) appear to be entirely different. It has been shown that the observed depression of TCR (x) corresponds to a reduction of the superconducting volume fraction and the formation of superconducting clusters, in total agreement with the picture described above (Section 3.1). We described in Section 2 an interesting study of the La-based compound (Iguchi et al., 2001) performed using the STM technique with magnetic imaging (Fig. 1) which has directly demonstrated the presence of diamagnetic “islands” embedded in a normal matrix and that demonstrates the percolative picture as T → Tc . As was indicated above, there are two possible scenarios of inhomogeneity, but the inhomogeneous distribution of pair-breakers is a dominant factor. Indeed, according to tunneling and infrared data, the energy gap is almost temperature
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independent; an increase in T only leads to a decrease in the depth of the dip in the density of states. The pair-breaking scenario corresponds to this result. We will discuss it in more detail below (Section 4.3). Note also, that according to NMR data (Bobroff et al., 2002), the charge distribution does not have a nanoscale variation. However, this does not exclude large scale variations (stripes or commensurate charge distribution (Haase et al., 2000; Haase and Slichter, 2003). Importance of pair-breaking was demonstrated by measurements of the microwave resistance of YBCO films (Prokohorov et al., 2001). Pair-breaking scattering appears to be strongly dependent on oxygen content, in an agreement with the theoretical model proposed here. The inhomogeneity that is observed is caused by a non-uniform distribution of pair-breakers. Qualitatively, this conclusion follows naturally from the statistical nature of the doping process. 3.4. Energy scales The name “pseudogap state” reflects the presence of a gap structure above the critical temperature. One should stress however, that this single fact, namely, the presence of an energy gap does not lead to a complete understanding of the nature of the “pseudogap” state. Indeed, there could be many reasons for the appearance of a gap structure (superconducting pairing, charge density waves (CDW), spin density waves (SDW), band gap, Coulomb disorder, etc.). Such complex systems as the cuprates might have different channels leading to the appearance of the energy gap. As a result, the presence of the energy gap structure above Tc (“pseudogap” state) we believe is caused by a combination of factors. In general, it is known that the energy gap structure could be affected by two factors: pairing and CDW instability (see, e.g., Balseiro and Falicov, 1979). This can be seen in the cuprates. For example, the presence of a chain structure in the YBCO compound means the existence of nesting states, and this leads to a CDW instability. At the same time the continuous transition of the gap structure through Tc as well as diamagnetism above Tc indicates that pairing is also essential. The real picture in the cuprates is complicated and we are dealing with three different energy scales (Kresin et al., 2004) and, correspondingly, with three characteristic temperatures (we denote them Tc , Tc∗ , and T ∗ ). Highest energy scale (T ∗ ) . The highest energy scale, which we have labeled T ∗ (a5 × 102 K) corresponds to the formation of the inhomogeneity and peculiar crystal structure of the compounds. For example, forYBCO, the formation of the chains occurs at T ∗ . An energy gap could open in the region below T ∗ . This gap is not related to the pairing, but, as was mentioned above, there are many other sources for the appearance of a gap. For example, the presence of a chain structure in YBCO is consistent with a charge density wave and, correspondingly, with a gap on part of the Fermi surface. Nesting of states might lead to a CDW instability in other compounds as well. Another important property of the compound below T ∗ is its intrinsic inhomogeneity; this is due to the statistical nature of doping and is manifested in phase separation (see above). This property implies the coexistence of metallic and insulating phases. The periodic stripe structure (Bianconi, 1994a,b; Bianconi et al., 1996; Tranquada et al., 1995, 1997; Zaanen, 1998) also appears below T ∗ . Phase separation is a key ingredient which determines T ∗ as a corresponding onset temperature. Its value can be determined by the NMR measurements (see above, Eq. (3.1), Gor’kov and Teitelbaum, 2004). Such a frustrated first order phase transition was described by Gor’kov (2001). Diamagnetic transition (Tc∗ ). If the compound is cooled down below T ∗ , then at some characteristic temperature we have labeled Tc∗ (Tc∗ ≈ 2 × 102 K) one can observe a transition into the diamagnetic state. The characteristic temperature Tc∗ corresponds to the appearance of superconducting regions embedded in a normal metallic matrix (Fig. 1). The presence of such superconducting clusters (“islands”) leads to a diamagnetic moment, whereas the resistance remains finite, because of the normal matrix. As for the energy gap, coexistence of pairing and a CDW determine its value below Tc∗ . It is remarkable that the superconducting state appears at a temperature Tc∗ which is much higher than the resistive Tc . This value of Tc∗ corresponds to the real transition to the superconducting state (one can call it an “intrinsic critical temperature”, see Kresin et al., 1996, 1997). Strictly speaking, the experimentally measured value of Tc∗ lies below the intrinsic critical temperature, because of the impact of the proximity effect. Nevertheless, Tc∗ is an important experimentally measured parameter. It corresponds to the appearance of diamagnetic “islands” and reflects the impact of pairing. At the same time, the value of Tc∗ ,
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Fig. 7. Energy scales.
unlike Tc , is not noticeably depressed by pair breaking. The superconducting phase appears, at first, as a set of isolated “islands”. The picture of different energy scales, T ∗ and Tc∗ just described is in total agreement with interesting experimental data by Kudo et al. (2005a,b). The impact of external magnetic field was studied by out-of-plane resistive measurements. According to the study, there are, indeed, two characteristic temperatures (Kudo et al. dubbed them as T ∗ and T ∗∗ ; T ∗ > T ∗∗ ). The behavior of the resistivity appears to be independent on magnetic field in the region T ∗ > T > T ∗∗ , but strongly affected by the field at T < T ∗∗ . According to Kudo et al., the state formed below T ∗∗ is related to superconductivity. The characteristic temperatures T ∗ , T ∗∗ directly correspond to the energy scales T ∗ and Tc∗ (in our notation T ∗∗ ≡ Tc∗ ) introduced above. Resistive transition (Tc ). As the temperature is lowered below Tc∗ , new superconducting clusters appear (Fig. 6) and existing clusters form larger “islands”. This is a typical percolation scenario. At some characteristic temperature (Tc ) the macroscopic superconducting phase is formed (“infinite” cluster in terms of the percolation theory, see, e.g., Shklovskii and Efros, 1984). The formation of a macroscopic phase at Tc leads to the appearance of a dissipationless state (R = 0). It is important also to stress, that in the region Tc∗ > T > Tc each “island” has its own phase, so that there is no phase coherence for the whole sample. Macroscopic phase coherence appears only below Tc . Therefore, there are three different energy scales and, correspondingly, three characteristic temperatures T ∗ , Tc∗ , Tc . (Fig. 7). The value of Tc is lower than Tc∗ because of local depressions caused by the pair-breaking effect and an inhomogeneous distribution of pair-breakers (dopants). It is interesting to note that the value of Tc∗ is close to an intrinsic value of the critical temperature. This value is noticeably higher than the resistive Tc . To conclude this section, let us stress again that the inhomogeneous distribution of pair-breakers (dopants) along with local depressions in the value of critical temperature leads to a spatial dependence of Tc , i.e. Tc (r) (Fig. 6). The value of Tc∗ is close to an “intrinsic” critical temperature. This temperature corresponds to the transition into the superconducting state in the absence of pair-breaking and has a value of order of 2 × 102 K (Kresin et al., 1996). It is important to note that the value of Tc∗ is much higher than Tc . 4. Theory In this section we are going to present the theoretical analysis of the main features of the pseudogap state: density of states and the appearance of a gap structure, diamagnetism, a.c. properties, and the “giant” Josephson proximity effect. The analysis is based on our model (Ovchinnikov et al., 1999, 2001, 2002; Kresin et al., 2000, 2003, 2004; Kresin and Wolf, 1994); the qualitative picture was described above (Section 3).
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4.1. General equations Inhomogeneity of the system is a key ingredient of the theory. Because of it, it is convenient to use a formalism describing the compound in real space. That’s why we employed the method of integrated Green’s function which was developed by Eilenberger (1968) and independently by Larkin and Ovchinnikov (1969), see also the review by Larkin and Ovchinnikov (1986). The main equations have the form − +
D (j2− − j2r ) = , 2
|2 | + ||2 = 1, . = 2T ||
(4.1)
(4.1 )
(4.1 )
>0
Here and are the usual and pairing Green’s functions (Gor’kov, 1959) integrated over energy, is the order parameter, ≡ −1 s is the spin-flip relaxation time (see also Ovchinnikov and Kresin, 1996), is the pairing coupling constant. Because of the inhomogeneity, all of these quantities are spatially dependent. In addition, j± = jr ± 2ieA, A is the vector potential, jr = (j/jr ). We consider the “dirty” case, so that D is the diffusion coefficient. These equations contain the spatially dependent functions , , . The method is very effective for treatment of spatially dependent properties. 4.2. Diamagnetism The Cu–O layers contain superconducting “islands” and their presence leads to an observed diamagnetic moment. Because of the dependence Tc (r), the size of the superconducting region occupied by the “islands” decreases as temperature is increased. As a result, diamagnetism which is strongly temperature dependent is observed. The evaluation of the diamagnetic moment (Ovchinnikov et al., 1999) will now be described. Based on Eqs. (4.1), one can calculate the order parameter (r) and then the current j (r). Then, one can calculate the magnetic moment, since the magnetic moment for an isolated cluster is MZ = L d [ j]z . (4.2) Here L is the effective thickness of the superconducting layer; axis z has chosen to be perpendicular to the layers, is perpendicular to OZ. Assume that the sample contains a sufficient amount of magnetic impurities so that s Tc0 >1; as a result Tc >Tc0 , where Tc is the average value of the critical temperature, and Tc0 corresponds to the transition temperature with no magnetic impurities. In this case, with the use of Eqs. (4.1), we obtain D 2 −1 D 2 2 2 = 2T ||
+ − j− − 0 |0 | + 0 jr |0 | . (4.3) 2 2 4 >0
The order parameter can be found in the form = C0 , where C ≡ C(T ) and 0 is the solution of the equation: [ − (D/2)j2− ]0 = ( ∞ + )0 . Here ∞ is the value of outside of the “island” and is the minimum eigenvalue. As a result, we arrive at the following equation: 2 C 2 (∗2
∞ + 0 , 0 ) ln(Tc0 /T ) = 0.5 + − (0.5) + . 2T 12 2∞ (∗0 , 0 )
(4.4)
(4.5)
Here is the Euler function, and the notation (f,g) corresponds to scalar product of the functions. The usual transition temperature Tc is determined by the equation which can be obtained from Eq. (4.5) if we insert C = 0 and = 0: ln(Tc0 /Tc ) = [0.5 + ( ∞ /2Tc )] − (0.5)
(4.6)
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which is the well-known pair-breaking equation (Abrikosov and Gor’kov, 1961). Eq. (4.5) is the generalization of Eq. (4.6) for the inhomogeneous case. The current density is described by the expression (Larkin and Ovchinnikov, 1969) j = −ieDT (∗ j− − j+ ∗ ). (4.7)
Here is the density of states. With the use of Eqs. (4.4) and (4.6), we obtain
∞ + ieDC 2
0.5 + (∗0 j− 0 − 0 j+ ∗0 ), j =− 2T 2T
(4.8)
where 0 is the solution of Eq. (4.4). As a result we can obtain the following expression for the magnetic moment of an isolated cluster Mz = L d [ j]z :
∞ + 1 Mz = −(e2 DC 2 H L/T ) 0.5 + K. (4.9) 2T Here K = d 2 20 , and the vector-potential has been chosen as A = 21 [H r]; L is the effective thickness of the superconducting layer. Note also that because the cluster size is smaller than the penetration depth, one can neglect the spatial variation of the magnetic field. Consider the most interesting case when the variation of the amplitude (r ) = ∞ − has the form (); < 0 , (r ) = 0; > 0 , 0 is the ‘island” radius. Then Eq. (4.8) can be written in the form D 1 j j 2 2 2 0 () = 0 (). () − −e H 2 j j A similar equation has been studied by Ovchinnikov (1980) and by Bezryadin et al. (1996). The solution is 1 M(+ )/2eH D;0 (eH 2 ), < 0 , 0 () =
2 > 0 . eH 2 C1 W/2eH D;0 (eH ),
(4.10)
(4.11)
Here W, (z) and M, (z) are the Whittaker functions. Finally, one can obtain the following expression for the magnetic moment: Mz = −A[(B˜ − (T /Tc )2 )]H .
(4.12)
Here A = (82 e2 DT 2c / ∞ )20 z0−4 ns (x˜3;2 x˜2;1 /x˜1;4 );
B˜ = B + 1,
B = −61 ∞ /(Tc )2
(4.12 )
1 = − + 0.5D(z0 /0 )2 ,
(4.12 )
and
z ns is the concentration of superconducting clusters, and x˜n;i = 0 0 dx · x n J0i (x). z0 is the lowest zero of the Bessel function, that is J0 (z0 ) = 0; z0 2.4. If ≈ ∞ , then Tc∗ ?Tc . ˜ depends on an interplay of two terms. The first term reflects the The value of 1 (and, therefore, the value of B) impact of pair-breaking, and the second term describes the proximity effect. It is natural that the impact of the proximity effect increases with a decrease in the size of the inhomogeneity 0 .
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One can see directly from Eq. (4.12), that it is possible to observe a noticeable diamagnetic moment. Indeed, if we ˚ (l is a mean free path: D = vF l/3), Tc = 10 K, ∞ = 102 K, assume realistic values: pF = 10−20 gcm-s−1 , l = 40 A ˚ and ns 0.1, we obtain the following values of the parameters: A 10−5 , B = 3, = 50 K (we put kB = 1), 0 = 80 A, = 5 K. Then, for example, at T = 11 K, one can observe D = MZ /H = −3 × 10−5 ; this contribution greatly exceeds the usual paramagnetic response of a normal metal, P 10−6 . ˜ This is natural, since the influence of the A diamagnetic response can be observed in the region (T /Tc ) < B. proximity effect (see, e.g., Gilabert, 1977) to depress the superconductivity grows with a decrease in the size 0 of the superconducting grain. As previously mentioned, Bergemann et al. (1998) described torque measurements performed on a Tl2 Ba2 CuO6 overdoped sample (Tc 15 K; the value of Tc 15 K was determined by resistive measurements, Mackenzie et al., 1993; Osofsky et al., 1993). A diamagnetic moment, proportional to the external magnetic field, has been observed at T > Tc . An analysis based on the theory described above is in very good agreement with the data (Fig. 3). 4.3. Density of states: gap structure Based on Eqs. (4.1), one can evaluate the density of states for inhomogeneous system (Fig. 8). As mentioned before, there are two possible scenarios for the appearance of an inhomogeneous structure: (1) an inhomogeneous distribution of pair-breakers (as we know, the presence of a pair breakers leads to a local depression in Tc ), and (2) an inhomogeneous distribution of carriers leading to a spatial dependence of the coupling constant ≡ (r). Let us focus on the first scenario, since the inhomogeneous distribution of pair-breakers appears to be a major factor which determines the spatial dependence of the temperature Tc ≡ Tc (r). The energy gap manifests itself as a dip in the low frequency region of the density of states ≡ ( ). The density of states is defined by the relation = Re , where is the usual Green’s function integrated over energy (see Eqs. (4.1)), Larkin and Ovchinnikov (1972a, b). The calculation of (Ovchinnikov et al., 2001) leads to the following expression for its average value:
= 1 − (ns C 2 /2)[˜2 − 2 + 2i˜ ](˜2 + 2 )−2 ,
(4.13)
where ˜ = + ∞ , and the eigenvalue and C(T ) is determined by Eqs. (4.12 ) and (4.5), respectively. The density of states for the inhomogeneous system of interest is plotted in Fig. 8. One can see directly that, indeed, there is a “softening” of the low-energy part of the density of states, and this is a clear manifestation of the “pseudogap” structure. If the temperature is above Tc and is increased towards Tc∗ then the difference = max − min → 0. At the same time the position of the peak is independent of T. This feature is very specific for the “pseudogap” phenomenon caused by an inhomogeneous distribution of pair breakers. Indeed the density of states and its temperature dependence were directly measured by tunneling spectroscopy by Renner et al. (1998). One can see directly from the data (see Fig. 4), that the gap structure (“pseudogap”) persists above Tc , but the peak position does not depend on temperature. This is in an agreement with the scenario discussed here and, therefore, for the Bi2 Sr 2 CaCu2 O8+ sample studied by Renner et al. (1998), see also Tao et al. (1997), the “pseudogap” phenomenon is caused by an inhomogeneous distribution of pair breakers. An interesting measurement of the interlayer tunneling spectroscopy for an overdoped Bi2 Sr 2 CaCu2 O8+ compound was described by Suzuki et al. (1999). The authors also observed the occurrence of the pseudogap below 150 K, that is at much higher temperatures than the resistive Tc 87 K. The data are also consistent with the picture of an inhomogeneous distribution of pair breakers. Infrared spectroscopy data discussed above (see Section 2.4) also is in agreement with the conclusion that the inhomogeneous distribution of pair-breakers is a major source of the inhomogeneity in the cuprates. Generally speaking, an inhomogeneous charge distribution and, correspondingly, the dependence (r) can also lead to a spatially dependent Tc . However, one can show that in this case the value of the gap decreases with increasing T in the region above Tc (Ovchinnikov et al., 2001). Therefore, based on various experimental data, see also Phillips et al., 1992; Niedermeier et al., 1993, one can conclude that the inhomogeneous distribution of pair-breakers caused by the statistical nature of the doping, plays a dominant role as a source of inhomogeneity.
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Fig. 8. The behavior of the density of states for different types of inhomogeneites. The dashed line corresponds to higher temperatures: (a) density of states for an inhomogeneous distribution of pair breakers; (b) density of states for an inhomogeneous distribution of carriers and, correspondingly, the coupling constants.
4.4. ac transport The dc transport properties of the inhomogeneous system above Tc are determined by the normal phase, since only this phase can provide a continuous path. The situation with ac transport is entirely different, and the superconducting “islands” make a direct contribution to the ac conductivity and to surface impedance. As we know, the real and imaginary parts of the surface impedance of a normal metal are almost equal (see e.g., Landau and Lifshitz, 1960). The situation is entirely different in superconductors (see e.g., Tinkham, 1996). To describe the
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“pseudogap” state, it is interesting to consider an inhomogeneous system which consists of normal and superconducting regions (Ovchinnikov and Kresin, 2002). The analysis is based on Eqs. (4.1); the system is in an external field A(r, ) = exp(−i 0 )A(r);
(r, ) = exp(−i 0 )(r).
(A and are the vector and scalar potentials, 0 is the frequency, here is an imaginary time; do not confuse it with the relaxation time). The formalism of thermodynamic Green’s functions is employed (see, e.g., Abrikosov et al., 1963; Ovchinnikov, 1977). One can formulate a complete general system of equations determining the ac response for an inhomogeneous system (Larkin and Ovchinnikov, 1973a, b). Based on these equations, one can calculate the ac conductivity, and then the surface impedance Z, since 1/2 Z= exp(−i/4) (4.14) 4 (see, e.g. Landau and Lifshitz, 1960). For normal metals the difference between Re Z and Im Z is negligibly small and is connected with the dependence: ( ) = 0 (1 − i tr )−1 ; in our case tr >1. The situation for the “pseudogap” state is different. One can show that a metallic compound which contains superconducting “islands”, is characterized by a strong inequality: Re Z = |Im Z|. In the temperature region close to Tc∗ , i.e. (T − Tc∗ )>Tc∗ , the expressions for the conductivity eff and, correspondingly, for the impedance, can be simplified and has the form Re Z = Z˜ n [1 − (2 /21 )], Im Z = −Z˜ n [1 + (2 /21 )].
(4.15)
2 /21 = (s/ )(Tc∗ − T ).
(4.15 )
˜ c∗ − T ). Re Z = −Im Z − −1 (2s Z)(T
(4.16)
Here
or
Here Z˜ n = ( /81 )1/2 , s ≡ s(ns ); the quantity s depends exponentially on ns , ns is the concentration of superconducting “islands”. Note that Re Z = |Im Z| at T Tc∗ . This can be seen directly from (4.15), (4.16). The inequality Re Z = |Im Z| at T Tc∗ is caused by the presence of superconducting “islands” and is described by the second term in Eq. (4.16). It is important to note that this term is proportional to −1/2 . A relatively small value of the frequency , e.g. in the microwave region, leads to a noticeable contribution to the impedance. In addition, the dependence ∝ −1/2 can be directly measured experimentally. Eqs. (4.15) and (4.16) describe the ac response of an inhomogeneous superconductor in the pseudogap region (Tc < T < Tc∗ ). The quantity s depends on a number of experimentally accessible parameters, including geometry (Ovchinnikov and Kresin, 2002). For example, if we assume the values: Tc∗ = 200 K, = 2.5,
Tc = 110 K;
= (D/Tc )1/2 ,
∞ = 160 K, nS 10−2 ,
= 21010 s−1 ,
(4.17)
we obtain: s 4 × 108 s−1 k −1 .
(4.18)
Measurements of Z for the HgBa2 Ca2 Cu3 O8− compound at T > Tc were performed by Kusco et al. (2002). It has indeed been observed that the slopes of the temperature dependencies are different meaning that Re Z = Im Z. Using experimentally determined value of the slope, one can calculate the parameter s, and it is close to the value we estimated above (4.18). This substantiates the choice of parameters we used.
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Fig. 9. Electronic terms.
