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SHOT NOISE IN MESOSCOPIC CONDUCTORS
Ya.M. BLANTER, M. BUG TTIKER DeH partement de Physique TheH orique, UniversiteH de Gene` ve, CH-1211, Gene` ve 4, Switzerland
AMSTERDAM } LAUSANNE } NEW YORK } OXFORD } SHANNON } TOKYO
Physics Reports 336 (2000) 1}166
Shot noise in mesoscopic conductors Ya.M. Blanter*, M. BuK ttiker De& partement de Physique The& orique, Universite& de Gene% ve, CH-1211, Gene% ve 4, Switzerland Received October 1999; editor: C.W.J. Beenakker Contents 1. Introduction 1.1. Purpose of this Review 1.2. Scope of the Review 1.3. Subjects not addressed in this Review 1.4. Fundamental sources of noise 1.5. Composition of the Review 2. Scattering theory of thermal and shot noise 2.1. Introduction 2.2. The Pauli principle 2.3. The scattering approach 2.4. General expressions for noise 2.5. Voltage #uctuations 2.6. Applications 2.7. Inelastic scattering. Phase breaking 3. Scattering theory of frequency-dependent noise spectra 3.1. Introduction: current conservation 3.2. Low-frequency noise for independent electrons: at equilibrium and in the presence of dc transport 3.3. Low-frequency noise for independent electrons: photon-assisted transport 3.4. Noise of a capacitor 3.5. Shot noise of a conductor observed at a gate 4. Shot noise in hybrid normal and superconducting structures 4.1. Shot noise of normal-superconductor interfaces 4.2. Noise of Josephson junctions 4.3. Noise of SNS hybrid structures
4 4 4 6 7 11 12 12 12 18 24 29 32 63 69 69
72 75 79 82 86 86 96 97
5. Langevin and master equation approach to noise: double-barrier structures 5.1. Quantum-mechanical versus classical theories of shot noise 5.2. Suppression of shot noise in double-barrier structures 5.3. Interaction e!ects and super-Poissonian noise enhancement 6. Boltzmann}Langevin approach to noise: disordered systems 6.1. Fluctuations and the Boltzmann equation 6.2. Metallic di!usive systems: classical theory of -noise suppression and multi-probe generalization 6.3. Interaction e!ects 6.4. Frequency dependence of shot noise 6.5. Shot noise in non-degenerate conductors 6.6. Boltzmann}Langevin method for shot noise suppression in chaotic cavities with di!usive boundary scattering 6.7. Minimal correlation approach to shot noise in deterministic chaotic cavities 7. Noise in strongly correlated systems 7.1. Coulomb blockade 7.2. Anderson and Kondo impurities 7.3. Tomonaga}Luttinger liquids and fractional quantum Hall edge states 7.4. Composite fermions
* Corresponding author. E-mail address:
[email protected] (Ya.M. Blanter). 0370-1573/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 1 2 3 - 4
101 101 103 108 113 113
116 119 123 125
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131 134 134 141 142 148
Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166 8. Concluding remarks, future prospects, and unsolved problems 8.1. General considerations 8.2. Summary for a lazy or impatient reader Acknowledgements
149 149 151 152
Note added in proof Appendix A. Counting statistics and optical analogies Appendix B. Spin e!ects and entanglement Appendix C. Noise induced by thermal transport References
3 153 153 156 158 158
Abstract Theoretical and experimental work concerned with dynamic #uctuations has developed into a very active and fascinating sub"eld of mesoscopic physics. We present a review of this development focusing on shot noise in small electric conductors. Shot noise is a consequence of the quantization of charge. It can be used to obtain information on a system which is not available through conductance measurements. In particular, shot noise experiments can determine the charge and statistics of the quasiparticles relevant for transport, and reveal information on the potential pro"le and internal energy scales of mesoscopic systems. Shot noise is generally more sensitive to the e!ects of electron}electron interactions than the average conductance. We present a discussion based on the conceptually transparent scattering approach and on the classical Langevin and Boltzmann}Langevin methods; in addition a discussion of results which cannot be obtained by these methods is provided. We conclude the review by pointing out a number of unsolved problems and an outlook on the likely future development of the "eld. 2000 Elsevier Science B.V. All rights reserved. PACS: 72.70.#m; 05.40.#j; 73.50.Td; 74.40.#k Keywords: Fluctuations; Shot noise; Mesoscopic systems; Kinetic theory
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1. Introduction 1.1. Purpose of this Review During the past two decades mesoscopic physics has developed into a fascinating sub"eld of condensed matter physics. In this article, we review a special topic of this "eld: We are concerned with the dynamical noise properties of mesoscopic conductors. After a modest start, a little more than a decade ago, both theories and experiments have matured. There is now already a substantial theoretical literature and there are a number of interesting experiments with which theoretical predictions can be compared. Some experiments ask for additional theoretical work. The "eld has thus reached a stage in which a review might be useful as a "rst orientation for researchers who wish to enter the "eld. Also researchers which are already active in the "eld might appreciate a review to help them keep an overview over the rapid development which has occurred. Any review, of course, re#ects the authors' preferences and prejudices and in any case cannot replace the study of the original literature. Presently, there are no reviews covering the actual state of development of the "eld. The only article which provides a considerable list of references, and gives a description of many essential features of shot noise in mesoscopic systems, has been written in 1996 by de Jong and Beenakker [1]. It is useful as a "rst introduction to the subject, but since then the "eld has developed considerably, and a broader review is clearly desirable. An additional brief review has been written by Martin [2]. The subject has been touched in books with broader scopes by Kogan [3] (Chapter 5) and Imry [4] (Chapter 8). These reviews, and in particular, the work of de Jong and Beenakker [1], provided a considerable help in starting this project. 1.2. Scope of the Review Our intention is to present a review on shot noise in mesoscopic conductors. An e!ort is made to collect a complete list of references, and if not comprehensively re-derive, then at least to mention results relevant to the "eld. We do not cite conference proceedings and brief commentaries, unless we feel that they contain new results or bring some understanding which cannot be found elsewhere. Certainly, it is very possible that some papers, for various reasons, have not come to our attention. We apologize to the authors whose papers we might have overlooked. Trying to classify the already large literature, we chose to divide the Review into sections according to the methods by which the results are derived and not according to the systems we describe. Many results can be obtained in the framework of the scattering approach and/or by classical methods. We deliberately avoid an explanation of the Green's function method, the master equation approach, and the bosonization technique. An attempt to explain how any one of these approaches work would probably double the size of this Review, which is already long enough. Consequently, we make an e!ort to re-derive the results existing in the literature by either the scattering or one of the classical (but not the master equation) approaches, and to present a unifying description. Certainly, for some systems these simple methods just do not work. In particular, this concerns Section 7, which describes shot noise in strongly correlated systems. Results obtained with more sophisticated methods are discussed only brie#y and without an attempt to re-derive them. We incorporate a number of original results in the
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text; they are usually extensions of results available in the literature, and are not specially marked as original. A good review article should resemble a textbook useful to learn the subject but should also be a handbook for an expert. From this perspective this Review is more a textbook than a handbook. We encourage the reader who wants to enter the "eld to read the Review from the beginning to the end. On the other hand, we also try to help the experts, who only take this Review to look for the results concerning some particular phenomenon. It is for this reason that we have included Table 1 with references to the di!erent subsections in which the subject is discussed. The Review is concluded with a brief summary.
Table 1 The results reviewed in this article arranged by subject Subject
Section
Ballistic conductors Electron}phonon interactions Electron}electron interactions in non-degenerate ballistic conductors Hanbury Brown}Twiss e!ects Aharonov}Bohm e!ect
6.5 2.6.8, 2.6.9, 4.1, Appendix B 2.6.10
Tunnel barriers Normal barriers Barriers in di!usive conductors Coulomb blockade regime Frequency dependence of noise Barriers of oscillating random height NS interfaces Josephson junctions Barriers in Luttinger liquids Counting statistics for normal and NS barriers
2.6.1 2.6.4 7.1 3.2, 3.3 3.3 4.1 4.2 7.3 Appendix A
Quantum point contacts Normal quantum point contacts SNS contacts
2.6.2 4.3
Double-barrier structures Resonant tunneling; linear regime Double-barrier suppression Counting statistics Double wells and crossover to the di!usive regime Quantum wells in the non-linear regime: super-Poissonian shot noise enhancement Interaction e!ects in quantum wells Coulomb blockade in quantum dots (normal, superconducting, or ferromagnetic electrodes)
6.3
2.6.3 2.6.3, 5.2 Appendix A 2.6.3, 5.2 5.3 2.7, 5.2, 5.3 7.1
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Table 1 (Continued) Subject
Section
Frequency dependence of noise Resonant tunneling through localized states; Anderson and Kondo models NINIS junctions
5.2, 5.3, 7.1
Disordered conductors Noise suppression in metallic di!usive wires Counting statistics Multi-terminal generalization and Hanbury Brown}Twiss e!ects Interaction e!ects Frequency dependence of noise Disordered contacts and interfaces Disordered NS and SNS contacts Crossover to the ballistic regime Localized regime Non-degenerate di!usive conductors Composite fermions with disorder Noise induced by thermal transport Chaotic cavities Noise suppression in two-terminal chaotic cavities Cavities with di!usive boundary scattering Multi-terminal generalization and Hanbury Brown}Twiss e!ects Counting statistics Quantum Hall ewect IQHE edge channels Hanbury Brown}Twiss e!ects with IQHE edge channels FQHE edge channels Composite fermions Systems with purely capacitive coupling Frequency dependence of noise
2.6.3, 7.2 4.1
2.6.4, 6.2 Appendix A 2.6.9, 6.2 2.7, 6.3 3.2, 6.4 2.6.4 4.1, 4.3 2.6.4 2.6.4 6.5 7.4 Appendix C
2.6.5, 6.6, 6.7 6.6 2.6.5, 6.7 Appendix A
2.6.6, 2.6.10 2.6.7, 2.6.9 7.3 7.4
3.4, 3.5
1.3. Subjects not addressed in this Review First of all, we emphasize that this is not a general review of mesoscopic physics. In many cases, it is necessary to describe brie#y the systems before addressing their noise properties. Only a few references are provided to the general physical background. These references are not systematic; we cite review articles where this is possible, and give references to the original papers only in cases
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such reviews are not readily available. This Review is not intended to be a tool to study mesoscopic physics, though possibly many important ideas are present here. A multitude of sources typically contribute to the noise of electrical conductors. In particular, some of the early experimental e!orts in this "eld su!ered from 1/f-noise which is observed at low frequencies. There is still to our knowledge no established theory of 1/f-noise. The present article focuses on fundamental, unavoidable sources of noise, which are the thermal (Nyquist}Johnson noise) and the shot noise due to the granularity of charge. The reader may "nd an account of other sources of noise in the book by Kogan [3]. We mention also that, though we try to be very accurate in references, this Review cannot be regarded as a historical chronicle of shot noise research. We re-derive a number of results by other methods than were used in the original papers, and even though we make an e!ort to mention all achievements, we do not emphasize historical priorities. 1.4. Fundamental sources of noise 1.4.1. Thermal noise At non-zero temperature, thermal #uctuations are an unavoidable source of noise. Thermal agitation causes the occupation number of the states of a system to #uctuate. Typically, in electric conductors, we can characterize the occupation of a state by an occupation number n which is either zero or one. The thermodynamic average of the occupation number 1n2 is determined by the Fermi distribution function f and we have simply 1n2"f. In an equilibrium state the probability that the state is empty is on the average given by 1!f, and the probability that the state is occupied is on the average given by f. The #uctuations away from this average occupation number are (n!1n2)"n!2n1n2#1n2. Taking into account that for a Fermi system n"n, we "nd immediately that the #uctuations of the occupation number at equilibrium away from its thermal average are given by 1(n!1n2)2"f (1!f ) .
(1)
The mean-squared #uctuations vanish in the zero temperature limit. At high temperatures and high enough energies the Fermi distribution function is much smaller than one and thus the factor 1!f in Eq. (1) can be replaced by 1. The #uctuations are determined by the (Maxwell)}Boltzmann distribution. The #uctuations in the occupation number give rise to equilibrium current #uctuations in the external circuit which are via the #uctuation-dissipation theorem related to the conductance of the system. Thus, investigation of equilibrium current #uctuations provides the same information as investigation of the conductance. This is not so with the shot noise of electrical conductors which provides information which cannot be obtained from a conductance measurement. We next brie#y discuss the source of shot noise. 1.4.2. Shot noise Shot noise in an electrical conductor is a consequence of the quantization of the charge. Unlike for thermal noise, to observe shot noise, we have to investigate the non-equilibrium (transport) state of the system.
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To explain the origin of shot noise we consider "rst a "ctitious experiment in which only one particle is incident upon a barrier. At the barrier the particle is either transmitted with probability ¹ or re#ected with probability R"1!¹. We now introduce the occupation numbers for this experiment. The incident state is characterized by an occupation number n , the transmitted state is characterized by an occupation number n and the re#ected state by an occupation number n . If 2 0 we repeat this experiment many times, we can investigate the average of these occupation numbers and their #uctuations away from the average behavior. In our experiments the incident beam is occupied with probability 1 and thus 1n 2"1. However, the transmitted state is occupied only with probability ¹ and is empty with probability R. Thus 1n 2"¹ and 1n 2"R. The #uctu2 0 ations away from the average occupation number, can be obtained very easily in the following way: The mean-squared #uctuations in the incident beam vanish, (n !1n 2)"0. To "nd the mean-squared #uctuations in the transmitted and re#ected state, we consider the average of the product of the occupation numbers of the transmitted and re#ected beam 1n n 2. Since in each 2 0 event the particle is either transmitted or re#ected, the product n n vanishes for each experiment, 2 0 and hence the average vanishes also, 1n n 2"0. Using this we "nd easily that the mean squares of 2 0 the transmitted and re#ected beam and their correlation is given by 1(*n )2"1(*n )2"!1*n *n 2"¹R , 2 0 2 0
(2)
where we have used the abbreviations *n "n !1n 2 and *n "n !1n 2. Such #uctuations 2 2 2 0 0 0 are called partition noise since the scatterer divides the incident carrier stream into two streams. The partition noise vanishes both in the limit when the transmission probability is 1 and in the limit when the transmission probability vanishes, ¹"0. In these limiting cases no partitioning takes place. The partition noise is maximal if the transmission probability is ¹". Let us next consider a slightly more sophisticated, but still "ctitious, experiment. We assume that the incident beam is now occupied only with probability f. Eventually f will just be taken to be the Fermi distribution function. The initial state is empty with probability 1!f. Apparently, in this experiment the average incident occupation number is 1n 2"f, and since the particle is transmit ted only with probability f ¹ and re#ected with probability fR, we have 1n 2"f ¹ and 1n 2"f R. 2 0 Since we are still only considering a single particle, we have as before that in each event the product n n vanishes. Thus we can repeat the above calculation to "nd 2 0 1(*n )2"¹f (1!¹f ) , 2
(3)
1(*n )2"Rf (1!Rf ) , 0
(4)
1*n *n 2"!¹Rf . 2 0
(5)
If we are in the zero-temperature limit, f"1, we recover the results discussed above. Note that now even in the limit ¹"1 the #uctuations in the transmitted state do not vanish, but #uctuate like the incident state. For the transmitted stream, the factor (1!¹f ) can be replaced by 1, if either the transmission probability is small or if the occupation probability of the incident carrier stream is small. We can relate the above results to the #uctuations of the current in a conductor. To do this we have to put aside the fact that in a conductor we deal not as above just with events of a single
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carrier but with a state which may involve many (indistinguishable) carriers. We imagine a perfect conductor which guides the incident carriers to the barrier, and imagine that we have two additional conductors which guide the transmitted and re#ected carriers away from the conductor, such that we can discuss, as before, incident, transmitted and re#ected currents separately. Furthermore, we want to assume that we have to consider only carriers moving in one direction with a velocity v(E) which is uniquely determined by the energy E of the carrier. Consider next the average incident current. In a narrow energy interval dE, the incident current is dI (E)"ev(E) do(E), where do(E) is the density of carriers per unit length in this energy range. The density in an energy interval dE is determined by the density of states (per unit length) l(E)"do/dE times the occupation factor n (E) of the state at energy E. We thus have do(E)"n (E)l(E) dE. The density of states in our perfect conductors is l(E)"1/(2p v(E)). Thus the incident current in a narrow energy interval is simply e n (E) dE . dI (E)" 2p
(6)
This result shows that there is a direct link between currents and the occupation numbers. The total incident current is I "(e/2p )n (E) dE and on the average is given by 1I 2"(e/2p ) f (E) dE. Similar considerations give for the average transmitted current 1I 2"(e/2p ) f (E)¹ dE and for 2 the re#ected current 1I 2"(e/2p ) f (E)R dE. Current #uctuations are dynamic phenomena. The 0 importance of the above consideration is that it can now easily be applied to investigate timedependent current #uctuations. For occupation numbers which vary slowly in time, Eq. (6) still holds. The current #uctuations in a narrow energy interval are at long times determined by dI (E, t)"(e/2p )n (E, t) dE where n (E, t) is the time-dependent occupation number of states with energy E. A detailed derivation of the connection between currents and occupation numbers is the subject of an entire section of this Review. We are interested in the low-frequency current noise and thus we can Fourier transform this equation. In the low-frequency limit we obtain I(u)"(e/2p )dE n(E, E# u). As a consequence the #uctuations in current and the #uctuations in occupation number are directly related. In the zero frequency limit the current noise power is S "edE S (E). In each small energy interval particles arrive at a rate dE/(2p ) and contribute, '' LL with a mean square #uctuation, as given by one of Eqs. (3)}(5), to the noise power. We have S (E)"(1/p )1*n*n2. Thus the #uctuation spectra of the incident, transmitted, and re#ected LL currents are e dE f (1!f ) , S "2 ' ' 2p
(7)
(8)
(9)
e dE ¹f (1!¹f ) , S 2 2 "2 '' 2p
e dE Rf (1!Rf ) . S 0 0 "2 '' 2p
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The transmitted and re#ected current are correlated,
e S 2 0 "!2 dE ¹f Rf . '' 2p
(10)
In the limit that either ¹ is very small or f is small, the factor (1!¹f ) in Eq. (8) can be replaced by one. In this limit, since the average current through the barrier is 1I2"(e/2p )dE ¹f, the spectrum, Eq. (8), is Schottky's result [5] for shot noise, S 2 2 "2e1I2 . ''
(11)
Schottky's result corresponds to the uncorrelated arrival of particles with a distribution function of time intervals between arrival times which is Poissonian, P(*t)"q\ exp(!*t/q), with q being the mean time interval between carriers. Alternatively, Eq. (11) is also referred to in the literature as the Poisson value of shot noise. The result, Eq. (8), is markedly di!erent from Eq. (11) since it contains, in comparison to Schottky's expression, the extra factor (1!¹f ). This factor has the consequence that the shot noise (8) is always smaller than the Poisson value. For truly ballistic systems (¹"1) the shot noise even vanishes in the zero-temperature limit. As the temperature increases, in such a conductor (¹"1) there is shot noise due to the #uctuation in the incident beam arising from the thermal #uctuations. Eventually, at high temperatures the factor 1!f can be replaced by 1, and the ballistic conductor exhibits Poisson noise, in accordance with Schottky's formula, Eq. (11). The full Poisson noise given by Schottky's formula is also reached for a scatterer with very small transparency ¹;1. We emphasize that the above statements refer to the transmitted current. In the limit ¹;1 the re#ected current remains nearly noiseless up to high temperatures when (1!Rf ) can be replaced by 1. We also remark that even though electron motion in vacuum tubes (the Schottky problem [5]) is often referred to as ballistic, it is in fact a problem in which carriers have been emitted by a source into vacuum either through thermal activation over or by tunneling through a barrier with very small transparency. Our discussion makes it clear that out of equilibrium, and at "nite temperatures, the noise described by Eq. (8) contains the e!ect of both the #uctuations in the incident carrier beam as well as the partition noise. In a transport state, noise in mesoscopic conductors has two distinct sources which manifest themselves in the #uctuations of the occupation numbers of states: (i) thermal #uctuations; (ii) partition noise due to the discrete nature of carriers. Both the thermal and shot noise at low frequencies and low voltages re#ect in many situations independent quasi-particle transport. Electrons are, however, interacting entities and both the #uctuations at "nite frequencies and the #uctuation properties far from equilibrium require in general a discussion of the role of the long-range Coulomb interaction. A quasi-particle picture is no longer su$cient and collective properties of the electron system come into play.
Note that this terminology, common in mesoscopic physics, is di!erent from that used in the older literature [6], where shot noise (due to random injection of particles into a system) and partition noise (due to random division of the particle stream between di!erent electrodes, or by potential barriers) are two distinct independent sources of #uctuations.
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The above considerations are, of course, rather simplistic and should not be considered as a quantitative theory. Since statistical e!ects play a role, one would like to see a derivation which relates the noise to the symmetry of the wave functions. Since we deal with many indistinguishable particles, exchange e!ects can come into play. The following sections will treat these questions in detail. 1.5. Composition of the Review The review starts with a discussion of the scattering approach (Section 2). This is a fully quantum-mechanical theory which applies to phase-coherent transport. It is useful to take this approach as a starting point because of its conceptual clarity. The discussion proceeds with a number of speci"c examples, like quantum point contacts, resonant double barriers, metallic di!usive wires, chaotic cavities, and quantum Hall conductors. We are interested not only in current #uctuations at one contact of a mesoscopic sample but also in the correlations of currents at di!erent contacts. Predictions and experiments on such correlations are particularly interesting since current correlations are sensitive to the statistical properties of the system. Comparison of such experiments with optical analogs is particularly instructive. Section 3 describes the frequency dependence of noise via the scattering approach. The main complication is that generally one has to include electron}electron interactions to obtain #uctuation spectra which are current conserving. For this reason, not many results are currently available on frequency-dependent noise, though the possibility to probe in this way the inner energy scales and collective response times of the system looks very promising. We proceed in Section 4 with the description of superconducting and hybrid structures, to which a generalized scattering approach can be applied. New noise features appear from the fact that the Cooper pairs in superconductors have the charge 2e. Then in Sections 5 and 6 we review recent discussions which apply more traditional classical approaches to the #uctuations of currents in mesoscopic conductors. Speci"cally, for a number of systems, as far as one is concerned only with ensemble-averaged quantities, the Langevin and Boltzmann}Langevin approaches provide a useful discussion, especially since it is known how to include inelastic scattering and e!ects of interactions. Section 7 is devoted to shot noise in strongly correlated systems. This section di!ers in many respects from the rest of the Review, mainly because strongly correlated systems are mostly too complicated to be described by the scattering or Langevin approaches. We resort to a brief description and commentary of the results, rather than to a comprehensive demonstration how they are derived. We conclude the Review (Section 8) with an outlook. We give our opinion concerning possible future directions along which the research on shot noise will develop. In the concluding section, we provide a very concise summary of the state of the "eld and we list some (possibly) important unsolved problems. Some topics are treated in a number of Appendices mostly to provide a better organization of the paper. The Appendices report important results on topics which are relatively well rounded, are relevant for the connections between di!erent sub-"elds, and might very well become the subject of much further research.