Let us make several comments related to the dc transport. As we mentioned above, (Section 2), the normal resistivity above Tc displays a specific non-linear behavior. Such an effect can be understood as a result of the interplay between normal matrix and superconducting “islands”. A decrease in temperature leads to an increase in the superconducting fraction, and this affects the total normal resistivity. A relevant analysis of the normal resistivity for the Bi2 Sr 2 Ca1−x Cu2 O8+ compound, based on the inhomogeneous picture, has been carried out by Dos Santos et al. (2003). An interesting study of the in-plane transport for manganite—YBCO heterostructures (Soltan et al., 2005) has shown that the in-plane resistance drops as a result of the spin-injection into the Cu–O plane in the c-direction. This effect can be explained by an increase in the number of pair-breakers caused by the injection; this leads to an increase in the “normal” fraction. The impact of the “pseudogap” on thermal conductivity (Minami et al., 2003). Minami et al. (2003) has demonstrated that the “pseudogap” phenomenon does not arise from fluctuations. This conclusion follows also from the paper by Bergeal et al. (2006), see discussion in Section 2.5. As a whole, one should note that transport phenomenon deserve additional theoretical and experimental study. For example, it would be interesting to perform additional measurements of the ac conductivity and surface impedance in the ‘pseudogap’ region for different doping levels. 4.5. Isotope effect The isotope effect on Tc∗ observed experimentally (Section 2.6) also reflects the fact that the superconducting pairing persists above the resistive transition. It is interesting to note that the isotope coefficient has a negative sign. This unusual feature is consistent with the microscopic model of the isotope effect in the high Tc oxides (Kresin and Wolf, 1994). Indeed, a strong non-adiabaticity (axial oxygen in YBCO is in such state) results in a peculiar polaronic isotope effect. The cuprates are doped materials, and the charge transfer is playing a very important role. It turns out that the doping, and therefore, the carrier concentration are affected by the isotope substitution. Since the value of Tc strongly depends on carrier concentration (Tc ≡ Tc (n)), we are dealing with a peculiar isotopic dependence of Tc . If the charge transfer occurs in the framework of the usual adiabatic picture, so that only the carrier motion is involved, then the isotope substitution does not affect the forces and therefore does not change the charge transfer dynamics. However, the situation of strong non-adiabaticity is different and does not allow the separation of electronic and nuclear motion; in this case charge transfer appears as a more complex phenomenon which does involve nuclear motion, and this leads to a dependence of the doping on isotopic mass. Consider the case when the lattice configuration corresponds to the situation when some definite ion (e.g., axial oxygen for YBCO) is in (or near) a degenerate state; this means that the degree of freedom describing its motion corresponds to electronic terms crossing (see Fig. 9). Then the ion has two close equilibrium positions (double-well structure). Consider the axial oxygen in YBCO. Its dynamics is described by the double-well structure, and such a structure has been observed experimentally with use of the X-ray absorption fine structure technique (Haskel et al., 1997), Fig. 10.
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Fig. 10. Occupancy of the O(2) apical oxygen.
The charge transfer in this case is described by polaronic motion (dynamic polaron). Note that a similar effect leads to the isotope effect in manganites (see Gor’kov and Kresin, 2004). Qualitatively the charge transfer for such non-adiabaticity can be visualized as a multistep process; first the carrier makes the transition from the chain site to the axial oxygen O(2), then the axial oxygen transfers to another term, and this is finally followed by the transition of the carriers to the plane. The second step is affected by the isotope substitution. For a whole crystal it can be viewed as a motion of the dynamic polaron. In order to describe this phenomenon, it is convenient to use a so-called “diabatic” representation (see, e.g., O’Malley, 1967; Smith, 1969; Kresin and Lester, 1984; Dateo et al., 1987; Kresin and Bill, 1998). In this representation we are dealing directly with the crossing of electronic terms. The operator Hel. = Tˆr + V (r , R). is a total potential energy, r and R are the electronic and nuclear coordinates, (Tˆr is a kinetic energy operator, V (r , R) correspondingly) has non-diagonal terms (unlike the usual adiabatic picture when Hˆ el is diagonal). The charge transfer in this picture is accompanied by the transition to another electronic term. Such a process is analogous to the Landau–Zener effect (see, e.g., Landau and Lifshits, 1977). The total wave function can be written in the form ¯ t) = a(t)1 (r , R) + b(t)2 (r , R). (r , R,
(4.19)
Here = i (r , R) i (R), i (r , R)
i = {1, 2},
i (R) are the electronic and vibrational wave functions that correspond to two different electronic terms
i (r , R), (see Fig. 10). In the diabatic representation the transition between the terms are described by the matrix element V12 , where V ≡ Hˆ r . One can show that V12 L0 F12 , where
L0 =
Hˆ r 1 (r , R)| R0 dr ∗2 (r , R)
(4.20)
(4.20 )
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is the electronic constant (R0 correspond to the crossing configuration), and ¯ 1 (R) ¯ dZ F12 = ∗2 (R)
(4.20 )
is the Franck–Condon factor. The presence of the Franck–Condon factor is a key ingredient of our analysis. Its value strongly depends on the ionic mass and, therefore is affected by the isotope substitution. The calculation (Kresin and Wolf, 1994) leads to the following expression for the isotope coefficient: =
n jTc , Tc jn
(4.21)
where has a weak logarithmic dependence on ionic mass M. Therefore, the polaronic isotope effect is determined by the dependence Tc (n), n is the carrier concentration. The analysis described above can be applied to the “pseudogap’ region, since Tc∗ is the intrinsic value of the critical temperature. In this case we have =
n jTc∗ . Tc∗ jn
(4.22)
Based on Eq. (4.22), one can explain why the isotope coefficient has a negative value. It is important to note that ∝ jTc∗ /jn. The isotope coefficient is a negative, because (jTc∗ /jn) < 0. Indeed, increase in doping in the underdoped region leads to decrease in the value of Tc∗ (at optimum doping Tc∗ Tc ), and this is due to an increase in a number of dopants that is, the pair-breakers, so that the distribution of pair-breakers becomes more uniform. 5. “Giant” Josephson proximity effect We mentioned above (see Section 2.4) an interesting experimental study of S-N-S Josephson junctions (Bozovic et al., 2004). This phenomenon cannot be explained by the usual theory of S-N-S proximity junctions. We focus on the especially interesting case of S-N -S junctions (for a general discussion see review by Devin and Kleinsasser, 1996) where the electrodes are the high Tc superconducting films (e.g., La0.85 Sr 0.15 CuO4 , or YBa2 Cu3 O7 ), and the barrier N is made of the underdoped cuprate, Tc is the critical temperature of the underdoped barrier, and Tc is the critical temperature of the electrode. The generally accepted notation N emphasizes a difference between SN S and a typical SNS junction (then TcN = 0 K), so that Tc < Tc . Here we consider temperatures when the barrier is in the normal resistive state because T > Tc . The use of the underdoped cuprate as a barrier is beneficial for various device applications because the structural similarities between the electrodes S and the barrier N eliminate many interface problems. The “giant” phenomenon is manifested in a finite superconducting current through the S-N -S Josephson junction with a thick N barrier, so that L?N (L is the thickness of the barrier, N is the proximity coherence length). The configuration is such that the layers forming the barrier N are parallel to the electrodes so that the Josephson current flows in the c-direction. Then the coherence length is very short c ≈ 4A, so that we are dealing with the “clean” limit. This type of junction using the LaSrCuO material was studied by Bozovic et al. (2004). The films of La0.85 Sr 0.15 CuO4 (Tc ≈ 45 K) were used as electrodes, whereas the underdoped LaCuO compound (Tc ≈ 25 K) formed the barrier. The atomic-layer-by-layer molecular beam epitaxy technique was used for these junctions and it provides atomically smooth interfaces (Ekstein and Bozovic, 1995; Bozovic, 2001). The barrier was prepared in the c-axis geometry. As ˚ The measurements were performed at Tc < T < 35 K. The Josephson was noted above, the coherence length c ≈ 4 A. ˚ current was observed for thickness of L up to 200 A(!). Such a “giant” effect cannot be explained with use of the conventional theory. Indeed, as we know, (see, e.g., Barone and Paterno, 1982) the amplitude of the Josephson current for the “clean” limit is jm = j0 exp(−L/N ).
(5.1)
The thickness of the barrier L should be comparable with the barrier coherence length N , and this condition is satisfied for conventional Josephson junctions. The picture described above for the SN S junctions with the cuprates is entirely different, since L?N . The superconducting current in the c-direction occurs via an intrinsic Josephson effect between
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the neighboring layers (see Kleiner et al., 1992). If the barrier contains several homogeneous normal layers, then the Josephson current through such a barrier cannot flow. To understand the nature of the “giant” Josephson proximity effect it is very important to stress that the barriers we are considering are formed by underdoped cuprates. As a result, the barriers are not in the usual normal state, but in the “pseudogap” state; indeed, Tc < T < Tc∗ . For example, the study (Iguchi et al., 2001) of the compound La2−x Sr x CuO4 (x 0.1; Tc 18 K) in which the stoichiometry of the compound La2−x Sr x CuO4 (x 0.1; Tc 18 K), used by Iguchi et al. (2001) is close to that used by Bozovic et al. (2004) as the barrier. According to Iguchi et al., this compound has a value of Tc∗ 80 K whereas Tc = Tcres 20 K describes the resistive transition to the dissipationless state. Since the diamagnetic moment measured by Iguchi et al. (2001) persists up to Tc∗ , the question of the origin of the “giant” proximity effect is directly related to the general problem of the nature of the “pseudogap” state and this makes the question of the nature of the “giant” Josephson proximity effect particularly interesting. This effect can be explained (Kresin et al., 2003) by the approach described here and based on the intrinsically inhomogeneous structure of the compound. According to our model, the CuO layers forming the N -barrier contain superconducting “islands”, and these “islands” form the path for the Josephson tunneling current. For typical SNS junctions the propagation of a Josephson current requires the overlap of the pairing functions FL and FR (see, e.g., Kresin, 1986); FR and FL are pairing Gor’kov functions for left and right-side electrodes). This overlap is caused by the penetration of FL and FR (“proximity”) to the N -barrier. For the system of interest here the situation is quite different. Each “island” has its own pairing function with its own phase. As a result, the Josephson current is caused by the overlap of FL and F1 , F1 and F2 , etc., where F1 corresponds to the “island” located at the layer nearest to the left electrode, etc. The superconducting “islands” form the network with the path for the superconducting current. The propagation of a Josephson current through the S-N -S junction requires the formation of a channel between the electrodes. The transport of the charge in superconducting cuprates in the c-direction is provided by the interlayer Josephson tunneling (intrinsic Josephson effect see, e.g., Kleiner et al., 1992; Scheekga et al., 1998). Therefore, the Josephson current through the barrier is measurable because of the superconducting state present in the layers. The transfer of the Josephson current in our model implies that the electrons tunnel inside of the layers between the superconducting “islands” until one of them appears to be close to some “island” in the neighboring layer. Then the next step, namely the interlayer charge transfer via the intrinsic Josephson effect occurs, etc. As a result, the chain formed by the superconducting “islands” provides the Josephson tunneling between the electrodes and the path represents a sequence of superconducting links. It is important to note that the amplitude of the total current is determined by the “weakest” link in the chain. The density of the critical current is determined by the equation: j = A exp(−r/) dP (5.2) or
j = (A/)
∞
dRP exp(−R/), 0
here ≡ 11 is the in-plane coherence length, R is the distance between the “islands’ on the same layer, A ∝ ns jc⊥ , ns ≡ ns (T ) is the concentration of the superconducting region, so that Ssup . = ns S is the area occupied by the superconducting phase, S is the total area of the layer, jC⊥ is the amplitude of the Josephson interlayer transition, and P is a probability of formation of chain with length R for the links, so that P = p m−1 (see, e.g., Stuffer and Aharony, 1992), m is the number of layers forming the barrier, and p is the probability for two neighboring in-plane “islands” to be separated by distance r < R. Assume that p is described by a Gaussian distribution, that is R drr exp{−−1 }[(r/) − C −1/2 ]2 , (5.3) p = (c/)1/2 −2 0
is the width of the distribution, c = const. Then the integral (5.3) can be calculated by the method of steepest descent, and we obtain ˜ (m) exp(−1/n1/2 jm = Af s ),
(5.4)
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Fig. 11. Dependence of the Josephson current on temperature: dotted line—experimental data; solid line theory.
A˜ = const. Eq. (5.4) can be written in the form jmax = j0 (−1/(T )).
(5.5)
Here 1/2 −1 j0 = jmax (Tc ), (T ) = n1/2 s (1 − ns ) ,
˜ f (m)∼ ¯ exp[− ln1/2 (ml)],
l˜ = ()−1/2 .
We assumed that >1, m?1. One can see directly from Eqs. (5.4) and (5.5) that the current amplitude depends strongly on temperature and it is determined mainly, by the dependence of the area occupied by the superconducting “islands” on T: ns ≡ ns (T ). In addition, there is a weak dependence of jmax on the barrier thickness. The dependence ns (T ) is different for various systems and is determined by the function Tc (r), that is, by the nature of the doping. Note that near Tc∗ the value of ns (T ) is very small and the current amplitude is negligibly small. However, the situation is different in the intermediate temperature region and in the region T >Tc∗ which is not far from Tc . This is true for the data by Bozovic et al. (2004) where Tc = 25 K and Tc < T < 35 K. For example, the value T = 30 K is relatively close to Tc but much below T ∗ = 80–100 K. At T = 30 K there are many superconducting “ islands”, so that the value of ns (T ) is relatively large. At temperatures close to Tc one can use Eq. (5.5) with = a(t − 1)v ; t = T /Tc and a = const (we have chosen a = 10). One can see (Fig. 11) that such a dependence with = 1.3 is in a good agreement with the experimental data. Note that this value for is close to the value of the critical index for the correlation radius in the percolation theory (see, e.g., Shklovskii and Efros, 1984). In principle, one can use a junction with a barrier grown in the ab direction, so that the c-axis is parallel to the S electrodes. Then the path contains SNS junctions formed by the “islands” with metallic N barriers. Since ab ?c (ab 20–30 å), one should expect even a larger scale for the “giant” Josephson proximity effect with thickness L ˚ up to 103 A. Therefore, the “giant” Josephson proximity effect is also caused by intrinsic inhomogeneity of the cuprates. The “giant” scale of the phenomenon is provided by the presence of “superconducting” islands embedded in the metallic matrix and forming the chain transferring the current. The use of superconductors in the “pseudogap” state as barriers represents an interesting opportunity for “tuning” the Josephson junction on a “giant” scale. 6. Other systems Intrinsic inhomogeneity is an essential feature of the high Tc oxides, and this feature is manifested in a peculiar “pseudogap” behavior. However, the scenario is a more general one and the “pseudogap” state can be observed in other inhomogeneous systems as well. Let us describe some of these systems.
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Fig. 12. Pb/Ag proximity system.
6.1. Borocarbides Borocarbides represent an interesting family of novel superconductors, because they allow us to study an interesting interplay between superconductivity and magnetism (see, e.g., Canfield et al., 1998; Schmiedeshoff et al., 2001). According to data by Lascialfari et al. (2003), the borocarbides YNi2 B2 C display precursor diamagnetism above Tc (Tc =15.25 K). Analysis of magnetization data taken with a high resolution SQUID magnetometer led to the conclusion that the unusual results were caused by an inhomogeneity of the compound and by the presence of superconducting droplets with a local value of the critical temperature higher than the usual Tc corresponding to macroscopic dissipationless current. It was shown that fluctuating magnetization cannot lead to the observed dependence. The presence of the superconducting isolated droplets is due to an inhomogeneity described above. Indeed, the pair-breaking effect in borocarbides as a major source of inhomogeneity (see Section 2) was analyzed by Ovchinnikov and Kresin (2002). An unconventional temperature dependence of the critical fields Hc and Hc2 observed experimentally (Schmiedeshoff et al., 2001) was explained by the presence of pair-breakers. Pair-breaking also leads to depression of Tc . A statistical distribution of pair-breakers leads to spatial dependence of the critical temperature: Tc ≡ Tc (r) and to the inhomogeneous “pseudogap” picture (Fig. 6). 6.2. W O 3 + N a compound Another complex system, the Na-doped WO3 compound has been studied by Reich and Tsabba (1999a,b) and later by Shengelaya et al. (1999), Reich et al. (1999). The material displays a small diamagnetic moment and a concomitant decrease in resistivity. STM spectroscopy of this material (Reich et al., 2000) has revealed a dip in the density of states, that is, a gap structure. Probably, this system is inhomogeneous and contains superconducting “islands”. 6.3. Granular superconductors; Pb+Ag system Granular superconducting films have been studied intensively before the discovery of high Tc superconductivity (see, e.g., Simon et al., 1987; Dynes and Garno, 1981). These films also represent inhomogeneous superconducting systems. Such inhomogeneous films could display a diamagnetic moment above Tc . It would be interesting to carry out a study of their magnetic properties. An interesting example of an inhomogeneous conventional superconducting system was described by Elsinga and Uher (1985). They studied a Sn-doped Bi sample. The sample has Sn-grains embedded in a semimetallic matrix which provides the proximity charge transfer. An interesting study of the Pb + Ag system was described recently by Merchant et al. (2001). An electrically discontinuous (insulating) Pb film was covered with increasing thickness of Ag (Fig. 12). The Ag act to couple the superconducting Pb grains via the proximity effect. The resistive transition, as well as tunneling spectra, has been taken on a series of these films. The most insulating film has no resistive transition but a full Pb gap as revealed by the tunneling spectra. This gap is reduced as silver is added reflecting the decrease in the mean-field Tc of the Pb grains. At some point, the composite film becomes continuous and superconducting with a low resistive transition temperature. The evolution of the mean-field transition temperature and the resistive transition temperature with increasing Ag thickness mimics the phase diagram of the cuprates with doping. The mean-field transition temperature resembles the pseudogap onset temperature and the resistive Tc resembles the superconducting transition temperature, with the mean-field transition temperature lying above the resistive transition.
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We believe that the results of the Pb/Ag artificial inhomogeneous superconductor model the behavior of the cuprates. The cuprates are doped substitutionally and inhomogenously. At some concentration of doping there are regions with a high enough concentration of carriers to locally superconduct and therefore reduce the low energy density of states. The evolution of these islands into a percolating dissipationless state would resemble the percolating proximity coupling described above. It is not then surprising that the phase diagrams would be nearly identical. 7. Conclusion Inhomogeneity is an important feature of novel superconducting systems and, most importantly, of high Tc cuprates. This property is caused by the statistical nature of the doping combined with pair-breaking effect. As a result, the critical temperature is spatially dependent: Tc = Tc (r). It is important to note that the “intrinsic” critical temperature (Tc∗ ) which corresponds to the formation of Cooper pairs inside of the diamagnetic “islands” (Fig. 6) is much higher that the temperature of the transition into the macroscopic dissipationless and coherent state (at Tc ; “resistive” transition). Our approach contains two important ingredients. First of all, the compound is intrinsically inhomogeneous and at T < Tc∗ contains superconducting regions (“islands”) embedded in a non-superconducting matrix. The superconducting regions form a percolation network, so that the transition (at T = Tc ) to the dissipationless, macroscopically phase coherent state is of a percolative nature. This aspect is general and totally irrelevant to the nature of superconductivity in the cuprates. We introduced this picture in 1999 (Ovchinnikov et al., 1999). Later, this picture was employed by Mihailovic et al. (2002) and Alvarez et al. (2005). In addition, and this is a second aspect of our scenario, we calculated the density of states (energy gap spectrum) and described several specific phenomena (anomalous diamagnetism, ac transport, “giant” Josephson proximity effect). These calculations performed for the Cooper paired superconducting state in the “islands” are in good agreement with the experimental data. This approach is still rather general in the sense that it is valid for any pairing force (phonons, magnons, plasmons, excitons, etc.), but, nevertheless, we are dealing (in the region Tc < T < Tc∗ ) with real Cooper pairs, so that Tc∗ is the “intrinsic” critical temperature. Moreover, each superconducting “island” has its own phase. It is an essential feature of this picture that the superconducting regions are embedded in a normal metallic matrix which provides normal dc transport. As a result, the proximity effect between the superconducting regions and the normal metal plays an important role. Until recently, the presence of inhomogeneites was considered as a signature of a poor quality sample (except for the “pinning” problem). However, we think that the situation is similar to that in the history of semiconductors. Indeed, initially the presence of impurities in these materials was considered as a negative factor (they were called “dirty” semiconductors). But later, when scientists developed tools allowing the precise control of the impact of various impurities (donors and acceptors), it became clear that the presence of impurity atoms is a critical ingredient; even the language had changed and sounds more “respectful” (“doped” semiconductors) The analogy between inhomogeneous novel superconductors and semiconductors is even stronger, because we are dealing with doping for both classes of materials. The “pseudogap” state is intrinsically inhomogeneous. Since a number of superconducting properties (diamagnetism, a.c. response, etc.) persists above Tc up to Tc∗ , one might think about interesting applications at temperatures higher than Tc . Microwave properties, Josephson effects with thick barriers (“Giant” effect, Section 5) are especially promising. In connection with this it would be interesting to carry out more detailed experimental and theoretical studies of transport properties (especially thermal transport), ac response, etc. One can expect many new and interesting results. References Abrikosov, A., Gor’kov, L., 1961. JETP 12, 1243. Abrikosov, A., Gor’kov, L., Dzyaloshinski, I., 1963. Methods of Quantum Field Theory in Statistical Physics. Dover, New York. Alloul, H., Ohno, T., Mendels, P., 1989. Phys. Rev. Lett. 67, 3140. Alvarez, G., Mayr, M., Moreo, A., Dagotte, E., 2005. Phys. Rev. B 71, 014514. Ashcroff, N., Merman, N., 1976. Solid State Physics. Holt, New York. Balatsky, A., Vekhter, I., Zhu, J., 2006. Rev. Mod. Phys. 78, 373. Balseiro, C., Falicov, L., 1979. Phys. Rev. 20, 4457. Barone, A., Paterno, G., 1982. Physics and Applications of the Josephson Effect. Wiley, New York. Bednorz, G., Muller, K., 1986. Z. Phys. 64, 189. Bergeal, N., Lesueur, J., Aprili, M., Faini, G., Contour, J., Leridon, B., 2006. arXiv: cond-mat/0601265.
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Physics Reports 431 (2006) 261 – 302 www.elsevier.com/locate/physrep
Review on the symmetry-related properties of carbon nanotubes Eduardo B. Barrosa, f , Ado Joriob , Georgii G. Samsonidzef , Rodrigo B. Capazc, d , Antônio G. Souza Filhoa , Josué Mendes Filhoa , Gene Dresselhause , Mildred S. Dresselhausf ,∗ a Departamento de Física, Universidade Federal do Ceará, Fortaleza, Ceará, CEP 60455-760, Brazil b Departamento de Física, Universidade Federal de Minas Gerais, Belo Horizonte, MG, 30123-970 Brazil c Instituto de Física, Universidade Federal do Rio de Janeiro, Caixa Postal 68528, Rio de Janeiro, RJ 21941-972, Brazil d Divisão de Metrologia de Materiais, Instituto Nacional de Metrologia, Normalização e Qualidade Industrial - Inmetro,
R. Nossa Senhora das Graças 50, Xerém, Duque de Caxias, RJ 25245-020, Brazil e Francis Bitter Magnet Lab, Massachusetts Institute of Technology, Cambridge, MA 02139-4307, USA f Departament of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA 02139-4307, USA
Accepted 24 May 2006 Available online 14 August 2006 editor: D.L. Mills
Abstract In this work we review the basic properties of carbon nanotubes from the standpoint of group theory. The zone folding scheme is reviewed in the light of the helical symmetry of the nanotube. The group theory for chiral and achiral nanotubes is reviewed, and the representations of the factor group of the wavevector k are obtained. The similarities and differences between the formalism of the group of the wavevector and that of line groups are addressed with respect to the irreducible representations and quantum numbers associated with linear and angular momenta. Finally, we extend the results of group theory to illuminate the electronic and vibrational properties of carbon nanotubes. Selection rules for the optical absorption and double resonance Raman scattering are discussed for the case where the electron–electron interaction is negligible (metallic nanotubes) and for the case where exciton binding energies are strong and cannot be neglected. © 2006 Published by Elsevier B.V. PACS: 61.46.Fg; 61.50.Ah; 78.67.Ch
Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Carbon nanotube structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Nanotube geometry and the (n, m) indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Lattice vectors in real space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Lattice vectors in reciprocal space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Compound operations and tube helicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1. Helical–helical construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2. Linear–helical construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∗ Corresponding author.
E-mail addresses: ebarros@fisica.ufc.br (E.B. Barros),
[email protected] (M.S. Dresselhaus). 0370-1573/$ - see front matter © 2006 Published by Elsevier B.V. doi:10.1016/j.physrep.2006.05.007
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2.4.3. Helical–angular construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.4. Linear–angular construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Group theory for carbon nanotubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Symmetries for chiral carbon nanotubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1. Group of the wave vector at the point (k = 0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2. Group of the wave vector at a general point (0 < k < /T ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Group theory for achiral carbon nanotubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Eigenvectors and the irreducible representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Quantum numbers and crystal momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Line groups vs. group of the wavevector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Line group of chiral carbon nanotubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Eigenvectors and the irreducible representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Quantum numbers and crystal momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Electronic band structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Symmetry of electronic states from the line group approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Phonons in carbon nanotubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Selection rules for optical phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1. Selection rules for optical absorption from space group theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. Selection rules for Raman and infrared spectroscopy from group theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3. Selection rules for double resonance Raman processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4. Selection rules from zone folding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Excitons in carbon nanotubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1. The exciton symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1. Chiral nanotubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.2. Zigzag nanotubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.3. Armchair nanotubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2. Selection rules for optical absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3. Selection rules for Raman scattering processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
269 271 271 272 273 274 275 277 278 279 280 282 283 283 286 287 289 289 289 290 291 293 293 294 297 297 298 299 300 300 301
1. Introduction The symmetry properties of carbon nanotubes have been a subject of intense discussion from the very beginning of the study of carbon nanotubes [1–3]. However, the earlier description was not complete, and much work was required to achieve a full description of the nanotube symmetry-related properties [4–7]. Also, Saito et al. used the zone-folding scheme to obtain, as a first approximation, the electronic and vibrational properties of carbon nanotubes [3]. Within this scheme, the properties of the nanotube are obtained directly from that of a graphene sheet by imposing the confinement of the wavefunctions in the circumferential direction, neglecting curvature effects. The symmetry properties of periodic lattices are usually described in terms of the group of the wavevector [8]. However, since nanotubes can be viewed as quasi-1D systems, the line group approach is suited to describe nanotube properties as well. The line group approach to nanotubes was described by Damnjanovi´c et al. [4]. Much confusion can arise from the use of these two different formalisms to describe the symmetry of the nanotubes. It is thus imperative to develop a full description of the carbon nanotube symmetry from the standpoint of the factor group of the wavevector and then to make a direct comparison with the symmetry properties of carbon nanotubes obtained using the formalism of line groups. To obtain a detailed group theoretical analysis of carbon nanotubes we begin in Section 2 by giving a summary of the structural properties of carbon nanotubes in real and reciprocal space. In Section 3, we describe the symmetry operations of chiral and achiral nanotubes and obtain the representations for each of the symmetry operations in view of factor group of the translational sub-group of the wavevector, as is common in general space group theory. In this section, the irreducible representations and the quantum numbers which are used to label them are compared to the symmetry properties and selection rules obtained from “zone-folding” of the reciprocal space of 2D graphite, as described in Section 2. In Section 4, the review continues with a comparison between the symmetry properties of nanotubes obtained using the formalism of the group of the wavevector and the formalism of line groups. The focus
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of this section is on chiral nanotubes. For a better comprehension of the symmetry-related properties of nanotubes, in Section 5 we discuss the symmetry of the electronic eigenstates using both the group of the wavevector and the approach of line groups. Furthermore, the symmetry-related properties of phonons in carbon nanotubes are discussed in Section 6, from the standpoint of the group of the wavevector developed in Section 3, and within the “zone-folding” scheme. These sections are followed by a description of the selection rules for optical phenomena in Section 7, neglecting the exciton interaction. The excitonic interaction is then described in Section 8 within a simple model that gives insight into the symmetry properties of the excitonic states and how they affect the selection rules for optical phenomena. The paper is concluded in Section 9 with a summary of the main results obtained here and the conclusions that can be drawn from this review about the symmetry of carbon nanotubes. 2. Carbon nanotube structure Carbon nanotubes can be viewed as a graphene sheet (a single layer from a 3D graphite crystal) rolled up into a cylinder, one atomic layer in thickness. The nanotube physical properties depend on how the graphene sheet is rolled up, and from a symmetry point of view, two types of nanotubes can be formed, namely the achiral armchair or zigzag tubes, as shown in Figs. 1(a) and (b), respectively, and the chiral tubes, as shown in Fig. 1(c). Because of the small diameter of a carbon nanotube (∼ 1 nm) and the large length-to-diameter ratio (> 104 ), carbon nanotubes are an important system for studying one-dimensional physics, both theoretically and experimentally. Therefore, in discussing the symmetry of carbon nanotubes, it is assumed that the nanotube length is much larger than its diameter, so that the nanotube ends (see Fig. 1) can be neglected when discussing the electronic and lattice properties of the nanotubes. Thus from a symmetry standpoint, a carbon nanotube is a one-dimensional crystal with a translation vector T along the cylinder axis and a small number of carbon hexagons associated with the circumferential direction. 2.1. Nanotube geometry and the (n, m) indices A single wall carbon nanotube (SWNT) is constructed starting from a strip of a graphene layer (see Fig. 2) by rolling it up into a seamless cylinder. The nanotube structure is uniquely determined by the chiral vector Ch which spans the
(a)
(b)
(c) Fig. 1. Schematic theoretical model for examples of the three different types of single-wall carbon nanotubes: (a) the “armchair” nanotube; (b) the “zigzag” nanotube; and (c) the “chiral” nanotube. The actual nanotubes shown in the figure correspond to (n, m) values of: (a) (5, 5), (b) (9, 0), and (c) (10, 5)—see text [3].
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y B B
x
T
θ R
A a1
O
Ch a2
Fig. 2. An unrolled nanotube unit cell projected on the graphene layer. When the nanotube is rolled up, the chiral vector Ch turns into the circumference of the cylinder, and the translation vector T is aligned along the cylinder axis. R is the symmetry vector (Section 2.4) and is the chiral angle. The unit vectors ( a1 , a2 ) of the graphene layer are indicated in the figure along with the inequivalent A and B sites within the unit cell of the graphene layer given by a hexagon. The unit cell of the nanaotube is defined by the rectangle delimited by the vectors Ch and T [3].
circumference of the cylinder when the graphene layer is rolled up into a tube. The chiral vector can be written in the form Ch = n a1 + m a2 , where the vectors a1 and a2 bound the unit cell of the graphene layer, which contains the two distinct carbon atom sites A and B. The values of n and m are arbitrary integer numbers which uniquely characterize the nanotube structure. In the shortened (n, m) form, the chiral vector is written as a pair of integers, and the same notation is widely used to characterize the geometry of each distinct (n, m) nanotube [3]. The nanotube can also be characterized by its diameter dt and chiral angle , which determine the length Ch =|Ch |=dt of the chiral vector and its orientation on √ the graphene layer (see Fig. 2). Both √ dt and are expressed in terms of the indices n and m √ by the relations dt = a n2 + nm + m2 / and tan = 3m/(2n + m), as one can derive from Fig. 2, where a = 3aC−C = 0.246 nm is the lattice constant for the graphene layer and aC−C = 0.142 nm is the nearest neighbor C.C distance. As an example, the chiral vector Ch shown in Fig. 2 is given by Ch = 4 a1 + 2 a2 , and thus the corresponding nanotube can be identified by the pair of integers (n, m) = (4, 2). Due to the six-fold symmetry of the graphene layer, all non-equivalent nanotubes can be characterized by the (n, m) pairs of integers where 0 m n. It is also possible to define nanotubes with opposite handedness, for which 0 n m [9]. The nanotubes are classified as chiral (0 < m < n) and achiral (m = 0 or m = n), which in turn are known as zigzag (m = 0) and armchair (m = n) nanotubes (see Figs. 1 and 2). A (4, 2) nanotube is one of the smallest diameter nanotubes ever synthesized [10]. It should be mentioned here that, for small diameter nanotubes (dt < 1 nm), the geometrical structure of the nanotube will be slightly different from that of a rolled up graphene layer. For a correct description of the nanotube properties, it is necessary to take the geometrical structure relaxation due to the curvature effect into account [11,12]. 2.2. Lattice vectors in real space To specify the symmetry properties of carbon nanotubes as 1D systems, it is necessary to define the lattice vector or translation vector T along the nanotube axis and normal to the chiral vector Ch defined in Fig. 2. The vectors T and Ch define the unit cell of the 1D nanotube. The translation vector T , of a general chiral nanotube as a function of n and m, can be written √ as T = t1 a1 + t2 a2 , where t1 = (2m + n)/dR and t2 = −(2n + m)/dR . The length of the translation vector is T = 3Ch /dR , where d is the greatest common divisor of (n, m) (denoted by gcd(n, m)), and dR is the greatest common divisor of 2n + m and 2m + n. Then d and dR are related by [3] d if n − m is not a multiple of 3d, (1) dR = 3d if n − m is a multiple of 3d.
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A
K
B a1
b1
K b2
a2 (a)
265
(b)
Fig. 3. (a) Real space lattice of a graphene layer. The rhombus represents the graphene unit cell with lattice vectors a1 and a2 delimiting it. Note that this area encloses a total of 2 atoms, one A atom and one B atom. (b) Reciprocal space lattice of the a graphene layer showing the unit vectors b1 and b2 . Note also that the reciprocal space structure has two inequivalent points K and K .
For the (4, 2) nanotube shown in Fig. 2, we have dR = d = 2 and (t1 , t2 ) = (4, −5). For armchair and zigzag achiral √ tubes, T = a and 3a, respectively. The unit cell of an unrolled nanotube on a graphene layer is a rectangle bounded by the vectors Ch and T (see the rectangle shown in Fig. 2 for the (4, 2) nanotube). The area of the nanotube unit cell √ h × T | = 3a 2 (n2 + nm + m2 )/dR . Dividing this can be easily calculated as a vector-product of these two vectors, |C √ product by the area of the unit cell of a graphene layer | a1 × a2 | = 3a 2 /2, one can get the number of hexagons in the unit cell of a nanotube, N=
2(n2 + nm + m2 ) . dR
(2)
For the (4, 2) nanotube we have N = 28, so that the unit cell of the (4, 2) nanotube (see the rectangle shown in Fig. 2) contains 28 hexagons, or 2 × 28 = 56 carbon atoms. For armchair (n, n) and zigzag (n, 0) tubes, N = 2n. 2.3. Lattice vectors in reciprocal space The graphene reciprocal lattice unit vectors b1 and b2 can be constructed from the real space lattice vectors a1 and a2 using the standard definition ai · bj = 2ij , where ij is the Kroneker delta symbol. In Figs. 3(a) and (b) we show the real and reciprocal space lattice of a graphene sheet, correspondingly. Note the difference of 30◦ in the orientation of the hexagons in real space (Fig. 3(a)) with respect to those in reciprocal space (Fig. 3(b)). In a similar fashion, the reciprocal space of a nanotube can be constructed, if we consider the nanotube as a 1D system with an internal structure that is composed of the 2N atoms in its unit cell and with a translational symmetry given by the translation vector T . The reciprocal space of the nanotube can be constructed by finding a pair of reciprocal lattice vectors K 1 and K 2 which satisfy: Ch · K 1 = T · K 2 = 2 and Ch · K 2 = T · K 1 = 0. Due to the spatial confinement of the nanotube in the circumferential direction, the vector Ch does not play the role of a translation vector, rather it acts as a generator of pure rotations. In this sense, the relation Ch · K 1 = 2 can only be satisfied for K 1 being integer multiples of 2/dt , where dt is the diameter of the nanotube. This approach was used by Saito et al. to obtain the properties of nanotubes from those of a graphene layer by means of “zone-folding” and “zone-unfolding” [3]. Although the “zonefolding” approach is easy to understand in its geometrical construction, the literature lacks a rigorous mathematical formalism to explain this approach. As we discuss in Section 2.4, the mathematical basis of the zone-folding scheme lies in the presence of the screw operations of the nanotubes, which allows for the nanotube to be represented in terms of a reduced unit cell composed of only 2 atoms, similar to a graphene layer unit cell. Thus, by using the screw translations, the properties of the nanotube can be directly related to those of a graphene layer. 2.4. Compound operations and tube helicity All multiples of the translation vector T will be translational symmetry operations of the nanotube [13]. However, to be more general, it is necessary to consider that any lattice vector tp,q = p a1 + q a2 ,
(3)
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with p and q integers, of the unfolded graphene layer will also be a symmetry operation of the tube. In fact, the symmetry operation that arises from tp,q will appear as a screw translation of the nanotube. Screw translations are combinations of a rotation by an angle (R ) and a translation in the axial direction of the nanotube. The screw translation can be written as {R |}, using a notation common for space group operations [3,8]. The translation vector tp,q can also be written in terms of components of the nanotube lattice vectors, T and Ch , as tp,q = tu,v = (u/N )Ch + (v/N )T ,
(4)
where u and v are given by u=
(2n + m)p + (2m + n)q dR
(5)
and v = mp − nq.
(6)
Both u and v are integer numbers which can assume either negative or positive values. The screw translation of the nanotube which is associated with the graphene lattice vector tu,v can then be written as u tu,v = {CN |vT /N },
(7)
u is a rotation of u(2/N ) around the nanotube axis, and {E|vT /N } is a translation of vT /N along the where CN nanotube axis, with T being the magnitude of the primitive translation vector T . It is clear that if a screw vector u |vT /N } is a symmetry operation of the nanotube, then the vectors {C u |vT /N }s , for any integer value of s, are {CN N also symmetry operations of the nanotube. The number of hexagons in the unit-cell N assumes the role of the “order” u |vT /N }N = {E|vT }, where E is the identity operator, and v T is of the screw axis, since the symmetry operation {CN a pure translation of the nanotube. The nanotube structure can be obtained from a small number of atoms (between 2 and 2N ) by using any choice u1 u2 of two non-colinear screw vectors {CN |v1 T /N } and {CN |v2 T /N }. Two vectors are colinear if there exists a pair of integers s and l different from 1, for which lu1 = su2 + N , and lv 1 = sv 2 + T , where and are two arbitrary integers. The area of the nanotube cylindrical surface delimited by these two non-colinear vectors can be regarded as a reduced unit cell. Note that the number of atoms in this reduced unit cell is given by the ratio between the area delimited by these vectors (|tu1 ,v1 × tu2 ,v2 |) and the area of the unit cell of a graphene sheet (| a1 × a2 |) multiplied by 2, for A and B atoms in the graphene unit cell. Thus the number of atoms in the reduced unit cell defined by tu1 ,v1 and tu2 ,v2 is given by
2
|tu1 ,v1 × tu2 ,v2 | |v2 u1 − u2 v1 | =2 . | a1 × a2 | N
(8)
It is important to point out that, in this case, the nanotube ceases to be described as a quasi-1D system, but as a system with two quasi-translational dimensions, which are generated by the two screw vectors. There are many combinations of screw vectors that can be used to construct the nanotube. These combinations can be divided into four categories: helical–helical, linear–helical, helical–angular and linear–angular, as described below. 2.4.1. Helical–helical construction The helical–helical construction is pertinent when two general non-colinear screw axes are used to construct the nanotube, denoted by the non-colinear vectors tu1 ,v1 and tu2 ,v2 , after Eq. (4). With a convenient choice of screw vectors, it is possible to obtain the nanotube structure from a motif that has only two atoms, as in the unit-cell of graphene. The most obvious choice of screw vectors that will define a reduced unit cell with only two atoms is the screw vectors corresponding to the lattice vectors a1 and a2 of the graphene unit cell. It can be seen from Eqs. (5) and (6) that these screw vectors will be obtained by choosing u1 = (2n + m)/dR , v1 = m, u2 = (2m + n)/dR , v2 = −n. In the case of the 5 |T /14} and {C 4 | − T /7}. Fig. 4(a) shows the shape of the (4, 2) nanotube, these screw vectors can be written as {C28 28 2-atom reduced unit cell of a (4, 2) nanotube. The shape of this reduced unit cell can be understood as the projection of the unit cell of the graphene layer onto the cylindrical surface of the nanotube.
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T
tu1,v1 tu,v (a)
tu
2,v2
(b)
tu,v
(c)
Ch/2
Fig. 4. 2-atom reduced unit cell for the (a) helical–helical construction, (b) linear–helical construction and (c) helical–angular construction for a (4, 2) nanotube.
The 2D reciprocal lattice vectors of the nanotube 1 and 2 , which follow the relation tu1 ,v1 · 1 = tu2 ,v2 · 2 = 2 and tu1 ,v1 · 2 = tu2 ,v2 · 1 = 0 can be written in terms of the reciprocal lattice vectors K 1 and K 2 as 1 =
Nv 2 N u2 K 1 − K 2 v2 u 1 − u 2 v1 v 2 u 1 − u 2 v1
(9)
and 2 = −
Nv 1 N u1 K 1 + K 2 . v 2 u1 − u 2 v 1 v2 u 1 − u 2 v1
(10)
From Eq. (8), we see that the number of atoms in the reduced unit cell is proportional to v2 u1 − u2 v1 and thus this number will not be zero for any choice of screw vectors that yield the nanotube structure. In the case where (u1 , v1 ) and (u2 , v2 ) are chosen so that the number of atoms in the reduced unit cell is 2, the reciprocal lattice vectors of the nanotube become the lattice vectors b1 and b2 of the graphene reciprocal space. The highest order symmetry operation of the nanotube, which maintains this unit cell invariant is a rotation of around an axis perpendicular to the nanotube axis and which goes through the center of a hexagon, or between the two inequivalent atoms. For achiral nanotubes, there are other symmetry operations such as mirror planes and inversion centers, but they are all of order 2. With the exception of the case of zigzag tubes, neither of the graphene reciprocal space unit vectors b1 and b2 are aligned with the translation vector T nor with the pure rotations of the nanotube Ch . Therefore, in the case of a general chiral nanotube, the helical–helical construction will not be the best choice to describe the nanotube properties. 2.4.2. Linear–helical construction The linear–helical construction is obtained when one of the screw vectors is chosen to be the translational vector u |vT /N } in the screw axis notation. The T while the other screw vector is a general one, which will be written as {CN u |vT /N } plays the translational vector T plays the role of the generator of translations, while the screw vector {CN role of a quasi-rotational symmetry. Substituting T as one of the screw vectors in Eq. (8), the number of atoms in the reduced unit cell of the nanotube is found to be 2u. Therefore, by choosing a screw vector with u = 1, it is also possible to construct the nanotube from only two atoms. Substituting u = 1 in Eqs. (5) and (6) gives the relation for v: v=
N md R − q, (2n + m) (2n + m)
(11)
where q has to be conveniently chosen so that v is an integer and is within the first unit-cell of the nanotube. Due to the symmetry of the nanotube upon a rotation perpendicular to the nanotube axis, for any given (n, m) nanotube there are two equivalent vectors which follow Eq. (11) within the first unit-cell, one for q > 0 and another for q < 0. These
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NR
MT
T R O
Ch
u |vT /N }, with u = 1 and v = M, is shown on the cylindrical surface, starting from O. The effect of applying R N times Fig. 5. The vector R = {CN u N (N R = {CN |vT /N } ) is depicted as an helicoidal path. After rotating by 2 around the tube, the vector N R reaches a lattice point equivalent to point O, but separated from O by M T . In the figure we show the case Ch = (4, 2) where v = M = 6.
vectors are related by |v + | = N − |v − | and, to avoid ambiguity, the smallest value of v should always be used. By choosing this vector, we reproduce the symmetry vector R defined by Saito et al. [3] for which the value v was denoted by M. Note that M can be regarded as the number of unit cells spanned by the screw vector R when it is applied N times, as shown in Fig. 5. Another characteristic of R is that the nanotube is rotated by exactly 2 when the screw vector R is applied N times. The 2-atom reduced unit cell, in this case, has a rhombic shape projected on the cylindrical surface The highest with one of the sides coinciding with the translational vector T and other aligned with the screw vector R. symmetry operation of this unit cell is also the rotation perpendicular to the nanotube axis. In Fig. 4(b) we show the 2-atom reduced unit-cell of the (4, 2) nanotube for the linear–helical construction, where the screw vector R has we show in Fig. 6(a) a diagram of components u = 1 and v = M = 6. To better illustrate the action of the screw vectorR, the (4, 2) nanotube. The dark atoms in the bottom represent a 2-atom motif. We also show another set of atoms in dark which is equivalent to this motif due to a rotation of 2/d, with d = 2, around the nanotube axis. The dark-grey helix of atoms is composed of the atoms in the nanotube unit cell which can be obtained by consecutive applications of the screw vector R to the atoms in the motif, while the light-grey atoms are obtained by successive operations of the screw vector R followed by a pure translation which brings the motif back to the original unit cell. For better visualization, a 2D diagram of the unfolded unit cell of the (4, 2) nanotube is shown in Fig. 6(b). In this diagram, the 2-atom motif is depicted as an ellipse and each helix in the nanotube is depicted as a dashed line. Note that the translation vector T connects one motif in this unit cell (full dark ellipse) to an equivalent motif in the next unit cell (open ellipse). By substituting the two unit vectors of the linear–helical construction (the translation vector T and a general screw vector tu,v ) into Eqs. (9) and (10), we find the reciprocal lattice vectors to be 1 = (N/u)K 1 ,
(12)
2 = K 2 − (v/u)K 1 .