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2. Scattering theory of thermal and shot noise 2.1. Introduction In this section we present a theory of thermal and shot noise for fully phase-coherent mesoscopic conductors. The discussion is based on the scattering approach to electrical conductance. This approach, as we will show, is conceptually simple and transparent. A phase-coherent description is needed if we consider an individual sample, like an Aharonov}Bohm ring, or a quantum point contact. Often, however, we are interested in characterizing not a single sample but rather an ensemble of samples in which individual members di!er only in the microscopic arrangement of impurities or small variations in the shape. The ensemble-averaged conductance is typically, up to a small correction, determined by a classical expression like a Drude conductance formula. Similarly, noise spectra, after ensemble averaging, are, up to small corrections, determined by purely classical expressions. In these case, there is no need to keep information about phases of wave functions, and shot noise expressions may be obtained by classical methods. Nevertheless, the generality of the scattering approach and its conceptual clarity, make it the desired starting point of a discussion of noise in electrical conductors. Below we emphasize a discussion based on second quantization. This permits a concise treatment of the many-particle problem. Rather than introducing the Pauli principle by hand, in this approach it is a consequence of the underlying symmetry of the wave functions. It lends itself to a discussion of the e!ects related to the quantum mechanical indistinguishability of identical particles. In fact, it is an interesting question to what extent we can directly probe the fact that exchange of particles leaves the wave function invariant up to a sign. Thus, an important part of our discussion will focus on exchange ewects in current}current correlation spectra. We start this section with a review of #uctuations in idealized one- and two-particle scattering problems. This simple discussion highlights the connection between symmetry of the wave functions (the Pauli principle) and the #uctuation properties. It introduces in a simple manner some of the basic concepts and it will be interesting to compare the results of the one- and two-particle scattering problems with the many-particle problem which we face in mesoscopic conductors. 2.2. The Pauli principle The investigation of the noise properties of a system is interesting because it is fundamentally connected with the statistical properties of the entities which generate the noise. We are concerned with systems which contain a large number of indistinguishable particles. The fact that in quantum mechanics we cannot label di!erent particles implies that the wave function must be invariant, up to a phase, if two particles are exchanged. The invariance of the wave function under exchange of two particles implies that we deal with wave functions which are either symmetric or antisymmetric under particle exchange. (In strictly two-dimensional systems more exotic possibilities are permitted). These symmetry statements are known as the Pauli principle. Systems with symmetric (antisymmetric) wave functions are described by Bose}Einstein (Fermi}Dirac) statistics, respectively. Prior to the discussion of the noise properties in electrical conductors, which is our central subject, in this subsection we illustrate in a simple manner the fundamental connection between the
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symmetry of the wave function and the statistical properties of scattering experiments. We deal with open systems similarly to a scattering experiment in which particles are incident on a target at which they are scattered. The simplest case in which the symmetry of the wave function matters is the case of two identical particles. Here we present a discussion of idealized two-particle scattering experiments. We consider the arrangement shown in Fig. 1, which contains two sources 1 and 2 which can emit particles and two detectors 3 and 4 which respond ideally with a signal each time a particle enters a detector. An arrangement similar to that shown in Fig. 1 is used in optical experiments. In this "eld experiments which invoke one or two incoming particle streams (photons) and two detectors are known as Hanbury Brown}Twiss experiments [7], after the pioneering experiments carried out by these two researchers to investigate the statistical properties of light. In Fig. 1 the scattering is, much as in an optical table top experiment, provided by a half-silvered mirror (beam-splitter), which permits transmission from the input-channel 1 through the mirror with probability amplitude s "t to the detector at arm 4 and generates re#ected particles with amplitude s "r into detector 3. We assume that particles from source 2 are scattered likewise and have probability amplitudes s "t and s "r. The elements s , when written as a matrix, GH form the scattering matrix s. The elements of the scattering matrix satisfy "r"#"t""1 and trH#rtH"0, stating that the scattering matrix is unitary. A simple example of a scattering matrix often employed to describe scattering at a mirror in optics is r"!i/(2 and t"1/(2. We are interested in describing various input states emanating from the two sources. To be interesting, these input states contain one or two particles. We could describe these states in terms of Slater determinants, but it is more elegant to employ the second quantization approach. The incident states are described by annihilation operators a( or creation operators a( R in arm i, i"1, 2. G G The outgoing states are, in turn, described by annihilation operators bK and creation operators G bK R, i"3, 4. The operators of the input states and output states are not independent but are related G
Fig. 1. An arrangement of a scattering experiment with two sources (1 and 2) and two detectors (3 and 4).
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by a unitary transformation which is just the scattering matrix,
bK a( "s . (12) bK a( Similarly, the creation operators a( R and bK R are related by the adjoint of the scattering matrix sR. G G Note that the mirror generates quantum mechanical superpositions of the input states. The coe$cients of these superpositions are determined by the elements of the scattering matrix. For bosons, the a( obey the commutation relations G [a( , a( R]"d . (13) G H GH Since the scattering matrix is unitary, the bK obey the same commutation relations. In contrast, for G fermions, the a( and bK obey anti-commutation relations G G +a( , a( R,"d . (14) G H GH The di!erent commutation relations for fermions and bosons assure that multi-particle states re#ect the underlying symmetry of the wave function. The occupation numbers of the incident and transmitted states are found as n( "a( Ra( and G G G n( "bK RbK , G G G 1n( 2 R ¹ 1n( 2 " , (15) 1n( 2 ¹ R 1n( 2 where we introduced transmission and re#ection probabilities, ¹""t" and R""r".
2.2.1. Single, independent particle scattering Before treating two-particle states it is useful to consider brie#y a series of scattering experiments in each of which only one particle is incident on the mirror. Let us suppose that a particle is incident in arm 1. Since we know that in each scattering experiment there is one incident particle, the average occupation number in the incident arm is thus 1n 2"1. The #uctuations away from the average occupation number *n "n !1n 2 vanish identically (not only on the average). In particular, we have 1(*n )2"0. Particles are transmitted into arm 4 with probability ¹, and thus the mean occupation number in the transmitted beam is 1n 2"¹. Similarly, particles are re#ected into arm 3 with probability R, and the average occupation number in our series of experiments is thus 1n 2"R. Consider now the correlation 1n n 2 between the occupation numbers in arms 3 and 4. Since each particle is either re#ected or transmitted, it means that in this product one of the occupation numbers is always 1 and one is zero in each experiment. Consequently, the correlation of the occupation numbers vanishes, 1n n 2"0. Using this result, we obtain for the #uctuations of the occupation numbers *n "n !1n 2 in the re#ected beam and *n "n !1n 2 in the transmitted beam, 1(*n )2"1(*n )2"!1*n *n 2"¹R . (16) The #uctuations in the occupation numbers in the transmitted and re#ected beams and their correlation are a consequence of the fact that a carrier can "nally only be either re#ected or transmitted. These #uctuations are known as partition noise. The partition noise vanishes for
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a completely transparent scatterer ¹"1 and for a completely re#ecting scatterer R"1 and is maximal for ¹"R"1/2. We emphasize that the partition noise is the same, whether we use a fermion or boson in the series of experiments. To detect the sensitivity to the symmetry of the wave function, we need to consider at least two particles. 2.2.2. Two-particle scattering We now consider two particles incident on the mirror of Fig. 1, focusing on the case that one particle is incident in each arm. We follow here closely a discussion by Loudon [8] and refer the reader to this work for additional information. The empty (vacuum) state of the system (input arms and output arms) is denoted by "02. Consider now an input state which consists of two particles with a de"nite momentum, one incident in arm 1 and one incident in arm 2. For simplicity, we assume that the momentum of both particles is the same. With the help of the creation operators given above we can generate the input state "W2"a( R a( R "02. The probability that both particles appear in output arm 3 is P(2, 0)"1W"n( n( "W2, the probability that in each output arm there is one particle is P(1, 1)"1W"n( n( "W2, and the probability that both particles are scattered into arm 4 is P(0, 2)"1W"n( n( "W2. Considering speci"cally the probability P(1, 1), we have to "nd (17) P(1, 1)"1W"n( n( "W2"10"a( a( bK R bK bK R bK a( R a( R "02 . First we notice that in the sequence of bK -operators bK and bK R anti-commute (commute), and we can thus write bK R bK bK R bK also in the sequence GbK R bK R bK bK . Then, by inserting a complete set of states with "xed number of particles "n21n" into this product, GbK R bK R "n21n"bK bK , it is only the state with n"0 which contributes since to the right of 1n" we have two creation and two annihilation operators. Thus, the probability P(1, 1) is given by the absolute square of a probability amplitude (18) P(1, 1)""10"bK bK a( R a( R "02" . To complete the evaluation of this probability, we express a( R and a( R in terms of the output operators bK R and bK R using the adjoint of Eq. (12). This gives (19) a( R a( R "rtbK R bK R #rbK R bK R #tbK R bK R #rtbK R bK R . Now, using the commutation relations, we pull the annihilation operators to the right until one of them acts on the vacuum and the corresponding term vanishes. A little algebra gives P(1, 1)"(¹$R) .
(20)
Eq. (20) is a concise statement of the Pauli principle. For bosons, P(1, 1) depends on the transmission and re#ection probability of the scatterer, and vanishes for an ideal mirror ¹"R". The two particles are preferentially scattered into the same output branch. For fermions, P(1, 1) is independent of the transmission and re#ection probability and given by P(1, 1)"1. Thus fermions are scattered with probability one into the di!erent output branches. It is instructive to compare these results with the one for classical particles, P(1, 1)"¹#R. We see thus that the probability to "nd two bosons (fermions) in two di!erent detectors is suppressed (enhanced) in comparison with the same probability for classical particles. A similar consideration also gives for the probabilities P(2, 0)"P(0, 2)"2R¹
(21)
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for bosons, whereas for fermions the two probabilities vanish P(2, 0)"P(0, 2)"0. For classical partition of carriers the probability to "nd the two particles in the same detector is R¹, which is just one-half of the probability for bosons. The average occupation numbers are 1n 2"1n 2"1, since we have now two particles in branch 3 with probability P(2, 0) and one particle with probability P(1, 1). Consequently, the correlations of the #uctuations in the occupation numbers *n( "n( !1n 2 are given by G G G 1*n( *n( 2"!4R¹ ,
(22)
for bosons and by 1*n( *n( 2"0 for fermions. For bosons the correlation is negative due to the enhanced probability that both photons end up in the same output branch. For fermions there are no #uctuations in the occupation number and the correlation function thus vanishes. 2.2.3. Two-particle scattering: Wave packet overlap The discussion given above implicitly assumes that both `particlesa or `wavesa arrive simultaneously at the mirror and `seea each other. Clearly, if the two particles arrive at the mirror with a time-delay which is large enough such that there is no overlap, the outcome of the experiments described above is entirely di!erent. If we have only a sequence of individual photons or electrons arriving at the mirror we have for the expectation values of the occupation numbers 1n 2"1n 2"R#¹"1, and the correlation of the occupation number 1n n 2"0 vanishes. Consequently, the correlation of the #uctuations of the occupation number is 1*n *n 2"!1. Without any special sources at hand it is impossible to time the carriers such that they arrive simultaneously at the mirror, and we should consider all possibilities. To do this, we must consider the states at the input in more detail. Let us assume that a state in input arm i can be written with the help of plane waves W (k, x )"exp(!ikx ) with x the G G G G coordinate along arm i normalized such that it grows as we move away from the arm toward the source. Similarly, let y be the coordinates along the output arms such that y vanishes at the mirror G G and grows as we move away from the splitter. A plane wave W (k, x )"exp(!ikx ) incident from arm 1 thus leads to a re#ected wave in output arm 3 given by W (k, y )"r exp(!iky ) and to a transmitted wave W (k, y )"t exp(!iky ) in output arm 4. We call such a state a `scattering statea. It can be regarded as the limit of a wave packet with a spatial width that tends towards in"nity and an energy width that tends to zero. To build up a particle that is localized in space at a given time we now invoke superpositions of such scattering states. Thus, let the incident particle in arm i be described by W (x , t)"dk a (k) exp(!ikx ) exp(!iE(k)t/ ), where a (k) is a function G G G G G such that
dk "a (k)""1 , G
(23)
and E(k) is the energy of the carriers as a function of the wave vector k. In second quantization the incident states are written with the help of the operators
AK R(x , t)" G G
dk a (k )W (k , x )a( R(k ) exp(!iE(k )t/ ) , G G G G G G G G G
(24)
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and the initial state of our two-particle scattering experiment is thus AK R (x , t) AK R (x , t)"02. We are again interested in determining the probabilities that two particles appear in an output branch or that one particle appears in each output branch, P(2, 0)"dk dk 1n( (k )n( (k )2, Let P(1, 1)"dk dk 1n( (k )n( (k )2, and P(0, 2)"dk dk 1n( (k )n( (k )2. us again consider as in the case of pure scattering states. We P(1, 1). Its evaluation proceeds in much the same way re-write P(1, 1) in terms of the absolute square of an amplitude,
dk dk "10"bK (k )bK (k )"W2" . (25) We then write the a( operators in the AK in terms of the output operators bK . Instead of Eq. (20), we obtain P(1, 1)"
P(1, 1)"¹#R$2¹R"J" ,
(26)
where
J"
dk aH(k)a (k) exp(ik(x !x ))
(27) is the overlap integral of the two particles. For the case of complete overlap "J""1 we obtain Eq. (20). For the case that we have no overlap we obtain the classical result P(1, 1)"¹#R which is independent of whether a boson or fermion is incident on the scatterer. In the general case, the overlap depends on the form of the wave packet. If two Gaussian wave packets of spatial width d and central velocity v are timed to arrive at time q and q at the scatterer, the overlap integral is "J""exp[!v(q !q )/2d] . (28) A signi"cant overlap occurs only during the time d/v. For wave packets separated in time by more than this time interval the Pauli principle is not e!ective. Complete overlap occurs in two simple cases. We can assume that the two wave packets are identical and are timed to arrive exactly at the same instant at the scatterer. Another case, in which we have complete overlap, is in the basis of scattering states. In this case "a (k)""d(k!k ) for G G a scattering state with wave vector k and consequently for the two particles with k and k we have G G H J"d . The "rst option of timed wave packets seems arti"cial for the thermal sources which we GH want to describe. Thus in the following we will work with scattering states. The probabilities P(2, 0), P(1, 1) and P(0, 2) for these scattering experiments are shown in Table 2. The considerations which lead to these results now should be extended to take the polarization of photons or spin of electrons into account. 2.2.4. Two-particle scattering: spin Consider the case of fermions and let us investigate a sequence of experiments in each of which we have an equal probability of having electrons with spin up or down in an incident state. Assuming that the scattering matrix is independent of the spin state of the electrons, the results discussed above describe the two cases when both spins point in the same direction (to be denoted as P(1!, 1!) and P(1 , 1 )). Thus what remains is to consider the case in which one incident particle has spin up and one incident particle has spin down. If the detection is also spin sensitive, the probability which we determine is P(1!, 1 ). But in such an experiment we can tell which of the two
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Table 2 Output probabilities for one particle incident in each input arm (from Ref. [8]) Probability
Classical
Bosons
Fermions
P(2, 0) P(1, 1) P(0, 2)
R¹ R#¹ R¹
R¹(1#"J") R#¹!2R¹"J" R¹(1#"J")
R¹(1!"J") R#¹#2R¹"J" R¹(1!"J")
particles went which way at the scatterer and there is thus no interference. The outcome is classical: P(1!, 1 )"R for an initial state with a spin up in arm 1 and a spin down in arm 2. For the same state we have P(1 , 1!)"¹. Now let us assume that there is no way of detecting the spin state of the outgoing particles. For a given initial state we have P(1, 1)" P(1p, 1p ) where p and p are spin variables. If we consider N NY all possible incident states with equal probability, we "nd P(1, 1)"¹#R#¹R"J" i.e. a result with an interference contribution which is only half as large as given in Eq. (26). For further discussion, see Appendix B. The scattering experiments considered above assume that we can produce one or two particle states either in a single mode or by exciting many modes. Below we will show that thermal sources, the electron reservoirs which are of the main interest here, cannot be described in this way: In the discussion given here the ground state is the vacuum (a state without carriers) whereas in an electrical conductor the ground state is a many-electron state. 2.3. The scattering approach The idea of the scattering approach (also referred to as Landauer approach) is to relate transport properties of the system (in particular, current #uctuations) to its scattering properties, which are assumed to be known from a quantum-mechanical calculation. In its traditional form the method applies to non-interacting systems in the stationary regime. The system may be either at equilibrium or in a non-equilibrium state; this information is introduced through the distribution functions of the contacts of the sample. To be clear, we consider "rst a two-probe geometry and particles obeying Fermi statistics (having in mind electrons in mesoscopic systems). Eventually, the generalization to many probes and Bose statistics is given; extensions to interacting problems are discussed at the end of this Section. In the derivation we essentially follow Ref. [9]. 2.3.1. Two-terminal case; current operator We consider a mesoscopic sample connected to two reservoirs (terminals, probes), to be referred to as `lefta (L) and `righta (R). It is assumed that the reservoirs are so large that they can be To avoid a possible misunderstanding, we stress that the long-range Coulomb interaction needs to be taken into account when one tries to apply the scattering approach for the description of systems in time-dependent external "elds, or "nite-frequency #uctuation spectra in stationary "elds. On the other hand, for the description of zero-frequency #uctuation spectra in stationary xelds, a consistent theory can be given without including Coulomb e!ects, even though the #uctuations themselves are, of course, time-dependent and random.
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characterized by a temperature ¹ and a chemical potential k ; the distribution functions * 0 * 0 of electrons in the reservoirs, de"ned via these parameters, are then Fermi distribution functions f (E)"[exp[(E!k )/k ¹ ]#1]\, a"L, R ? ? ? (see Fig. 2). We must note at this stage, that, although there are no inelastic processes in the sample, a strict equilibrium state in the reservoirs can be established only via inelastic processes. However, we consider the reservoirs (the leads) to be wide compared to the typical cross-section of the mesoscopic conductor. Consequently, as far as the reservoirs are concerned, the mesoscopic conductor represents only a small perturbation, and describing their local properties in terms of an equilibrium state is thus justi"ed. We emphasize here, that even though the dynamics of the scattering problem is described in terms of a Hamiltonian, the problem which we consider is irreversible. Irreversibility is introduced in the discussion, since the processes of a carrier leaving the mesoscopic conductor and entering the mesoscopic conductor are unrelated, uncorrelated events. The reservoirs act as sources of carriers determined by the Fermi distribution but also act as perfect sinks of carriers irrespective of the energy of the carrier that is leaving the conductor. Far from the sample, we can, without loss of generality, assume that transverse (across the leads) and longitudinal (along the leads) motion of electrons are separable. In the longitudinal (from left to right) direction the system is open, and is characterized by the continuous wave vector kl . It is advantageous to separate incoming (to the sample) and outgoing states, and to introduce the longitudinal energy El " kl /2m as a quantum number. Transverse motion is quantized and described by the discrete index n (corresponding to transverse energies E , which can be * 0_L di!erent for the left and right leads). These states are in the following referred to as transverse (quantum) channels. We write thus E"E #El . Since El needs to be positive, for a given total L energy E only a "nite number of channels exists. The number of incoming channels is denoted N (E) in the left and right lead, respectively. * 0 We now introduce creation and annihilation operators of electrons in the scattering states. In principle, we could have used the operators which refer to particles in the states described by the quantum numbers n, kl . However, the scattering matrix which we introduce below, relates current amplitudes and not wave function amplitudes. Thus, we introduce operators a( R (E) and a( (E) *L *L which create and annihilate electrons with total energy E in the transverse channel n in the left lead, which are incident upon the sample. In the same way, the creation bK R (E) and annihilation bK (E) *L *L
Fig. 2. Example of a two-terminal scattering problem for the case of one transverse channel.
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operators describe electrons in the outgoing states. They obey anticommutation relations a( R (E)a( (E )#a( (E )a( R (E)"d d(E!E ) , *L LLY *L *LY *LY a( (E)a( (E )#a( (E )a( (E)"0 , *L *LY *LY *L a( R (E)a( R (E )#a( R (E )a( R (E)"0 . *L *LY *LY *L Similarly, we introduce creation and annihilation operators a( R (E) and a( (E) in incoming states 0L 0L and bK R (E) and bK (E) in outgoing states in the right lead (Fig. 2). 0L 0L The operators a( and bK are related via the scattering matrix s,
bK a( * * 2 2 bK * a( *, "s *,* . (29) bK a( 0 0 2 2 bK 0 a( 0 0, 0, The creation operators a( R and bK R obey the same relation with the Hermitian conjugated matrix sR. The matrix s has dimensions (N #N );(N #N ). Its size, as well as the matrix elements, * 0 * 0 depends on the total energy E. It has the block structure
s"
r t t
r
.
(30)
Here the square diagonal blocks r (size N ;N ) and r (size N ;N ) describe electron re#ection * * 0 0 back to the left and right reservoirs, respectively. The o!-diagonal, rectangular blocks t (size N ;N ) and t (size N ;N ) are responsible for the electron transmission through the sample. 0 * * 0 The #ux conservation in the scattering process implies that the matrix s is quite generally unitary. In the presence of time-reversal symmetry the scattering matrix is also symmetric. The current operator in the left lead (far from the sample) is expressed in a standard way,
R R
e dr WK R (r, t) WK (r, t)! WK R (r, t) WK (r, t) , IK (z, t)" , * * * Rz * Rz * 2im where the "eld operators WK and WK R are de"ned as
,* # s (r ) *L , [a( e I*L X#bK e\ I*L X] WK (r, t)" dE e\ #R *L * (2p v (E)) *L *L L and
sH (r ) ,* # *L , [a( R e\ I*L X#bK R e I*L X] . WK R (r, t)" dE e #R *L * (2p v (E)) *L *L L Here r is the transverse coordinate(s) and z is the coordinate along the leads (measured from , left to right); s* are the transverse wave functions, and we have introduced the wave vector, L
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k " \[2m(E!E )] (the summation only includes channels with real k ), and the velocity *L *L *L of carriers v (E)" k /m in the nth transverse channel. L *L After some algebra, the expression for the current can be cast into the form
1 e dE dE e #\#YR
IK (z, t)" * 4p
(v (E)v (E ) L *L *L +[v (E)#v (E )][exp[i(k (E )!k (E))z]a( R (E)a( (E ) *L *L *L *L *L *L ! exp[i(k (E)!k (E ))z]bK R (E)bK (E )] *L *L *L *L # [v (E)!v (E )][exp[!i(k (E)#k (E ))z]a( R (E)bK (E ) *L *L *L *L *L *L ! exp[i(k (E)#k (E ))z]bK R (E)a( (E )], . (31) *L *L *L *L This expression is cumbersome, and, in addition, depends explicitly on the coordinate z. However, it can be considerably simpli"ed. The key point is that for all observable quantities (average current, noise, or higher moments of the current distribution) the energies E and E in Eq. (31) either coincide, or are close to each other. On the other hand, the velocities v (E) vary with energy quite L slowly, typically on the scale of the Fermi energy. Therefore, one can neglect their energy dependence, and reduce the expression (31) to a much simpler form,
e (32) IK (t)" dE dE e #\#YR [a( R (E)a( (E )!bK R (E)bK (E )] . *L *L * *L *L 2p
L Note that n( > (E)"a( R (E)a( (E) is the operator of the occupation number of the incident carriers in *L *L *L lead L in channel n. Similarly, n( \ (E)"bK R (E)bK (E) is the operator of the occupation number of *L *L *L the out-going carriers in lead L in channel n. Setting E "E# u and carrying out the integral over u gives
e IK (t)" dE [n( > (E, t)!n( \ (E, t)] . (33) * *L *L 2p
L Here n( ! (E, t) are the time-dependent occupation numbers for the left and right moving carriers at *L energy E. Thus, Eq. (33) states that the current at time t is simply determined by the di!erence in occupation number between the left and right movers in each channel. We made use of this intuitively appealing result already in the introduction. Using Eq. (29) we can express the current in terms of the a( and a( R operators alone,
e dE dE e #\#YR a( R (E)AKL(¸; E, E )a( (E ) . (34) IK (t)" ?K ?@ @L * 2p
?@ KL Here the indices a and b label the reservoirs and may assume values L or R. The matrix A is de"ned as (E)s (E ) , AKL(¸; E, E )"d d d ! sR *?_KI *@_IL ?@ KL ?* @* I
A discussion of the limitations of Eq. (32) is given in Ref. [10].