(13)
and
The Brillouin zone of this reciprocal space is delimited by the vectors 1 and 2 . However, since 1 is perpendicular to the cutting lines, it is convenient to define the Brillouin zone as a rectangle composed of N/u cutting lines of length 2/T which can be translated by the vectors 1 and 2 to cover all of reciprocal space. It is important to notice that
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T
R
R
(a)
(b)
T
Z (c)
Z (d)
Ch
Fig. 6. Unit cell of the (4, 2) nanotube, with its 28 atoms colored for a better visualization of the linear–helical (a) and helical–angular (c) constructions from a 2-atom motif (dark atoms in the tubes). Diagrams for the unfolded unit cell of the (4, 2) nanotube where, for better visualization, the 2-atom motif is depicted as a filled ellipse. The dark grey atoms and ellipses represent those atoms in the unit cell which can be directly obtained by the for (c) and (d), while the light grey atoms can be obtained from the latter by applying application of the R vector, for (a) and (b), or by the vector Z, other symmetry operations, such as the translation vector T , for (a) and (b), or the rotations of 2/d, for (c) and (d) [3].
each of the N/u lines will be associated with a number of solutions of the Hamiltonian which are always a multiple of 2u, and thus are equal to the number of atoms in the reduced unit cell, so that the number of solutions of the Hamiltonian can be invariant. Furthermore, N/u should be an integer, and therefore, the number of different possible choices of screw vectors that can be used to construct the nanotube is limited by the number of integer factors of N. By which will define a reduced unit cell with two atoms, the reciprocal appropriately choosing the screw vector tu,v = R, space structure of the nanotube can be superimposed on that of a graphene layer and all the properties of the nanotube can be obtained from that of 2D graphite. Thus, the “zone-folding” scheme can be understood as an application of the compound operations of the nanotube. Fig. 7(a) shows the reciprocal space of the (4, 2) nanotube constructed using the The reciprocal space lattice vectors for the (4, 2) nanotubes linear–helical representation and choosing the vector R. ( 1 = 28K1 and 2 = K2 − 6K1 ) are shown in Fig. 7(a) as compared to the graphene reciprocal space. The 1st Brillouin zone, which is shown in dark-grey, can be translated to the adjacent Brillouin zones, shown in light-grey, by applying reciprocal lattice vectors. 2.4.3. Helical–angular construction The helical–angular construction (see Fig. 4(c)) is obtained by choosing one of the screw vectors used to construct the nanotube to be parallel to Ch . An obvious choice for that is the vector given by Ch /d, where d is the greatest common divisor of n and m, thus dividing Ch = n a1 + m a2 into integer units of a1 and a2 . The other vector used to construct the u |vT /N }. In the helical–angular construction, nanotube can be any general screw vector, which can be written as {CN the screw vector plays the role of a generator of translations, while the vector in the circumferential direction generates
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K1
K2 κ1 κ2
(a)
K1
κ2
K2
κ1
(b) Fig. 7. Reciprocal space of the graphene layer. Parallel equidistant solid lines represent the cutting lines for the (4, 2) nanotube. The linear–helical (a) and helical–angular (b) 1st Brillouin zones are shown in dark-grey. The light grey rectangles are the Brillouin zones obtained by the unit vectors 2 and 1 of the linear–helical (a) and the helical–angular (b) reciprocal space structures, respectively [14].
pure rotations. This approach also allows the choice of a general screw vector for which the number of atoms in the reduced unit cell is 2, corresponding to the choice of a screw vector with v = d. In this case, u will follow the relation: u=
2n + m d N + q, m dR m
(14)
for which there will be 2d different values of q that lead to integer values of u. These vectors are equivalent, and therefore, to avoid confusion, the shortest vector should always be chosen. In their work, Damnjanovi´c et al. defined related to the quantity r, which is defined in Ref. [5]. The value r can be regarded as the number the screw vector Z, is applied N times. In their publication, it was pointed out of times the nanotube is rotated by 2 when the vector Z that any multiple of N/d could be added to r without causing any changes to the symmetry properties of the nanotube. This will result in d possible values for u. However, the values of u obtained with both q 0 and q < 0 should be taken into account, and thus 2d choices of u can be used without loss of generality. In Fig. 4(c) we show again the 2-atom reduced unit cell for a (4, 2) nanotube. For this nanotube the smallest value u = 5 in the helical–angular construction defined by Damnjanovi´c et al. used u = 9 (or actually u = −9) can be obtained by choosing q = 0. The screw vector Z for the (4, 2) nanotube which is obtained by choosing q = −1 in Eq. (14). This discrepancy originates from the fact that the arithmetical equation developed in Ref. [5] for obtaining the screw vectors only considered values of q < 0. vector obtained here with u = 5 is equivalent to the one obtained by Damnjanovi´c It should be pointed out that the Z et al. [5] and the choice of one vector or another is only a matter of convention. As expected, the reduced unit cell also has a rhombic shape projected onto the cylindrical surface of the nanotube. However, in this case one side of the rhombus is aligned with the circumferential direction and has a magnitude of Ch /d while the other side coincides with To help the visualization of the effect of the screw vector Z, we show a diagram of the (4, 2) unit the screw vector Z. cell in Fig. 6(c). The vector Z was obtained using the smallest value of u = 5. The darker atoms represent a 2-atom motif, while the dark-grey helix represents the atoms in the unit cell which are obtained from the 2-atom motif by The light-grey atoms are obtained by applying the vector Ch /d, d = 2 for the consecutive applications of the vector Z.
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(4, 2) nanotube, to one of the atoms in the dark-grey helix. For a better visualization, the diagram in Fig. 6(d) shows the unit cell of the (4, 2) nanotube unfolded, and with the 2-atom motif represented by an ellipse. u |vT /N } Considering that in the helical–angular construction one of the unit vectors is a general screw translation {CN and the other is the vector Ch /d in the circumferential direction, the reciprocal space vectors are found to be 1 = (d)K 1 − (ud/v)K 2
(15)
2 = (N/v)K 2 .
(16)
and In this formulation, the vector 2 is parallel to the cutting lines, and thus the Brillouin zone of the nanotube can be defined as a rectangle formed by d cutting lines of length (N/v)2/T . Each of the cutting lines will be associated with a number of solutions of the Hamiltonian which is a multiple of the number of atoms in the reduced unit cell (2v/d). Each of the d lines can be associated with pure angular momenta while the points within the line will be associated with a helical momentum. By choosing a screw vector with v = d, the number of atoms in the reduced unit cell is 2, and therefore, the reciprocal space can be superimposed on that of a graphene layer. Fig. 7(b), shows the reciprocal lattice vectors 1 = 2K 1 − 5K 2 and 2 = 14K 2 obtained for the (4, 2) nanotube from Eqs. (15) and (16). It is important to note that, since the vector Ch /d is a finite symmetry operation and does not play the role of a translation in this construction, it is convenient to consider a different reduced unit cell which is composed of the 2d atoms. This reduced unit cell is obtained by applying the Ch /d operation on the original 2-atom motif, or equivalently by choosing the vector in the circumferential direction to be Ch instead of Ch /d. By making this choice, the nanotube restores its quasi-1D-system character and the reduced unit cell can be represented by a cylinder of height given by T d/N which includes 2d atoms. The advantage of using this 2d-atom reduced unit cell is the fact that it exhibits all the point group operations of the nanotube and the highest order symmetry operation is the rotation of 2/d around the nanotube axis. Thus, the nanotube will have a 1D reciprocal lattice, spaced by a 2N /(T d) length. The number of independent solutions of any nanotube property will be a multiple of 2d. Since, in this case, the reduced unit cell has more than two atoms, the nanotube reciprocal space structure cannot be directly compared with that of a graphene layer. 2.4.4. Linear–angular construction The last case is the linear–angular construction where the translation vector T and a vector parallel to Ch are used to construct the nanotube. In the case where the vector in the circumferential direction is chosen to be Ch /d, the reduced unit cell is formed by one of the d sections of the nanotube translational unit cell, and thus it has 2N/d atoms. This construction can be regarded as a specific case of either the linear–helical or the helical–angular construction. Therefore, all the properties obtained generally for either of these two constructions independently should coincide as they approach the linear–angular case. In the linear–angular construction, the unit vectors are T and Ch /d. Thus the reciprocal space vectors 1 and 2 are given by: 1 = (d)K 1
(17)
2 = K 2 .
(18)
and
Since the number of atoms in the unit cell is 2N/d, the reciprocal space structure of the nanotube, as defined here, cannot be mapped onto that of a graphene layer. In the case where the vector in the circumferential direction is chosen to be the chiral vector Ch , instead of Ch /d, we restore the full nanotube unit cell, with a total of 2N atoms, as it was defined in Section 2.2. The nanotube can then be regarded as a 1D system, which will have a reciprocal lattice vector given by K 2 , as defined in Section 2.3, and thus the 1st Brillouin zone has a length 2/T . 3. Group theory for carbon nanotubes As discussed in Section 2.4, chiral carbon nanotubes exhibit compound operations and therefore belong to nonsymmorphic space groups. In this section we will start by developing the symmetry of a general chiral nanotube within
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the framework of space group theory. The discussion is then extended to the more symmetric achiral (armchair and zigzag) carbon nanotubes, which are also non-symmorphic, due to the presence of glide planes. The space group theory presented here is based on the formalism of the group of the wavevector, which takes into consideration the full symmetry of the purely translational unit cell, which has 2N atoms and is compatible with the linear–angular construction. However, the linear–helical construction of the nanotube, in which the translational vector T and a general screw vector tu,v are used to obtain the nanotube structure, also maintains the pure translational symmetry of the nanotube. Therefore, space group theory within the framework of the group of the wavevector is also compatible with the linear–helical construction. As we will show in Section 3.4, this compatibility allows for the direct comparison between the symmetry properties obtained from group theory and from the “zone-folding” scheme. 3.1. Symmetries for chiral carbon nanotubes The chiral nanotube symmetry operations can be separated into two sets. The first set, which we shall call the symmorphic set, is formed by the translation operations of the nanotube and the point group operations. The symmorphic set forms a sub-group of the total space-group of the nanotube, and thus it can be used to obtain some of the symmetryrelated properties. This group can be obtained as a weak direct-product [15] between the translational sub-group of the nanotube, composed of all the primitive translations, and the point group operations of the nanotube. To obtain the point group of the nanotube, we observe that in Section 2.4.4 we have shown that the nanotube can be rotated by an angle 2/d without changing its structure. Therefore, the Cd operation is a point group operation of the nanotube. Also, by choosing an appropriate axis perpendicular to the nanotube axis, the rotation by around this axis (C2 or C2 ) will also be a symmetry operation of the nanotube (see Fig. 8(a)). There are two different classes of rotations perpendicular to the nanotube axis (C2 and C2 ). For one of the classes (C2 ), the axis goes through the center of bonds between two equivalent atoms (shown in Fig. 8(a)). For the other class (C2 ), the axis goes through the centers of the hexagons. The point group of the nanotube can thus be obtained to be the axial point group Dd . In achiral nanotubes, besides the screw vectors, there are also mirror planes perpendicular to the nanotube axis and parallel to it, which are shown, respectively, in Figs. 8(b) and (c) for the (3, 3) nanotube. The second set of symmetry operations, which we shall refer to as the non-symmorphic set, is formed by the compound operations of the space group of the nanotube, which cannot be decomposed into pure translations of multiples of T and point group operations. In the case of chiral nanotubes, all the screw vectors tu,v , with the exception of multiples of T and Ch /d, are part of this set of operations. In achiral tubes, glide planes are also part of this set. Crystalline structures (both symmorphic and non-symmorphic) are described by infinite groups since, ideally, these structures have an infinite number of translations. It is easier to work the group theory of finite groups, and thus, it is
Cd
Cd
v
h
C2′
(4,2)
(a)
CHIRAL
(3,3)
(b)
(3,3)
ACHIRAL (c)
Fig. 8. (a) Unit cell of the chiral (4, 2) nanotube, showing the Cd , with d = 2, rotation around the nanotube axis and one of the C2 rotations perpendicular to the tube axis. A different class of rotations (C2 ), which is also present for chiral and achiral nanotubes, is not shown here. (b) A section of an achiral armchair (3, 3) nanotube is shown with the horizontal mirror plane h and the symmetry operation Cd , with d = 3. (c) The same (3, 3) armchair nanotube is shown with one of the vertical mirror planes v .
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necessary to factor out the translations by use of the concept of factor groups [8,16]. This concept is based on the fact that for any group G which has an invariant sub-group T, a factor group G/T can be defined by separating the elements of G into cosets of T. Each coset of T is going to act as an element of the factor group G/T, that is a finite group which can be isomorphic to a point group. In the case of materials exhibiting only symmorphic operations, the factorization is obvious and we end up with the point group of the space group (primitive translation = 0) and its sub-groups. In contrast, for non-symmorphic structures, the factor group is usually not a sub-group of the space group. Therefore, the separation between the translation and the point group operations is not obvious and the factor group has to be found by obtaining the cosets of the translational sub-group T. The group of the wavevector k for non-symmorphic space groups is obtained by defining the invariant subgroup Tk of the primitive translations that follow the relation: exp(ik) = 1.
(19)
Note here, that both the wavevector k and the translation vector are being regarded as scalars, instead of vectors since all translations in nanotubes occur in the axial direction. An arbitrary k = 0 can be written in the form k = ± /T , where and are both integers and coprime [17]. In this case, any vector of the form = T will satisfy Eq. (19), with being an arbitrary integer, and therefore will be an element of the subgroup Tk . In the cases of k = 0 and /T , the invariant subgroup Tk is exactly the group of all primitive translations of the nanotube which can be written as = T , with being an arbitrary integer. The next step is to determine the different cosets of Tk which are going to be the elements of the factor group and to organize them into classes. This can be done by observing two properties of cosets of invariant subgroups [18] which will not be proven here: • Given two different symmetry elements (g1 and g2 ) of the group G, if there is at least one pair of elements of the invariant subgroup Tk ( and ) which satisfy the condition: g1 = g2 , then the cosets g1 Tk and g2 Tk are identical, and therefore it can be said that both g1 and g2 generate the same coset of Tk . • Given two different cosets of the factor group G/Tk generated by different elements of G (g1 and g2 ), if there is an element X of G for which g2 = X−1 g1 X, the two cosets (g1 Tk and g2 Tk ) will belong to the same class in the factor group G/Tk . 3.1.1. Group of the wave vector at the point (k = 0) For easier comprehension, we will first obtain the factor group for k = 0, in which case the elements of T can be written in the form = T . u |vT /N }, {C u |vT /N }2 , {C u |vT /N }3 . . . which we Consider two arbitrary elements of a set of screw translations {CN N N u u s l will denote by {CN |vT /N} and {CN |vT /N} . Taking the product between each of these elements with two arbitrary members of the invariant translational subgroup of the wavevector k = 0, namely {E| T } and {E| T }, we obtain u su |vT /N }s {E| T } = {CN |svT /N + T }, {CN
(20)
u lu |vT /N }l {E| T } = {CN |lvT /N + T }. {CN
(21)
If there is at least one pair ( , ) which satisfies su lu |svT /N + T } = {CN |lvT /N + T }, {CN
(22)
u |vT /N}s T and {C u |vT /N }l T are identical. then the cosets formed by the products {CN N Relation (22) will only be true if
l = s + N/u
(23)
sv/N + = lv/N + ,
(24)
and
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Table 1 Character table for the group of the wavevectors k = 0 and /T for chiral tubes. This group is isomorphic to the point group DN DN
u |vT /N } {E|0} 2{CN
u |vT /N }2 2{CN
u |vT /N }(N/2)−1 . . . 2{CN
A1 A2 B1 B2 E1 E2 .. . E(N/2−1)
1 1 1 1 2 2 .. . 2
1 1 1 1 2 cos 4/N 2 cos 8/N .. . 2 cos 4(N/2 − 1)/N
... ... ... ... ... ... .. . ...
1 1 −1 −1 2 cos 2/N 2 cos 4/N .. . 2 cos 2(N/2 − 1)/N
1 1 (−1)(N/2−1) (−1)(N/2−1) 2 cos 2(N/2 − 1)/N 2 cos 4(N/2 − 1)/N .. . 2 cos 2(N/2 − 1)2 /N
u |vT /N }N/2 {CN
(N/2){C2 |0} (N/2){C2 |0}
1 1 (−1)N/2 (−1)N/2 −2 2 .. . 2 cos(N/2 − 1)
1 −1 1 −1 0 0 .. . 0
1 −1 −1 1 0 0 .. . 0
where is an integer. Substituting for the value of l, we obtain: v = ( − )u.
(25)
Since and can assume any integer values, it is always possible to find a pair of values for which the difference is an integer multiple of v, namely − = v. Substituting this relations back into the value of l in Eq. (23), we obtain l = s + N ,
(26)
where is an arbitrary integer. Therefore, l and s have N independent values (1, . . . , N) and the factor group of chiral nanotubes will have N distinct cosets related to the screw axis operation. Each of these cosets can be associated with one of the screw translations of order N which, in turn, is associated with specific lattice vectors of the graphene layer. The space group of the nanotube should not depend on the choice of the screw vector. It can be easily shown that any choice of screw vectors of order N will yield the same cosets. To obtain the full symmetry of the factor group of the wavevector k = 0 we also have to consider the point group symmetries, and how they can form classes. It can be shown that the C2 and C2 rotations perpendicular to the nanotube axis generate two independent cosets {C2 |0}T and {C2 |0}T, each belonging to a different class. Also, these symmetry u |vT /N}s into {C −u | − vT /N }s = {C u |vT /N }−s ≡ {C u |vT /N }N−s . In operations take the screw operation {CN N N N general, these two operations (s and −s) will originate different cosets, with the exception of the cases where s = 0 and N/2. Using the second property in Section 3.1, it can be shown that C2 and C2 operations make the cosets of s and −s belong to the same class. s = 0 will generate the same coset as the identity and s = N/2 and −N/2 will generate the same coset, and the class to which this coset belongs has only one element. It should be emphasized that N is always even, and therefore N/2 is always an integer. Finally, it is necessary to show the effect of the primitive rotational operation (Cd ), which is a symmetry operation in nanotubes where d = gcd(n, m) is larger than 1. The rotational symmetry in this case is a d-fold axis Cd which brings lu |lvT /N} into {C lu+N/d |lvT /N }. The operation {C |0} generates the same coset as the arbitrary screw translation {CN d N u N/d the screw vector {CN |vT /N} , and thus, it does not increase the symmetry of the nanotubes any further. We can conclude that the symmetry of the factor group is isomorphic to the DN point group, for which the character table is shown in Table 1. However, it is important to mention that although the factor group of chiral nanotubes is isomorphic to DN , the symmetry operations of the nanotube may be different from the symmetry elements of the DN s symmetry operations in the D point group correspond to the cosets of the screw operations point group. The CN N u s {CN |vT /N } T. In the same way, the C2 and C2 operations correspond to the cosets {C2 |0}T and {C2 |0}T, respectively. ˜ which should Thus, it is clear that each of the representations of the DN group can be labeled with a quantum number h, ˜ be associated with a helical momentum. The nature of this helical quantum number h will be discussed in more detail in Section 3.4. 3.1.2. Group of the wave vector at a general point (0 < k < /T ) For k = 0 one must find the star of k, which is the set of wavevectors that are obtained by carrying out all the point group operations on the wavevector k [8]. The rotational symmetry of chiral nanotubes (Cd ) will not affect the values of
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k in the axial direction, and therefore the symmetry operation Cd has no effect on the star of k. However, it is clear that the presence of a perpendicular axis C2 takes −k into k and, therefore, the star of k will include both the positive and the negative value of k. Thus, it is only necessary to find the group of the wavevector k for either positive or negative values of k. For simplicity, we shall only consider positive values of k. The group of the wavevector k is the group of all symmetry operations which leave k invariant or takes k to an equivalent value (different by a primitive translation in reciprocal space). Rotations around the nanotube axis will leave k invariant, and thus will be part of the group of the wavevector k. However, in the case of 0 < k < /T , the C2 and C2 operations do not leave k invariant, and therefore they should not be considered symmetry operations the group of the wavevector k. To obtain the factor group of the wavevector k, it is convenient to write k as /T with and being coprimes, as introduced in Section 3.1. We then write the elements of the translational subgroup of the wavevector k, Tk , in the form {E| T }, with being an arbitrary integer and E being the identity operation. The value represents the number of unit cells of the nanotube which have to be spanned so that the modified Bloch relation in Eq. (19) is satisfied for k = 0. We can repeat the same procedure of Section 3.1.1 to find the symmetry of the factor group for k = 0. The product u |vT /N}s and {C u |vT /N}l of a set of screw vectors with two arbitrary members of the between two elements {CN N invariant translational subgroup of the wavevector k > 0, namely {E| T } and {E| T } will only be equal if l = s + N/u
(27)
sv/N + = lv/N + ,
(28)
and
which leads to v = ( − )u. As before, and can assume any integer values, and therefore there can always be a pair ( , ) which satisfies: − = v, for being an integer. This leads to the relation l = s + N.
(29)
The group of the wavevector at 0 < k < /T is, therefore, isomorphic to a CN point group, which is Abelian and has each symmetry element in a class by itself. The only point group operation of k in the axial direction is the rotational operation Cd , which, as we have shown in Section 3.1.1, does not increase the symmetry of the factor group any further. The CN point group has N 1D irreducible representations. However, it can be shown that due to the compatibility relations between two arbitrary wavevectors k, the only irreducible representations of this group which can be related to solutions of the Hamiltonian are the ones which transform as the representations of the CN point group. The extra representations originate from a usual result in the factor group analysis of the group of the wavevector, that is the generation of irrelevant representations related to pure translations [8]. Thus, for any physical problem, the symmetry properties of nanotubes for 0 < k < /T can be fully described by using a factor group which is isomorphic to CN . In Table 2, we show the character table for the representations of the CN point group. Note that from the N irreducible representations of the CN group, there are (N/2 − 1) representations which are doubly degenerate due to time reversal symmetry. These representations are in fact 1D and to differentiate them from the 2D irreducible representations of the DN group we use different fonts in Tables 2 and 1 to label the representations. The case of k = /T , known as the X point in the Brillouin zone boundary [3], can be obtained by considering
= = 1 and including the C2 and C2 symmetry operations, since the k = /T and −/T can be translated into each other by a reciprocal lattice vector 2 = 2/T . Thus, because of the presence of the C2 symmetry operations, the group of the wavevector at k = /T is also isomorphic to DN . For example, in the case of the (4, 2) nanotube shown in Fig. 8(a), where N = 28, the group of the wavevector at a general point 0 < k < /T is isomorphic to the D28 point group and the number of irreducible representations and of classes is 28, and while at k = 0 and /T the group of the wavevector is isomorphic to C28 and the number of irreducible representations and classes is 13. 3.2. Group theory for achiral carbon nanotubes The group theory of achiral nanotubes can be obtained in a similar manner as was done for chiral tubes in Sections 3.1.1 and 3.1.2. Both armchair (n, n) and zigzag (n, 0) carbon nanotubes exhibit all the symmetry operations that were u |vT /N }s , where N = 2n, the rotation around the nanotube observed for chiral nanotubes, namely the screw axes {CN
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Table 2 Character table for the group of the wavevector 0 < k < /T for chiral tubesa CN
{E|0}
u |vT /N }1 {CN
u |vT /N }2 {CN
···
u |vT /N } {CN
···
u |vT /N }N−1 {CN
A B
1 1
1 −1
1 1
··· ···
1 (−1)
··· ···
1 −1
E±1 E±2 .. .
E±( N −1) 2
1 1 1 1 .. . 1 1
∗ 2 ∗2
.. .
N 2 −1 N ∗ 2 −1
2 ∗2 4 ∗4
∗ 2 ∗2
··· ···
.. .
.. .
N 2( 2 −1) N ∗2( 2 −1)
.. .
N ( 2 −1) N ∗( 2 −1)
···
··· ··· .. . ···
N−1 ∗(N−1) 2(N−1) ∗2(N −1)
.. .
N
(N−1)( 2 −1) N ∗(N−1)( 2 −1)
a This group is isomorphic to the point group C . The ± signs label the different 1D representations (E) with characters which are complex N conjugates of each other. These representations are degenerate due to time reversal symmetry. The complex number is ei2/N and ∗ is the complex conjugate of .
{ v|0} { v|T/2} tu,v={CNu|vT/N} T/2
{ h|0}
I Cn /2
Cn /4 Fig. 9. Extra symmetry operations of an armchair achiral nanotube. Short-dashed arrows correspond to vertical mirror planes (heavy dashes) and glide planes (light dashes) and the horizontal long-dashed line corresponds to a horizontal mirror plane. The solid arrow corresponds to the screw vector which is also present in achiral tubes. The large open dot indicates the position of an inversion center I [6,7].
axis Cd , where d = n, and the rotations perpendicular to the nanotube axis C2 and C2 . However, achiral nanotubes also exhibit other symmetry operations such as inversion centers, mirror planes and glide planes. The horizontal mirror plane h and one of the vertical mirror planes v are shown in Figs. 8(b) and (c), respectively. There is also an inversion center at the intersection of the h plane and the nanotube axis. The glide planes, which can be represented as { v |T /2}, are shown schematically in Fig. 9 for an armchair nanotube. To obtain the factor group of the wavevector k = 0, it is necessary to obtain all the non-equivalent cosets of the translational subgroup T, and then to separate them into classes, in the same way that is done for point groups [8]. It is known from Section 3.1.1 that the factor group of the wavevector k = 0 exhibits all the symmetry operations of the group D2n , since for achiral tubes N = 2n. The other symmetries of the nanotube can be obtained from the direct product group D2n ⊗ C1h which results in a group which is isomorphic to D2nh . The same point group was obtained by Alon for the rod [19] group of achiral nanotubes [6,7] for which the screw vectors were written in terms of improper axes S2n . The character tables for the D2nh group is shown in Table 3, where the C2n classes correspond to screw vectors of the nanotube, while the v and v classes correspond, respectively, to mirror and glide planes containing the nanotube axis.
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Table 3 Character table for the group of the wavevectors k = 0 and /T for achiral tubesa u |vT /2n}s · · · {C u |vT /2n}n n{C |0} n{C |0} {I |0} · · · 2{I C u |vT /2n}s · · · { |0} D2nh {E|0} · · · 2{C2n h 2 2 2n 2n
n{ v |0} n{ v |T /2}
A1g A2g B1g B2g .. . Eg .. .
1 1 1 1 .. . 2 .. .
··· ··· ··· ··· .. . ··· .. .
1 1 (−1)s (−1)s .. . 2 cos(s /n) .. .
··· ··· ··· ··· .. . ··· .. .
1 1 −1 −1 .. . 2(−1) .. .
1 −1 1 −1 .. . 0 .. .
1 −1 −1 1 .. . 0 .. .
1 1 1 1 .. . 2 .. .
··· ··· ··· ··· .. . ··· .. .
1 1 (−1)s (−1)s .. . 2 cos(s /n) .. .
··· ··· ··· ··· .. . ··· .. .
1 1 −1 −1 .. . 2(−1) .. .
1 −1 1 −1 .. . 0 .. .
1 −1 −1 1
A1u A2u B1u B2u .. . Eu .. .
1 1 1 1 .. . 2 .. .
··· ··· ··· ··· .. . ··· .. .
1 1 (−1)s (−1)s .. . 2 cos(s /n) .. .
··· ··· ··· ··· .. . ··· .. .
1 1 −1 −1 .. . 2(−1) .. .
1 −1 1 −1 .. . 0 .. .
1 −1 −1 1 .. . 0 .. .
−1 −1 −1 −1 .. . –2 .. .
··· ··· ··· ··· .. . ··· .. .
−1 −1 −(−1)s −(−1)s .. . −2 cos(s /n) .. .
··· ··· ··· ··· .. . ··· .. .
−1 −1 1 1
−1 1 −1 1
−1 1 1 −1
−2(−1) .. .
0 .. .
0 .. .
a This group is isomorphic to the point group D
0 .. .
2nh . The values of s and span the integer values between 1 and n − 1.
Table 4 Character table for the group of the wavevectors 0 < k < /T for achiral tubesa C2nv
{E|0}
u |vT /2n}1 2{C2n
u |vT /2n}2 {C2n
...
u |vT /2n}n−1 2{C2n
u |vT /2n}n {C2n
n{ v |0}
n{ v |T /2}
A A B B E1 E2 .. . E(n−1)
1 1 1 1 2 2 .. . 2
1 1 −1 −1 2 cos /n 2 cos 2/n .. . 2 cos(n − 1)/n
1 1 1 1 2 cos 2/n 2 cos 4/n .. . 2 cos 2(n − 1)/n
... ... ... ... ... ... .. . ...
1 1 (−1)(n−1) (−1)(n−1) 2 cos 2(n − 1)/n 2 cos 4(n − 1)/n .. . 2 cos(n − 1)2 /n
1 1 (−1)n (−1)n −2 2 .. . 2 cos(n − 1)
1 −1 1 −1 0 0 .. . 0
1 −1 −1 1 0 0 .. . 0
a This group is isomorphic to the point group C
2nv .
For 0 < k < /T the only symmetry operations which maintain k invariant are the screw vectors and the mirror and glide planes which contain the nanotube axis ( v and v ). The factor group of the wavevector k will then be isomorphic to the C2nv point group, for which the character table is shown in Table 4. In the case of the (3, 3) nanotube (see Figs. 8(b) and (c)), the group of the wavevector at a general point 0 < k < /T is isomorphic to the C6v point group while at k = 0 and /T the group of the wavevector is isomorphic to the D6h point group. 3.3. Eigenvectors and the irreducible representations Having the irreducible representations of the wavevector k, it is possible to obtain the symmetries of the eigenvectors used to describe the electronic and vibrational properties for all the points of the first Brillouin zone. The first Brillouin zone of the nanotube will extend from k = −/T to /T . According to space group theory, for each point k in the 1st Brillouin zone, the eigenvectors will have a symmetry which is given by one of the irreducible representations of the group of the wavevector k. Therefore, we can associate each solution of the Hamiltonian which transforms as the space group of the nanotube with an irreducible representation D (k) of the group of the wavevector k, and therefore we can label the solution with the quantum numbers k and , and the wavefunction can be denoted by (k).
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Table 5 Compatibility relations between the irreducible representations of the DN and CN groups, where = 1 . . . (N/2 − 1) CN 0 < k < /T A
DN k = 0, /T A1 A2
B
B1 B2
E±˜
E˜
Table 6 Compatibility relations between the irreducible representations of the D2nh and C2nv groups, where = 1 . . . (N/2 − 1) C2nv 0 < k < /T A
A B
B
E
D2nh k = 0, /T A1g A2u A1u A2g B1g B2u B1u B2g E g E u
As shown in Section 3.1.2, for the case of chiral nanotubes, the representations at 0 < k < /T will be given by group CN . In the CN group there are N 1D irreducible representations, from which N − 2 representations are brought together in pairs (E˜ and E−˜ ) by time reversal symmetry making −k to be equivalent to k. The remaining symmetries u |vT /N }s , for s odd. At k = 0 and A and B differ in relation to their parity on the application of the screw vector {CN /T , there are four 1D representations and [(N/2) − 1] 2D representations. The 1D representations A1 and A2 differ regarding parity with respect to the C2 and C2 operations, which exchange the two inequivalent atoms in the 2-atom reduced unit cell. The same difference applies to the B1 and B2 irreducible representations. The compatibility relations between the group of the wavevector at k = 0 and at k = /T (DN ) and at k = 0 (CN ) are summarized in Table 5. In the case of achiral tubes, the factor group of the wavevector 0 < k < /T is isomorphic to C2nv , which has four 1D-representations and (n − 1) 2D-representations. In this group, the B representations are related to states which are u |vT /N }s with s odd, while the A representations relate to antisymmetric upon the application of the screw vector {CN the corresponding symmetric states. The representations with a single prime ( ) are related to states which are symmetric under the v operation, while the double prime representations ( ) are antisymmetric under the same set of operations. At k = 0 and /T , the factor group is isomorphic to the D2nh point group. The D2nh group has eight 1D representations and (2n − 2) 2D representations. The 1D representations are labeled by A1(g,u) , A2(g,u) , B1(g,u) and B2(g,u) . Again, A and B differ in parity with respect to the application of the screw vector, while the subscripts 1 and 2 are related to the parity under rotations perpendicular to the nanotube axis (C2 and C2 ), 1 for even and 2 for odd. The u and g subscripts are related to the parity upon the inversion of the coordinate system. The compatibility relations between the representations of the D2nh and the C2nv groups are summarized in Table 6. 3.4. Quantum numbers and crystal momentum As explained in Section 3.1.1, the quantum number that labels the irreducible representations of the factor group of the wavevector k is associated with a helical momentum labeled by a helical quantum number h˜ which can be understood in
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terms of an helical coordinate . The variable specifies the position on a helicoidal orbit around the nanotube cylinder which can be defined so that it follows two conditions: first, that any point on the cylinder that delimits the nanotube can be determined by the z coordinate along the nanotube axis and the coordinate . Second, that the application of the u |vT /N}, used to construct the nanotube, changes only the coordinate , leaving the coordinate z screw operation {CN invariant. In this sense, the helical wavevector h˜ can be associated with a helical momentum which will be conserved. However, to avoid the discomfort of working with helical coordinates, it is possible to use only the projection of this coordinate in the direction of the pure rotations of the nanotube, and to disregard its translational component. In this case, the helical quantum number is reduced to its angular projection, denoted as , ˜ which is not a conserved quantity when crossing the Brillouin zone boundary, but still can be used to label the irreducible representations. We can understand the nature of the quantum number ˜ by comparing it to the reciprocal lattice vectors in the linear–helical construction of the nanotube structure (see Section 2.4.2). Within this construction, the nanotube structure in reciprocal space is obtained from the reciprocal lattice vectors 1 = (N/u)K 1 and 2 = K 2 − (v/u)K 1 , where K 1 and K 2 are the reciprocal lattice vectors of the nanotube structure as defined in Section 2.3. Since the reciprocal lattice vector K 1 can only assume integer values of 2/Ch , for a given value of k along the cutting lines, there are N/u inequivalent values of . ˜ In the case of a 2-atom reduced unit cell, obtained for u = 1 within this construction, each of the N inequivalent cutting lines can be directly associated with one of the N irreducible representations of the factor group of the wavevector k (for chiral tubes). Therefore, we have a direct relation between the quantum number ˜ and the cutting line indices which have been referred to in the literature as [3]. It is important to emphasize that this connection can only be obtained by choosing the 2-atom reduced unit cell. Following this approach, the linear quantum number k can be directly associated with K 2 , since both originate from the translational symmetry vector T . However, it should be noted that if the value of k crosses the boundary of the 1st Brillouin zone, it is necessary to use the vectors 1 and 2 to translate it back to the 1st Brillouin zone (see Fig. 7). By doing this, the quasi-angular momentum ˜ will be corrected by a multiple of v/u, which, in the case of the 2-atom unit cell is denoted by the quantity M. This correction ˜ and it will be of great importance for arises from the fact that ˜ is only a projection of the helical quantum number h, the evaluation of the selection rules in the case where the operator couples k states in different Brillouin zones. This will happen in the case of double resonance Raman scattering processes and in the formation of excitonic eigenstates, which will be discussed in Sections 7.3 and 8.1, respectively. For most cases, the selection rules for the interaction between two different k, ˜ states will be easily obtained directly from the conservation of linear momenta on the graphene reciprocal lattice structure, in terms of the cutting lines (see Section 7.4) [14]. This approach has been widely used in the literature and is useful for better physical understanding. The limitations of the cutting lines approach for obtaining the selection rules appear when the group of the wavevector k is of higher symmetry, such as at k = 0 for chiral tubes and at any value of k for achiral tubes. In this case, the selection rules for the coupling between states of different parity, such as A and B symmetry states or Eg and Eu states, cannot be obtained from the cutting lines approach. Thus, although many of the selection rules can be obtained from the conservation of momentum in the “unfolded” graphene layer, this approach is not complete and in many cases it has to be refined by a more detailed analysis. 4. Line groups vs. group of the wavevector The symmetry of systems exhibiting translational periodicity in one dimension, such as stereoregular polymers and carbon nanotubes, have been described in terms of line groups [4,20]. Also, some three-dimensional crystals can be highly anisotropic, as for example ferromagnetic and ferroelectric systems, chain-type crystals, and these crystals have line groups as subgroups of their space groups. Whenever only one direction is relevant for some physical properties of a three-dimensional system, one can expect to derive useful information by applying suitable line groups only. Line groups are infinite groups for which the irreducible representations contain all the symmetry properties of the material in 1 direction. Line groups can be represented as a weak-direct product L = Z · P , where Z and P are, respectively, the generalized translation group and the axial point group symmetry of the system. The product must be a weak-direct product [15] (indicated here by “·”) because all elements of Z, except for the identity, have a nonzero translational part, while no point group element in P has translations. The intersection is, therefore, only the identity operation [21]. Thus, the main difference between the formalism used in Section 3 and the formalism which will be presented here is the fact that, within the framework of the group of the wavevector k, the infinite translational part of the space group operations is factored out, and the remaining operations are viewed as “generalized” point group operations, while in the line group
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formalism, the point group operations, and the remaining operations are viewed as “generalized” translations, forming an infinite group. The line groups are then constructed as a weak-direct [15] product between the point group and the group of generalized translations. It is important to notice that the axial point group P has to be composed solely of symmetry operations which maintain the line axis invariant, and we will conveniently choose the line axis to be the z axis. Therefore, the axial point groups P can be: Cn , S2n , Cnh , Cnv , Dn , Dnh and Dnd , where n = 1, 2, 3, . . . is the order of the principal rotation axis. There are two types of axial point groups, those for which all operations leave a point z along the axis invariant and those for which there is at least one operation in the group which takes z into −z [4]. By the generalized translation group, we mean that group Z denotes infinite cyclic groups composed of general translational operations along the line axis, and these operations may include screw axes or glide planes. The symmetry u of u times 2/N around the nanotube axis elements of Z can be represented by {R|}, where R is either a rotation CN or a mirror plane which contains the line axis ( v ), while is a translation vT /N along the line axis of a fraction of the translational vector of the system denoted by T . In the case of the glide plane vT /N = T /2. The values of u and v are integers, which in the case of nanotubes will be defined as in Eqs. (5) and (6). There are three different types of generalized translation groups: • Those formed by simple translations, T = {E|T }. The translational period is T; u |vT /N }. The translational period is vT ; • Those with the occurrence of a screw axis, TNu = {CN • Those with the occurrence of a glide plane, Tc = { v |T /2}. The translational period is T /2. However, a product Z ·P forms a group only if Z and P commute (this is always the case only for Z =T ). Furthermore, some products with different factors are identical. There is an infinite number of line groups, and they are classified into 13 families [21]. The line group of chiral nanotubes belongs to the family LNv 2 or LNv 22 depending on whether d is even or odd. The LNv 2 and LNv 22 families of line groups, as defined in Ref. [20], can be factorized as TNu · Dd , where Dd is the point group of the chiral nanotubes. For achiral nanotubes, the line group will belong to the family denoted by L2dd /mcm which can be factorized as T2d · Ddh [20,21]. In Ref. [21], a table of all the 13 families is shown together with their respective factorizations. It should be noted that, for the sake of consistency within our work, the symbols used here to label the different families are different from the symbols used in Ref. [21]. Also, the irreducible representations for all the line group families are described in the literature [22,23]. For clarity, we will obtain here the irreducible representations of the line group of chiral carbon nanotubes from the properties of the “generalized” translational group. The line group approach can only be applied using the pure translational group, which will lead to irreducible representations and quantum numbers equivalent to those obtained using the approach of the group of the wavevector [24]. The results obtained in this section will be used later for a comparison between the line group formalism and the formalism of the group of the wavevector for obtaining the symmetry-related properties of carbon nanotubes. 4.1. Line group of chiral carbon nanotubes In this section, we will derive the line group of chiral carbon nanotubes using the formalism first developed by Vujiˇci´c et al. [20]. The corresponding derivation for achiral nanotubes follows the same pattern and will not be shown here. To construct the line group for chiral carbon nanotubes, it is necessary to first define the group of generalized translations Z. In the case of chiral nanotubes, the group Z can be obtained by choosing one of the general screw vectors of the u |vT /N }. It will not be explicitly shown here, but any nanotube which, as defined in Section 2.4, can be written as {CN choice of screw vector of the nanotube will yield line groups which belong to the same family. We can construct the irreducible representations of the line group of generalized translations by writing the wavefunctions in terms of (, ), u |vT /N }s = + 2s and is a rotation around the nanotube axis. where the helical coordinate is defined so that {CN The irreducible representations of the generalized translational group should then obey a modified Bloch relation: u ˜ {CN |vT /N }s (, ) = ( + 2s, ) = exp(ih)(, ) = (, ).