(35)
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and s (E) is the element of the scattering matrix relating bK (E) to a( (E). Note that Eq. (34) is *?_KI *K ?I independent of the coordinate z along the lead. 2.3.2. Average current Before we proceed in the next subsection with the calculation of current}current correlations, it is instructive to derive the average current from Eq. (34). For a system at thermal equilibrium the quantum statistical average of the product of an electron creation operator and annihilation operator of a Fermi gas is 1a( R (E)a( (E )2"d d d(E!E ) f (E) . (36) ?K @L ?@ KL ? Using Eqs. (34) and (36) and taking into account the unitarity of the scattering matrix s, we obtain
e dE Tr[tR(E)t(E)][ f (E)!f (E)] . 1I 2" * 0 * 2p
(37)
Here the matrix t is the o!-diagonal block of the scattering matrix (30), t "s . In the KL 0*_KL zero-temperature limit and for a small applied voltage, Eq. (37) gives a conductance e G" Tr[tR(E )t(E )] . $ $ 2p
(38)
Eq. (38) establishes the relation between the scattering matrix evaluated at the Fermi energy and the conductance. It is a basis invariant expression. The matrix tRt can be diagonalized; it has a real set of eigenvalues (transmission probabilities) ¹ (E) (not to be confused with temperature), each L of them assumes a value between zero and one. In the basis of eigen-channels we have instead of Eq. (37)
e 1I 2" dE ¹ (E)[ f (E)!f (E)] . * L * 0 2p
L and thus for the conductance
(39)
e G" ¹ . (40) L 2p
L Eq. (40) is known as a multi-channel generalization of the Landauer formula. Still another version of this result expresses the conductance in terms of the transmission probabilities ¹ ""s " 0* KL 0* KL for carriers incident in channel n in the left lead L and transmitted into channel m in the right lead R. In this basis the Hamiltonians of the left and right lead (the reservoirs) are diagonal and the conductance is given by e G" ¹ . (41) KL 2p
KL We refer to this basis as the natural basis. We remark already here that, independently of the choice of basis, the conductance can be expressed in terms of transmission probabilities only. This is not case for the shot noise to be discussed subsequently. Thus the scattering matrix rather then transmission probabilities represents the fundamental object governing the kinetics of carriers.
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2.3.3. Multi-terminal case We consider now a sample connected by ideal leads to a number of reservoirs labeled by an index a, with the Fermi distribution functions f (E). At a given energy E the lead a supports N (E) ? ? transverse channels. We introduce, as before, creation and annihilation operators of electrons in an incoming a( R , a( and outgoing bK R , bK state of lead a in the transverse channel n. These operators ?L ?L ?L ?L are again related via the scattering matrix. We write down this relation, similar to Eq. (29), in components, bK (E)" s (E)a( (E) . (42) ?K ?@_KL @L @L The matrix s is again unitary, and, in the presence of time-reversal symmetry, symmetric. Proceeding similarly to the derivation presented above, we obtain the multi-terminal generalization of Eq. (34) for the current through the lead a,
e IK (t)" dE dE e #\#YR a( R (E)AKL(a; E, E )a( (E ) , ? @K @A AL 2p
@A KL with the notation
(43)
AKL(a; E, E )"d d d ! sR (E)s (E ) . (44) @A KL ?@ ?A ?@_KI ?A_IL I The signs of currents are chosen to be positive for incoming electrons. Imagine that a voltage < is applied to the reservoir b, that is, the electro-chemical potential is @ k "k#e< , where k can be taken to be the equilibrium chemical potential. From Eq. (43) we "nd @ @ the average current,
Rf e < dE ! [N d !Tr(sR s )] , (45) 1I 2" @ ? ?@ ?@ ?@ ? RE 2p
@ where the trace is taken over channel indices in lead a. As usual, we de"ne the conductance matrix G via G "d1I 2/d< " @ . In the linear regime this gives ?@ ?@ ? @ 4 1I 2" G < ? ?@ @ @
(46)
with
Rf e dE ! [N d !Tr(sR s )] . G " ? ?@ ?@ ?@ ?@ 2p
RE
(47)
The scattering matrix is evaluated at the Fermi energy. Eq. (47) has been successfully applied to a wide range of problems from ballistic transport to the quantum Hall e!ect. 2.3.4. Current conservation, gauge invariance, and reciprocity Any reasonable theory of electron transport must be current-conserving and gauge invariant. Current conservation means that the sum of currents entering the sample from all terminals is equal to zero at each instant of time. For the multi-terminal geometry discussed here this means I "0. ? ?
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The current is taken to be positive if it #ows from the reservoir towards the mesoscopic structure. For the average current in the two-terminal geometry, we have I #I "0. We emphasize that * 0 current conservation must hold not only on the average but at each instant of time. In particular, current conservation must also hold for the #uctuation spectra which we discuss subsequently. In general, for time-dependent currents, we have to consider not only contacts which permit carrier exchange with the conductor, but also other nearby metallic structures, for instance gates, against which the conductor can be polarized. The requirement that the results are gauge invariant means in this context, that no current arises if voltages at all reservoirs are simultaneously shifted by the same value (and no temperature gradient is applied). For the average currents (see Eqs. (39), (47)) both properties are a direct consequence of the unitarity of the scattering matrix. For the conductance matrix G current conservation and gauge invariance require that the ?@ elements of this matrix in each row and in each column add up to zero, G " G "0 . (48) ?@ ?@ ? @ Note that for the two terminal case this implies G,G "G "!G "!G . In the two ** 00 *0 0* terminal case, it is thus su$cient to evaluate one conductance to determine the conductance matrix. In multi-probe samples the number of elements one has to determine to "nd the conductance matrix is given by the constraints (48) and by the fact that the conductance matrix is a susceptibility and obeys the Onsager}Casimir symmetries G (B)"G (!B) . ?@ @? In the scattering approach the Onsager}Casimir symmetries are again a direct consequence of the reciprocity symmetry of the scattering matrix under "eld reversal. In the stationary case, the current conservation and the gauge invariance of the results are a direct consequence of the unitarity of the scattering matrix. In general, for non-linear and non-stationary problems, current conservation and gauge invariance are not automatically ful"lled. Indeed, in ac-transport a direct calculation of average particle currents does not yield a current conserving theory. Only the introduction of displacement currents, determined by the long-range Coulomb interaction, leads to a theory which satis"es these basic requirements. We will discuss these issues for noise problems in Section 3. 2.4. General expressions for noise We are concerned with #uctuations of the current away from their average value. We thus introduce the operators *IK (t),IK (t)!1I 2. We de"ne the correlation function S (t!t ) of the ? ? ? ?@ current in contact a and the current in contact b as S (t!t ),1*IK (t)*IK (t )#*IK (t )*IK (t)2 . ? @ @ ? ?@
(49)
Note that several de"nitions, di!ering by numerical factors, can be found in the literature. The one we use corresponds to the general de"nition of time-dependent #uctuations found in Ref. [11]. We de"ne the Fourier transform with the coe$cient 2 in front of it, then our normalization yields the equilibrium (Nyquist}Johnson) noise S"4k ¹G and is in accordance with Ref. [1], see below. The standard de"nition of Fourier transform would yield the Nyquist}Johnson noise S"2k ¹G. Ref. [9] de"nes the spectral function which is multiplied by the width of the frequency interval where noise is measured.
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Note that in the absence of time-dependent external "elds the correlation function must be a function of only t!t . Its Fourier transform, 2pd(u#u )S (u),1*IK (u)*IK (u )#*IK (u )*IK (u)2 , ?@ ? @ @ ? is sometimes referred to as noise power. To "nd the noise power we need the quantum statistical expectation value of products of four operators a( . For a Fermi gas (or a Bose gas) at equilibrium this expectation value is 1a( R (E )a( (E )a( R (E )a( (E )2!1a( R (E )a( (E )21a( R (E )a( (E )2 ?I @J AK BL ?I @J AK BL " d d d d d(E !E )d(E !E ) f (E )[1Gf (E )] . (50) ?B @A IL KJ ? @ (The upper sign corresponds to Fermi statistics, and the lower sign corresponds to Bose statistics. This convention will be maintained whenever we compare systems with di!ering statistics. It is also understood that for Fermi statistics f (E) is a Fermi distribution and for Bose statistics f (E) is ? ? a Bose distribution function.) Making use of Eq. (43) and of the expectation value (50), we obtain the expression for the noise power [9],
e S (u)" dE AKL(a; E, E# u)ALK(b; E# u, E) ?@ AB BA 2p
AB KL ;+ f (E)[1Gf (E# u)]#[1Gf (E)] f (E# u), . (51) A B A B Note that with respect to frequency, it has the symmetry properties S (u)"S (!u). For ?@ @? arbitrary frequencies and an arbitrary s-matrix, Eq. (51) is neither current conserving nor gauge invariant and additional considerations are needed to obtain a physically meaningful result. In the reminder of this section, we will only be interested in the zero-frequency noise. For the noise power at u"0 we obtain [9]
e dE AKL(a; E, E)ALK(b; E, E) S ,S (0)" AB BA ?@ ?@ 2p
AB KL ;+f (E)[1Gf (E)]#[1Gf (E)] f (E), . (52) A B A B Eqs. (52) are current conserving and gauge invariant. Eq. (52) can now be used to predict the low-frequency noise properties of arbitrary multi-channel and multi-probe, phase-coherent conductors. We "rst elucidate the general properties of this result, and later on analyze it for various physical situations. 2.4.1. Equilibrium noise If the system is in thermal equilibrium at temperature ¹, the distribution functions in all reservoirs coincide and are equal to f (E). Using the property f (1Gf )"!k ¹Rf/RE and employing the unitarity of the scattering matrix, which enables us to write Tr(sR s sR s )"d N ?@ ? ?A ?B @B @A AB
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(where as before the trace is taken over transverse channel indices, and N is the number of ? channels in the lead a), we "nd
ek ¹ Rf S " dE ! [2N d !Tr(sR s #sR s )] . @? @? ?@ ?@ ?@ ? ?@ p
RE
(53)
This is the equilibrium, or Nyquist}Johnson noise. In the approach discussed here it is a consequence of the thermal #uctuations of occupation numbers in the reservoirs. Comparing Eqs. (45) and (53), we see that S "2k ¹(G #G ) . (54) ?@ ?@ @? This is the manifestation of the #uctuation-dissipation theorem: equilibrium #uctuations are proportional to the corresponding generalized susceptibility, in this case to the conductance. For the time-reversal case (no magnetic "eld) the conductance matrix is symmetric, and Eq. (54) takes the form S "4k ¹G , ?@ ?@ which is familiar for the two-terminal case, S"4k ¹G, with G being the conductance. From Eq. (54) we see that the #uctuation spectrum of the mean squared current at a contact a is positive (since G '0) but that the current}current correlations of the #uctuations at di!erent probes are ?? negative (since G (0). The sign of the equilibrium current}current #uctuations is independent ?@ of statistics: Intensity}intensity #uctuations for a system of bosons in which the electron reservoirs are replaced by black-body radiators are also negative. We thus see that equilibrium noise does not provide any information of the system beyond that already known from conductance measurements. Nevertheless, the equilibrium noise is important, if only to calibrate experiments and as a simple test for theoretical discussions. Experimentally, a careful study of thermal noise in a multi-terminal structure (a quantum Hall bar with a constriction) was recently performed by Henny et al. [12]. Within the experimental accuracy, the results agree with the theoretical predictions. 2.4.2. Shot noise: zero temperature We now consider noise in a system of fermions in a transport state. In the zero temperature limit the Fermi distribution in each reservoir is a step function f (E)"h(k !E). Using this we can ? ? rewrite Eq. (52) as
e S " dE Tr[sR s sR s ] + f (E)[1!f (E)]#f (E)[1!f (E)], . (55) ?@ 2p
?A ?B @B @A A B B A A$B We are now prepared to make two general statements. First, correlations of the current at the same lead, S , are positive. This is easy to see, since their signs are determined by positively de"ned ?? quantities Tr[sR s sR s ]. The second statement is that the correlations at diwerent leads, S with ?A ?B ?B ?A ?@
For bosons at zero temperature one needs to take into account Bose condensation e!ects. These quantities are called `noise conductancesa in Ref. [9].
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aOb, are negative. This becomes clear if we use the property s sR "0 and rewrite Eq. (55) as B ?B @B e S "! dE Tr s sR f (E) s sR f (E) . ?@ @A ?A A ?B @B B p
A B The integrand is now positively de"ned. Of course, current conservation implies that if all cross-correlations S are negative for all b di!erent from a, the spectral function S must be ?@ ?? positive. Actually, these statements are even more general. One can prove that cross-correlations in the system of fermions are generally negative at any temperature, see Ref. [9] for details. On the other hand, this is not correct for a system of bosons, where under certain conditions cross-correlations can be positive.
2.4.3. Two-terminal conductors Let us now consider the zero-temperature shot noise of a two-terminal conductor. Again we denote the leads as left (L) and right (R). Due to current conservation, we have S,S " ** S "!S "!S . Utilizing the representation of the scattering matrix (30), and taking into 00 *0 0* account that the unitarity of the matrix s implies rRr#tRt"1, we obtain after some algebra e S" Tr(rRrtRt)e"<" , p
(56)
where the scattering matrix elements are evaluated at the Fermi level. This is the basis invariant relation between the scattering matrix and the shot noise at zero temperature. Like the expression of the conductance, Eq. (38), we can express this result in the basis of eigen-channels with the help of the transmission probabilities ¹ and re#ection probabilities R "1!¹ , L L L e"<" ¹ (1!¹ ) . (57) S " L L ** p
L We see that the non-equilibrium (shot) noise is not simply determined by the conductance of the sample. Instead, it is determined by a sum of products of transmission and re#ection probabilities of the eigen-channels. Only in the limit of low-transparency ¹ ;1 in all eigen-channels is the shot L noise given by the Poisson value, discussed by Schottky, e"<" S " ¹ "2e1I2 . (58) . L p
L It is clear that zero-temperature shot noise is always suppressed in comparison with the Poisson value. In particular, neither closed (¹ "0) nor open (¹ "1) channels contribute to shot L L noise; the maximal contribution comes from channels with ¹ "1/2. The suppression below the L This statement is only valid for non-interacting systems. Interactions may cause instabilities in the system, driving the noise to super-Poissonian values. Noise in systems with multi-stable current}voltage characteristics (caused, for example, by a non-trivial structure of the energy bands, like in the Esaki diode) may also be super-Poissonian. These features are discussed in Section 5.
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Poissonian limit given by Eq. (58) was one of the aspects of noise in mesoscopic systems which triggered many of the subsequent theoretical and experimental works. A convenient measure of sub-Poissonian shot noise is the Fano factor F which is the ratio of the actual shot noise and the Poisson noise that would be measured if the system produced noise due to single independent electrons, F"S /S . ** . For energy-independent transmission and/or in the linear regime the Fano factor is
(59)
¹ (1!¹ ) L . (60) F" L L ¹ L L The Fano factor assumes values between zero (all channels are transparent) and one (Poissonian noise). In particular, for one channel it becomes (1!¹). Unlike the conductance, which can be expressed in terms of (transmission) probabilities independent of the choice of basis, the shot noise, even for the two terminal conductors considered here, cannot be expressed in terms of probabilities. The trace of Eq. (56) is a sum over k, l, m, n of terms rH r tH t , which by themselves are not real-valued if mOn (in contrast to Eq. (41) for the IL IK JK JL conductance). This is a signature that carriers from di!erent quantum channels interfere and must remain indistinguishable. It is very interesting to examine whether it is possible to "nd experimental arrangements which directly probe such exchange interference e!ects, and we return to this question later on. In the remaining part of this subsection we will use the eigen-channel basis which o!ers the most compact representation of the results. The general result for the noise power of the current #uctuations in a two-terminal conductor is
e (61) S" dE+¹ (E)[ f (1Gf )#f (1Gf )]$ ¹ (E)[1!¹ (E)]( f !f ), . L * * 0 0 L L * 0 p
L Here the "rst two terms are the equilibrium noise contributions, and the third term, which changes sign if we change statistics from fermions to bosons, is the non-equilibrium or shot noise contribution to the power spectrum. Note that this term is second order in the distribution function. At high energies, in the range where both the Fermi and Bose distribution function are well approximated by a Maxwell}Boltzmann distribution, it is negligible compared to the equilibrium noise described by the "rst two terms. According to Eq. (61) the shot noise term enhances the noise power compared to the equilibrium noise for fermions but diminishes the noise power for bosons. In the practically important case, when the scale of the energy dependence of transmission coe$cients ¹ (E) is much larger than both the temperature and applied voltage, these quantities in L Eq. (61) may be replaced by their values taken at the Fermi energy. We obtain then (only fermions are considered henceforth)
e< e 2k ¹ ¹#e
L L where < is again the voltage applied between the left and right reservoirs. The full noise is a complicated function of temperature and applied voltage rather than a simple superposition of
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equilibrium and shot noise. For low voltages e<;k ¹ we obtain S"4k ¹G, in accordance with the general result (54). Note that, since coth x'1/x for any x'0, the actual noise (62) for any voltage is higher than the equilibrium noise. This is not generally correct if the transmission coe$cients are strongly energy dependent. As pointed out by Lesovik and Loosen [13], in certain situations (for instance, when the transmission coe$cients sharply peak as functions of energy) the total non-equilibrium noise may be actually lower than the equilibrium noise at the same temperature. We conclude this subsection with some historical remarks. Already Kulik and Omel'yanchuk [14] noticed that the shot noise in ballistic contacts (modeled as an ori"ce in an insulating layer between two metallic reservoirs) vanishes if there is no elastic impurity scattering. Subsequently, Khlus [15] considered such a point contact with elastic scattering and derived Eq. (62) by means of a Keldysh Green's function technique. The papers by Kulik and Omel'yanchuk and by Khlus remained unknown, they were either not or only poorly cited even in the Russian literature. Later Lesovik [16] derived Eq. (61) in the framework of the scattering approach (for the case of fermions). Independently, Yurke and Kochanski [17] investigated the momentum noise of a tunneling microscopic tip, also based on the scattering approach, treating only the one-channel case; Ref. [16] treated the multi-channel case, but assumed at the outset that the scattering matrix is diagonal and that the diagonal channels are independent. A generalization for many-channel conductors described by an arbitrary scattering matrix (without assumption of independence) and for the many-terminal case was given in Refs. [18,19]. The same results were later discussed by Landauer and Martin [20] and Martin and Landauer [21] appealing to wave packets. The treatment of wave-packet overlap (see Section 2.2) is avoided by assuming that wave packets are identical and timed to arrive at the same instant. Ref. [9], which we followed in this subsection, is a long version of the papers [18,19]. 2.5. Voltage yuctuations 2.5.1. Role of external circuit Thus far all the results which we have presented are based on the assumption that the sample is part of an external circuit with zero impedance. In this case the voltage (voltages) applied to the sample can be viewed to be a "xed non-#uctuating quantity and the noise properties are determined by the current correlations which we have discussed. The idealized notion of a zeroimpedance external circuit does often not apply. Fig. 3 shows a simple example: The sample S is part of an electrical circuit with resistance R and a voltage source which generates a voltage < . (In general, the external circuit is described by a frequency-dependent impedance Z (u).) As a consequence, in such a circuit, we deal with both current #uctuations and voltage #uctuations. The current #uctuations through the sample are now governed by the #uctuations *<(t) of the voltage across the sample, which generate a #uctuating current *I (u)"G(u)*<(u), where G(u) is 4 frequency-dependent conductance (admittance) of the sample. In addition, there is the contribution
The full noise can be divided into equilibrium-like and transport parts, see Ref. [9]. This division is, of course, arbitrary.
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Fig. 3. Noise measurements in an external circuit. The sample is denoted by S.
of the spontaneous current #uctuations dI(u) themselves. The total #uctuating current through the sample is thus given by *I(u)"G(u)*<(u)#dI(u) .
(63)
Eq. (63) has the form of a Langevin equation with a #uctuating source term given by the spontaneous current #uctuations determined by the noise power spectrum S of a two terminal conductor. To complete these equations we must now relate the current through the sample to the external voltage. The total current I is related to the external voltage < and the voltage across the sample by the Kircho! law <#R I"< . Here < is a constant, and the voltage and current #uctuations are thus related by *<#R *I"0, or *<(u)"!R *I(u) . (64) For R "0 (zero external impedance) we have the case of a voltage controlled external circuit, while for R PR (in"nite external impedance) we have the case of a current-controlled external circuit. The Langevin approach assumes that the mesoscopic sample and the external circuit can be treated as separate entities, each of which might be governed by quantum e!ects, but that there are no phase coherent e!ects which would require the treatment of the sample and the circuit as one quantum mechanical entity. In such a case the distinction between sample and external circuit would presumably be meaningless. Eliminating the voltage #uctuations in Eqs. (63) and (64) gives for the current #uctuations *I(u)(1#G(u)R )"dI(u) and with the resistance of the sample R"1/G(0) and the noise power spectrum S we obtain in the zero-frequency limit S . S " '' (1#R /R)
(65)
The quantity S is de"ned by Eq. (49), as before. The quantity S is de"ned by the same expression where current '' 44 #uctuations *I are replaced by the voltage #uctuations *<.
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Eq. (65) shows that the external impedance becomes important if it is comparable or larger than the resistance of the sample. Eliminating the current we obtain (1/R #G(u))*<(u)"!dI(u), and thus a voltage #uctuation spectrum given by S . (66) S " 44 (1/R #1/R) At equilibrium, where the current noise power is given by S"4k ¹G, Eq. (66) gives S " 44 4k ¹/R(1/R #1/R) which reduces in the limit R PR to the familiar Johnson}Nyquist result S "4k ¹R for the voltage #uctuations in an in"nite external impedance circuit. The procedure 44 described above can also be applied to shot noise as long as we are only concerned with e!ects linear in the voltage <. Far from equilibrium, this approach applies if we replace the conductance (resistance) by the di!erential conductance (resistance) and if a linear #uctuation theory is su$cient. 2.5.2. External circuit: multi-probe conductors For a multi-probe geometry the consideration of the external circuit is similarly based on the Langevin equation [9], *I " G *< #dI , (67) ? ?@ @ ? @ where G is an element of the conductance matrix and dI is a #uctuating current with the noise ?@ ? power spectrum S . The external circuit loops connecting to a multi-probe conductor can have ?@ di!erent impedances: the external impedance is thus also represented by a matrix which connects voltages and currents at the contacts of the multi-probe conductor. Ideally, the current source and sink contacts are connected to a zero impedance external circuit, whereas the voltage probes are connected to an external loop with in"nite impedance. At equilibrium, in the limiting case that all probes are connected to in"nite external impedance loops, the voltage #uctuations can be expressed in terms of multi-probe resistances. Consider "rst a four-probe conductor. A four-probe resistance is obtained by injecting current in contact a taking it out at contact b and using two additional contacts c and d to measure the voltage di!erence < !< . The four-probe resistance is de"ned as R "(< !< )/I. Using the conductance A B ?@ AB A B matrix of a four-probe conductor, a little algebra shows that R "D\(G G !G G ) , (68) ?@ AB A? B@ A@ B? where D is any sub-determinant of rank three of the conductance matrix. (Due to current conservation and gauge invariance all possible subdeterminants of rank three of the conductance matrix are identical and are even functions of the applied magnetic "eld). Eqs. (68) can be applied to a conductor with any number of contacts larger than four, since the conductance matrix of any dimension can be reduced to a conductance matrix of dimension four, if the additional contacts not involved in the measurement are taken to be connected to in"nite external impedance loops. Similarly, there exists an e!ective conductance matrix of dimension three which permits to de"ne a three-probe measurement. In such a measurement one of the voltages is measured at the current source contact or current sink contact and thus two of the indices in R are identical, a"c or ?@ AB b"d. Finally, if the conductance matrix is reduced to a 2;2 matrix, we obtain a resistance for
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which two pairs of indices are identical, R or R "!R . With these resistances we can ?@ ?@ ?@ @? ?@ ?@ now generalize the familiar Johnson}Nyquist relation S "4k ¹R for two-probe conductors, to 44 the case of a multi-probe conductor. For the correlation of a voltage di!erence < !< measured ? @ between contacts a and b with a voltage #uctuation < !< measured between contacts c and d, A B Eq. (67) leads to 1(< !< )(< !< )2"2k ¹(R #R ). (69) ? @ A B ?@ AB AB ?@ The mean-squared voltage #uctuations a"c and b"d are determined by the two-terminal resistances R of the multi-probe conductor. The correlations of voltage #uctuations (in the case ?@ ?@ when all four indices di!er) are related to symmetrized four-probe resistances. If shot noise is generated, for instance, by a current incident at contact a and taken out at contact b (in a zero external impedance loop) and with all other contacts connected to an in"nite impedance circuit, the voltage #uctuations are [22] 1(d< !d< )(d< !d< )2" R R S , (70) A B C D ?@ CD E@ AB ?E ?E where S is the noise power spectrum of the current correlations at contacts a and g, and b is an ?E arbitrary index. These examples demonstrate that the #uctuations in a conductor are in general a complicated expression of the noise power spectrum determined for the zero-impedance case, the resistances (or far from equilibrium the di!erential resistances) and the external impedance (matrix). These considerations are of importance since in experiments it is the voltage #uctuations which are actually measured and which eventually are converted to current #uctuations. 2.6. Applications In this subsection, we give some simple applications of the general formulae derived above, and illustrate them with experimental results. We consider only the zero-frequency limit. As we explained in the Introduction, we do not intend to give here a review of all results concerning a speci"c system. Instead, we focus on the application of the scattering approach. For results derived for these systems with other methods, the reader is addressed to Table 1. 2.6.1. Tunnel barriers For a tunnel barrier, which can be realized, for example, as a layer of insulator separating two normal metal electrodes, all the transmission coe$cients ¹ are small, ¹ ;1 for any n. Separating L L terms linear in ¹ in Eq. (62) and taking into account the de"nition of the Poisson noise, Eq. (58), L we obtain
e< e"<" e< coth ¹ "coth S . (71) S" L 2k ¹ 2k ¹ . p
L At a given temperature, Eq. (71) describes the crossover from thermal noise at voltages e"<";k ¹ to shot noise at voltages e"<"
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Eq. (71) is also obtained in the zero-frequency, zero charging energy limit of microscopic theories of low transparency normal tunnel junctions or Josephson junctions [23}28]. These theories employ the tunneling Hamiltonian approach and typically only keep the terms of lowest nonvanishing order in the tunneling amplitude. Poissonian shot noise was measured experimentally in semiconductor diodes, see e.g. Ref. [29]; these devices, however, could hardly be called mesoscopic, and it is not always easy to separate various sources of noise. More recently, Birk et al. [30] presented measurements of noise in a tunnel barrier formed between an STM tip and a metallic surface. Speci"cally addressing the crossover between the thermal and shot noise, they found an excellent agreement with Eq. (71). Their experimental results are shown in Fig. 4. 2.6.2. Quantum point contacts A point contact is usually de"ned as a constriction between two metallic reservoirs. Experimentally, it is typically realized by depleting of a two-dimensional electron gas formed with the help of a number of gates. Changing the gate voltage < leads to the variation of the width of the channel, and consequently of the electron concentration. All the sizes of the constriction are assumed to be shorter than the mean-free path due to any type of scattering, and thus transport through the point contact is ballistic. In a quantum point contact the width of the constriction is comparable to the Fermi wavelength. Quantum point contacts have drawn wide attention after experimental investigations [31,32] showed steps in the dependence of the conductance on the gate voltage. This stepwise dependence is illustrated in Fig. 5, curve 1. An explanation was provided by Glazman et al. [33], who modeled the quantum point contact as a ballistic channel between two in"nitely high potential walls
Fig. 4. Crossover from thermal to shot noise measured by Birk et al. [30]. Solid curves correspond to Eq. (71); triangles show experimental data for the two samples, with lower (a) and higher (b) resistance. Copyright 1995 by the American Physical Society. Fig. 5. Conductance in units of e/2p (curve 1) and zero-frequency shot noise power in units of e"<"/6p (curve 2) for a quantum point contact with u "4u as a function of the gate voltage. Here m"(E !< )/ u is a dimensionless W V $ V energy.
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Fig. 6. Geometry of the quantum point contact in the hard-wall model (a) and the e!ective potential for one-dimensional motion (b). Fig. 7. Conductance (upper plot) and shot noise (lower plot) as functions of the gate voltage, as measured by Reznikov et al. [42]. Di!erent curves correspond to "ve di!erent bias voltages. Copyright 1995 by the American Physical Society.
(Fig. 6a). If the distance between the walls d(x) (width of the contact) is changing slowly in comparison with the wavelength, transverse and longitudinal motion can be approximately separated. The problem is then e!ectively reduced to one-dimensional motion in the adiabatic potential ;(x)"pn /2md(x), which depends on the width pro"le and the transverse channel number n. Changing the gate voltage leads to the modi"cation of the potential pro"le. Theoretically it is easier to "x the geometry of the sample, i.e. the form of the potential, and vary the Fermi energy in the channel E (Fig. 6b). The external potential is smooth, and therefore may be treated $ semi-classically. This means that the channels with n(k d /p (here k ,(2mE ) , and d is $ $ $
the minimal width of the contact) are open and transparent, ¹ "1, while the others are closed, L ¹ "0. The conductance (40) is proportional to the number of open channels and therefore L exhibits plateaus as a function of the gate voltage. At the plateaus, shot noise is equal to zero, since all the channels are either open or closed. The semi-classical description fails when the Fermi energy lies close to the top of the potential in one of the transverse channels. Then the transmission coe$cient for this channel increases from zero to one due to quantum tunneling through the barrier and quantum re#ection at the barrier. The transition from one plateau to the next is associated with a spike in the shot noise as we will now discuss. A more realistic description of the quantum point contact takes into account that the potential in the transverse direction y is smooth [34]. The constriction can then be thought of as a bottleneck with an electrostatic potential of the form of a saddle. Quite generally, the potential can be expanded in the directions away from the center of the constriction, <(x, y)"< !mux#muy , W V where the constant < denotes the potential at the saddle point. Experimentally it is a function of the gate voltage. The transmission probabilities are given by Ref. [34], ¹ (E)"[1#exp(!ne )]\, L L
e ,2[E! u (n#)!< ]/ u . V L W
(72)
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The transmission probability ¹ (E) exhibits a crossover from zero to one as the energy E passes the L value < # u (n#1/2). The resulting zero temperature shot noise as a function of < , using W Eq. (57), is illustrated in Fig. 5 for the case u "4u (curve 2). The conductance of this quantum W V point contact is shown in Fig. 5 as curve 1. As expected, the shot noise dependence is a set of identical spikes between the plateaus. The height of each spike is e"<"/4p up to exponential accuracy. At the plateaus shot noise is exponentially suppressed. This behavior of shot noise in a quantum point contact was predicted by Lesovik [16]. The shot noise of a saddle point model of a quantum point contact was presented in Ref. [1]. Scherbakov et al. [35] thoroughly analyze and compare shot noise for various models of quantum point contacts. Using a classical (master equation) approach, shot noise suppression was also con"rmed by Chen and Ying [36]. If a magnetic "eld is applied in the transverse direction, the energies e are pushed up. Shot noise L is thus an oscillating function of the magnetic "eld for a "xed gate voltage. Strong magnetic "elds may even drive the quantum point contact to the regime E (e , suppressing the shot noise $ completely [35]. Using Eq. (61), it is easy also to study shot noise in the non-linear regime as a function of the applied bias voltage <. We obtain S+(eu /(2p))N , N "e"<"/ u , (73) V 4 4 W where N is the number of channels which open in the energy interval between zero and e"<". 4 Eq. (73) applies when this number is large, N <1. For even higher voltages e"<"'< , the noise 4 becomes voltage independent, as found by Larkin and Reznikov [37]. They also discuss selfconsistent interactions and found that the non-linear shot noise is suppressed as compared to the non-interacting value. As the number of open channels becomes large, so that the width of the constriction is much wider than the Fermi wavelength (classical point contact), the shot noise stays the same, while the conductance grows proportional to the number of channels. Thus, shot noise becomes small in comparison with the Poisson value (at the top of the nth spike this suppression equals (4n)\), and in this sense shot vanishes for a classical point contact, as found by Kulik and Omel'yanchuk [14]. Note, however, that the shot noise really disappears only when inelastic scattering becomes signi"cant (see Section 2.7). Experimentally, sub-Poissonian shot noise suppression was observed by Akimenko et al. [38] (see their Fig. 9) in a slightly di!erent system, a metallic quantum point contact, which is essentially an ori"ce in a thin insulating layer between two metallic reservoirs. In this system, however, it is di$cult to separate di!erent sources of noise. In ballistic quantum point contacts sub-Poisson suppression was observed in an early experiment by Li et al. [39], and later by Dekker et al. [40,41]. Reznikov et al. [42] found clearly formed peaks in the shot noise as a function of the gate voltage. A considerable improvement in the experimental technique was obtained by measuring noise in the MHz range at frequencies far above the range where 1/f-noise contributes. The results of Reznikov et al. [42] are shown in Fig. 7. Compared to the theory the experimental noise peaks exhibit a slight asymmetry around the transition point. Kumar et al. [43] developed a di!erent low-frequency technique based on voltage correlation measurements to "lter out unwanted noise. They found that `the agreement 1of experimental results2 with theoretical expectations, within the calculable statistical deviations, is nearly perfecta. Recently, van den Brom and van Ruitenbeek [44] demonstrated that shot noise measurements can be used to extract information on the transmission
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Fig. 8. Resonant double barriers. The case of low voltage is illustrated; resonant levels inside the well are indicated by dashed lines.
probabilities of the eigen-channels of nanoscopic metallic point contacts. Subsequently, BuK rki and Sta!ord [45] were able to reproduce their results quantitatively based on a simple theoretical model which takes into account only two features of the contacts, the con"nement of electrons and the coherent backscattering from imperfections. 2.6.3. Resonant tunnel barriers The transport through two consecutive tunnel barriers allows already to discuss many aspects of shot noise suppression. Let us "rst consider the case of purely one-dimensional electron motion through two potential barriers with transmission probabilities ¹ and ¹ , separated * 0 by a distance w, as shown in Fig. 8. Eventually, we will assume that the transmission of each barrier is low, ¹ ;1 and ¹ ;1. An exact expression for the transmission coe$cient of the whole * 0 structure is ¹ ¹ * 0 , ¹(E)" 1#(1!¹ )(1!¹ )!2(1!¹ ) (1!¹ ) cos (E) * 0 * 0
(74)
with (E) being the phase accumulated during motion between the barriers; in our particular case
(E)"2w(2mE) / . Eq. (74) has a set of maxima at the resonant energies EP such that the phase L
(EP ) equals 2pn. Expanding the function (E) around EP , and neglecting the energy dependence of L L the transmission coe$cients, we obtain the Breit}Wigner formula [46,47] C/4 L , ¹(E)"¹ L (E!EP )#C/4 L L
4C C ¹ " *L 0L . L C L
(75)
¹ is the maximal transmission probability at resonance. We have introduced the partial decay L widths C " l ¹ . The attempt frequency l of the nth resonant level is given by *L 0L L * 0 L l\"( /2)(d /dEP )"w/v , v "(2EP /m) . C ,C #C is the total decay width of the resonL L L L L L *L 0L ant level. Eq. (75) is, strictly speaking, only valid when the energy E is close to one of the resonant
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energies EP . In many situations, however, the Lorentz tails of ¹(E) far from the resonances are L not important, and one can write C/4 L . ¹(E)" ¹ L (E!EP )#C/4 L L L Despite the fact that the transparencies of both barriers are low, we see that the total transmission coe$cient shows sharp peaks around resonant energies. This e!ect is a consequence of constructive interference and is known as resonant tunneling. The transmission coe$cient at the top of a peak equals ¹ ; for a symmetric resonance C "C the transmission is ideal, ¹ "1. This L *L 0L L dependence may be probed by applying a gate voltage. The gate voltage moves the positions of the resonant levels, and the conductance exhibits peaks around each resonance. In the linear regime the shot noise is determined by the transmission coe$cient evaluated at the Fermi level, and is thus an oscillating function of the gate voltage, vanishing almost completely between the peaks. The Fano factor (60) at the top of each peak is equal to F"(C !C )/C. *L 0L L It vanishes for a symmetric barrier. For a resonance with ¹ '1/2 the Fano factor reaches L a maximum each time when the transmission probability passes through ¹"1/2; for a resonance with ¹ (1/2 the shot noise is maximal at resonance. L One-dimensional problem, non-linear regime. For arbitrary voltage, direct evaluation of the expressions (39) and (61) gives an average current, e ,4 C C I" *L 0L , C
L L and a zero-temperature shot noise,
(76)
2e ,4 C C (C #C ) 0L . *L 0L *L (77) S" C
L L Here N is the number of resonant levels in the energy strip e"<" between the chemical potentials of 4 the left and right reservoirs. Eqs. (76) and (77) are only valid when this number is well de"ned } the energy di!erence between any resonant level and the chemical potential of any reservoir must be much greater than C. Under this condition both the current and the shot noise are independent of the applied voltage. The dependence of both the current and the shot noise on the bias voltage < is thus a set of plateaus, the height of each plateau being proportional to the number of resonant levels through which transmission is possible. Outside this regime, when one of the resonant levels is close to the chemical potential of left and/or right reservoir, a smooth transition with a width of order C from one plateau to the next occurs. Consider now for a moment, a structure with a single resonance. If the applied voltage is large enough, such that the resonance is between the Fermi level of the source contact and that of the sink contact, the Fano factor is F"(C #C )/C . * 0 We assume that the resonances are well separated, C; /2mw. For discussion of experimental realizations, see below.
(78)
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It varies between 1/2 (symmetric barrier) and 1 (very asymmetric barrier). Expression (78) was obtained by Chen and Ting [48] using a non-equilibrium Green's functions technique, and independently in Ref. [19] using the scattering approach. It was con"rmed in Monte Carlo simulations performed by Reklaitis and Reggiani [49,50]. If the width of the resonance is comparable to the applied voltage, Eq. (78) has to be supplemented by correction terms due to the Lorentz tails of the Breit}Wigner formula, as found in Ref. [19] and later by Averin [51]. It is also worthwhile to point out that our quantum-mechanical derivation assumes that the electron preserves full quantum coherence during the tunneling process (coherent tunneling model). Another limiting case occurs when the electron completely loses phase coherence once it is inside the well (sequential tunneling model). This latter situation can be described both classically (usually, by means of a master equation) and quantum-mechanically (e.g., by connecting to the well one or several "ctitious voltage probes which serve as `dephasinga leads). These issues are addressed in Section 5, where we show that the result for the Fano factor, Eq. (78), remains independent of whether we deal with a coherent process or a fully incoherent process. The Fano factor Eq. (78) is thus insensitive to dephasing. Quantum wells. The double-barrier problem is also relevant for quantum wells, which are two- or three-dimensional structures, consisting of two planar (linear in two dimensions) potential barriers. Of interest is transport in the direction perpendicular to the barriers (across the quantum well, axis z). These systems have drawn attention already in the 1970s, when resonant tunneling was investigated both theoretically [52] and experimentally [53]. If the area of the barriers (in the plane xy) A is very large, the summation over the transverse channels in Eqs. (39), (61) can be replaced by integration, and we obtain for the average current
el A dE dE ¹(E )+ f (E #E )!f (E #E ), I" , X X * X , 0 X , 2p
and the shot noise
el A S" dE dE ¹(E )[1!¹(E )]+ f (E #E )!f (E #E ), , , X X X * X , 0 X , p
with l "m/2p the density of states of the two-dimensional electron gas (per spin). The key point is that the transmission coe$cient depends only on the energy of the longitudinal motion E , X and thus is given by the solution of the one-dimensional double-barrier problem, discussed above. Denoting k "E #e<, k "E , and integrating over dE , we write (the temperature is set to * $ 0 $ , zero)
#$ #$ >C4 el A dE ¹(E )# e< dE (E #e
$ #
For simplicity, we use a three-dimensional notation. Specialization to two dimensions is trivial. Along the current, not along the well.
(79)
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and
#$ #$ >C4 el A dE ¹(E )[1!¹(E )]# e< dE (E #e
#$ (80) Expressions (79) and (80) are valid in the linear and non-linear regimes, provided interactions are not important. We will consider the noise in the non-linear regime in Section 5, where it will be shown that e!ects of charging of the well may play an important role. Here, specializing on the regime linear in the bias voltage <, we obtain a current 1I2"G< determined by the conductance el AN C C $ * 0 , G"
C
(81)
and a shot noise power S"2e
(C #C ) * 0 1I2 . C
(82)
Here N is the number of resonant states in the one-dimensional problem, which lie below E . $ $ Eqs. (81) and (82) are valid only when the distance between all resonant levels and the Fermi level is much greater than C. This dependence may be probed again, like in a one-dimensional structure, with the help of a gate. Both the current and the shot noise exhibit plateaus as a function of gate voltage; these plateaus are smoothly joined over a width of order C. This dependence resembles that of the non-linear one-dimensional regime, but it clearly is a consequence of di!erent physics. Nevertheless, in the plateau regime the Fano factor is the same, Eq. (78). This fact was noted by Davies et al. [54], who presented both quantum and classical derivations of this result. Classical theories of shot noise suppression in quantum wells are discussed in Section 5. Averaging. Another point of view was taken by Melsen and Beenakker [55] and independently by Melnikov [56], who investigated not a single resonant tunneling structure but an ensemble of systems. Imagine an ensemble of quasi-one-dimensional double-barrier systems, in which some parameter is random. For de"niteness, we assume that the systems are subject to a random gate voltage. Then in some of them the Fermi level is close to one of the resonant energies, and in others it lies between two resonant levels. Therefore, on average, shot noise must be "nite even in the linear regime. To quantify this argument, we turn to the exact expression (74) for the transmission coe$cient and assume that the phase is a random variable, uniformly distributed on the interval (0, 2p). We only consider the linear regime. First, we calculate the conductance,
eN ¹ ¹ eN p e * 0 , ¹( ) d " ¹ " G" L (2p)
2p ¹ #¹ 2p
* 0 L
The Breit}Wigner formula (75) cannot be used for this purpose, since it is not exact far from the resonance.
(83)
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where N is the number of transverse channels, and we have taken into account ¹ ;1, ¹ ;1. * 0 The same calculation for the shot noise yields [56,1]
e"<"N ¹ ¹ (¹ #¹ ) e"<"N p * 0 * 0 . ¹( )[1!¹( )] d " (84) S" p
(¹ #¹ ) 2p
* 0 The Fano factor is given again by Eq. (78). We can learn two lessons from this simple model. First, the Fano factor (78) F"(C #C )/C * 0 appears each time when there is some kind of averaging in the system which involves onedimensional motion across two barriers. In the two examples we considered, the one-dimensional non-linear problem, and the quantum well in the linear regime, this averaging is provided by the summation over all the levels between the chemical potentials of the two reservoirs (one dimension), or the summation over the transverse channels at the given total energy (quantum well). Thus, both problems prove to be self-averaging. At the same time, this averaging is absent in the one-dimensional linear problem, and the Fano factor has nothing to do with Eq. (78) even between the resonances. The second lesson is provided by the distribution function of the transmission coe$cients of the eigen-channels in the one-dimensional problem [55,56]. Without giving details, we mention only that it has a bimodal form. The transmission coe$cients assume values between ¹ "¹ ¹ /4;1 and ¹ ; those close to ¹ and ¹ have higher probability than those
* 0
lying in between. The Fano factor is very sensitive to the appearance of transmission coe$cients close to one, since it is these values which cause the sub-Poisson suppression. Thus, for a symmetric barrier ¹ "1, and the probability to "nd the transmission coe$cient close to 1 is high. This
yields the lowest possible Fano factor which is possible in this situation. We will return in more detail to the distribution of transmission probabilities for metallic di!usive conductors and chaotic cavities. Related work. Here we mention brie#y additional theoretical results on noise in double-barrier and similar structures. Runge [57] investigates noise in double-barrier quantum wells, allowing for elastic scattering inside the well. He employs a non-equilibrium Green's function technique and a coherent potential approximation, and arrives at rather cumbersome expressions for the average current and noise power. In the limit of zero temperature, however, his results yield the same Fano factor (78), despite the fact that both current and noise are sensitive to impurity scattering. Lund B+ and Galperin [58,59] consider a resonant quantum well in a strong magnetic "eld perpendicular to the interfaces (along the axis z). They "nd that the shot noise power (in the linear and non-linear regimes, but without charging e!ects taken into account) shows peaks each time when the new Landau level in the well crosses the chemical potential in the right reservoir. Xiong [60] analyzes noise in superlattices of "nite size (several consecutive barriers) using the transfer matrix method. His numerical results clearly show shot noise suppression with respect to the Poisson value, but, unfortunately, the Fano factor is not plotted. Resonant tunneling through localized states. The following problem was discussed by Nazarov and Struben [61]. Consider now non-linear transport through one, one-dimensional, symmetric
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barrier, situated in the region !w/2(z(w/2. We assume that there are resonant states which are randomly distributed inside the barrier and strongly localized. Applying the model suggested for this situation by Larkin and Matveev [62], we assume that these resonant states are provided by impurities inside the barrier; the localization radius of each state is denoted by m, m;w. Transition rates are exponentially sensitive to the position of these impurities inside the barrier. In the regime of low impurity concentration, only those situated close to the center of the barrier contribute to the transport properties. Thus, the problem is e!ectively mapped onto a doublebarrier problem, where the impurity region near the center of the barrier serves as a potential well, and is separated by two `barriersa from the left and right reservoirs. The tunneling rates through these `barriersa to a resonant state depend on the position z of the impurity which provides this resonant state. We have [62] C (z)"C exp[$z/m], "z"(w/2 . * 0 We have assumed that this amplitude is energy independent, and that impurities are uniformly distributed in energy and space. Thus, our expressions (76) and (77) hold, and must be averaged over impurity con"gurations. We write
dz e I" n C 2 cosh(z/m)
\ and
2e cosh(2z/m) S" n C , dz
4 cosh(z/m) \ where n is the spatial concentration of impurities. By extending the integration to in"nity, we have taken into account m;w. Performing the average and calculating the Fano factor, we "nd [61] F" , which is markedly di!erent form the usual double-barrier suppression F". The model can be generalized to include Coulomb correlations [63]. Imagine that each resonant center has two degenerate electron states available for tunneling, corresponding to two di!erent spin states. However, if one of the states is "lled, the other one is shifted up by the Coulomb energy ;. We assume that the Coulomb energy is very large, so that once one electron has tunneled, the tunneling of the second one is suppressed. Then the e!ective tunneling rate through the `lefta barrier is 2C (we assume that voltage is applied from left to the right), and * instead of Eqs. (76) and (77) we write for the current [63] and shot noise power per spin [61] C C 2eN C C (4C #C ) eN * 0 , S" * 0 . 4 * 0 I" 4 (2C #C )
2C #C * 0 * 0 Averaging over impurity con"gurations and calculating the Fano factor, we "nd again F" [61]. This example concerns interacting systems, and is included in this section only as an exception.