(30)
u |vT /N } will be written as Thus the irreducible representations of the line group of screw translations {CN
˜
h˜ T = exp(ih),
(31)
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where the notation h˜ T is introduced to avoid confusion with the space group notation Tk . Note that, in this notation, the quantum number related to the generalized translation appears as a subscript on the left-hand side of the character ˜ which is a helical which labels the representation (T). In this purely helical formulation, the quantum number h, wavevector that can only be represented in two dimensions, is completely conserved. The variables and can be written in terms of cylindrical coordinates z and and thus the irreducible representations will be obtained in terms of the projections of h˜ on the translational and rotational parts (k˜ and ) ˜ as 2su u s ˜ svT ˜ . (32) k˜ T˜ ({CN |vT /N } ) = exp ik N + i N u |0} are symmetry operations However, in this notation, neither the pure translation {E|vT /N } nor the pure rotation {CN of the nanotube. Thus the quantum numbers that label the irreducible representations, k˜ and , ˜ are not conserved separately. By using Eq. (32), the modified Bloch relation for the wavefunctions, given by Eq. (30) can be rewritten in terms of z and : svT 2su 2su svT u s ˜ {CN |vT /N } (z, ) = z + , + = exp i k + ˜ (z, ) = (z, ), (33) N N N N
where the value of k˜ varies within the range of (−N /(vT ), N /(vT )], while ˜ assumes integer values between −N/(2u) + 1 and N/(2u), provided that u is a divisor of N/2. Since we are constructing the irreducible representation of the translational group, it is convenient to take only the ˜ This projection assumes the values of k˜ which vary from [−N /(vT ), N /(vT )]. It translational projection of h. should be noted that in Section 3.4, the same coordinate was introduced to obtain the irreducible representations of the space group of nanotubes. Note that in the formulation of the group of the wavevector, the angular projection of h˜ was used to label the representations (see Section 3.4). To obtain the irreducible representations of the line group of chiral nanotubes, it is necessary to apply the operations of the axial point group of the nanotube to the irreducible representations of the group Z. The axial point group of chiral nanotubes is the Dd group, where d is defined in Section 2 as d = gcd(n, m). The Dd point group can be represented as Dd = Cd + C2 Cd where Cd is the Abelian group of the d-fold rotations around the z axis and C2 is a rotation perpendicular to the z axis. The irreducible representations of the group of the nanotube can also be obtained by first getting the irreducible representations of the product Z · Cd and then constructing the final representations by applying Z · Dd = Z · Cd + C2 (Z · Cd ). The irreducible representations of the group are easily obtained from Z·Cd , where Z is the group of screw translations. The operations of Cd leave any point on the z axis invariant. Therefore, the representations of Z · Cd will be obtained by multiplying the representations of Z, given in Eq. (32), by the irreducible representations of group Cd . The point group Cd is an Abelian group in which all representations are 1D. The th representation of the symmetry operation Cdl of group Cd is given by (Cdl ) = exp(il2/d), where l and can assume integer values between 0 and d. Here, labels the representations of the nanotube point group, and is related to pure rotations, in contrast with , ˜ which is a ˜ projection of h. The representations of the group Z · Cd will then be given by 2 l u ˜ , (34) D (C {C |vT /N}) = exp i h + il N h˜ d d where h˜ D is used to denote the irreducible representations of line groups. Here, h˜ and are quantum numbers ˜ in the labeling these representations. It is convenient to use only the projection of h˜ in the translational direction (k) representations, ignoring the rotational part (). ˜ Thus, the representations will be written as vT 2 u+Nl/d ˜ D ({C |vT /N}) = exp i k + il . (35) N k˜ N d However, it can be understood that the C2 symmetry operation not only takes z to −z, but also causes the two u |vT /N}s and {C u |vT /N }−s to be equivalent and therefore they should have intrinsically different screw vectors {CN N the same irreducible representation, and thus belong to the same class. Note that for k = 0 the representations
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and k˜ ({CN |vT /N}−s ) are complex conjugates of each other, and thus they are of the same irreducible representation. Therefore, to construct the representations of the group Z · Dd it is necessary to reorganize the representations for either k˜ or different from zero into 2-dimensional representations. The procedure for the construction of these representation is described in the work of Vujiˇci´c et al. and Bozovi´c et al. [20,22,23]. These representations will then assume the form 0 u+lN/d −k˜ − s , (36) E ({CN |vT /N} ) = k˜ 0 ∗
u+lN/d |vT /N }s ) k˜ ({CN
u+lN/d
and u+lN/d −k˜ − E ({C2 CN |vT /N}s ) = k˜
0
∗ 0
,
(37)
˜ /N +isl2/d), and the notation −k˜ E− , with the left-hand side super and subscripts, is introduced where =exp(is kvT k˜ to differentiate this line group representation from the notation used for the representations of the group of the wavevector k, which are labeled only by ˜ as a subscript on the right-hand side. For k˜ = 0 and = 0, we have = ∗ = 1 and the representations 00 E00 will be reducible. Therefore, these representations will be rewritten as 1D representations 0 A+ 0 and 0 A− 0 , where the “+” and “−” signs refer to the parity of the representation upon the C2 operation, “+” for even and “−” for odd. The notation 0 A0 is also introduced to differentiate it from the A symmetries used for the irreducible representations of the group of the wavevector. The subscripts 0 indicate that the labels k˜ and are both equal to zero. The same happens for k˜ = N /(vT ) and = 0, for which the irreducible representations will be rewritten as N /(vT ) A+ 0 and N /(vT ) A− 0. A similar procedure can be used to construct the line group of achiral nanotubes, which will belong to the L2dd /mcm family. The irreducible representations, Bloch functions and selection rules for chiral and achiral nanotubes are described in detail in the literature [25]. 4.2. Eigenvectors and the irreducible representations In space group theory, the representations of the translation group are obtained of the pure translational symmetry, and thus are independent of the rotations of the system. Therefore, as is common in group theory, the irreducible representations of this translation group are all 1D and do not have any influence on the dimension of the irreducible representations of the factor group. In contrast, in the line group formalism, the generator of translations is a screw vector, which also applies a rotation to the system. Thus, the separation between rotations and translations is not completely achieved. The representations of the line group are not specifically of rotations or translations, but a mixture of both. Although only the projection of the helical wavevector h˜ is used to label the representations, the representations have to reflect the mixed character between the translational and rotational operations. Therefore, the representations of the group of “generalized” translations will have some properties of point group representations. For example, the symmetry operation C2 takes all translations in the z direction to − and all rotations by around the z axis to rotations by −. In the line group formalism, the screw vectors {R |} and {R− | − } ≡ {R |}−1 will belong to the same class and have the same characters for each irreducible representation. The relation between irreducible representations and the eigenvectors within the line group formalism becomes more clear if we take into consideration the 2d-atom reduced unit cell introduced in Section 2.4.3. The 2d-atom reduced unit cell exhibits all the symmetry of the nanotube, and can be regarded as a “generalized” unit cell, from which the nanotube is constructed by successive applications of a “generalized” translation. Thus, there will be a multiple of 2d independent solutions to a Hamiltonian with the nanotube symmetry. For chiral nanotubes there are d 2D irreducible representations for 0 < k˜ < N/(vT ). Thus, each of the d representations will be associated with two different eigenstates, one eigenstate related to bonding states and one to anti-bonding states, in a similar fashion to that − discussed in Section 3.3. At k˜ = 0 there are two 1D irreducible representations (namely, 0 A+ 0 and 0 A0 ) and also two − 1D representations for k˜ = N /(vT ) (namely, N /(vT ) A+ 0 and N /(vT ) A0 ). In these representations the “+” signs are related to bonding states while the “−” signs can be associated with anti-bonding states, as in Section 3.3. Within the line group approach, both helical momentum and angular momentum conservation will arise directly from the selection rules. Thus, the formalism of line groups is very convenient when there is a coupling between states
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Table 7 Quantum numbers used to label the irreducible representations in the line group and in the group of the wavevector formalisms Quantum number
Symmetry
Group theory
Range
k
Translational Rotational Translational Rotational
Wavevector Line Line Wavevector
(−/T , /T ] 0, . . . , d − 1 [0, N/(dT )] 1 − N/2, . . . , N/2
k˜ h˜ ˜
˜ can be used in either of the formalisms. However, the use of its projections k˜ and ˜ is more common. The helical quantum number (h)
which are associated with different linear and angular momentum quantum numbers. However, it should be noted here that the formalism of the group of the wavevector yields the same symmetry-related properties, and the choice of the formalism is more dependent on the acquaintance of the researcher with one of the formalisms than on any intrinsic difference between them. In Section 5.1 we will compare the results for the group of the wavevector and for line groups regarding the symmetry of electronic states. 4.3. Quantum numbers and crystal momentum The irreducible representations of the line group of nanotubes are labeled by the quasi-linear quantum number k˜ and the angular momentum quantum number . The fact that the quasi-linear wavevector k˜ is not a conserved quantity ˜ This case is opposite comes directly from its definition as the translational projection of the helical quantum number h. to that of the group of the wavevector, for which the label k corresponds to a pure linear quantum number and ˜ is ˜ In Table 7, we show a summary of the quantum numbers used to label the irreducible the angular projection of h. representations of line groups and the group of the wavevector, specifying where each quantum number can be applied and what is the range of inequivalent values that each quantum number can assume. To better comprehend the conservation of the quasi-linear quantum number k˜ and of the angular momentum , we follow the same procedure as was used in Section 3.4 and we compare the quantum numbers k˜ and with the lattice vectors 1 and 2 for the 2-atom unit cell with the helical–angular reciprocal space structure. As shown in Section 2.4.3, the nanotube lattice vectors will be given by 1 = d K1 + (ud/v)K2 and 2 = N/dK 2 , with v = d. Thus, for a given cutting line , the inequivalent values of K2 lie in the (0, (N/d)/T ) range. Each representation of the line group of chiral nanotubes labeled by k˜ and can be associated with different points of the “unfolded” reciprocal space of the nanotube. In the case when the quantum numbers k˜ and exceed the limits of the 1st Brillouin zone of the helical–angular reciprocal space, it is necessary to use the reciprocal lattice vectors 1 and 2 to translate them back into the 1st Brillouin zone. This reciprocal space operation corrects the quasi-linear wavevector by a multiple of u. The correction of the quasi-linear quantum number k˜ also arises from the fact that this quantum number is only a projection ˜ As in the group of the wavevector formalism, this correction will affect the evaluation of the helical quantum number h. of the selection rules for coupling between states from different Brillouin zones. In this sense, the approach of using the cutting lines to evaluate the selection rules may prove to be more convenient, although, it should always be followed by a more detailed analysis to determine if the parity of the eigenstates influences the selection rules. For this part of the analysis, the researcher can choose between the line group formalism and that of the group of the wavevector, whichever he (she) finds more convenient. 5. Electronic band structure For large diameter nanotubes (dt > 1.5 nm) the electronic band structure can be derived from that of a flat graphene layer by using the concept of cutting lines [14]. Using the zone-folding scheme, the electronic band structure of the nanotube can be easily obtained by superimposing the cutting lines in the linear–helical representation of reciprocal space on the band structure of a graphene layer. The two carbon atoms A and B per unit cell of the graphene layer (see Fig. 2) have one free -electron each, resulting in the appearance of the valence and conduction bands in the electronic band structure of the graphene layer, where the valence band corresponds to a bonding state between the two atoms, and the conduction ∗ band corresponds to an anti-bonding state. The valence and conduction bands are degenerate at the K and K points in reciprocal space, which occur at the Fermi level (see Fig. 10(a)), while the other electronic
15
15
10
10
5
5
0
0
-5
-5
-π/T
(a)
(b)
0 Wave vector
Energy (eV)
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Energy (eV)
284
π/T 0.0 0.5 1.0 1. 5 DOS (states/C-atom/eV)
(c)
Fig. 10. (a) The conduction and valence bands of the graphene layer in the first Brillouin zone calculated according to the -band nearest-neighbor tight-binding model. Solid curves show the cutting lines for the (4, 2) nanotube. Solid dots show the ends of the cutting lines in the 2-atom linear–helical representation. (b) Electronic energy band diagram for the (4, 2) nanotube obtained by zone-folding from (a). (c) Density of electronic states for the electronic energy band diagram shown in (b) [14].
energy bands correspond to sp 2 -hybridized electrons involved in covalent bonds, and these sp 2 electrons lie far away in energy from the Fermi level. The energy–momentum contours for the valence and conduction bands of the graphene layer in the first Brillouin zone, calculated by using the -band nearest-neighbor tight-binding approximation [3], are shown in Fig. 10(a). The solid curves plotted in Fig. 10(a) on the energy–momentum contours indicate the cutting lines for our (4, 2) sample nanotube. The solid dots stand for the ends of the cutting lines obtained from the 2-atom linear–helical construction, for which u = 1 and v = M = 6, which we translated to the 1st Brillouin zone for a better visualization. Following the zonefolding scheme, we now place the E(k) dispersion along all the cutting lines from Fig. 10(a) in the one-dimensional Brillouin zone of the (4, 2) nanotube, as shown in Fig. 10(b). The corresponding density of electronic states (DOS) for the energy band diagram in Fig. 10(b) is shown in Fig. 10(c). One can clearly observe the presence of spikes, or van Hove singularities (VHSs), in the DOS of the (4, 2) nanotube in Fig. 10(c). The electronic structure of the (4, 2) nanotube is shown here only for illustrative purposes, since for small diameter nanotubes (dt < 0.6 nm), the nanotube curvature mixes and bands, and thus, the zone-folding scheme does not apply. The total number of cutting lines in the 2-atom linear–helical construction of reciprocal space is equal to the number of hexagons in the nanotube translational unit cell, N (Section 2.4.2). Therefore, by superimposing N cutting lines on the two energy–momentum contours of the valence and conduction bands in the two-dimensional reciprocal space of the graphene layer (see Fig. 10(a)), one will obtain 2N electronic energy subbands in the one-dimensional reciprocal space of the nanotube (see Fig. 10(b)). Since there are two atoms per unit cell in the graphene layer, the total number of atoms in the nanotube unit cell is 2N . Therefore, the total number of electronic energy subbands in the one-dimensional reciprocal space of the nanotube is equal to the total number of atoms in the nanotube unit cell, in agreement with the fact that there is one free -electron per sp2 -hybridized carbon atom. It is possible to relate each of the cutting lines in chiral nanotubes (and thus each of the energy bands) to an irreducible representation of the factor group of nanotubes for a general point k. This factor group is isomorphic to CN , and therefore has N 1D irreducible representations. For this reason, the cutting line index relates perfectly with the quasi-angular quantum number ˜ which labels the irreducible representations in the space group formalism. In this sense, the cutting lines with index ˜ = 0 and N/2 can be associated with the A and B symmetry states, respectively. Since each cutting line is associated with two different energy bands, one in the valence band and one in the conduction band of the graphene band structure, each irreducible representation will also label two different eigenfunctions. At k = 0 the number of irreducible representations is changed to N/2 + 2 due to the fact that the (N − 2) E representations of the CN group become doubly degenerate. The A symmetry representation in the valence band will become an A1 representation, following the symmetry of the graphene valence band at the center of the Brillouin zone, while the A symmetry representation in the conduction band will become an A2 representation [3].
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Table 8 Irreducible representations for the electronic conduction and valence bands of chiral and achiral nanotubes Valence
Chiral
Armchair
Zigzag
⎧ ⎨ ˜ = 0 0 < ˜ < N/2 ⎩ ˜ = N/2 ⎧ ⎨ ˜ = 0 0 < ˜ < n ⎩ ˜ = n ⎧ ⎨ ˜ = 0 0 < ˜ < n ⎩ ˜ = n
Conduction
k = 0, /T
0 < k < /T
k = 0, /T
0 < k < /T
A1 E˜ B1
A
A
B
A2 E˜ B2
A1g E˜ g B1g
A E˜ B
A2g E˜ u B2g
A E˜ B
A1g E˜ u,˜g a B1g
A E˜ B
A2u E˜ g,˜u a B2u
A E˜ B
E±˜
E±˜ B
a For zigzag nanotubes, if ˜ < 2n/3 the valence (conduction) band at k = 0 belongs to the E˜ g (E˜ u ) representation for ˜ even and E˜ u (E˜ g ) for ˜ odd, while if ˜ > 2n/3 the its the opposite.
In the case of achiral nanotubes, the factor group for 0 < k < /T is isomorphic to the group C2nv , for which the character table is given in Table 4. This group has (n + 2) irreducible representations, from which four are 1D and (n − 1) are 2D. The cutting lines with opposite indices +˜ and −, ˜ for ˜ varying from 1 to (n − 1), will be associated with the same 2D irreducible representation labeled by . ˜ The conduction and valence bands associated with the same index 0 < ˜ < n will have the same symmetry E˜ . The ˜ = 0 cutting line will cross the point of the graphene layer and thus will be associated with A symmetry states, while the cutting line with ˜ = n will be associated with B symmetries. For armchair tubes, the anti-bonding states which constitute the conduction band will have A and B symmetries, for ˜ = 0 and n, respectively, while the valence band will be associated with A and B states. For zigzag nanotubes, the v operation does not exchange the adjacent atoms, and thus both valence and conduction bands have A and B representations at 0 < k < /T , where A states correspond to the cutting lines with index =0 ˜ and B states correspond to ˜ = n. To obtain the symmetries of the valence and conduction bands for both zigzag and armchair nanotubes at k =0, for which the group of the wavevector is isomorphic to D2nh , it is necessary to apply the compatibility relations in Table 6. For zigzag nanotubes, eigenstates at k = 0 in the valence (conduction) band associated with cutting line index ˜ < 2n/3 belong to irreducible representations which are even (odd) under the h mirror operation, which exchanges the A and B atoms. However, cutting lines with index ˜ > 2n/3 are out of the first Brillouin zone in the unfolded (graphene) reciprocal space, and thus a phase shift causes an exchange in the parity of the k = 0 eigenfunctions under the h operation. The irreducible representations of the electronic states of chiral nanotubes and achiral nanotubes are summarized in Table 8. The cutting line approach has been successfully used to predict the metallic or semiconducting nature of the nanotubes. Since the graphene valence and conduction bands cross at the K and K points in the Brillouin zone, carbon nanotubes for which one of the cutting lines crosses the K or K points should have a metallic behavior. In contrast, carbon nanotubes for which none of the cutting line passes through the K or K points, show a band gap, and thus show semiconducting behavior. The cutting lines in the vicinity of the K point are shown in Fig. 11 for three different cases, mod(2n + m, 3) = 0, mod(2n + m, 3) = 1, and mod(2n + m, 3) = 2. The first case, mod(2n + m, 3) = 0, corresponds to the cutting line crossing the K point, resulting in metallic behavior, as discussed above. However, it has been reported that only armchair nanotubes are pure metallic nanotubes, whereas the other tubes for which the cutting lines cross the K and K points are, in fact, tiny gap semiconductors [13,25–28]. Kleiner et al. used a simple model based on the band nearest neighbor tight-binding approximation to explain these results based on symmetry-breaking between bonding and anti-bonding states in the valence and conduction bands due to the curvature of the nanotube [28]. This effect can be understood in terms of group theory. For both the valence and the conduction bands in chiral nanotubes, the cutting lines which cross the K and K points are associated with the same representation E . Therefore, the valence and conduction bands cannot cross at the Fermi energy, and the nanotube has to exhibit a small band gap [25]. For armchair nanotubes, the bands which cross the K and K points are the bands associated with ˜ = n and thus have B (valence band) and B (conduction band) symmetries. Since these bands
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mod (2n + m, 3) = 0
K
metal M0
mod (2n + m, 3) = 1
K
semiconductor S1
mod (2n + m, 3) = 2
K
semiconductor S2
Fig. 11. Three different configurations of the cutting lines in the vicinity of the K point. The first configuration mod(2n + m, 3) = 0 corresponds to the case of metallic nanotubes M0, and the last two configurations, mod(2n + m, 3) = 1 and mod(2n + m, 3) = 2, correspond to the case of semiconducting nanotubes of types S1 and S2, respectively [12].
have different symmetries, group theory does not forbid them to cross, and thus the presence of a mini-gap is not necessary. The two cases of semiconducting nanotubes, mod(2n + m, 3) = 1 and mod(2n + m, 3) = 2, are also different from each other, depending on which side of the K point (K to or K to M), in the unfolded two-dimensional Brillouin of a graphene sheet, the first VHS (van Hove singularity) in the DOS appears. We classify these two types of semiconducting nanotubes as S1 and S2, respectively. In a similar fashion, we can classify metallic nanotubes by what ratio the K point divides the cutting line in the 2-atom linear–helical representation. By projecting the vector K on the direction K 2 , one can find (K · K 2 )/(K 2 · K 2 ) = m/dR . Thus, we can classify metallic nanotubes as M1 and M2 for dR = d and dR = 3d, correspondingly (see Section 2.2) [29]. For M1 metallic nanotubes, m/dR = m/d is an integer and therefore the K point appears at the wave vector k = 0 in the one-dimensional Brillouin zone of the nanotube (the point). For M2 metallic nanotubes, m/dR = (m/d)/3 is one third of an integer, and therefore the K point appears at the wave vector k = ±(2/3)(/T ) in the one-dimensional Brillouin zone of the nanotube (two thirds of the distance from the central point to the edge, X point, of the one-dimensional Brillouin zone). While armchair nanotubes (n, n) are always M2 type metallic tubes, and zigzag nanotubes can be either M1 type metallic (3, 0) or S1 type (3 + 1, 0) or S2 type (3 + 2, 0) semiconducting tubes, while chiral nanotubes can be of each of the four types, M1, M2, S1, or S2. See Ref. [29] for more information on the classification of different types of nanotubes. Although these considerations could be obtained in view of the simple “zone-folding” procedure, the symmetry of nanotubes, as described here, can be used to obtain the electronic band structure of carbon nanotubes within the approximations of the extended tight-binding model [11,12]. In fact, Samsonidze et al. used extended tight-binding calculations to obtain the electronic transition energies considering the structural relaxation that becomes significant in small diameter nanotubes [12]. The symmetry-adapted extended tight-binding model was able to obtain the dependence of the electronic transitions on the nanotube geometrical structure which could fit the experimental results with great accuracy [12]. However, a constant energy difference was found between the experimental and calculated results, which could be attributed to many-body corrections [30]. In Fig. 12 we show the electronic energy band structure of a (4, 2) nanotube, calculated using the extended tight-binding model. 5.1. Symmetry of electronic states from the line group approach For a better comprehension of the relation between the group of the wavevector and the line group approaches, we will now discuss the symmetry of the electronic bands of chiral nanotubes from the standpoint of line groups. As explained in Section 4.2, each of the 2D irreducible representations can be associated with two different states, one for bonding states, which is related to the valence band, and one for anti-bonding states, related to the conduction band. In the case − of k˜ =0 and =0, ˜ there are two 1D representations 0 A+ 0 and 0 A0 . Thus, the states in the valence band will be associated + with the 0 A0 representation, while the states in the conduction band will have 0 A− 0 symmetry. These representations are equivalent to the A1 and A2 representations obtained for k = 0 in space group theory. At k˜ = N /(vT ) there are − also two 1D-representations, N /(vT ) A+ 0 and N /(vT ) A0 , which will be associated with the valence and conduction bands, respectively. Again, the relation between these symmetries and the B1 and B2 symmetries obtained at k = /T in space group theory is obvious. However, the representation of the line groups for a general wavevector k˜ cannot be
E.B. Barros et al. / Physics Reports 431 (2006) 261 – 302
Electron Energy (eV)
~ µ=-10
287
~ µ=-10
~ µ=-9
~ µ=-9
~ µ=-9
~ µ=-9
~ µ=-10 -π/T
~ µ=-10 π/T
0 Electron Wavevector (k)
Fig. 12. Electronic bands for a (4, 2) nanotube, calculated using an extended tight-binding model [12], for the conduction and valence band states with quantum number ˜ = ±9 and ˜ = ±10. For clarity, the bands with negative indices (˜ ) are shown by grey lines.
Electron Energy (eV)
6 4 2
µ=1
µ=0
0 -2 -4 0
7π/T
14π/T
~ Electron quasi-linear wavevector (k)
Fig. 13. Band structure of a (4, 2) nanotube for a 2d-atom reduced unit-cell calculated using an extended tight-binding model. Also shown on the left is the electronic band structure for the 2N -atom unit cell [12].
directly related to representations of the group of the wavevector k, since for these values of k˜ the translational part of the symmetry operation cannot be factored out of the representation. To obtain the relation between the representations obtained from the line group formalism and that of the group of the wavevector, it is necessary to consider the total ˜ helical quantum number h. As an example, in Fig. 13 we show the band structure of a (4, 2) chiral nanotube for the 2d-atom reduced unit cell ˜ (see Section 2.4.3). As discussed above, each k-point on these electronic bands will be associated with a representation of the line group of chiral nanotubes labeled by the quantum numbers k˜ and . The band structure for the 2N -atom unit cell can be obtained by “folding” the band structure of the 2d-atom unit cell, as shown on the left-hand side of Fig. 13. This “folding” process can be understood from Section 2.4.3 as a change from the helical–angular construction to the linear–angular construction (Section 2.4.4). 6. Phonons in carbon nanotubes The phonon dispersion relations of the graphene layer can be calculated within a force constant model, or by ab initio methods [31–34]. Two atoms A and B in the unit cell of the graphene layer (see Fig. 2) give rise to six phonon branches,
1600
1600
1200
1200
800
800
400
400
0
0 −π/T
0 Wave vector
(a)
Frequency (cm-1)
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Frequency (cm-1)
288
(b)
π/T 0.00
0.02
0.04
DOS (states/C-atom/cm-1)
(c)
Fig. 14. (a) The phonon dispersion relations of the graphene layer in the first Brillouin zone calculated with the force constants fitted to the Raman scattering data for various graphitic materials. Solid curves show the cutting lines for the (4, 2) nanotube. Solid dots show the ends of the cutting lines in the 2-atom unit cell linear–helical representation. (b) Phonon modes for the (4, 2) nanotube obtained by zone-folding from (a). (c) Density of states for the phonon modes shown in (b). Spikes appear in the density of phonon states of the carbon nanotube, similar to the spikes (VHSs) appearing in the electronic DOS discussed in Section 5.
because of the three degrees of freedom per atom (see Fig. 14(a)). Superimposing the N cutting lines in the 2-atom linear–helical reciprocal structure of the nanotube on the six phonon frequency surfaces in the reciprocal space of the graphene layer yields 6N phonon branches for each carbon nanotube (see Fig. 14(b)), according to the zone-folding scheme as described in Section 2.4.2. Group theory provides accurate information on both the number of phonons and the irreducible representations for the lattice modes Dlat. mod. that are obtained by taking the direct product of the irreducible representations for the atomic sites Da.s. with that of the vector [8] (Dlat. mod. = Da.s. ⊗ Dvec. ). For k = 0 phonons in achiral tubes, the factor group is isomorphic to the D2nh group, where Dvec = A2u + E1u . The Da.s. for zigzag SWNTs is [6]: Da.s. zigzag = A1g + B2g + A2u + B1u +
n−1
(Ejg + Ej u ),
(38)
j =1
giving rise to the following irreducible representations for the lattice modes [6]: mod. Dlat. zigzag = 2A1g + A2g + B1g + 2B2g + A1u + 2A2u + 2B1u + B2u +
n−1
(3Ejg + 3Ej u ).