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Experiments. The simplest experimental system one can imagine which should exhibit the features of a two-barrier structure is just a one-dimensional channel constrained by two potential barriers. If the barriers are close to each other, the region between the two barriers can be considered as a zero-dimensional system and is called a quantum dot. In addition, one usually places one more electrode (gate), which couples only capacitively to the dot. Roughly speaking, the voltage applied to the gate shifts all electron levels in the dot with respect to the chemical potential of the reservoirs, and may tune them to the resonance position. However, typically quantum dots are so small that Coulomb interaction e!ects (Coulomb blockade) become important, and the theoretical picture described above is no longer valid. If the space between the barriers is large and one-dimensional (one channel), interaction e!ects are also important, and a Luttinger liquid state is formed. For a more extensive discussion of noise in interacting systems, the reader is addressed to Section 7. Quantum wells, however, are macroscopic objects, and hence are less sensitive to interactions. Thus, experiments carried out on quantum wells may probe the non-interacting theory of noise suppression in a double-barrier system. Sub-Poissonian shot noise suppression in quantum wells was observed by Li et al. [64] even before a theory of this suppression was available. Li et al. noted that the suppression is maximal for symmetric barriers, and is insigni"cant for very asymmetric structures (Fig. 9). This suppression was later observed by van de Roer et al. [65], Ciambrone et al. [66], Liu et al. [67], and Przadka et al. [68]. Liu et al. compared their experimental data with the results of numerical simulations attempting to take into account speci"c features of their sample, and found that theory and experiment are in a reasonable agreement. Yau et al. [69] observed shot noise suppression in double quantum wells (triple barrier structures). We should note, however, that in all experimental data available, the Fano factor depends considerably on the applied voltage in the whole range of voltages. Apparently, this happens because already relatively low voltages drive the system out of the linear regime. To the best of our knowledge, this issue has not been addressed systematically, although some results, especially concerning the negative di!erential resistance range exist. They are summarized in Section 5. 2.6.4. Metallic diwusive wires 1/3-suppression. We consider now transport in multi-channel di!usive wires in the metallic regime. This means that, on one hand, the length of the wire ¸ is much longer than the meanfree path l due to disorder. On the other hand, in a quasi-one-dimensional geometry all electron states are localized in the presence of arbitrarily weak disorder; the localization length equals ¸ "N l, where N is the number of transverse channels. Thus for a wire to be metallic K , , we must have ¸;¸ (which of course implies N <1). As everywhere so far, we ignore inelastic K , processes. Comparison between the Drude}Sommerfeld formula for conductance, 2 enq w (85) G" p m ¸ We use below two-dimensional notations: for a strip of width w the number of transverse channels is equal to N "p w/p . All results expressed through N remain valid also for a three-dimensional (wire) geometry. , $ , The factor 2/p which might look unusual to some readers only re#ects a di!erent de"nition of the mean-free path, and is not essential for any results which we describe below.
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Fig. 9. The Fano factor observed experimentally by Li et al. [64] as a function of current for three quantum wells, which di!er by their asymmetry. The solid line represents the Poisson shot noise value. Copyright 1990 by the American Physical Society.
(n is the electron concentration, and q"l/v is the momentum relaxation time), and the Landauer $ formula (40) yields the expression for the average transmission coezcient, 1¹2"l/¸ .
(86)
In the di!usive regime we have 1¹2;1. A naive point of view would be to assume that all the transmission coe$cients of the wire are of the order of the average transmission eigenvalue 1¹2 and thus, that all transmission probabilities are small. From our previous consideration it would then follow that the Fano factor is very close to one: a metallic di!usive wire would exhibit full Poissonian shot noise. On the other hand, it is well known that a macroscopic metallic conductor exhibits no shot noise. Using this information as a guide one might equally naively expect that a mesoscopic metallic di!usive conductor also exhibits no shot noise. In fact, these naive assumptions are incorrect. In particular, the fact that the transmission eigenvalues of a metallic conductor are not all small has long been recognized: In the metallic regime, for any energy open channels (with ¹&1) coexist with closed ones (¹;1). The distribution function of transmission coe$cients has in fact a bimodal form. This bimodal distribution leads to sub-Poissonian shot noise. Quantitatively, this situation can be described by random matrix theory of one-dimensional transport. It implies [73] that the channel-dependent inverse localization lengths f , related to the transmission coe$cients by ¹ "cosh\(¸/f ), are uniformly L L L distributed between 0 and l\. This statement can be transformed into the following expression for
This statement has a long history, and we only cite two early papers on the subject by Dorokhov [70] and Imry [71]. For a modern discussion we refer to Ref. [72].
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Fig. 10. Distribution function of transmission coe$cients (87) for ¸/l"10.
the distribution function of transmission coe$cients: l 1 P(¹)" , ¹ (¹(1,
2¸ ¹(1!¹
¹ "4 exp(!2¸/l) ,
(87)
and P(¹)"0 otherwise. As discussed, it has a bimodal form: almost open and almost close channels are preferred. The dependence P(¹) is illustrated in Fig. 10. The distribution function P(¹) must be used now to average expressions (40) and (57) over impurity con"gurations. Direct calculation con"rms Eq. (86), and, thus, the distribution function (87) yields the Drude-Sommerfeld formula (85) for the average conductance. Furthermore, we obtain l , 1¹(1!¹)2" 3¸ which implies that the zero-temperature shot noise power is e"<" N l 1 , " S . S" 3 . 3p ¸
(88)
The shot noise suppression factor for metallic di!usive wires is equal to F"1/3. The remarkable feature is that this result is universal: As long as the geometry of the wire is quasi-one-dimensional and l;¸;¸ (metallic di!usive regime), the Fano factor does not depend on the degree of K disorder, the number of transverse channels, and any other individual features of the sample. This result was "rst obtained by Beenakker and one of the authors [74] using the approach described above. Independently, Nagaev [75] derived the same suppression factor by using a classical theory based on a Boltzmann equation with Langevin sources. This theory and subsequent developments are described in Section 6. It seems that the question whether the Fano factor depends on the type of disorder has never been addressed. In all cases disorder is assumed to be Gaussian white noise, i.e. 1;(r);(r )2Jd(r!r ).
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Later on, the suppression of shot noise became a subject of a number of microscopic derivations. Altshuler et al. [76] recovered the Fano factor by direct microscopic calculation using the Green's function technique. Nazarov [77] proved that the distribution (87) holds for an arbitrary (not necessarily quasi-one-dimensional) geometry; thus, the 1/3-suppression is `superuniversala. He used a slightly di!erent technique, expressing scattering matrices through Green's functions and then performing disorder averages. The same technique, in more elaborated form, was used in Ref. [78], which also obtains the -suppression. It is clear that the quantum-mechanical theories of Refs. [74,76}78] are equivalent for the quasi-one-dimensional geometry, since they deal with disorder averages basically in the same way. On the other hand, their equivalence to the classical consideration of Ref. [75] is less obvious. Experimentally, shot noise in metallic di!usive wires was investigated by Liefrink et al. [79], who observed that it is suppressed with respect to the Poisson value. The suppression factor in this experiment lies between 0.2 and 0.4 (depending on gate voltage). More precise experiments were performed by Steinbach et al. [80] who analyzed silver wires of di!erent length. In the shortest wires examined they found a shot noise slightly larger than and explained this larger value as due to electron}electron interaction. A very accurate measurement of the noise suppression was performed by Henny et al. [81]. Special care was taken to avoid electron heating e!ects by attaching very large reservoirs to the wire. Their results are displayed in Fig. 11. Localized regime. In quasi-one-dimensional wires with length ¸<¸ the transmission coe$cients K are `crystallizeda around exponentially small values [72]. This leads to a conductance and a shot noise power which decay exponentially with the length ¸. Shot noise is not suppressed with respect to the Poissonian value: F"1. The shot noise in the one-dimensional case was analyzed by Melnikov [56], who obtained di!erent suppression factors for various models of disorder in a one-channel wires. In particular, the model of Gaussian delta-correlated one-dimensional disorder leads to the suppression factor . Weak localization and mesoscopic yuctuations. In the metallic regime, quantum interference e!ects due to disorder, which eventually drive the system into the localized regime, manifest themselves in the form of weak localization corrections. For the shot noise, the weak localization correction was studied by de Jong and Beenakker [82], and later by Mace( do [83], Mace( do and Chalker [84], and Mace( do [85]. They found
1 e"<" N l , ! . S" 45 p 3¸ Comparison with a similar expression for the conductance,
e N l 1 , ! G" 2p ¸ 3
,
In long wires, interaction e!ects play a role; this is addressed in Section 6.
(89)
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Fig. 11. Shot noise measurements by Henny et al. [81] on three di!erent samples. The lower solid line is the -suppression, the upper line is the hot-electron result F"(3/4 (see Section 6). The samples (b) and (c) are short, and clearly display -suppression. The sample (a) is longer (has lower resistance), and the shot noise deviates from the non-interacting suppression value due to inelastic processes. Copyright 1999 by the American Physical Society.
yields the Fano factor 1 4 ¸ F" # . (90) 3 45 N l , The second terms represent weak localization corrections (¸;N l). These expressions are valid , for the case of preserved time-reversal symmetry (orthogonal symmetry). In the case of broken time-reversal symmetry (unitary symmetry; technically, this means that a weak magnetic "eld is applied) weak localization corrections are absent and the Fano factor stays at . Thus, we see that weak localization e!ects suppress noise, but enhance the Fano factor, in agreement with the general expectation that it lies above in the localized regime. The crossover from the metallic to the localized regime for shot noise has not been investigated. De Jong and Beenakker [82], Mace( do [83], and Mace( do and Chalker [84] studied also mesoscopic #uctuations of shot noise, which are an analog of the universal conductance #uctuations. For the root mean square of the shot noise power, they found e"<" r.m.s. S" p
46 , 2835b
where the parameter b equals 1 and 2 for the orthogonal and unitary symmetry, respectively. These #uctuations are independent of the number of transverse channels, length of the wire, or degree of disorder, and may be called [82] `universal noise #uctuationsa. The picture which emerges is, therefore, that like the conductance, the ensemble-averaged shot noise is a classical quantity. Quantum e!ects in the shot noise manifest themselves only if
This requires the knowledge of the joint distribution function of two transmission eigenvalues.
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we include weak localization e!ects or if we ask about #uctuations away from the average. With these results it is thus no longer surprising that the noise suppression factor derived quantum mechanically and from a classical Boltzmann equation for the #uctuating distribution are in fact the same. The same picture holds of course not only for metallic di!usive wires but whenever we ensemble average. We have already discussed this for the resonant double barrier and below will learn this very same lesson again for chaotic cavities. Chiral symmetry. Mudry et al. [86,87] studied the transport properties of disordered wires with chiral symmetry (i.e. when the system consists of two or several sublattices, and only transitions between di!erent sublattices are allowed). Examples of these models include tight-binding hopping models with disorder or the random magnetic #ux problem. Chiral models exhibit properties usually di!erent from those of standard disordered wires, for instance, the conductance at the band center scales not exponentially with the length of the wire ¸, but rather as a power law [88]. Ref. [87], however, "nds that in the di!usive regime (l;¸;N l) the Fano factor equals precisely , , like for ordinary symmetry. The only feature which appears due to the chiral symmetry is the absence of weak localization corrections in the zero order in N . Weak localization corrections, , both for conductance and shot noise, scale as ¸/(lN ) and discriminate between chiral unitary and , chiral orthogonal symmetries. Transition to the ballistic regime. In the zero-temperature limit, a perfect wire does not exhibit shot noise. For this reason, one should anticipate that in the ballistic regime, l'¸, shot noise is suppressed below . The crossover between metallic and ballistic regimes in disordered wires was studied by de Jong and Beenakker [82] (see their Eq. (A.10)), who found for the noise suppression factor
1 1 F" 1! . 3 (1#¸/l)
(91)
It, indeed, interpolates between F" for l;¸ and F"0 for ¸
An extensive list of references is provided by Ref. [86]. This must be considered as a toy model, since the classical theory ignores localization e!ects. Single-channel wires are in reality either ballistic or localized, but never metallic.
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Eq. (92) describes a crossover from F" (the metallic regime) to F"1 (classical point contact between metallic di!usive banks) and actually follows [92] from Eq. (91). Disordered interfaces. Schep and Bauer [93] considered transport through disordered interfaces, modeled as a con"guration of short-ranged scatterers randomly distributed in the plane perpendicular to the direction of transport. In the limit g;N , with g and N being the dimensionless , , conductance and the number of transverse channels, respectively, they found the following distribution function of transmission coe$cients:
1 pN \ g , , 1# (¹(1 , (93) P(¹)" 2g pN ¹ (1!¹ , and zero otherwise. Eq. (93) accidentally has the same form as the distribution function of transmission coe$cients for the symmetric opaque double-barrier structure. The noise suppression factor for this system equals , e.g. the suppression is weaker than for metallic di!usive wires. 2.6.5. Chaotic cavities 1/4-suppression. Chaotic cavities are quantum systems which in the classical limit would exhibit chaotic electron motion. We consider ballistic chaotic systems without any disorder inside the cavity; the chaotic nature of classical motion is a consequence of the shape of the cavity or due to surface disorder. The results presented below are averages over ensembles of cavities. The ensemble can consist of a collection of cavities with slightly di!erent shape or a variation in the surface disorder, or it can consist of cavities investigated at slightly di!erent energies. Experimentally, chaotic cavities are usually realized as quantum dots, formed in the 2D electron gas by back-gates. They may be open or almost closed; we discuss "rst the case of open chaotic quantum dots, shown in Fig. 12a. We neglect charging e!ects. One more standard assumption, which we use here, is that there is no direct transmission: electrons incident from one lead cannot enter another lead without being re#ected from the surface of the cavity (like Fig. 12a). The description of transport properties of open chaotic cavities based on the random matrix theory was proposed independently by Baranger and Mello [96] and Jalabert et al. [97]. They assumed that the scattering matrix of the chaotic cavity is a member of Dyson's circular ensemble of random matrices, uniformly distributed over the unitary group. For the cavity where both left and right leads support the same number of transverse channels N <1, this conjecture implies the , following distribution function of transmission eigenvalues: P(¹)"1/p(¹(1!¹) ,
(94)
shown in Fig. 12b. As a consequence of the assumption underlying random matrix theory, this distribution is universal: It does not depend on any features of the system. Taking into account that 1¹2" and 1¹(1!¹)2", we obtain for conductance, G"eN /4p , , In the case when the cavity is open, i.e. connected by ideal leads to the electron reservoirs, charging e!ects may still play a role. This e!ect, called mesoscopic charge quantization [94], was recently shown to a!ect very weakly the conductance of open chaotic cavities [95]. Results for shot noise are currently unavailable.
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Fig. 12. (a) An example of a chaotic cavity. (b) Distribution function of transmission eigenvalues (94).
and for the zero-temperature shot noise, S"e"<"N /8p "S . , . The Fano factor equals F", and is, of course, also universal. For the one-channel case, the whole distribution function of shot noise may be found analytically [98],
>
\E>\(\E , orthogonal symmetry , ( E\E (95) P(S)" , unitary symmetry , ( \E where we de"ned g"p S/(e"<"). In the general case when the numbers of transverse channels in the left N and right N lead are * 0 not equal, but still N <1 and N <1, the distribution function of transmission eigenvalues has * 0 been calculated by Nazarov [100]. Using this result, a calculation of the shot noise gives for the Fano factor [100,72] (
F"N N /(N #N ) . (96) * 0 * 0 This suppression factor has as its maximal value for the symmetric case N "N , and shot noise * 0 is suppressed down to zero in the very asymmetric case N ;N or N
(97)
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Calculating the averages 1 1¹2" ¹I G 2N G and 1 1¹(1!¹)2" ¹I (2!¹I ) , G G 8N G we "nd the Fano factor, 1¹(1!¹)2 1 ¹I (2!¹I ) G . F" " G G (98) 1¹2 ¹I 4 G G In particular, if all the transmission probabilities ¹I are the same and equal to ¹I , we obtain G F"(2!¹I )/4. This expression reproduces the limiting cases F" for ¹I "1 (no barriers } open quantum cavity) and F" for ¹I P0 (double-barrier suppression in a symmetric system). Thus, Eq. (98) describes the crossover between the behavior characteristic for an open cavity and the situation when the barriers are so high that the dynamics inside the cavity does not play a role. 2.6.6. Edge channels in the quantum Hall ewect regime Now we turn to the description of e!ects which are inherently multi-terminal. The calculation of the scattering matrix is in general a di$cult problem. However, in some special situations the scattering matrix can be deduced immediately even for multi-terminal conductors. We consider a four-terminal conductor (Fig. 13) made by patterning a two-dimensional electron gas. The conductor is brought into the quantum Hall regime by a strong transverse magnetic "eld. In a region with integer "lling of Landau levels the only extended states at the Fermi energy which connect contacts [102] are edge states, the quantum mechanical equivalent of classical skipping orbits. Since the net current at a contact is determined by the states near the Fermi surface, transport in such a system can be described by considering the edge states. Note that this fact makes no statement on the spatial distribution of the current density. In particular, a description based on edge states does not mean that the current density vanishes away from the edges. This point which has caused considerable confusion and generated a number of publications is well understood, and we refer the reader here only to one particularly perceptive discussion [103]. Edge states are uni-directional; if the sample is wide enough, backscattering from one edge state to another one is suppressed [102]. In the plateau regime of the integer quantum Hall e!ect, the number of edge channels is equal to the number of "lled Landau levels. For the discussion given here, we assume, for simplicity, that we have only one edge state. In a quantum Hall conductor wide enough so that there is no backscattering, there is no shot noise [18]. Hence, we introduce a constriction (Fig. 13) and allow scattering between di!erent edge states at the constriction [18]: the probability of scattering from contact 4 to the contact 3 is ¹, while that from 4 to 1 is 1!¹. In We do not give a microscopic description of edge states. Coulomb e!ects in the integer quantum Hall e!ect regime lead to a spatial decomposition into compressible and incompressible regions. Edge channels in the fractional quantum Hall e!ect regime will be discussed in Section 7.
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Fig. 13. Four-probe quantum Hall conductor. Solid lines indicate edge channels; dashed lines show the additional scattering probability ¹ through the quantum point contact.
the following, we will focus on the situation when the chemical potentials of all the four reservoirs are arranged so that k "k "k, k "k "k#e<. The scattering matrix of this system has the form
s"
0
s
0 0
0
s
s
0
s
0
s
0
.
(99)
s 0 0 0 Here the elements s "r, s "r , s "t, s "t form the 2;2 scattering matrix at the constriction: "t"""t ""¹, "r"""r ""1!¹, rHt #tHr "rHt#r t H"0. The two remaining elements, s "exp(ih ) and s "exp(ih ) describe propagation along the edges without scatter ing. It is straightforward to check that the matrix (99) is unitary. We consider "rst the shot noise at zero temperature [18,9]. Using the general Eq. (55), we see immediately that the only non-zero components of the shot noise power tensor are S "S " !S "!S , with S "!(e"<"/p )¹(1!¹) . (100) Indeed, if there is no scattering between the edge states, there is no shot noise in the system. The same is true for the case when this scattering is too strong: all the current from 2 #ows to 1, and from 4 to 3. For "nite temperature all components become non-zero (except for S ) and can be found [9] from Eq. (52). We only give the result for S "S , e dE ¹(1!¹)( f !f ) , (101) S "! p
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where f and f are Fermi distribution functions in the reservoirs 1 and 3, respectively. This result is remarkable since it vanishes for ¹"1 at any temperature: The correlation function (101) is always `shot-noise-likea. One more necessary remark is that all the shot noise components in this example are actually expressed only through absolute values of the scattering matrix elements: phases are not important for noise in this simple edge channel problem. Early experiments on noise in quantum Hall systems were oriented to other sources of noise (see e.g. Refs. [104,105]), and are not discussed here. Shot noise in the quantum Hall regime was studied by Washburn et al. [106], who measured voltage #uctuations in a six-terminal geometry with a constriction, which, in principle, allows for direct comparison with the above theory. They obtained results in two magnetic "elds, corresponding to the "lling factors l"1 and l"4. Although their results were dominated by 1/f-noise, Washburn et al. were able to "nd that shot noise is very much reduced below the Poisson value, and the order of magnitude corresponds to theoretical results. 2.6.7. Hanbury Brown}Twiss ewects with edge channels The conductor of Fig. 13 is an electrical analog of the scattering of photons at a half-silvered mirror (see Fig. 1). Like in the table top experiment of Hanbury Brown and Twiss [7,8], there is the possibility of two sources which send particles to an object (here the quantum point contact) permitting scattering into transmitted and re#ected channels which can be detected separately. Bose statistical e!ects have been exploited by Hanbury Brown and Twiss [7] to measure the diameter of stars. The electrical geometry of Fig. 13 was implemented by Henny et al. [12], and the power spectrum of the current correlation between contact 1 (re#ected current) and contact 3 (transmitted current) was measured in a situation where current is incident from contact 4 only. Contact 2 was closed such that e!ectively only a three-terminal structure resulted. For the edgechannel situation considered here, in the zero-temperature limit, this does not a!ect the correlation between transmitted and re#ected current. (At "nite temperature, the presence of the fourth contact would even be advantageous, as it avoids, as described above, the `contaminationa due to thermal noise of the correlation function of re#ected and transmitted currents, see Eq. (101)). The experiment by Henny et al. "nds good agreement with the predictions of Ref. [18], i.e. Eq. (100). With experimental accuracy the current correlation S is negative and equal in magnitude to the mean-square current #uctuations S "S in the transmitted and re#ected beam. The experiment "nds thus complete anti-correlation. This outcome is related to the Fermi statistics only indirectly: If the incident carrier stream is noiseless, current conservation alone leads to Eq. (100). As pointed out by Henny et al. [12], the experiment is in essence a demonstration that in Fermi systems the incident carrier stream is noiseless. The pioneering character of the experiment by Henny et al. [12] and an experiment by Oliver et al. [107] which we discuss below lies in the demonstration of the possibility of measuring current}current correlation in electrical conductors [108]. Henny et al. [12] measured not only the shot noise but used the four-terminal geometry of Fig. 13 to provide an elegant and interesting demonstration of the #uctuation-dissipation relation, Eq. (54). Now it is interesting to ask, what happens if this complete population of the available states is destroyed (the incident carrier stream is not noiseless any more). This can be achieved [355] by inserting an additional quantum point contact in the path of the incident carrier beam (see Fig. 14).
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Fig. 14. Three-probe geometry illustrating the experiment by Oberholzer et al. [355].
We denote the transmission and re#ection probability of this "rst quantum point contact by i and j"1!i, and the transmission and re#ection probability of the second quantum point contact by ¹ and R"1!¹ as above. The scattering matrix of this system has the form
s"
!ij (i¹)
!i(iR)
0
i
!iR !i(j¹) . ¹
!(jR)
(102)
The phases in this experiment play no role and here have been chosen to ensure the unitarity of the scattering matrix. In the zero-temperature limit with a voltage di!erence < between contact 1 and contacts 2 and 3 (which are at the same potential) the noise power spectra are
ij !ij¹ !ijR e"<" S" !ij¹ i¹(1!i¹) !i¹R . p
!ijR !i¹R iR(1!iR)
(103)
The correlation function between transmitted and re#ected beams S "S "!iR¹ is pro portional to the square of the transmission probability in the "rst quantum point contact. For i"1 the incident beam is completely "lled, and the results of Henny et al. [12] are recovered. In the opposite limit, as i tends to zero, almost all states in the incident carrier stream are empty, and the anti-correlation between transmitted and re#ected beams also tends to zero. 2.6.8. Three-terminal structures in zero magnetic xeld A current}current correlation was also measured in an experiment by Oliver et al. [107] in a three-probe structure in zero magnetic "eld. This experiment follows more closely the suggestion of Martin and Landauer [21] to consider the current}current correlations in a Y-structure. Ref. [21], like Ref. [18], analyzes the noise power spectrum in the zero-frequency limit. Early experiments on a three-probe structure by Kurdak et al. [109] were dominated by 1/f-noise and did not show any e!ect. Here the following remark is appropriate. Strictly speaking, the Hanbury Brown}Twiss (HBT) e!ect is a coincidence measurement. In the optical experiment the intensity #uctuation dI (t) is ?
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measured and correlated with the intensity #uctuation dI (t#q), where q is a short time smaller @ than the response time q of the detector. The coincidence rate C is thus ?@ O C "(1/2q) dq1dIK (t)dIK (t#q)#dIK (t#q)dIK (t)2 . (104) ?@ ? @ @ ? The coincidence rate is related to the frequency-dependent noise power spectrum by
C "(1/q) ?@
O
(105) dq du e SOS (u) . ?@ In Section 3, we discuss the frequency dependence of the noise power spectrum in more detail. Typically, its lowest characteristic frequencies are given by RC-times. In principle, such a measurement should, therefore, be able to give information on the frequency dependence of the noise power spectrum. In the experiment of Oliver et al. the resolution time q is probably long compared to such intrinsic time scales, and thus the experiment is e!ectively determined by the white-noise limit of the power spectrum. Let us now brie#y consider a Y-shaped conductor [21] and discuss its correlations in the white-noise limit. We assume that the same voltage < is applied between the terminals 1 and 2, and 1 and 3: k "k #e<"k #e<. For zero temperature, the general formula (55) yields the following expression for the cross-correlations of currents in leads 2 and 3: S "!(e"<"/p )Tr[sR s sR s ] , (106) which is negative in accordance with the general considerations. Note the formal similarity of this result to the shot-noise formula in the two terminal geometry given by Eq. (56). In the singlechannel limit, if we assume that there is no re#ection back into contact 1, Eq. (106) becomes S "!(e"<"/p )¹(1!¹), where ¹ is the transmission probability from 1 to 2. This simple result underlines (the formal) equivalence of scattering at a QPC with separation of transmitted and re#ected streams and scattering at a re#ectionless Y-structure. The experiments by Oliver et al. [107] con"rm these theoretical predictions. The experiments by Henny et al. [12] and Oliver et al. [107] test the partitioning of a current stream. If the incident carrier stream is noiseless, the resulting current correlation is negative already due to current conservation alone. Therefore, experiments are desirable, which test electron statistical e!ects (and the sign of correlations) in situations where current conservation plays a much less stringent role. 2.6.9. Exchange Hanbury Brown}Twiss ewects Another HBT experiment was proposed in Ref. [19] (see also Ref. [9]). It is based on the comparison of the noise generated in the presence of two incident currents with the noise generated by one source only. We "rst present a general discussion and later consider a number of applications. Consider the four-terminal structure of Fig. 15a. We will be interested in crosscorrelations of currents at the contacts 2 and 4, S,!S
. The quantity S de"ned in this way is always positive. Now we discuss three di!erent ways of applying voltage. The "rst one (to be referred to as experiment A) is to apply a voltage to the
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reservoir 1, k !e<"k "k "k . In the next one (experiment B) the voltage is applied to 3, k "k "k !e<"k . Finally, in the experiment C the identical voltages are applied to 1 and 3, k !e<"k "k !e<"k . Results for zero temperature are readily derived from Eq. (55) and read S "(e"<"/p )N , S "(e"<"/p )N , S "(e"<"/p )(N #N #N #N ) , ! where
(107)
N "Tr[sR s sR s ], N "Tr[sR s sR s ] , N "Tr[sR s sR s ], N "Tr[sR s sR s ] . (108) The quantities S and S are determined by transmission probabilities from 2 and 4 to 1 and 3, respectively, and are not especially interesting. New information is contained in S . In systems ! obeying classical statistics, the experiment C would be just a direct superposition of the experiments A and B, S "S #S . The additional terms N and N on the rhs of Eq. (107) are due to ! quantum (Fermi) statistics of the electrons. These terms now invoke products of scattering matrices which are in general not real-valued. These terms are not products of two pairs of scattering matrices as in Eq. (56) or Eq. (106) but contain four scattering matrices in such a way that we are not able to distinguish from which of the two current carrying contacts a carrier was incident. For future convenience, we de"ne the quantity *S"S !S !S , which indicates the fermionic ! analog of the HBT e!ect. One can show [9] that for "nite temperatures the corresponding correction for bosons is of the same form but has the opposite sign, hence it will be called `exchange contributiona. One more remarkable feature of the result (107) is that the exchange correction *S is phase sensitive. Indeed, it represents the contribution of trajectories indicated by the dashed line in Fig. 15a (traversed in both directions), and thus is proportional to exp($i ), with being the phase accumulated during the motion along the trajectories. For this reason, one cannot generally predict the sign of *S: the only restrictions are that all the quantities S , S and S need to be ! positive. Gramespacher and one of the authors [110}112] considered a particular geometry where the leads 2 and 4 are tunneling contacts locally coupled to the sample (e.g. scanning tunneling microscope tips). In this case, the exchange contribution can directly be expressed in terms of the wave functions (scattering states) 1 1 (109) Re+t (r)tH (r )tH (r)t (r ), , *S" L L K K p v v K L KL where the sum is over all transverse channels m in the lead 1 and n in the lead 3; r and r are the points to which the contacts 2 and 4 couple, respectively, and t is the wave function of the ?I corresponding scattering state. Thus, the exchange contribution explicitly depends on phases of the wave functions. We note in passing that if diwerent voltages are applied to 1 and 3, the correlation functions (107) imply that S cannot be an analytic function of the two voltages. Thus, our four-terminal conductor is a non-linear circuit element due to exchange e!ects.
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Fig. 15. Four-terminal conductors for Hanbury Brown}Twiss exchange e!ects (a), quantum point contact geometry (b). Terminals are numbered by digits. The dashed line in (a) indicates a phase-sensitive trajectory which contributes to exchange terms in Eq. (107). Arrows in (b) indicate non-zero transmission probabilities.
We investigate now the general expression for the four-terminal phase-sensitive HBT e!ect (107) for various systems. Our concern will be the sign and relative magnitude of the exchange contribution *S"S !S !S . ! Disordered systems. Naively, one might assume that in a disordered medium the phase accumulated along the trajectory, indicated by the dashed line in Fig. 15a, is random. Then the phasesensitive exchange contribution would be zero after being averaged over disorder. Thus, this view implies S "S #S . ! A quantitative analysis of these questions was provided by the authors of this review in Ref. [78]. In this work the scattering matrices in Eq. (108) are expressed through Green's functions to which disorder averaging was applied using the diagram technique. The key result found in Ref. [78] is that the naive picture mentioned above, according to which one might expect no exchange e!ects after disorder averaging, is completely wrong. Exchange ewects survive disorder averaging. The reason can be understood if the principal diagrams (which contain four di!usion propagators) are translated back into the language of electron trajectories. One sees then that the typical trajectory does not look like the dashed line in Fig. 15a. Instead, it looks like a collection of dashed lines shown in Fig. 16a: the electron di!uses from contact 1 to some intermediate point 5 in the bulk of the sample (eventually, the result is integrated over the coordinate of point 5), then it di!uses from 5 to 2 and back from 2 to 5 precisely along the same trajectory, and so on, until it returns from 5 to 1 along the same di!usive trajectory as it started. Thus, there is no phase enclosed by the trajectory. This explains why the exchange contribution survives averaging over disorder; apparently, there is a classical contribution to the exchange correlations which requires knowledge only of Fermi statistics, but no information about phases of scattering matrices. Indeed, a classical theory of ensemble-averaged exchange e!ects was subsequently proposed by Sukhorukov and Loss [113,114]. Once we determined that exchange correction *S exists in di!usive conductors, we must evaluate its sign and relative magnitude. We only describe the results qualitatively; details can be
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Fig. 16. Examples of four-terminal disordered conductors. Disordered area is shaded. Dashed line denotes di!usive motion between its ends.
found in Refs. [78,114]. Two speci"c geometries have been investigated: the disordered box (Fig. 16a) and the disordered cross (Fig. 16b). For the box, the exchange correction *S is negative, i.e. exchange suppresses noise (S (S #S ). The e!ect is quite considerable: The correction is of ! the same order of magnitude as the classical contributions in S and S , and is suppressed only by a numerical factor. For the cross, the exchange contribution is positive } exchange enhances noise } but the magnitude is by powers of l/¸ smaller than S and S . Here l and ¸ are the mean free path and the length of the disordered arms, respectively. Thus, neither sign nor magnitude of the exchange e!ects is predetermined in di!usive systems: they are geometry and disorder dependent, and the only limitation is S '0. ! Gramespacher and one of the authors [110,111] considered a geometry of a disordered wire (along the axis z) between the contacts 1 (z"0) and 3 (z"¸), coupled locally at the points z and z to the contacts 2 and 4, respectively, via high tunnel barriers (these latter can be viewed as scanning tunneling microscope tips) and evaluated Eq. (109). It was found that the exchange e!ect is positive in the case, i.e. it enhances noise, irrespectively of the position of the contacts 2 and 4. For the particular case when both tunnel contacts are situated symmetrically around the center of the wire at a distance d, z"(¸!d)/2 and z "(¸#d)/2, the relative strength of the exchange term is
d d *S 1 " 2# !2 , ¸ 3 ¸ S ! and reaches its maximum for d"¸/4. We see that the exchange e!ect in this case generally has the same order of magnitude as the classical terms S and S . Chaotic cavities. A similar problem in chaotic cavities was addressed in Ref. [115] (see also Ref. [116]). Similarly to disordered systems, it was discovered that exchange e!ects survive on average. An additional feature is however that the exchange e!ects in chaotic cavities are universal. Ref. [111] also considers a three-terminal structure (a (disordered) wire with a single STM tip attached to it). The #uctuations of the current through the tip are in this case proportional to the local distribution function of electrons at the coupling point, see Eq. (249) below.
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That of course is a consequence of the assumption that the cavity can be described by using Dyson's circular ensemble. For open cavities, one "nds N "N "!3N "!3N "N /(16N !1) , , ,
(110)
where we assumed that all leads are identical and support N transverse channels. This implies , S "S , S "4S /3, or *S"!2S /3. Thus, exchange e!ects suppress noise in open chaotic ! cavities. The situation changes if the cavity is separated from the leads by tunnel barriers. Assuming that the transmission coe$cients of all barriers in all transverse channels are identical and equal ¹, N ¹<1, Ref. [115] "nds ,
N "N ¹#2 "N, ¹ . 64 !3¹#2 N "N
Thus, for ¹" the exchange e!ect changes sign: if the barriers separating the cavity from the reservoirs are high enough, the exchange enhances the correlations. In the limit of very opaque barriers, ¹P0, we have S "2(S #S ): the exchange correction is the same as the classical ! contributions S and S .
Fig. 17. Hanbury Brown}Twiss e!ect with edge states. Fig. 18. Experimental results of Liu et al. [118]. Upward and downward triangles correspond to the situations when only one of the two input contacts is open (values S and S , respectively); squares indicate the case when both input contacts are e!ective (S ). Copyright 1998 by Nature. !
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Edge channels in the quantum Hall ewect regime. Interesting tests of exchange (interference) e!ects can also be obtained in high magnetic "elds using edge channels. In fact, this leads to a simple example where the phase dependence of the exchange e!ect is indeed essential [9]. Imagine that the system is placed into a strong magnetic "eld, and the transport is only due to edge channels. We assume that there is only one edge state #owing from 1 through 2 and 3 to 4 and back to 1 (Fig. 17). Furthermore, for simplicity we assume that all the leads are identical, and the transmission probability to enter from the lead to the edge state is ¹. A direct calculation gives [9] ¹(1!¹) , N "N " [1#(1!¹)!2(1!¹) cos ] ¹(1!¹) exp(i ) N "NH" , [1#(1!¹)!2(1!¹) cos ] where is the phase accumulated along the whole trajectory, and the phase dependence in the denominator appears due to the possibility of multiple traversals of the full circle. We have 2e"<" ¹(1!¹) S " (1!cos ) . ! p [1#(1!¹)!2(1!¹) cos ]
(111)
Here the term with 1 represents the `classicala contributions S #S , while that with cos
accounts for the exchange e!ect *S. We see that, depending on , exchange e!ects may either suppress (down to zero, for "0) or enhance (up to 2(S #S ), for "n) the total noise. This is an example illustrating the maximal phase sensitivity which the exchange e!ect can exhibit. This simple example gives also some insight on how exchange e!ects survive ensemble averaging. Since the phase occurs not only in the numerator but also in the denominator, the average of Eq. (111) over an ensemble of cavities with the phase uniformly distributed in the interval from 0 to 2p is non-zero and given by ¹(1!¹) 2e"<" . 1S 2" ! p [1!(1!¹)][1#(1!¹)]
(112)
The ensemble-averaged exchange contribution vanishes both in the limit ¹"0 and in the limit ¹"1. Note that if another order of contacts is chosen, 1P3P2P4, the whole situation changes: the exchange term is now phase insensitive and has a de"nite sign (negative, i.e. exchange suppresses noise) [9]. This is because the trajectories responsible for exchange terms do not form closed loops in this case. Experiments. The phase-sensitive Hanbury Brown}Twiss e!ect discussed above has not so far been probed in experiments. However, a related experiment was carried out by Liu et al. [117,118], who measured the mean-squared #uctuations S of the current in the lead 3, of a four-probe structure, applying voltages in the same three-fold ways that we have discussed.
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Prior to the description of experimental results, we discuss brie#y a measurement of S on the quantum Hall conductor of Fig. 13. If current is incident from contact 4 (experiment A), or contact 2 (experiment B) alone we have for the current #uctuations at contact 3 S "S "(e"<"/p )¹(1!¹). On the other hand, if currents are incident both from contact 4 (experiment C) and contact 2, all states are now completely "lled and thus in the zero-temperature limit S! "0. Thus, in comparison to experiments A or B there is a complete reduction of the shot noise at contact 3 in experiment C: the spectral density S! is suppressed down to zero. In the experiment of Liu et al. [117,118] the mean-square current #uctuations are measured in zero magnetic "eld in a conductor in which a left input (1) and output contact (2) are separated by a thin barrier from a right input contact (3) and output contact (4) (see Fig. 15b). The input contacts form QPCs and are adjusted to provide transmission close to 1. The output contacts support a number of channels. In this experiment it thus not possible to "ll all outgoing states in contact 3 completely and there is thus only a limited reduction of noise in experiment C compared to experiments A and B. The experimentally observed ratio S /(S #S )"0.56. ! It is also useful to compare the experiment of Liu et al. [117,118] (Fig. 18) simply with a chaotic cavity connected to point contacts which are fully transparent ¹"1 [115]. Then using Eq. (110) one "nds a ratio S /(S #S )" which is surprisingly close to what was observed in the ! experiment. 2.6.10. Aharonov}Bohm ewect The Aharonov}Bohm (AB) e!ect tests the sensitivity to a magnetic #ux U of electrons on a trajectory which enclose this #ux. In the pure AB-e!ect the electron does not experience the magnetic "eld, the electron trajectory is entirely in a "eld-free region. It is a genuine quantum e!ect, which is a direct consequence of the gauge invariance of the velocity and the wave nature of electrons. The simplest geometry demonstrating the AB e!ect in electric transport is a ring coupled to two reservoirs and threaded by a magnetic #ux, as shown in Fig. 19a. Then, the AB e!ect is manifest in a periodic #ux dependence of all the transport properties. Qualitatively di!erent phenomena arise in weak and strong magnetic "elds, and these two cases need to be considered separately. Weak magnetic xelds. In this regime, the transmission coe$cient(s) (and, subsequently, conductance) of the two-terminal structure shown in Fig. 19a, is a periodic function of the external #ux, with the period U "2p c/e. The resulting conductance is sample-speci"c, and, in particular, it is very sensitive to the phase of the trajectory enclosing the #ux. In phase coherent many-channel conductors the AB-oscillations in the conductance represent a small correction to a #ux insensitive (classical) background conductance. Thus the AB e!ect is most dramatic in single-channel rings [120,121] where the #ux-induced modulations of the conductance are of the order of the conductance itself. Shot noise in such a structure was studied by Davidovich and Anda [122] using the non-equilibrium Green's functions technique. They considered a one-channel ring and used In disordered systems, the ensemble-averaged conductance exhibits AB oscillations with the period of U /2. These oscillations are, like the weak localization correction, associated with the interference of two electron trajectories running in opposite directions [119]. Weak localization e!ects and shot noise have not so far been investigated.
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Fig. 19. (a) Geometry of a ring threaded by a #ux U, demonstrating the Aharonov}Bohm e!ect in a weak magnetic "eld. (b) Conductance in units of e/2p (1) and shot noise power in units of e
a tight-binding description of the ring and leads. Here we will give another derivation, based on the scattering approach. A related issue was discussed by Iannaccone et al. [123], who studied noise in a multiply connected geometry using the scattering approach, and found that if there is no transmission from the left part of the ring to the right part and vice versa (Fig. 19a), noise of the left and right parts add up classically. In particular, this means that such a system would not exhibit an AB e!ect. We follow Refs. [120,121] which study the transmission coe$cient of single-channel rings connected to external leads (Fig. 19a). Our purpose here is to illustrate only the principal e!ect, and therefore we consider the simple case without scattering in the arms of the ring. We also assume that the ring is symmetric. Formulae for shot noise in more complicated situations can be readily produced from Refs. [120,121], though, to the best of our knowledge, they have never been written down explicitly. We describe the `beam splittersa, separating the leads from the ring (black triangles in Fig. 19a) by the scattering matrix [121]
!(a#b) e e
s " @
e
a
b
e
b
a
,
(113)
where the parameter e, 0(e(, is responsible for the coupling of the ring to the lead, and a"(1/(2)((1!2e!1),
b"!(1/(2)((1!2e#1) .
Specializing to the case of the ring which is ideally coupled to the leads, e", we obtain for the transmission coe$cient [121] (1#cos h) sin
, ¹(U)" (1#cos h!cos 2 )#(1/2) sin 2
(114)
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where h"2pU/U , and is the phase accumulated during the motion along a half of the ring (without magnetic "eld). Now the conductance (40) G"(e/2p )¹ and the shot noise (57) S"(e"<"/p )¹(1!¹) are immediately expressed as functions of the applied magnetic #ux. They are strongly dependent on the phase , which is sample-speci"c. In particular, both the conductance and the shot noise vanish for "0 or "p. This is a consequence of the symmetry assumed here: if the leads are attached asymmetrically to the ring the transmission coe$cient stays "nite for any value of the phase [124]. The dependence of conductance and shot noise on #ux U for a particular value "p/2 is shown in Fig. 19b. We re-emphasize that the #ux dependence shown depends strongly on the sample-speci"c phase . Strong magnetic xelds. Now we turn to the situation of the quantum Hall e!ect, where transport current is carried by the edge states. A remarkable feature of this regime is that a two-terminal ring without backscattering does not exhibit the AB e!ect. The edge states (Fig. 20a) exist in di!erent regions of space, and thus do not interfere. Indeed, the absence of backscattering, which precludes the AB-e!ect, is just the condition for conductance quantization [102]. We cannot have both a quantized conductance and an AB-e!ect. The question which we want now to address is the following: can one observe AB e!ects in the noise, which is a fourth-order interference e!ect, in a situation when they do not exist in the conductance which is only a second-order interference e!ect? We noticed already that shot noise is a phase-sensitive e!ect, and contains interference terms, absent in the conductance. The shot noise contains non-real terms composed of four scattering matrix elements. This is the case already in the two-terminal shot noise formula when it is expressed in the natural basis. In the two-terminal case the appearance of such products depends, however, on the basis we chose: the shot noise is a function of transmission probabilities only, if it is evaluated in the eigen-channel basis. However, in a multi-terminal geometry, such products appear naturally if we consider current}current cross-correlations. We call these non-real products exchange interference terms, since they are
Fig. 20. (a) Geometry of a ring in strong magnetic "eld. Edge states are shown. (b) Four-terminal geometry which facilitates separation of scattering and AB e!ects.
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a manifestation of interference e!ects in multi-particle wave functions (Slater determinants of single-particle wave functions) which result from the indistinguishability of carriers. In contrast, the e!ects we have already seen in weak "elds, which contribute to the conductance as well as to the shot noise are a consequence of second order or direct interference. The two-terminal geometry of Fig. 20a is not appropriate for the observation of the exchange interference e!ects, since shot noise vanishes without scattering between edge channels. If scattering is introduced, shot noise becomes "nite, but at the same time the conductance becomes sensitive to the #ux, due to direct interference. One can try to separate AB e!ects in the shot noise due to direct and exchange interference, but this is awkward. A possible way out was proposed in Ref. [112], which suggested four-terminal geometries with two weak coupling contacts. We follow here a subsequent, clearer discussion given in Ref. [111]. The geometry is shown in Fig. 20b. This is a quantum dot in a strong magnetic "eld coupled via two quantum point contacts to reservoirs. The ring geometry is actually not needed in the experiment and serves only for conceptual clarity. Current #ows between contacts 1 and 2, and the contacts 3 and 4 are inserted locally at the quantum point contact between the edge states in the leads. The magnetic "eld is such that the two-probe conductance is quantized, but weak enough such that at the quantum point contact the left- and right-going wave functions of the two edge channels overlap. As is well known, the fact that the wave functions in the quantum point contact overlap, does not destroy the quantization, as long as the potential of the quantum point contact is smooth. We take the scattering matrix relating the amplitudes of carriers in the contact 3 (or 4) with those in the edge channels nearby to be of the same form (113) as for the `beam splittera discussed above. Here it is now essential to assume that coupling is weak thus we take e;1. In the experiment proposed in Refs. [112,111] the same voltage < is applied simultaneously to the contacts 1 and 2. Then we have S "S "(2e"<"/p )e , S "!(2e"<"/p )e[1#cos( #2pU/U )] , where is a certain phase. The relative value of S as compared to S is e. At the same time, the corrections to the conductance and the shot noise due to direct interference are proportional to e. Thus, in our geometry up to the terms of e conductance is not renormalized by the AB e!ect, while shot noise feels it due to its two-particle nature. This is thus a geometry where the AB e!ect manifests itself in the fourth-order interference and modulates the Hanbury Brown}Twiss e!ect (the current}current cross-correlation at contacts 3 and 4). 2.7. Inelastic scattering. Phase breaking Throughout this section, we treated the mesoscopic systems as completely phase coherent. In reality, there is always at least some inelastic or phase-breaking scattering present. The scattering approach as it was used here, is based on the carrier transmission at a de"nite energy. In contrast, electron}electron or electron}phonon interactions can change the energy of a carrier. Thus, a scattering theory of such processes has to be based on scattering amplitudes which permit incoming and outgoing particles to have di!erent energies. To our knowledge, the extension of
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scattering theory of electrical transport within such a generalized scattering matrix approach has not been worked out. It is, however, possible to make progress even within the scattering approach used so far: to treat phase breaking theoretically we often proceed by inventing a Hamiltonian system with many degrees of freedom while we are interested in the behavior of only a subsystem. Similarly, it is possible to arrive at an approach which describes inelastic transitions and phase breaking by "rst considering a completely phase-coherent conductor with one or a continuum of additional voltage probes which are purely "ctitious [125,47]. The additional "ctitious voltage probes act as dephasers on the actual conductor of interest. This approach has been widely used to investigate the e!ect of dephasing on conductance. We refer the reader here only to a few early works [125}127,47]. In this subsection we illustrate the application of these ideas to noise. Other approaches, based on Green's function techniques, have also been invoked to derive results for strongly correlated systems (see Section 7). Furthermore, on the purely classical level, it proved to be rather simple to extend the #uctuating Boltzmann equation approach to include interactions. For the results on interaction and noise in double-barrier resonant tunneling structures and metallic di!usive conductors the reader is addressed to Sections 5 and 6, respectively. The approach which uses voltage probes as dephasers is interesting because of its conceptual clarity and because of its close relation to experiments: the e!ect of additional voltage probes can easily be tested experimentally with the help of gates which permit to switch o! or on a connection to a voltage probe (see e.g. Ref. [128]). Voltage probes as dephasers. Consider a mesoscopic conductor connected to N (real) contacts. To introduce inelastic scattering, we attach a number M of purely "ctitious voltage probes to this conductor. The entire conductor with its N#M contacts is phase coherent and exhibits the noise of a purely phase coherent conductor. However, elimination of the M "ctitious voltage probes leads to an e!ective conduction problem for which the conductance and the noise depend on inelastic scattering processes [19,74,129}131]. Depending on the properties of the "ctitious voltage probes, three di!erent types of inelastic scattering can be realized, which de Jong and Beenakker [90] classify as `quasi-elastic scatteringa (phase breaking), `electron heatinga, and `inelastic scatteringa. Now we describe these types of probes separately. This division corresponds to the distinction of q , q , and q . We emphasize that only a microscopic theory can give explicit ( CC GL expressions for these times. What the approach based on "ctitious voltage probes can do is to "nd the functional dependence of the conductance or the noise on these times. The results for interaction e!ects in double-barrier structures seem to be well established by now. In contrast, for di!usive metallic wires with interactions the situation is less clear. For discussion, the reader is addressed to Section 6. In this subsection, we assume that the system is charge neutral, i.e. there is no pile-up of charge. This charge neutrality is normally provided by Coulomb interactions, which thus play an important role. If this is not the case, one can get di!erent results, like for resonant tunneling quantum wells with charging (Section 5) or quantum dots in the Coulomb blockade regime (Section 7). Inelastic scattering. We begin the discussion with the strongest scattering processes which lead to carrier energy relaxation and consequently also energy dissipation. Physically, this may correspond to electron}phonon scattering. To simulate this process, we consider a two-terminal structure
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Fig. 21. (a) Setup with intermediate electrode. (b) Equivalent circuit for the case when there is no direct transmission from 1 to 2.