(39)
j =1
The phonon modes of carbon nanotubes can also be associated with the irreducible representations of the factor groups by applying the zone-folding scheme to the two-dimensional phonon dispersion relations of the graphene layer (see Figs. 14(a) and (b)), in a similar fashion as has been used to describe the electronic dispersion relations in Section 5. In such an analysis, the 6((N/2) − 1) pairs of phonon modes arising from the cutting lines with indices ˜ and −, ˜ where ˜ = 1, . . . , ((N/2) − 1), are expected to be doubly degenerate, while the phonon modes associated with cutting lines with indices ˜ = 0 and N/2 are non-degenerate. However, the zone-folding scheme in Fig. 14(a) neglects the curvature of the nanotube wall, as was already mentioned in Section 5. Meanwhile, the curvature couples the in-plane and out-of-plane phonon modes of the graphene layer to each other, especially affecting the low-frequency acoustic phonon modes. Among the three acoustic phonon modes of the graphene layer, only one of the two in-plane modes results in an acoustic phonon mode of the nanotube corresponding to the vibrational motion along the nanotube axis. The two other in-plane and out-of-plane acoustic phonon modes give rise to the twisting mode (TW, the vibrational motion in the circumferential direction of the nanotube) and to the radial breathing mode (RBM, the vibrational motion in the radial direction of the nanotube), correspondingly. The two related acoustic phonon modes of the nanotube (the vibrational motion in two orthogonal
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directions perpendicular to the nanotube axis) can be constructed as linear combinations of the RBM and TW modes. The zone-folding scheme predicts zero frequencies for the A1g (or A1 ) perfectly symmetric RBM and TW phonon modes of the nanotube at the center of the Brillouin zone, since they arise from the zone folding of the acoustic phonon modes of the graphene layer. However, the curvature of the graphene layer affects the frequency of the RBM phonons resulting in their having non-zero frequencies. In fact, the frequency of the perfectly symmetric RBM is inversely proportional to the nanotube diameter, varying from around 100 to 250 cm−1 for typical nanotube diameters of 1–2 nm. 7. Selection rules for optical phenomena In spite of the large number of electronic and phonon subbands in carbon nanotubes (see Figs. 10 and 14), very few of them participate in light absorption, Raman scattering, or infrared spectroscopy, because of symmetry restrictions. The selection rules governing the above mentioned processes are commonly derived from group theory. At the same time, the selection rules are closely related to the concept of cutting lines, and they can be easily obtained from the zone-folding procedure. In the present section, we focus on the selection rules for the electron–photon interaction in chiral and achiral carbon nanotubes in view of the space group theory developed in Section 3, but without considering the presence of excitons. Because of the importance of the excitonic effects in the optical properties of carbon nanotubes [35–37], we will discuss excitonic effects from the standpoint of group theory in Section 8. 7.1. Selection rules for optical absorption from space group theory The electromagnetic interaction giving rise to electric dipole transitions is Hem =−
e p · A, 2mc
(40)
in which p is the momentum of the electron and A is the vector potential of an external electromagnetic field. The momentum operator is part of the physical system under consideration, while the vector A acts like the bath or reservoir in a ther modynamic sense. Thus, p acts like an operator with respect to Schrödinger’s equation, but A does not. Therefore, Hem for the electromagnetic interaction transforms like a vector in the context of the group of Schrödinger’s equation for the unperturbed system H0 =E. It is then clear that the Hamiltonian describing the electron–photon interaction will transform as a vector in the direction of the polarization of the light. Therefore for light polarized along the nanotube axis, the electron–photon interaction at k =0 will transform as the vector z. For achiral (chiral) nanotubes, the vector z transforms as the A2u (A2 ) irreducible representation of the factor group at k = 0 which is isomorphic to the D2nh (DN ) point group (see Table 3(1)). To obtain the selection rules for the optical absorption by the nanotube, it is necessary to take a direct product between the initial representation and the representation of the electron–photon interaction. It can be seen that the electronic energy bands that are in the energy range of the optical absorption have E˜ symmetries. Therefore, since the direct product leads to A2 ⊗ E˜ = E˜ , the selection rules require that ˜ = 0 for light polarized along the tube axis. The effect of an operation with A2u symmetry is to reverse the parity of the E˜ representation, and therefore, for achiral nanotubes, the state with E˜ u symmetry is coupled to a state with E˜ g symmetry and vice-versa by the electromagnetic interaction. In the case of light polarized perpendicular to the nanotube axis, the electron–photon interaction transforms as the vectors x and y, which correspond to the E1 representation of the factor group of chiral nanotubes and the E1u representation for achiral carbon nanotubes. The selection rules then require ˜ = ±1, since E1 ⊗ E˜ = E˜ ±1 . 7.2. Selection rules for Raman and infrared spectroscopy from group theory The optical activity of phonons in a first-order Raman or infrared scattering process is easily obtained from the basis functions related to the irreducible representations of the lattice modes. The Raman-active modes are those transforming like symmetric combinations of quadratic functions (xx, yy, zz, xy, yz, zx), and the infrared-active modes are those transforming like vectors (x, y, z).
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The list of Raman and infrared-active modes are given below [6,7]: DRaman zigzag = 2A1g + 3E1g + 3E2g → 8 modes;
(41)
Dinfrared zigzag = A2u + 2E1u → 3 modes;
(42)
DRaman armchair = 2A1g + 2E1g + 4E2g → 8 modes;
(43)
Dinfrared armchair = 3E1u → 3 modes;
(44)
DRaman chiral = 3A1 + 5E1 + 6E2 → 14 modes;
(45)
Dinfrared chiral = A2 + 5E1 → 6 modes.
(46)
A more detailed analysis of the Raman active modes for chiral and achiral nanotubes is provided in Ref. [7]. To illustrate the usage of the selection rules introduced by the electron–photon and electron–phonon interaction processes, we consider the first-order resonance Raman scattering process in carbon nanotubes [38,39]. The first-order Raman scattering process involves the following steps: creation of an electron–hole pair, scattering by a phonon, and light emission by an electron–hole recombination process [38,39]. The Raman signal is greatly enhanced when the electron scatters between VHSs in the valence and conduction band DOS, so that we can consider only the transitions between the two VHSs in the DOS as a first approximation. By utilizing the selection rules introduced above, we come up with the following five cases for the allowed first-order resonance Raman scattering processes in chiral SWNTs (v) (c) between the electronic energy VHSs in the valence and conduction bands denoted by E and E for a general k point in the Brillouin zone [40]: (v) Z
(c) A
(c) Z
(v)
(I)E˜ −→ E˜ −→ E˜ −→ E˜ , (v) X
A
(c)
X
(v)
(v) Z
(c) E1
(c)
X
(v)
(v) X
(c)
(c) Z
(v)
X
(v)
(c)
(II)E˜ −→ E˜ ±1 −→ E˜ ±1 −→ E˜ , (III)E˜ −→ E˜ −→ E˜ ±1 −→ E˜ , E1
(IV)E˜ −→ E˜ ±1 −→ E˜ −→ E˜ , (v) X
(c)
E2
(c)
(V)E˜ −→ E˜ ±1 −→ E˜ ∓1 −→ E˜ ,
(47)
where A, E1 , and E2 denote the symmetries of the phonon modes at k = 0, which are associated with the ˜ = 0, ˜ = ±1, and ˜ = ±2 cutting lines, respectively. The XZ plane is parallel to the substrate on which the nanotubes lie, the Z axis is directed along the nanotube axis, and the Y axis is directed along the light propagation direction, so that Z and X in Eq. (47) stand for the light polarized parallel and perpendicular to the nanotube axis, respectively. The five processes of Eq. (47) result in different polarization configurations for different phonon modes, ZZ and XX for A; ZX and XZ for E1 ; and XX for E2 , in perfect agreement with the basis functions predicted by group theory. Also, Eq. (47) predicts different resonance conditions for different phonon modes. The A and E1 modes can be observed (v) (c) (v) (c) in resonance for the E˜ → E˜ and the E˜ → E˜ ±1 processes, corresponding to Eii and Eij (j = i) transitions, (v)
(c)
respectively, the E2 modes can only be observed in resonance for the E˜ → E˜ ±1 process. Experimentally observed Raman scattering spectra follow the predicted polarization configurations and resonance conditions [40,41]. 7.3. Selection rules for double resonance Raman processes As explained in Section 7.2, only a few of the phonon branches in carbon nanotubes are Raman active. However, many peaks are observed experimentally in the carbon nanotube Raman spectra which are present due to a process known as double resonance Raman scattering (DRRS) [42,43]. In this process, the electron in the valence band is excited to a conduction band state, from which it is scattered by either a phonon or a lattice defect into another electronic state
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291
Electron Energy
Eµ
−k0
0
+k0
Electron Wavevector
Fig. 15. Diagram showing a possible double resonance process for chiral nanotubes. The electron is excited to the E−˜ electronic band by a photon, scattered to the E˜ electronic band by a phonon with E2˜ symmetry and then scattered back to the E−˜ electronic band, for which it can recombine optically, emitting a photon [42].
and then scattered back to the same initial state in the conduction band (in the case of light polarized parallel to the nanotube axis) by either another phonon or by a lattice defect. The electron can then emit a photon and recombine with the photo-induced hole. In the DRRS process it is necessary that two out of the three states in the conduction band which are visited by the electron are real electronic states, giving rise to a resonance process. If that is the case, the Raman cross section is strongly enhanced, resulting in an efficient Raman process. The selection rules for the optical excitation and emission in parallel polarization determine that the initial and final k value of the electron should be the same, in order to guarantee vertical electronic transitions, which will be dipole allowed (see Section 7.1). Therefore, the electronic states which are visited in the conduction band are determined by the energy difference between the electronic bands, by the conservation of linear momentum, and by obeying the relevant selection rules. In the vicinity of the Fermi energy, the electronic states of a general chiral nanotube will have E±˜ symmetry. Thus, for a transition to occur between the electron in a state E˜ 1 and a state E˜ 2 it is necessary that the phonon which couples the two states has E˜ 2 −˜ 1 symmetry. In Fig. 15, we show a possible double resonance process between two electronic bands with symmetries E−˜ and E˜ , which have the same eigenenergies for wavevectors k and −k. The phonon which couples these two bands has either E2˜ or E−2˜ symmetry, and the superscript (p) is used in Fig. 15 to denote phonon. 7.4. Selection rules from zone folding In the previous section the selection rules for the quantum numbers were obtained by symmetry considerations. It is interesting to discuss how equivalent selection rules can be derived considering momentum conservation in the unfolded two-dimensional graphene-sheet, considering the concepts of cutting lines. The optical transition in the nanotube is vertical within the 1D Brillouin zone, i.e., the electronic wave vector along the nanotube axis (along the K 2 vector in the unfolded 2D Brillouin zone) does not change. In contrast to the case of the graphene layer, the polarization vector can be either parallel or perpendicular to the nanotube axis for light propagating perpendicular to the substrate on which the nanotubes lie. The dipole selection rules tell us that the optical transition in the nanotube conserves the electronic subband index (the cutting line index ) ˜ for light polarized parallel to the nanotube axis (see Section 7.1). Conservation of both the 1D wave vector and the subband index implies conservation of the 2D wave vector in the Brillouin zone of the graphene layer (unfolded Brillouin zone of the nanotube). As an example, we plot in Fig. 16 the schematic band diagram of the nanotube in the unfolded 2D Brillouin zone. If an electron in the valence subband V2U of Fig. 16 (see the parabolic curve in Fig. 16 labeled V2U) absorbs a photon, the electron goes vertically to the conduction subband C2L. If the electron starts from the VHS in the valence subband
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C3L´ C2L´
C3U´ C3U C2U´ C3L
C1U´
C2U
C2L C1L
C1U V1L´
V2U
K´ V1U´ V2U´
K V1U
C1L´
V2L´ V1L V2L
V3U´
V3L´
V3L
V3U
Fig. 16. The electronic subbands for zigzag metallic nanotubes in the vicinity of the K and K (near the Fermi energy) points in the first Brillouin zone. The VHSs are labeled by three symbols, the first denotes valence or conduction band (C/V), the second denotes the VHS index counted away from the Fermi energy or the cutting line index counted away from the K and K points, and the third denotes the lower and upper energy components (L/U) due to the trigonal warping effect that splits the energy of the VHSs for metallic SWNTs [44].
(a)
(b)
Fig. 17. Light polarization (a) parallel and (b) perpendicular to the nanotube axis, shown for both a rolled-up SWNT (left), and a SWNT unrolled into the graphene layer (right). The arrows show the light polarization vector, and the dashed lines show the light propagation direction [14].
V2U (see the solid dot on the subband V2U in Fig. 16), this electron goes to the VHS in the conduction subband C2L, and the optical absorption is enhanced substantially because of the extremely high DOS at the VHSs in the valence and conduction subbands, V2U and C2L. If an electron in the valence subband V2U in Fig. 16 absorbs a photon polarized perpendicular to the nanotube axis (i.e., polarized along the K 1 vector), it can scatter to one of the two conduction subbands, either C1L or C3L. This implies a different set of VHSs in the JDOS for perpendicular polarization, E˜ ,˜ ±1 . While the optical transition is vertical for the light polarized parallel to the nanotube axis, it involves a wave vector change of ±K 1 (the distance between two adjacent cutting lines) for the perpendicular polarization. This wave vector change can be easily understood by considering an unrolled nanotube, as shown in Fig. 17. When the nanotube is unrolled into the graphene layer, the light polarized parallel to the nanotube axis transforms into light polarized parallel to the graphene layer, as shown in Fig. 17(a), resulting in a vertical interband optical transition in the unfolded 2D Brillouin zone, which is equivalent to the optical transition within the same subband ˜ in the folded 1D Brillouin zone of the nanotube, as is predicted by the dipole selection rules. However, perpendicular polarization in nanotubes becomes transformed into the in-plane and out-of-plane polarizations in the unfolded graphene layer, periodically modulated along the direction of the K 1 vector with the period dt (nanotube circumference), as shown in Fig. 17(b) [14]. The optical transitions induced by the out-of-plane polarization are expected to be much weaker compared to those induced by the in-plane polarization and are usually ignored, because of the much stronger in-plane interaction in the graphene layer [45]. This implies that the light polarization in the unrolled nanotube shown in Fig. 17(b) can be considered, as a first approximation, to be parallel to the graphene
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layer, with an additional phase factor describing oscillations of the in-plane polarization component, arising from the rotation of the polarization vector. The phase factor is given by cos(k · r) where the wave vector k has the direction of K 1 and a magnitude of 2/(dt ) = 2/dt , i.e., k = K 1 . By assuming wave vector conservation in the unfolded 2D Brillouin zone for the optical transition process, we come up with the selection rules kc = kv ± K 1 for light absorption and kv = kc ± K 1 for light emission, which correspond to an electronic transition to the adjacent cutting line in the unfolded 2D Brillouin zone, or the electronic transition to the adjacent subband in the 1D Brillouin zone of the nanotube. It is interesting to note that the photon wave vector ±K 1 in the unrolled graphene layer is much larger in magnitude than the photon wave vector in free space, K1 = 2/dt ? = 2/, because the nanotube diameter dt is much smaller than the optical wave length . Therefore, an optical photon in the unrolled graphene layer can be considered as an X-ray photon with respect to spatial considerations, yet the photon energy does not change when the nanotube is unrolled into the graphene layer. Such a “pseudo X-ray” photon is a source of breaking the optical selection rules in the case of perpendicular polarization. The selection rules for the scattering of electrons by phonons can also be obtained by momentum conservation in 2D graphite. Two cutting lines belonging to the irreducible representations E and E are separated from each other in the 2D Brillouin zone by k − k = (˜ − ˜ )2/dt , and this is the momentum that the phonon should transfer as a result of the transition. As explained in Section 6, the symmetry of the phonon with such a momentum can be obtained by rolling up the 2D graphene layer and this will yield a phonon with E˜ −˜ symmetry. 8. Excitons in carbon nanotubes Due to the strong spatial confinement of carriers in 1D materials, the electron–hole Coulomb interaction is strong. Therefore, it has been predicted that excitonic effects should dominate the optical transitions in semiconducting carbon nanotubes [35,46,47]. This fact has been confirmed experimentally by comparing the energies for 1-photon and 2photon excitations, which excite different excitonic states [36,37]. Rigorously, excitons originate from a many-body description of electron–electron and electron–hole interactions. However, to obtain the symmetry of the excitonic states, it is only necessary to have the correct symmetry for the Hamiltonian. Thus, although a detailed mathematical analysis of the many-body interactions is necessary to obtain the quantitative information, such as accurate values for energy levels, eigenfunctions, matrix elements, transition probabilities, etc., the proper form for the electron–hole interaction is sufficient to obtain the correct symmetry properties for excitons in carbon nanotubes. Also, for a full description of excitons in carbon nanotubes, it is necessary to consider the exchange of the electrons and holes involved in the formation of the excitonic states. The symmetry properties of the singlet and triplet excitonic states formed by the combinations of electrons and holes when considering the presence of exchange interaction can be obtained using the formalism of double groups [8]. Although the exchange interaction is known to be weak [48], implying a small splitting between the singlet and the triplet states, the selection rules for creating and annihilating triplet states is somewhat different from that of singlet states. Therefore, to fully understand the exciton dynamics from the point of view of symmetry, it is necessary to have a complete study of the electronic properties of carbon nanotubes within the formalism of double groups. However, the study of the formalism of double groups applied to nanotubes is not available in the literature. Therefore, we will focus the present work on the symmetry properties of singlet exciton states, which can be analyzed using the space group theory developed in Section 3, without taking the exchange between the electrons and holes into consideration. 8.1. The exciton symmetries The exciton wavefunction can be written as a linear combination of products of conduction (electron) and valence (hole) eigenstates as
(re , rh ) = Avc c (re )∗v (rh ), (48) v,c
where v and c stand for valence- and conduction-band states, respectively. To obtain an accurate solution for the excitonic eigenfunctions (the Avc coefficients) and eigenenergies, it is necessary to solve the Bethe–Salpeter equation [47,49], which includes many-body interactions and considers the mixing by the Coulomb interaction of electron and hole states with all the different wavevectors of all the different bands. The Coulomb interaction depends only on the
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relative distance between the electron and the hole, and thus the many-body Hamiltonian is invariant under the symmetry operations of the nanotube. Each excitonic eigenstate will then transform as one of the irreducible representations of the space group of the nanotube. In general the electron–hole interaction will mix states with all wavevectors and all bands, but for moderately small-diameter nanotubes (dt < 1.5 nm), the separation between singularities in the single-particle JDOS (joint density of states) is fairly large and it is reasonable to consider, as a first approximation, that only the electronic bands contributing to a given singularity will mix to form the excitonic states [47]. Within this approximation, it is possible to employ the usual effective-mass and envelope-function approximations (EMA) to obtain the exciton eigenfunctions [50]:
EMA (re , rh ) = Beh e (re )∗h (rh )F (ze − zh ). (49) v,c
The prime in the summation indicates that only the electron and hole states associated with the JDOS singularity are included. It is important to emphasize that the approximate wavefuntions EMA have the same symmetries as the full wavefunctions . The envelope function F (ze − zh ) provides an ad-hoc localization of the exciton in the relative coordinate ze − zh along the axis and labels the levels in the 1D hydrogenic series [51]. The envelope functions will be either even ( = 0, 2, 4 . . .) or odd ( = 1, 3, 5 . . .) upon z → −z operations. The use of such “hydrogenic” envelope-functions serves merely as a physically grounded guess to the ordering in which the different exciton states might appear. From Eq. (49), we can see that the irreducible representation of the excitonic state D(EMA ) will be given by the direct product between the irreducible representation of the envelope function and the irreducible representation of the electron and hole states at the band-edges [50]: D(EMA ) = D(e ) ⊗ D(h ) ⊗ D(F ),
(50)
where D(e ), D(h ) and D(F ) are the irreducible representations of the electron state, hole state and envelope function, respectively. To study the exciton symmetries in chiral, zigzag and armchair tubes we now apply Eq. (50) considering that the group of the wavevector for the exciton can be obtained from the excitonic center of mass wavevector K = ke + kh . 8.1.1. Chiral nanotubes Let us now consider the first optical transition (E11 ) in chiral nanotubes. As shown in Fig. 18(a), the minimum energy gap occurs at two inequivalent wavevectors k = ±k0 . The electronic states associated with a band ˜ and a wavevector k away from the zone center (k = 0) transform as the 1D representations E˜ (k) of the CN point group (see Section 5). Therefore, the electron and hole in states the VHS will, respectively, belong to the E±˜ (±k0 ) and E∓˜ (∓k0 ) irreducible representations. Both the electron and hole states are degenerate in energy at k = k0 and k = −k0 due to time reversal symmetry, and thus it is expected that the Coulomb interaction will strongly mix these two electrons and two hole states, resulting in four exciton states. In the case of the lowest-energy envelope function ( = 0), which is even and transforms as the totally symmetric representation A1 (0), Eq. (50), can be written for each possible electron–hole pair: ⎧ E˜ (k0 ) ⊗ E−˜ (−k0 ) ⊗ A1 (0), ⎪ ⎪ ⎪ ⎨ E (−k ) ⊗ E (k ) ⊗ A (0), 0 1 −˜ ˜ 0 (51) ⎪ E (k ) ⊗ E (k ) ⊗ A (0), 0 0 1 ˜ ˜ ⎪ ⎪ ⎩ E−˜ (−k0 ) ⊗ E−˜ (−k0 ) ⊗ A1 (0). It should be noted that the representations of the electron states, hole states and the envelope function are, in general, associated with different wavevectors (±k0 for electron and hole states and 0 for the envelope function), and thus belong to different point groups. Also, the final excitonic states will be associated with another different wavevector for the center of mass K = ke + kh . In order to obtain the results of the direct products between representations in different point groups, it is necessary to make use of the compatibility between the CN and DN point groups (see Table 5). The direct product between electrons and holes with opposite quantum numbers (ke , ˜ e ) = (−kh , −˜ h ) = ±(k0 , ) ˜ (first two lines in Eq. (51)) will result in two exciton states with quantum numbers (K, ˜ ) = (0, 0) and which will transform as the A irreducible representation of the CN point group. However, the group of the wavevector K = 0 is the
E.B. Barros et al. / Physics Reports 431 (2006) 261 – 302 Electron Wavevector +k0 -k0 0
Ε-~
+k0
0
−π/T
-k0
−π / T
Hole Wavevector
(a)
(d)
Electron Wavevector 0
−π/T
Hole Energy
Electron Energy
~ Eg
π/T
−π/T
0
A1
−k′
0
(e)
Electron Wavevector -k0 +k0 0
A2u A1u
E~
E~
E~
0
-k0
Hole Wavevector
−π/T
Exciton Energy
E~
+k0
π/ T
0 Exciton Wavevector (K)
π/T
Hole Energy
Electron Energy π/T
π/ T
E~′u
Eµ
(c)
+k′
Exciton Wavevector (K)
−π / T
Hole Wavevector
−π/T
~ Ε′
A2
π/T
~ Eu
(b)
~ Ε-′
Exciton Energy
Ε~
Exciton Energy
Ε~
Ε-~
π/T
π/T
Hole Energy
Electron Energy
−π/T
295
−π/ T
u,µ g
Eµ
Eµ A2u,2g
A1u,1g
B´, B´´
-k
0
B´, B´´
+k
π /T
Exciton Wavevector (K)
Fig. 18. Diagrams for the electronic bands and symmetries for: (a) chiral (n, m); (b) zigzag (n, 0); and (c) armchair (n, n) nanotubes and for their respective excitonic bands (d), (e) and (f). The electron, hole and exciton states at the band edges are indicated by a solid circle and labeled according to their irreducible representations. Different line types and colors in this figure are related to bands with different symmetries. Thick (black) solid lines correspond to the E˜ representation, the blue (thin) solid lines correspond to A1 excitons while the cyan (thin) dashed lines correspond to the A2 excitonic states. In the case of achiral nanotubes, we also have inversion and mirror plane symmetries. For a better visualization, the bands with different parities under the inversion and mirror plane symmetry operations were grouped together and appear with the same line color and pattern. In the case of armchair nanotubes, the bands which transform as the B and B representations are shown using a red dot-dash pattern. The electronic and excitonic band structures shown here are only pictorial. Group theory does not order the values for the eigenenergies and energy dispersions.