in the conceptually simple case where we add only one "ctitious voltage probe (marked as 3, see Fig. 21). As in our treatment of noise in multi-probe conductors we assume that all reservoirs (also the voltage probe reservoir 3) are characterized by Fermi distribution functions (for simplicity, we only consider zero-temperature case). We take k "0 and k "e<; the chemical potential k is found from the condition that at a voltage probe the current I vanishes at any moment of time. Then an electron which has left the conductor and escaped into the reservoir of the voltage probe must immediately be replaced by another electron that is reinjected from the voltage probe into the conductor with an energy and phase which are uncorrelated with that of the escaping electron. This approach to inelastic scattering was applied to noise in mesoscopic systems in Ref. [19] (where only the average current was taken to vanish), and the analysis for an instantaneously vanishing current was presented in Ref. [74]. To proceed, we introduce transmission probabilities from the lead a to the lead b, ¹ "Tr sR s , ¹ " ¹ "N , ?@ ?@ ?@ ?@ @? ? @ @ N being the number of transverse channels in the lead a. Currents can be then written as ? I "(e/2p )[¹ e<#¹ (e
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The "rst term on the right-hand side represents the average chemical potential 1k 2, while the second one is a #uctuating correction. The two-terminal conductance (de"ned according to 1I 2"G<) depends only on the average potential 1k 2 and is given by [125,47] ¹ ¹ e ¹ # . (116) G" 2p ¹ #¹ In Eq. (116) the probability ¹ describes coherent transmission, whereas the second term is the incoherent contribution. Due to the #uctuations of the chemical potential k , the random part of the current I is now *I "dI !(¹ /(¹ #¹ ))(dI #dI ) , and due to the current conservation the random part of the current I is the same with the opposite sign. Now the expression for the shot noise power S can be easily obtained from Eq. (55), but it is rather cumbersome in the general case. In the following, we consider only the fully incoherent case ¹ "0, when there is no direct transmission from 1 to 2: every carrier on its way from contact 1 to 2 enters the electrode 3 with the probability one. This condition also implies ¹ "¹ . Essential? ? ly, the fully incoherent case means that the two parts of the system, from 1 to 3, and from 3 to 1, are resistors which add classically. In particular, the conductance Eq. (116) contains now only the second term, which now just states that two consecutive incoherent scatterers exhibit a resistance which is equal to the series resistance. For the current correlations, we obtain S "S " !S "!S with e"<" ¹ Tr[s sR s sR ]#¹ Tr[s sR s sR ] . (117) S " (¹ #¹ ) p
This expression can be re-written in the following transparent manner: "rst, we de"ne the resistances of the parts of the system between 1 and 3, R "2p /(e¹ ), and between 2 and 3, R "2p /(e¹ ). The total resistance between 1 and 2 is given by R"R #R . Now the voltage drop between 1 and 3 is e
e"<" R Tr[s sR (1!s sR )] , S " p R and noise power measured between 2 and 3 is, e"<" R Tr[s sR (1!s sR )] . S " p R Now we can write Eq. (117) as [74] (S,S ) RS"R S #R S . (118) The meaning of Eq. (118) is obvious if we realize that RS is the voltage #uctuations (for an in"nite external impedance circuit) across the whole conductor. The right-hand side is just a sum of voltage #uctuations from 1 and 3, and from 3 to 2. Thus, Eq. (118) states nothing but that the voltage #uctuations are additive.
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Another form of Eq. (118) is useful [133]. We introduce the noise suppression factors F "S R/(2e"<") and F "S R/(2e"<") in the "rst and second resistor. For the Fano factor of the whole system we obtain F"(R F #R F )/(R #R ) .
(119)
First of all, we now evaluate Eq. (119) for the case of the double-barrier structure, where the intermediate electrode is placed between the barriers. Physically, this would correspond to strong inelastic scattering inside the quantum well } in contrast to the quantum-mechanical discussion of the previous subsection, which implicitly requires full-phase coherence. Taking into account that for high barriers F "F "1, and that the resistances R and R are inversely proportional to the tunneling rates C and C , respectively, we immediately arrive at Eq. (78), i.e. the result for the fully * 0 coherent case. We thus see that even though inelastic scattering modi"es both the conductance and shot noise of the resonant tunneling structure, it leaves the Fano factor unchanged. This statement, due to Chen and Ting [134] and Davies et al. [54], will be again demonstrated in Section 5, where the derivation of Eq. (78) based on a classical Langevin approach (which corresponds to the absence of quantum coherence) is presented. Next we consider a quasi-one-dimensional geometry and assume for a moment that the lead 3 divides the wire into two identical parts. Then in Eq. (119) R "R , F "F , and we obtain F"F /2. Thus, the Fano factor of the whole wire is one-half of the noise measured in each segment. This result, which describes local inelastic scattering in the middle of the wire, can be generalized to uniform inelastic scattering. For this purpose we introduce a certain length ¸ associated with G inelastic scattering; we assume that ¸ is much shorter than the total length of the wire ¸. One must G then consider initially a conductor with N "¸/¸ additional "ctitious voltage probes separated G G by distances ¸ along the conductor. In the fully incoherent case this picture is equivalent to G N identical classical resistors connected in series. The Fano factor of this system is then the Fano G factor of the phase-coherent segment divided by N . In particular, if the wire is di!usive, the G suppression factor is [74,82] F"(3N )\. Thus, the conclusion is the following: inelastic scattering G suppresses shot noise. A macroscopic system (large compared to an inelastic scattering length, N <1) exhibits no shot noise. This is a well-known fact, the absence of shot noise of macroscopic G conductors is used to stabilize lasers. Shimizu and Ueda [133] and Liu and Yamamoto [129,130] provided a similar discussion of noise suppression in the crossover regime between mesoscopic behavior and classical circuit theory (macroscopic behavior). Liu et al. and Yamamoto [91] performed Monte Carlo simulations of shot noise and included explicitly electron}phonon scattering.
Lund B+ and Galperin [59] performed microscopic calculation of noise in quantum wells in transverse magnetic "eld with the account of electron}phonon scattering (phonon-assisted tunneling). They found that the Fano factor is suppressed by inelastic scattering, as compared with non-interacting value. This result clearly contradicts to the conclusions of this subsection; we presently do not understand the reasons for this discrepancy. Of course, not only the average of the noise vanishes in macroscopic system, but also #uctuations. De Jong and Beenakker found r.m.s. SJN\ . The weak localization correction to shot noise decreases as N\. G G
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Quasi-elastic scattering. In contrast to inelastic scattering, dephasing processes leave the energy essentially invariant. To simulate a scattering process which destroys phase but leaves the energy invariant we now have to consider a special voltage probe. We require that the additional electrode conserves not only the total current, but also the current in each small energy interval [90]. Such a voltage probe will not give rise to energy relaxation and dissipates no energy. From the condition that the current in each energy interval vanishes we "nd that the distribution function in the reservoir of the voltage probe is given by G f (E)#G f (E) , (120) f (E)" G #G where f and f are the Fermi functions at the reservoirs 1 and 2, and the conductances are G "R\, G "R\. We have assumed again that there is no direct transmission between 1 and 2. Straightforward calculation gives for the Fano factor [90] R F #R F #R R #R R . (121) F" (R #R ) We analyze now this result for various situations. First, we see that for a ballistic wire divided by a dephasing electrode into two parts, F "F "0, shot noise does not vanish (unlike Eq. (119)). We obtain F"R R (R #R )\. Thus for a ballistic system, which is ideally noiseless, dephasing leads to the appearance of shot noise. For the strongly biased resonant double-barrier structure, we have F "F "1, and obtain again the result (78), which is thus insensitive to dephasing. For metallic di!usive wires, F "F ", Eq. (121) yields for the ensemble-averaged Fano factor F" independent on the location of the dephasing voltage probe, i.e. independent of the ratio of R and R . Thus, our consideration indicates that the noise suppression factor for metallic di!usive wires is also insensitive to dephasing, at least when the dephasing is local (in any point of the sample). This result hints that the Fano factor of an ensemble of metallic di!usive wires is not sensitive to dephasing even if the latter is uniformly distributed. Indeed, de Jong and Beenakker [82] checked this by coupling locally a dephasing reservoir to each point of the sample. Already at the intermediate stage their formulae coincide with those obtained classically by Nagaev [75]. This proves that introducing dephasing with "ctitious, energy-conserving voltage probes is in the limit of complete dephasing equivalent to the Boltzmann}Langevin approach. That inelastic scattering, and not dephasing, is responsible for the crossover to the macroscopic regime, has been recognized by Shimizu and Ueda [133]. The e!ect of phase breaking on the shot noise in chaotic cavities was investigated by van Langen and one of the authors [115]. For a chaotic cavity connected to reservoirs via quantum point contacts with N and N open quantum channels, the Fano factors vanish F "F "0, and since * 0 R "p /eN and R "p /eN , the resulting Fano factor is given by Eq. (96), i.e. it is identical * 0 with the result that is obtained from a completely phase coherent, quantum mechanical calculation. Thus for chaotic cavities, like for metallic di!usive wires, phase breaking has no e!ect on the ensemble-averaged noise power. Electron heating. This is the third kind of inelastic scattering, which implies that energy can be exchanged between electrons. Only the total energy of the electron subsystem is conserved.
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Physically, this corresponds to electron}electron scattering. Within the voltage probe approach, it is taken into account by including the reservoir 3, with chemical potential k determined to obtain zero (instantaneous) electrical current and a temperature ¹ , which is generally di!erent from the lattice temperature (or the temperature of the reservoirs 1 and 2), to obtain zero (instantaneous) energy #ux. For a detailed discussion we refer the reader to the paper by de Jong and Beenakker [82], here we only mention the result for two identical di!usive conductors at zero temperature. The Fano factor is in this case F+0.38, which is higher than the -suppression for the non interacting case. We will see in Section 6 that the classical theory also predicts shot noise enhancement for the case of electron heating. Intermediate summary. Here are the conclusions one can draw from the simple consideration we presented above. E Dephasing processes do not renormalize the ensemble-averaged shot noise power (apart from weak localization corrections, which are destroyed by dephasing). In particular, this statement applies to metallic di!usive wires, chaotic cavities, and resonant double-barrier structures. E Inelastic scattering renormalizes even the ensemble-averaged shot noise power: a macroscopic sample exhibits no shot noise. An exception is the resonant double-barrier structure, subject to a bias large compared to the resonant level width. Under this condition neither the conductance nor the shot noise of a double barrier are a!ected. E As demonstrated for metallic di!usive wires electron heating enhances noise. The last statement implies the following scenario for noise in metallic di!usive wires [80]. There exist three inelastic lengths, responsible for dephasing (¸ ), electron heating (¸ ) and inelastic scattering (¸ ). We expect ¸ (¸ (¸ . Indeed, requirements for dephasing (inelastic scattering) are stronger (weaker) than those for electron heating. Then for the wires with length ¸;¸ the Fano factor equals and is not a!ected by inelastic processes; for ¸ ;¸;¸ it is above , and for ¸<¸ it goes down and disappears as ¸PR. In Section 6 we will reconnect to the results presented here within the classical Boltzmann} Langevin approach.
3. Scattering theory of frequency-dependent noise spectra 3.1. Introduction: current conservation The investigation of frequency-dependent transport, in particular, noise, is important, since it can reveal information about internal energy scales of mesoscopic systems, not available from dc transport. On the other hand, the investigation of the dynamic noise is a more di$cult task than the investigation of quasi-static noise. This is true experimentally, since frequency-dependent
The temperature of the reservoirs 1 and 2 is zero. The temperature of the intermediate reservoir is in this case [82] k ¹ "((3/2p)e"<".
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measurements require a particularly careful control of the measurement apparatus (one wants to see the capacitance of the sample and not that of the coaxial cable connecting to the measurement apparatus), and it is also true theoretically. Addressing speci"cally mesoscopic systems, the conceptual di$culty is that generally it is meaningless to consider the dynamic response of non-interacting electrons. Since this point remains largely unappreciated in the literature, we give here a brief explanation of this statement. Consider the following system of equations of classical electrodynamics: 1 RE , E"! u , j"j # 4p Rt
(122)
u"!4po ,
(123)
Ro div j # "0 . Rt
(124)
Here j, u and o are the density of the electric current (particle current), the electric potential, and the charge density, respectively; E is the electric "eld. Eq. (122) states that the total current j is a sum of the particle current j and the displacement current, represented by the second term on the rhs. Eqs. (123) and (124) are the Poisson equation and the continuity equation, respectively; they must be supplemented by appropriate boundary conditions. We make the following observations. (i) Eqs. (122)}(124), taken together, yield div j"0: the total current density has neither sources nor sinks. This is a general statement, which follows entirely from the basic equations of electrodynamics, and has to be ful"lled in any system. Theories which fail to yield a source and sink-free total current density cannot be considered as correct. The equation div j"0 is a necessary condition for the current conservation, as it was de"ned above (Section 2). (ii) The particle current j is generally not divergenceless, in accordance with the continuity equation (124), and thus, is not necessarily conserved. To avoid a possible misunderstanding, we emphasize that it is not a mere di!erence in de"nitions: The experimentally measurable quantity is the total current, and not the particle current. Thus, experimentally, the fact that the particle current is not conserved is irrelevant. (iii) The Poisson equation (123), representing electron}electron interactions, is crucial to ensure the conservation of the total current. This means that the latter cannot be generally achieved in the free electron model, where the self-consistent potential u is replaced by the external electric potential. (iv) In the static case the displacement current is zero, and the particle current alone is conserved. In this case the self-consistent potential distribution u(r) is generally also di!erent from the external electrostatic potential, but to linear order in the applied voltage the conductance is determined only by the total potential di!erence. As a consequence of the Einstein relation the detailed spatial variation of the potential in the interior of the sample is irrelevant and has no e!ect on the total current. In mesoscopic physics the problem is complicated since a sample is always a part of a larger system. It interacts with the nearby gates (used to de"ne the geometry of the system and to control For simplicity, we assume the lattice dielectric constant to be uniform and equal to one.
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the number of charges in the system). Thus, a complete solution of the above system of equations is usually a hopeless task without some serious approximations. The theoretical task is to choose idealizations and approximations which are compatible with the basic conservation laws expressed by the above equations. For instance, we might want to describe interactions in terms of an e!ective (screened) interaction instead of the full long-range Coulomb interaction. It is then necessary to ensure that such an e!ective interaction indeed leads to the conservation of current [135,136]. Three frequency-dependent types of noise spectra should be distinguished: (i) "nite-frequency noise at equilibrium or in the presence of dc voltage, (ii) zero-frequency noise in the presence of an ac voltage; the resulting spectrum depends on the frequency of the ac-voltage, (iii) "nite-frequency noise in the presence of an ac voltage; this quantity depends on two frequencies. Here we are interested mostly in the "rst type of noise spectra; the second one is only addressed in Section 3.3. We re-iterate that, generally, one cannot "nd the ac conductance and the current #uctuations from a non-interacting model. Even the "nite frequency current}current correlations (noise) at equilibrium or in the presence of a dc voltage source, which are of primary interest in this section, cannot be treated without taking account interactions. A simple way to see this is to note that due to the #uctuation-dissipation theorem, the equilibrium correlation of currents in the leads a and b at "nite frequency, S (u), is related to the corresponding element of the conductance matrix, ?@ S (u)"2k ¹[G (u)#GH (u)]. The latter is the response of the average current in the lead a to ?@ ?@ @? the ac voltage applied to the lead b, and is generally interaction-sensitive. Thus, calculation of the quantity S (u) also requires a treatment of interactions to ensure current conservation. ?@ We can now be more speci"c and make the same point by looking at Eq. (51) which represents the #uctuations of the particle current at "nite frequency. Indeed, for u"0 the current conservation S "0 is guaranteed by the unitarity of the scattering matrix: the matrix A(a, E, E) ? ?@ (Eq. (44)) contains a product of two scattering matrices taken at the same energy, and therefore it obeys the property A(a, E, E)"0. On the other hand, for "nite frequency the same matrix ? A should be evaluated at two di!erent energies E and E# u, and contains now a product of two scattering matrices taken at diwerent energies. These scattering matrices generally do not obey the property sR (E)s (E# u)"d , and the current conservation is not ful"lled: S (u)O0. ? ?@ ?A @A ? ?@ Physically, this lack of conservation means that there is charge pile-up inside the sample, which gives rise to displacement currents. These displacement currents restore current conservation, and thus need to be taken into account. It is exactly at this stage that a treatment of interactions is required. Some progress in this direction is reviewed in this section later on. It is sometimes thought that there are situations when displacement currents are not important. Indeed, the argument goes, there is always a certain energy scale u , which determines the energy dependence of the scattering matrices. This energy scale is set by the level width (tunneling rate) C for resonant tunnel barriers, the Thouless energy E " D/¸ for metallic di!usive wires (D and ¸ are the di!usion coe$cient and length of the wire, respectively), and the inverse Ehrenfest time (i.e. the time for which an electron loses memory about its initial position in phase space) for chaotic cavities. The scattering matrices may be thought as energy independent for energies below u . Then for frequencies below u we have sR (E)s (E# u)&d , and the unitarity of the ? ?@ ?A @A scattering matrix assures current conservation, S "0. However, there are time-scales which ? ?@ are not set by the carrier kinetics, like RC-times which re#ect a collective charge response of the system. In fact, from the few examples for which the ac conductance has been examined, we know
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that it is the collective times which matter. Displacement currents can be neglected only, if we can assure that these collective times are much shorter than any of the kinetic time-scales discussed above. Furthermore, there are problems which can only be treated by taking interactions into account, even at arbitrarily low frequency: later on in this section we will discuss the RC-times of mesoscopic conductors capacitively coupled to a gate. The noise induced into a nearby capacitor is proportional to the square of the frequency. Naive discussions which do not consider the energy dependence of the scattering matrix cannot predict such currents. Let us at "rst consider the range of frequencies that are much smaller than any inverse kinetic time scale and smaller than any inverse collective response time. This case is in some sense trivial, since the energy dependence of the scattering matrices is neglected, and the system is now not probed on the scale u . As a consequence there is no novel information on the system compared to a zero-frequency noise measurement. In this case, as we will see, the entire frequency dependence of the noise is due to the frequency dependence of the Fermi functions. However, it is the lowfrequency measurements which are more easily carried out, and therefore there is some justi"cation to discuss noise spectra in this frequency interval. The rest of the section is organized as follows. Section 3.2 treats #uctuations of the particle current of independent electrons, either at equilibrium or in the presence of a dc bias, in the regime when the scattering matrices can be assumed to be frequency-independent. Section 3.3 generalizes the same notions for noise caused by an ac bias. Afterwards, we relax the approximation of the energy independence of scattering matrices, and in Sections 3.4 and 3.5 consider two simple examples. In both of them Coulomb interactions prove to be important. Though the theory of ac noise is far from being completed, we hope that these examples, representing the results available by now in the literature, can stimulate further research in this direction. In this section, we only review the quantum-mechanical description of frequency-dependent noise, based on the scattering approach. Alternatively, the frequency dependence of shot noise in di!usive conductors may be studied based on the classical Boltzmann}Langevin approach [140}145]. These developments are described in Section 6. 3.2. Low-frequency noise for independent electrons: at equilibrium and in the presence of dc transport 3.2.1. General consideration This subsection is devoted to low-frequency noise, in a regime where the scattering matrices are energy independent. We take the frequency, temperature and the voltage all below u , and below any frequencies associated with the collective response of the structure. For simplicity, we only consider the two-terminal case, k "e<, k "0, <50. We emphasize again that in this approach * 0 the internal energy scales of mesoscopic conductors cannot be probed, nor is there a manifestation of the collective modes. The frequency dependence of noise is entirely due to the Fermi functions.
Exceptions are frequency-dependent weak localization corrections [137] which depend in addition to the RC-time also on the dwell time [138], and perfect ballistic wires which have a charge neutral mode determined by the transit time as the lowest collective mode [139].
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Our starting point is Eq. (51), which in this form is given in Ref. [146]. Taking the scattering matrices to be energy independent, we write
e ¹ dE[ f (E, u)#f (E, u)] S(u),S (u)" L ** 00 ** 2p
L
# ¹ (1!¹ ) dE[ f (E, u)#f (E, u)] , L L *0 0* L
(125)
with the abbreviation f (E, u)"f (E)[1!f (E# u)]#[1!f (E)] f (E# u) . (126) ?@ ? @ ? @ Here the ¹ 's are, as before (energy independent) transmission coe$cients. Performing the L integration, we obtain
u
u#e< e 2 u coth ¹# ( u#e<) coth S(u)" L 2k ¹ 2k ¹ 2p
L
u!e< # ( u!e<) coth ¹ (1!¹ ) . (127) L L 2k ¹ L This formula expresses the noise spectral power for arbitrary frequencies, voltages, and temperatures (all of them are assumed to be below u ). The frequency-dependent functions in Eq. (127) are obtained already in the discussions of noise based on the tunneling Hamiltonian approach for junctions [24] (see also Refs. [26,147,27]). In this approach one expands in the tunneling probability, and consequently, to leading order, terms proportional to ¹ are disregarded. The full L L expression, Eq. (127), including the terms proportional to ¹ was derived by Khlus [15] assuming L from the outset that the scattering matrix is diagonal. It is a general result for an arbitrary scattering matrix, if the ¹ 's are taken to be the eigenvalues of tRt. Later the result of Khlus was L re-derived by Yang [148] in connection with the QPC; the many-channel case was discussed by Ueda and Shimizu [149], Liu and Yamamoto [130], and Schoelkopf et al. [150]. We note "rst that for u"0, Eq. (127) reproduces the results for thermal and shot noise presented in Section 2 (for the two-terminal case and energy-independent transmission coe$cients). Furthermore, at equilibrium (<"0) it gives the Nyquist noise,
u eu coth ¹ , (128) S (u)" L 2k ¹ p L as implied by the #uctuation-dissipation theorem. For zero temperature we obtain from Eq. (127)
e "u" L ¹L #e< L ¹L (1!¹L ), "u"(e< , (129) S(u)" p "u" ¹ ,
"u"'e< . L L The frequency dependence is given by a set of straight lines. For zero frequency, the result for shot noise is reproduced, S"(e
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equilibrium value, Eq. (128), determined by the zero-point quantum #uctuations, independent of the voltage. Finite temperature smears the singularities since now the Fermi functions are continuous. The noise spectrum is shown in Fig. 22. It can also be represented di!erently, if we de"ne excess noise (for k ¹"0) as the di!erence between the full noise power, Eq. (129), and equilibrium noise, S (u)"S(u)!e"u" ¹ /p. The excess noise is given by L L e S (u)" ¹ (1!¹ )(e
L and zero otherwise. Eq. (127) is general and valid for all systems under the conditions it was derived. Instead of discussing it for all the examples mentioned in Section 2, we consider only the application to a metallic di!usive wire. 3.2.2. Metallic diwusive wires Performing the disorder averages of transmission coe$cients with the distribution function (87), we "nd the result obtained earlier by Altshuler et al. [76],
u
u#e< 1 #( u#e<) coth S(u)" G 4 u coth 2k ¹ 2k ¹ 3 #( u!e<) coth
u!e< 2k ¹
,
(130)
where G"eN l/2p ¸ is the conductance of a wire with mean-free path l and length ¸. , An experimental investigation of the frequency-dependent noise in di!usive gold wires is presented by Schoelkopf et al. [150]; this paper reports one of the only two presently available measurements of ac noise. They "nd a good agreement with Eq. (130), where the electron temperature ¹ was used as a "tting parameter (the same for all frequencies). The results of Ref. [150] are plotted in Fig. 23 as a function of voltage for di!erent frequencies; theoretical curves are shown as solid lines. In the experiment, the highest frequency corresponded approximately to the Thouless energy, and in this regime Eq. (130) is well justi"ed. We have emphasized earlier that in this frequency regime no internal dynamics of the system is probed. The fact that Eq. (130) agrees with experiment is a consequence of the strong screening in metals. In poor metals, the RC-times might become long. A "nite value of the screening length may permit charge #uctuations and consequently modify the noise behavior even at relative low frequencies. We address this issue in Section 6. 3.2.3. Inelastic scattering Now one can ask: What is the e!ect of inelastic scattering on frequency-dependent noise in the regime, where the scattering matrices can be taken to be energy independent. This problem was studied by Ueda and Shimizu [149], and later by Zheng et al. [151], who included
Due to e!ects of electron heating; see Section 6.