DN point group, and thus one of these states will transform as the A1 representation and the other as the A2 representation of the DN point group. These states correspond, respectively, to states even and odd under the C2 rotation. The direct product with the A1 (0) irreducible representations of the envelope function will leave both irreducible representations unchanged.
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The mixing between electron and hole states with the same quantum numbers (ke , ˜ e ) = (kh , ˜ h ) = ±(k0 , ) ˜ (two last lines in Eq. (51)) will result in two exciton states with quantum numbers ±(2k0 , 2). ˜ We now need to observe that the group of the wavevector k is only defined for values of k within the 1st Brillouin zone and ˜ between −N/2 + 1 and N/2. If 2k0 crosses the boundary of the 1st Brillouin zone, the linear and quasi-angular quantum numbers of the excitonic states will be corrected to k = 2k0 ∓ 2/T , (52) ˜ = 2˜ ± M. Here, we use the value M, which was defined in Section 2.4.2. The ∓ corresponds to the cases where 2k0 > /T , while the ± sign corresponds to cases where 2k0 < − /T . If the quasi-angular momentum |˜ | is larger than N/2 it will also have to be corrected to ˜ = ˜ ∓ N ,
(53)
and here the “−” and “+” signs correspond to ˜ > N/2 and ˜ < − N/2, respectively. These corrections for k and ˜ for the excitonic states follow from the rules of translations in reciprocal space for the linear–helical approach (see Section 2.4.2). We see that the direct product between the electron and hole states with the same quantum numbers will lead to excitons with symmetries E±˜ (±k ), since, in general, the group of the wavevector k = 2k0 is the CN point group (except in the specific cases of k0 = 0 and /(2T )). Again, we note that the direct product with the envelope function leaves the irreducible representations unchanged. For a better understanding on how to obtain the correct representations for the excitonic states we will follow the example of correctly representing the (8, 2) nanotube, which is metallic, but still can have excitonic states. For this particular nanotube, the value of N is 28 and the lowest energy electronic transition occurs between electron and hole states with ˜ = ±7. Also, the states at the electronic band edges for this nanotube occur at k0 ∼ ±2/3T . Following Eq. (51), there will be two exciton states at K = 0, one with A1 symmetry and one with A2 symmetry, and two excitons with E symmetry with a band edge at k . The quantum numbers for the E symmetry exciton states would be 2˜ = ±14 and 2k0 ∼ ±4/3T . The irreducible representations for these excitons can be obtained by first translating the linear wavevector 2k0 back to the first Brillouin zone, which leads to k ∼ ±(4/3T − 2/T ) = ∓2/3T and ˜ = ±(2˜ + 10) = ±24, since M = 10 for the (8, 2) nanotube. Since |˜ | = 24 > N/2, we need to find the correct label ˜ for the irreducible representation by subtracting N = 28 from 24, which results in ˜ = ∓4. Therefore, we conclude that the excitons formed from the 1st VHS in the (6, 5) nanotube will have symmetries A1 (0), A2 (0), E−4 (−k0 ) and E4 (k0 ), where we have considered that k ∼ 2/3T = k0 . The symmetry of the exciton states away from the band edge can be obtained by applying the compatibility relations between the DN and CN point groups. The excitonic bands which have a band edge at K = 0 will both have symmetry A away from the center of the Brillouin zone, while the excitonic bands which have symmetries E±˜ at K = ±k will be brought together at the zone center (K = 0) to form a doubly degenerate 2-dimensional representation E2|˜ | (0) of the DN point group. Therefore, group theory shows that the lowest energy set of excitons is composed of four exciton bands, shown schematically in Fig. 18(d). Let us now consider higher-energy exciton states in chiral tubes. Those can be obtained, for instance, by considering the same VHS in the JDOS and higher values of . For even, the resulting decomposition is the same, since the envelope function also has A1 symmetry. For odd values of , the envelope function will transform as A2 , but this will only exchange the A1 and A2 symmetry excitons, and will leave E symmetry excitons unchanged. The result is still the same if one considers now higher-energy exciton states derived from higher singularities in the JDOS (for instance, the so-called E22 transitions). Therefore, we can write an equation which describes the symmetries of all exciton states associated with Eii transitions in chiral nanotubes as (E˜ (k0 ) + E−˜ (−k0 )) ⊗ (E−˜ (−k0 ) + E˜ (k0 )) ⊗ A1,2 (0) = A1 (0) + A2 (0) + E˜ (k ) + E−˜ (−k ), where A1 and A2 are related to even and odd envelope functions, respectively, and k and ˜ are the quantum numbers associated with the exciton linear and quasi-angular momenta, as obtained above. It should be mentioned that the values of ˜ and k0 will be different for each nanotube and also for each Eii transition. The situation differs for the so-called Eij transitions, with i = j . Specifically, the transitions for which the quasiangular momentum of the valence and conduction bands differ by ±1 are more relevant experimentally because,
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as we shall see later in Section 8.2, they may be observed for perpendicular polarization. Therefore, we focus on that particular case and Eq. (50) now can be written as
(E˜ (k0 ) + E−˜ (−k0 )) ⊗ E−(˜ ±1) (−k0 ) + E˜ ±1 (k0 ) ⊗ A1,2 (0) = E±1 (k ) + (E˜ (k ) + E−˜ (k )), where A1 (0) and A2 (0) are the irreducible representations of the envelope functions for even and odd, respectively. It is important to note that for a general chiral nanotube, the valence and conduction band extrema occur at different k = k0 and k0 points, leading to non-zero exciton wavevectors k and k for the all four exciton states. 8.1.2. Zigzag nanotubes For zigzag nanotubes, the VHSs for the electronic bands associated with all Eii transitions occur at k0 = 0, and the electronic valence and conduction bands transform as either the E˜ g or the E˜ u irreducible representations of the D2nh group. In contrast to chiral nanotubes, these irreducible representations for the electron and hole states are 2D, and thus correspond to degenerate states due to the inversion center symmetry. Although there is only one band edge in the 1D Brillouin zone, there are two electron and two hole states, which leads to four excitonic states. The symmetries of the excitonic states can be obtained form the direct product: E˜ g (0) ⊗ E˜ u (0) ⊗ A1g (0) = A1u (0) + A2u (0) + E˜ u (0)
(54)
for even and E˜ g (0) ⊗ E˜ u (0) ⊗ A2u (0) = A2g (0) + A1g (0) + E˜ g (0)
(55)
for odd. The corresponding band structure for = 0 (lowest exciton states) is shown in Fig. 18(e). It is interesting to note that, in this case, all four excitonic states have their band edges at the point (K = 0). The symmetries of the excitonic states away from the point for zigzag nanotubes can be obtained directly from the compatibility relations between the D2nh and the C2nv point groups. (See Table 6.) For Eij transitions with i = j and quasi-angular momentum differing by ±1 in zigzag tubes, the situation is slightly more complicated because valence and conduction bands can have either the same or opposite parity under inversion. For example, the valence (conduction) bands of the (10, 0) tube, transform as E7g (E7u ), E6g (E6u ) and E8u (E8g ) (listed in order of proximity to the gap). Therefore, transitions connecting ˜ = 6 and 7 are between states of different parities, whereas transitions connecting ˜ = 7 and 8 are between states of the same parity. Therefore, we need to consider the following cases: E ˜ (0) ⊗ E(˜ ±1) (0) ⊗ A1g,2u (0) = E˜ g,u + E1g,u ,
(56)
for valence and conduction states of the same parity, and: E ˜ (0) ⊗ E(˜ ±1)¯ (0) ⊗ A1g,2u (0) = E˜ u,g + E1u,g ,
(57)
for valence and conduction states of opposite parity. The envelope function label 1g or 2u correspond to even or odd, respectively. 8.1.3. Armchair nanotubes The optical transitions in armchair tubes are also excitonic, despite the metallic character of these tubes, because of symmetry gap effects [47]. As shown in Fig. 18(c), the Eii -derived excitons will be formed by two electron and hole states each with symmetry E˜ and at the band edges at k = ±k0 , where k0 ≈ 2/3a for the lowest electronic transition. Each of the bands is doubly degenerate, and therefore the Coulomb interaction is going to mix four different electron states with four different hole states which should result in 16 excitonic states. The symmetries of these excitons can be obtained using Eq. (50) and will result in (E˜ (k0 ) + E˜ (−k0 )) ⊗ (E˜ (k0 ) + E˜ (−k0 )) ⊗ A1g,2u = A1u (0) + A2u (0) + A1g (0) + A2u (0) + (B (k ) + B (−k )) + (B (k ) + B (−k )) + E˜ g (0) + E˜ u (0) + (En−˜ (k ) + En−˜ (−k )).
(58)
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Note that the same decomposition of symmetries is found for even (A1g ) and odd (A2u ) envelope functions. The excitons with band edges at the center of the Brillouin zone (K = 0) are obtained from the product between the electron and hole states with opposite wavevectors (ke =−kh =±k0 ), while the band edge states at K =k are obtained from electrons and holes with equal wavevectors ke = kh = ±k0 . It is important to mention that the direct product between E˜ states will always lead to A and E symmetry states. However, since k0 ∼ 2/3a, then 2k0 always crosses the boundary of the first Brillouin zone, and therefore the A and E symmetry states obtained from the direct product need to be translated back into the first Brillouin zone by the procedure described in Section 2.4.2 and explicitly applied to the (6, 5) nanotube in Section 8.1.1. The A and A states obtained from the direct product will have their parities exchanged and thus lead to B and B states while the E˜ states will have their quasi-angular quantum number ˜ corrected to n − ˜ , as indicated in Eq. (58). In Fig. 18(f) we show a schematic diagram for the 16exciton states obtained for = 0 in armchair SWNTs. For Eij transitions with i = j and quasi-angular momentum differing by ±1, we have
(E˜ (k0 ) + E˜ ±1 (−k0 )) ⊗ E˜ (k0 ) + E˜ ±1 (−k0 ) ⊗ A1g,2u = E1g (0) + E1u (0) + E˜ g (0) + E˜ u (0) + 2En−1 (±k ) + 2En−˜ (±k ), (59) for both A1g and A2u envelope functions. 8.2. Selection rules for optical absorption To obtain the selection rules for the optical absorption of the excitonic states, it is necessary to consider that the ground state of the nanotube transforms as a totally symmetric representation (A1 for chiral and A1g for achiral nanotubes) and that only K = 0 excitons can be created due to linear momentum conservation. For light polarized parallel to the nanotube axis the interaction between the electric field and the electric dipole in the nanotube transforms as the A2 (A2u ) representation for chiral (achiral) nanotubes. Thus, one photon excitation will create A2 (A2u ) excitonic states. The same is true for the emission of a photon by excitonic recombination. Since only A2 (A2u ) symmetry excitons are optically active for parallel polarized light, only one of the 4 excitons obtained for each envelope function and associated with an Eii transition is optically active, the remaining three being dark states. The same applies for the 16 exciton states in armchair tubes, described by Eq. (58). However, for zigzag tubes, one can see from Eq. (54) and Eq. (55) that only states with even (envelope functions even under z → −z) will have a bright exciton. The differences in the symmetry properties of chiral, zigzag and armchair nanotubes need to be taken into account when using specific nanotubes as representatives of optical properties occurring generally for carbon nanotubes. It is interesting to comment on the existence of a bright exciton for odd states in chiral and armchair tubes. From an analogy to the simple 1D hydrogen atom, one would expect odd envelope functions to give rise to dark states. However, an even wavefunction can also be constructed by the product of an odd Bloch function and an odd envelope function. Nevertheless, although being formally bright, we expect a very low oscillation strength for these excitons, since an odd envelope function should give a very low probability of finding electron and hole at the same position for recombination. Therefore, it is clear that the experimental Kataura plot, though initially constructed for band-band transitions [52,53], will nevertheless give a very similar picture when constructed for excitons. Although the excitonic picture is more complex, having up to 16 excitonic ground states for each band-band transition, plus excited excitonic states, most of the excitonic states are dark (not active optically) or have a very weak oscillator strength. The Kataura plot can, therefore, be interpreted as a plot of the energy of the bright exciton state with = 0 as a function of tube diameter. However, odd states are also important for the interpretation of the two-photon absorption experiments performed to prove the excitonic nature of the optical transition in carbon nanotubes [36,37]. In 2-photon excitation experiments, the incident photons should create A1 (A1g ) symmetry excitons, since A2 ⊗ A2 = A1 (or A2u ⊗ A2u = A1g for achiral tubes). Therefore, for zigzag tubes (see Eq. (55)), only the states with odd envelope functions will be accessible by two-photon transitions. For chiral (armchair) tubes, in principle all values of should contain an A1 (A1g ) symmetry exciton [54]. For light polarized perpendicular to the nanotube axis, the electron–photon interaction transforms as one of the E±1 (E1u ) representations. Thus, only excitonic states with E±1 (E1u ) symmetries can be accessed using perpendicularly polarized light. As described in Eqs. (54) and (59), such excitons are present for every value of in chiral and armchair
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Exciton Energy
E2µ A2
E-2µ
E-2µ
E2µ
A1
− 2k0
0
+ 2k0
Exciton Wave vector (K)
Fig. 19. Diagram showing a possible double resonance process. The A2 symmetry exciton is created by 1-photon absorption. This exciton is scattered to a state with E2˜ symmetry by a E2˜ symmetry phonon, and then scattered back to K = 0 by another phonon with E−2˜ symmetry. The last step can take the exciton either to states with A1 or A2 symmetries.
tubes as long as sub-bands of adjacent indices, namely ˜ and ˜ ± 1, are coupled. Once again the selection rules for zigzag tubes are more restrictive: Only states with even and coupling bands of opposite parity or odd and coupling bands of the same parity are bright under perpendicular polarization. However, with the exception of zigzag nanotubes, the excitons which are active for perpendicular polarized light have a band edge at a wavevector k = 0, and therefore, its optical excitation is, in principle, forbidden by the conservation of linear momenta. 8.3. Selection rules for Raman scattering processes The selection rules for first-order Raman scattering will not, as a first approximation, be affected by the excitonic interaction. The main effect of excitonic states in Raman spectroscopy is the fact that only the A2 (A2u ) symmetry excitons can resonantly absorb or emit photons. The double resonance process involves a transition between different exciton states assisted by a phonon. Thus, it is possible to use symmetry-based selection rules to determine the allowed transitions for a given phonon. To illustrate this, we show in Fig. 19 a possible double resonance process for a chiral nanotube. Here, the first optical transition occurs by the creation of an A2 symmetry exciton, which is off-resonance, as indicated in Fig. 19, and therefore A2 (p) in this case is a virtual state. This virtual excitonic state is then scattered by a phonon with E2˜ symmetry, where (p) stands for phonon, to an excitonic state with E2˜ symmetry. This E2˜ exciton, can then be scattered back to an A (p) symmetry exciton by another phonon with E−2˜ symmetry. As explained in Section 8.1.1, there are two exciton bands with A symmetry for each envelope function , one which has A1 symmetry at K = 0 and another which, at K = 0, has A2 symmetry. The energy difference between these two states is expected to be a few meV [46,47,55,56], and therefore the double resonance process should create excitons in both states with approximately equal probability. However, this energy difference depends on 1/dt2 [57] and can play an important role for smaller diameter nanotubes. Noted that, as explained in Section 8.1.1, the E2˜ exciton state discussed here is formed from the combination of the VHSs in the electronic bands associated with index . ˜ Therefore, this double resonance process is equivalent to the process discussed in Section 7.3 within the single particle approximation. It can be seen that, from the standpoint of group theory, the double resonance process within the excitonic picture is similar to that for one-electron bands since it gives the same quantum numbers for the phonon which couples the given states. However, for excitons formed through the mixing of E˜ and E−˜ symmetry electronic bands, the resulting E±2˜ symmetry excitons have different energies from that of A symmetry excitons. This energy difference can either enhance or quench the double resonance process for a specific phonon depending on whether the intermediate state of the double resonance process comes closer to the E±2˜ symmetry band edge, or farther from it. Thus, it is very important to take into consideration the energy
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splitting between the A and E symmetry excitons for evaluating the frequency and intensity of double resonance Raman peaks. Calculations of the excitonic binding energies have found a splitting between the A and E symmetry excitons ranging from 5 to 15 meV [46,47,55,56]. This energy difference will only affect double resonance processes for which one of the phonons involved in the process is an acoustic mode [58,59]. However, it is clear that by mixing different electronic bands, it is possible to create different excitons with very different symmetries and energies. Although optical transitions for these excitons will all be forbidden, they can be accessed through exciton–phonon coupling and other symmetry breaking interactions, and thus can contribute to double resonance processes. In fact, the Raman spectra of carbon nanotubes are populated with Raman peaks which cannot be directly associated with point phonons, such as the D-band, G -band, the intermediate frequency modes [58,59], and other combination modes [60]. A more accurate description of the processes giving rise to these Raman modes will have to take into account the excitonic nature of the optical transitions. However, to obtain the energy of these mixed excitons, and thus to calculate the resonance condition for these double resonance processes, it is necessary to solve the Bethe–Salpeter equation considering all the possible values of k and ˜ labeling the electron and hole eigenfunctions. 9. Summary and conclusions In this work we have reviewed the space group theory of carbon nanotubes giving a detailed summary of the symmetry operations exhibited by chiral and achiral nanotubes, as well as finding the irreducible representations within the formalism of factor groups. The irreducible representations of the groups of the wavevector of nanotubes were directly obtained and the quantum numbers were associated with linear and helical momenta. The electronic and vibrational properties of carbon nanotubes were discussed from the standpoint of group theory. We also discussed the symmetry properties of carbon nanotubes from the “zone-folding” standpoint, resulting from the presence of compound operations. The compound operations allow the geometrical construction of the nanotube as a 2D structure from a 2atom unit cell, which can be directly compared to the graphene layer. In this sense, most of the symmetry properties of carbon nanotubes can be understood as momentum conservation laws for the 2D nanotube structure. A comparison between the formalism of line groups and that of the factor group of the wavevector, which is commonly used for obtaining the irreducible representations of space groups, is performed in this review article to provide a bridge between the different analysis which have been used in the literature of SWNTs. The irreducible representations of the line groups are compared to those of the group of the wavevector, and the quantum numbers are associated with a helical momentum and a pure angular momentum. For completeness, we obtained the symmetry properties of electronic states of chiral nanotubes using the line group approach and compared them directly with the symmetries of the electronic states obtained from the formalism of the group of the wavevector. It was noted that a direct comparison could only be made for specific points of the nanotube 1D Brillouin zone. This limitation can be attributed to the 2D nature of the helical momentum, and can be overcome by using the “zone-folding” scheme to relate the two formalisms. The symmetries of the electron and phonon eigenstates and the selection rules for electron–photon and electron– phonon interactions were used to give insight into the dynamics of optical processes which are commonly used in the study of carbon nanotubes, such as optical absorption and Raman scattering processes. These properties were first analyzed neglecting the excitonic nature of the optical transitions. However, in view of the fact that the electron–electron and electron–hole interactions are significantly enhanced in a 1D system, we included a brief discussion of excitonic effects in SWNTs and obtained the exciton symmetries using a simple model for the electron–hole interaction. It is interesting to note that the symmetry properties of excitons in carbon nanotubes are different from those of excitons in typical semiconductors and organic conductors due to the presence of two degenerate states contributing to their formation. The selection rules for the optical absorption and Raman scattering, including double resonance processes, are discussed in terms of the excitonic symmetries. Acknowledgments The authors would like to Acknowledge Dr. R. Saito for his careful reading of the manuscript and very helpful discussions. The MIT authors acknowledge support under NSF Grant DMR04-05538. Authors E. B. B. and A. G. S. F. acknowledge financial support from FUNCAP-Brasil and CAPES-Brasil (PRODOC). Authors A. J. and R. B. C. acknowledge support from CNPq, Brasil and Faperj, respectively. Brazilian authors acknowledge support from Instituto de Nanotecnologia and Rede Nacional de Pesquisa em Nanotubos de Carbono.
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Contents of Volume 431 J.C. Parlebas, K. Asakura, A. Fujiwara, I. Harada, A. Kotani X-ray magnetic circular dichroism at rare-earth L23 absorption edges in various compounds and alloys
1
V. Schomerus Non-compact string backgrounds and non-rational CFT
39
J.P. Huang, K.W. Yu Enhanced nonlinear optical responses of materials: Composite effects
87
C. Noce The periodic Anderson model: Symmetry-based results and some exact solutions
173
V.Z. Kresin, Y.N. Ovchinnikov, S.A. Wolf Inhomogeneous superconductivity and the ‘‘pseudogap’’ state of novel superconductors
231
E.B. Barros, A. Jorio, G.G. Samsonidze, R.B. Capaz, A.G. Souza Filho, J. Mendes Filho, G. Dresselhaus, M.S. Dresselhaus Review on the symmetry-related properties of carbon nanotubes
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Contents of volume
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