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Fig. 22. Frequency dependence of noise for zero temperature (Eq. (129), solid line), and "nite temperatures (dash-dotted line, 2). Line 1 shows equilibrium noise, S"e"u" ¹ /p. Line 3 corresponds to the upper line of Eq. (129). In this "gure L L we set "1. Fig. 23. Experimental results of Schoelkopf et al. [150] for the frequency dependence of noise in metallic di!usive wires. Solid lines for each frequency indicate the theoretical result (130). The quantity shown on the vertical axis is essentially RS/R<. Copyright 1997 by the American Physical Society.
electron}phonon interaction directly into the scattering approach, and by Liu and Yamamoto [130], who used an approach based on dephasing voltage probes. The general conclusion is, that like in the case of the zero frequency, inelastic scattering suppresses noise. 3.3. Low-frequency noise for independent electrons: photon-assisted transport Now we generalize the results of the preceding subsection to the case when the applied voltage is time-dependent; the scattering matrices are still assumed not to depend on energy. The #uctuations in the presence of a potential generated by an ac magnetic #ux were treated by Lesovik and Levitov [152]; the #uctuation spectrum in the presence of oscillating voltages applied to the contacts of the sample was obtained in Ref. [153]. The results are essentially the same; below we follow the derivation of Ref. [153]. We consider a two-terminal conductor; the chemical potential of the right reservoir is kept "xed (we assume it to be zero), while the left reservoir is subject to a constant voltage <M plus the oscillating component ;(t)"<(X) cos Xt. One cannot simply use the stationary scattering theory as described in Section 2. Instead, the scattering states in the left lead are now solutions of the
In contrast, the e!ect of dephasing on the "nite frequency noise seems not to have been investigated.
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time-dependent SchroK dinger equation,
e<(X) t (r, E, t)"s (r )e I*L X\ #R J e\ JXR . (131) L *L , J X J\ Thus, in the presence of an oscillating voltage each state with the central energy E is split to in"nitely many subbands with energies E#l X, which have smaller spectral weight. Following the literature on tunneling, this phenomenon is called photon-assisted transport, since electrons with higher energies (l'0) have higher transmission probabilities, and might propagate through the sample owing to the additional energy. Now we use a formal trick, assuming that the oscillating potential only exists asymptotically far from the sample (and there Eq. (131) is valid), and decays slowly towards the sample. Thus, there is a certain portion of the left lead, where there is no oscillating potential, but still no scattering. The annihilation operators in this part of the left lead have thus the form
e<(X) a( (E!l X) , a( (E)" J *L J X *L J where the operators a( describe the states of the left reservoir. Instead of Eq. (34), we obtain
e e< e< ? J @ IK (t)" dE dE e #\#YR J * J X I X 2p
?@ KL JI ;a( R (E!l X)AKL(¸; E, E )a( (E !k X) , ?K ?@ @L
(132)
and we set < "<(X), < "0. Finally, we assume that the frequency is not too high, so that the left * 0 reservoir can be considered to be at (dynamic) equilibrium at any instant of time. Then the averages of the operators a( are essentially equilibrium averages, expressed through the Fermi functions, f "f (E!e<M ) and f "f (E). * $ 0 $ In the presence of a time-dependent voltage, the correlation function S (49) depends not only ** on the time di!erence t!t , but also on the absolute time q"(t#t )/2. In the following, we are interested in the noise spectra on a time scale long compared to X\. Then the noise power can be averaged over q,
1 O dq S(t!t , q), q"2p/X . S(t!t )" q Leaving more general cases aside, we only give an expression for the zero-frequency component of S [152,153], ** e e(< !< ) ? @ Tr[A (¸)A (¸)] f (E, l X) , S (u"0, X)" dE J (133) ** J ?@ @? ?@ 2p
X ?@ J where the matrices A are explicitly assumed to be energy independent. For zero external frequency, X"0, only the Bessel function with l"0 survives, and should be taken equal to one for any a and b; then we reproduce the zero-frequency expression (52). Performing the energy integration and
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introducing the transmission coe$cients, we obtain [154]
e<(X) e 4k ¹ ¹# J ¹ (1!¹ ) S(X)" L J X L L 2p
L J\ L l X#e<M l X!e<M ; (l X#e<M ) coth #(l X!e<M ) coth 2k ¹ 2k ¹
.
(134)
For zero temperature, Eq. (134) exhibits singularities at voltages <M "l X/e: The derivative RS/R<M is a set of steps. The height of each step depends on the ac voltage due to the Bessel function in Eq. (134). Lesovik and Levitov [152] considered a geometry of (an almost closed) one-channel loop of length ¸ connected to two reservoirs. The loop contains a scatterer, and is pierced by the time-dependent magnetic #ux U(t)"U sin Xt. The time-dependent #ux generates an electric "eld ? and thus an internal voltage ;(t)"; cos Xt with e; "2p(U /U )(¸/2pR) X, where ¸ is the ? ? ? length of the segment on the circle with radius R. In addition, a constant voltage ;M is applied. The magnetic #ux can be incorporated in the phase of the scattering matrix, and the previous analysis is easily generalized for this case. Ref. [152] found that for zero temperature RS/R;M is again a step function of voltage. Steps occur at ;M "l X/e, and the height of each step is j "¹(1!¹)J(e; / X), where ¹ is the transmission coe$cient. In Ref. [152] the argument of the J J ? Bessel function is written in terms of the ratio of #uxes 2pU /U and the e!ect is called a non? stationary Aharonov}Bohm ewect. However, we emphasize that what is investigated is the response to the external electric "eld generated by the oscillating #ux. This is a classical response, unrelated to any Aharonov}Bohm-type e!ect. It is remarkable that in the case of energy-independent transmission probabilities the response to the electric "eld considered in Ref. [152] is the same as that of an oscillating voltage <,; ? applied to a contact. Levinson and WoK l#e [155] considered a related problem: the noise for the transmission through a barrier with an oscillating random pro"le (originating, for instance, from the external irradiation). The latter is represented by a one-dimensional potential ;(x, t)"; (x)#d;(x, t) . The random component d; is assumed to be zero on average, and its second moment is a function of t!t . In this case, current}current correlations, 1dI(t)dI(t )2, for each particular realization of the random potential depend on both times t and t . However, after averaging over disorder realizations, the resulting noise only depends on t!t and can be Fourier transformed. The scattering matrices are energy independent for frequencies below the inverse time of #ight through the barrier, u;v /¸, with ¸ being the length of the barrier. $ A remarkable feature of this model is that if the barrier and the irradiation are symmetric, ; (x)"; (!x), and d;(x, t)"d;(!x, t), and no voltage is applied between the reservoirs, there is no current generated by the irradiation. On the other hand, a non-equilibrium contribution to noise exists. In particular, when the second moment of the random potential is d;(x, t)d;(x , t )"< d(x!x ), it happens to have the same frequency structure as the equilibrium one (128), but with the coe$cient proportional to < . The voltage applied to the reservoirs is, as
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Fig. 24. (a) Theoretical results (134) for various amplitudes of the ac voltage; dotted line shows the dc results. (b) Experimental results of Schoelkopf et al. [154] for the same parameters. (c) Experimental results plotted as RS/R<M for di!erent ac voltage amplitudes. The frequency X is "xed. Copyright 1998 by the American Physical Society.
usual, one more source of non-equilibrium noise. For further details, we refer the reader to Ref. [155]. Experimentally, noise in response to a simultaneous dc voltage and ac voltage applied to the contacts of the sample was studied by Schoelkopf et al. [154] in phase coherent metallic di!usive wires. They measured zero-frequency noise as a function of voltage <M in the GHz range. The results are presented in Fig. 24 as RS/R<M and RS/R<M . The latter quantity is expected to have sharp peaks at the resonant voltages <M "l X/e. Indeed, three peaks, corresponding to l"0,$1 are clearly seen; others are smeared by temperature and not so well pronounced. Ref. [153] emphasized the need for a self-consistent calculation of photon-assisted transport processes even in the case that the only quantity of interest are the currents or noise-spectra measured at zero frequency. The true electric "eld in the interior of the conductor is not the external "eld. The fact that the experimental results agree rather well with the simple results presented here (which do not invoke any self-consistency) is probably a consequence of the e!ective screening of the metallic conductor. The true potential is simply linear in the range of frequencies
As usual, for metallic di!usive wires ¹ and ¹ (1!¹ ) must be replaced by 2lN /3¸ and lN /3¸, respectively. L L L L L , ,
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investigated experimentally. A self-consistent spectrum for photon-assisted noise spectra can probably be developed along the lines of Ref. [153]. 3.4. Noise of a capacitor Now we turn to the problems where the energy dependence of the scattering matrices is essential. Rather than trying to give the general solution (which is only available for the case when the potential inside the system is spatially uniform [156,157]), we provide a number of examples which could serve as a basis for further investigations of "nite frequency noise. The simple case of shot noise in a ballistic wire was studied by Kuhn et al. [158,159]. They found signatures of the inverse #ight time v /¸. However, the interactions are taken into account $ only implicitly via what the authors call `quantum generalization of the Ramo}Shockley theorema. We do not know in which situations this approach is correct, and it certainly cannot be correct universally. Thus, even this simple case cannot be considered as solved and needs further consideration. We start from the simplest system } a mesoscopic capacitor (Fig. 25a), which is connected via two leads to equilibrium reservoirs. Instead of the full Poisson equation interactions are described with the help of a geometrical capacitance C. There is no transmission from the left to the right plate, and therefore there is no dc current from one reservoir to the other. Moreover, this system does not exhibit any noise even at "nite frequency if the scattering matrix is energy independent. Indeed, if only the matrices s and s are non-zero, we obtain from Eq. (51) ** 00 e d dE Tr +[1!sR (E)s (E# u)] S(u)" ?? ?? ?@ 2p ?@ ;[1!sR (E# u)s (E)],f (E)(1!f (E# u)) . (135) ?? ?? ? ? Here we used the superscript (0) to indicate that the #uctuations of the particle current, and not the total current, are discussed. If the scattering matrices are energy independent, Eq. (135) is identically zero due to unitarity. Another way to make the same point is to note that since S "0, *0 the only way to conserve current would be S "0. ** Before proceeding to solve this problem, we remark that Eq. (135) describes an equilibrium #uctuation spectrum and via the #uctuation dissipation theorem S(u)" ?? 2 ug (u) coth ( u/2k ¹) is related to the real part of a conductance g (u) given by ?? ?? f (E)!f (E# u) e ? . (136) g (u)" dE Tr+[1!sR (E)s (E# u)], ? ?? ?? ??
u
Fig. 25. A mesoscopic capacitor (a); a mesoscopic conductor vis-a`-vis a gate (b).
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In what follows, the #uctuation dissipation theorem for the non-interacting system also ensures this theorem for the interacting system. Current conservation is restored only if interactions are taken into account. Below we assume that charging e!ects are the only manifestation of interactions [157,131]. We "rst present the general result which is, within the limitations stated above, valid for arbitrary frequencies. We then consider in detail the low-frequency expansion of this result which can be expressed in physically appealing quantities: an electrochemical capacitance and a charge relaxation resistance. 3.4.1. General result Our starting point are the particle current operators in the left and right lead (43), IK (t) and * IK (t). Their #uctuation spectra are determined by Eq. (135). Now we must take into account that 0 the total currents are in fact not just the particle currents but contain an additional contribution generated by the #uctuating electrostatic potential on the capacitor plates. We introduce the operators of the potential on the left u( (t) and right u( (t) plate. The #uctuation of the total current * 0 through the lead a can be written in operator form as follows:
*IK (t)"dIK (t)# dt s (t!t )du( (t ), a"L, R . ? ? ? ?
(137)
Here we introduced dIK (t)"IK (t)!1I2 and du( (t)"u( (t)!1u 2. Furthermore, s is the ? ? ? ? ? ? ? response function which determines the current generated at contact a in response to an oscillating potential on the capacitor plate. For the simple case considered here, it can be shown [157] that this response function is directly related to the ac conductance, Eq. (136), for non-interacting electrons, which gives the current through the lead a in response to a voltage applied to the same lead, s (u)"!g (u). The minus sign is explained by noting that the current #uctuation is the ? ?? response to k !u rather than u . The operators *IK , and not dIK , determine the experimentally ? ? ? measured quantities. The total current in this system is the displacement current. In the capacitance model the charge of the capacitor QK is given by QK "C(u( !u( ). Note that this is just the Poisson equation expressed * 0 with the help of a geometrical capacitance. To the extent that the potential on the capacitor plate can be described by a uniform potential this equation is valid for all frequencies. Then the #uctuations of the current through the left and right leads are R R *IK (t)" dQK (t)"C [du( (t)!du( (t)] , * 0 * Rt Rt
(138)
and *IK (t)"!*IK (t). Thus, the conservation of the total current is assured. In contrast to the * 0 non-interacting problem, the currents to the left and right are now completely correlated. Eqs. (137) and (138) can be used to eliminate the voltage #uctuations. The result is conveniently expressed in the frequency representation, iuC *IK "!*IK " [g dIK !g dIK ] . * 0 g g !iuC(g #g ) ** 0 00 * ** 00 ** 00
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The noise power S,S becomes ** uC ["g "S #"g "S ] . (139) S(u)" ** 00 "g g !iuC(g #g )" 00 ** ** 00 ** 00 Eq. (139) expresses the frequency-dependent noise spectrum of the interacting system in terms of the conductances g (u), g (u) and the noise spectra S , S of the problem without interactions. ** 00 ** 00 Together with Eq. (135), this is the result of the "rst step of our calculation. Note that the noise of the capacitor depends only implicitly, through the scattering matrices and the Fermi functions, on the stationary (dc) voltage di!erence across the capacitor. Eq. (139) is thus valid independently on whether the potentials in the two reservoirs are the same or not. So far, our consideration is valid in the entire frequency range up to the frequencies at which the concept of a single potential no longer holds: u;e/ed, where e and d are the static susceptibility and the distance between the capacitor plates, respectively. 3.4.2. Low-frequency expansion Now we turn to the low-frequency expansion of Eq. (139), leaving only the leading term. The expansion of g can be easily obtained [157], ?? g (u)"!iuel A #O(u) , ?? ? ? Rf Rs (E) 1 dE ? Tr sR (E) ?? , (140) l A "! ?? ? ? RE RE 2pi
where l is the density of states per unit area and A is the area of the cross-section of the plate a. ? ? The fact that the density of states can be expressed in terms of the scattering matrix is well known [160,161]. Expanding Eq. (135), we write
Rf e uk ¹ dE ! ? S" ?? RE p
Rs (E) Tr sR (E) ?? #O("u") . ?? RE
Now we introduce the electrochemical capacitance, C\,C\#(el A )\#(el A )\ , I * * 0 0 and the charge relaxation resistances,
(141)
Rf Rs (E)
dE ! ? Tr sR (E) ?? . (142) R " ?? O? 4pelA RE RE ? ? These two quantities, which determine the RC-time of the mesoscopic structure, now completely specify the low-frequency noise of the capacitor. A little algebra gives [157,131] S"4k ¹uC(R #R )#O(u) . I O* O0
(143)
For zero temperature, the expansion of Eq. (135) starts with a term proportional to "u" rather than u. As a result, 4k ¹ is replaced by 2 "u" in the "nal expression (143) for noise S.
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We remark that the charge relaxation resistance is determined by half the resistance quantum p /e and not 2p /e, re#ecting the fact that each plate of the capacitor is coupled to one reservoir only. We note now that the low-frequency expansion of the ac conductance (admittance) of the same capacitor [157] has a form G(u)"!iuC #uC(R #R )#O(u) . I I O* O0
(144)
Thus, we see that the #uctuation-dissipation theorem is also obeyed for the interacting system. Again, the noise spectrum of this equilibrium system contains the same information as the admittance. We emphasize once more that it is not required that the electrochemical potential of the two plates of the capacitor are identical. The results given above also hold if there is a large dc voltage applied across the capacitor. The electrochemical capacitance and the charge relaxation resistance have been evaluated for a number of examples. In the limit of one quantum channel only, the charge relaxation resistance is universal, independent of the properties of the scattering matrix, and given by R "p /e. This is O astonishing in view of the fact that if a tunnel barrier is inserted in the channel connecting the capacitor plate to the reservoir one would expect a charge relaxation resistance that diverges as the tunnel barrier becomes more and more opaque. For a chaotic cavity connected via a perfect single-channel lead to a reservoir and coupled capacitively to a macroscopic gate, the distribution of the electrochemical capacitance has been given in Ref. [162] (see also Ref. [356]). In this case the charge relaxation resistance, as mentioned above, is universal and given by R "p /e. For O a chaotic cavity coupled via an N-channel quantum point contact to a reservoir and capacitively coupled to a macroscopic gate, the capacitance and charge relaxation resistance can be obtained from the results of Brouwer and one of the authors [138]. For large N, for an ensemble of chaotic cavities, the capacitance #uctuations are very small, and the averaged charge relaxation resistance is given by R "2p /eN. If a tunnel barrier is inserted into the contact, the ensemble-averaged O resistance is [163] R "2p /e¹N for a barrier which couples each state inside the cavity with O transmission probability ¹ to the reservoir. In accordance with our expectation the charge relaxation resistance is determined by the two-terminal tunnel barrier resistance R "2p /e¹N. R For additional examples we refer the reader to Ref. [164] which presents an overview of the known charge relaxation resistances R for mesoscopic conductors. O 3.5. Shot noise of a conductor observed at a gate It is interesting to ask what would be measured at a gate that couples capacitively to a conductor which is in a transport state. In such a situation, in the zero-temperature limit, the low-frequency noise in the conductor is the shot noise discussed in this Review. Thus we can ask, what are the current #uctuations capacitively induced into a gate due to the shot noise in the nearby conductor? To answer this question we consider a mesoscopic conductor vis-a`-vis a macroscopic gate [98]. The whole system is considered as a three-terminal structure, with L and R labeling the contacts of the conductor, and G denoting the gate (Fig. 25b). The gate and the conductor are coupled capacitively with a geometrical capacitance C. For a macroscopic gate the #uctuations of the potential within the gate are small and can be neglected. Finally, the most crucial assumption is that the potential inside the mesoscopic conductor is uniform and may be described by a single
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(#uctuating) value u. This assumption is often made in the discussion of the Coulomb blockade e!ect, but it is in reality almost never satis"ed. We provide a solution to this problem by extending the discussion of the mesoscopic capacitor. We start from the operators of the particle currents, IK . We have IK "0, since without interac? % tions there is no current through a macroscopic gate, and correlations of other current operators are given by Eq. (51). We introduce the operator of potential #uctuation du( (t) in the conductor; the charge #uctuation is dQK (t)"Cdu( (t). Since there is transmission from the left to the right, we write for the #uctuations of the total currents,
*IK (t)"dIK (t)# ? ?
R
dt s (t!t )du( (t ), a"L, R , ?
\ *IK (t)"!C(R/Rt)du( (t) , (145) % with the condition *IK #*IK "CRdu( (t)/Rt, which ensures the current conservation. It is impor* 0 tant that the quantities s , which determine the response of currents at the terminals to the ? potential inside the sample, must be evaluated at equilibrium, and, since this potential is time dependent, the ac current should be taken. The entire dependence on u of the average current is due to scattering matrices. In the semi-classical approximation they depend on the combination E!eu, and the derivative with respect to the internal potential is essentially the derivative with respect to energy. Retaining only the leading order in frequency, we obtain (see e.g. [136]) s (u)"iueN #O(u) , ? ? Rf Rs RsR 1 dE Tr sR ?@ ! ?@ s , (146) N "! ?@ ? RE RE RE ?@ 4pi @ which is the analog of Eq. (140) for a multi-probe conductor. The quantities N , called emittances ? in Ref. [136], obey the rule N #N "lA (with l and A being the density of states and the * 0 area/volume of the conductor, respectively). They have the meaning of a density of the scattering states which describes the electrons exiting eventually through the contact a, irrespectively of the contact they entered through. Combining Eqs. (145) and (146), we obtain
*IK "[1!eN K]dIK !eN KdIK , * * * * * *IK "!eN KdIK #[1!eN K]dIK , 0 0 * 0 0 *IK "!CK[dIK #dIK ] , % * 0 There are no quantum-mechanical calculations of frequency-dependent noise with the potential pro"le taken into account available in the literature. However, when the conductor is a perfect wire, with a nearby gate, the existing calculation of ac conductance [165] can be generalized to calculate noise. One has to start from the "eld operators, write a density operator, and solve the Poisson equation as an operator equation for the "eld operator of the (electro-chemical) potential. This lengthy calculation leads to an obvious result: there is no non-equilibrium noise in the absence of backscattering. Though this outcome is trivial, we hope that the same approach may serve as a starting point to solve other problems, like a wire with backscattering. A related discussion was developed classically for the frequency dependent noise in di!usive conductors [140}145], see Section 6.
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where K,(C#eN #eN )\ plays the role of an e!ective interaction which determines the * 0 change in the electrostatic potential inside the conductor in response to a variation of the charge inside the conductor. In the following, we are only interested in the #uctuations of the current through the gate. For this quantity, which vanishes for zero frequency, we obtain S "CK[S #S #S #S ] . (147) %% ** *0 0* 00 We note that the sum of all the current #uctuation spectra in Eq. (147) is just the #uctuation spectrum of the total charge on the conductor: The continuity equation gives [98] K (u) where N K is the operator of the charge in the mesoscopic conductor. From the IK (u)"iueN ? ? current operator, Eq. (43), we obtain
e dE dE e #\#YR a( R (E)NKL(E, E )a( (E ) , (148) N K (t)" @K @A AL 2p
?@A KL with the non-diagonal density of states elements NKL @A 1 NKL(E, E )" d d d ! sR (E)s (E ) , (149) @A KL ?@ ?A ?@_KI ?A_IL 2pi(E!E ) ? I or in matrix notation N "(1/2piu) A (a, E, E# u) with the current matrix A (a, E, E ) given ? @A @A @A by Eq. (44). Thus, instead of Eq. (147) we can also express the #uctuation spectrum of the current at the gate in terms of the charge #uctuation spectrum
with
S "eCKuS , %% ,,
(150)
e dE NKL(E, E# u)NLK(E# u, E) S (u)" AB BA ,, 2p
AB KL ;+ f (E)[1!f (E# u)]#[1!f (E)] f (E# u), . (151) A B A B Eq. (151) is, in the absence of interactions, the general #uctuation spectrum of the charge on a mesoscopic conductor. In the zero-temperature limit, we obtain [98]
R "u"#R (e
"u"'e< , O where <'0 is the voltage applied between left and right reservoirs. Here the electro-chemical capacitance is C\"C\#[e(N #N )]\ , I * 0 the charge relaxation resistance reads
(153)
p
\ R " , Tr(N NR ) Tr N O e AB AB AA AB* 0 A
(154)
In terms of the previous subsection, this is the charge relaxation resistance of the conductor. The charge relaxation resistance of the macroscopic gate is zero.
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and the non-equilibrium resistance is
2p
\ R " Tr(N NR ) Tr N , 4 *0 *0 AA e A with the notation
(155)
Rs 1 N " sR ?B . ?A RE AB 2pi ? It can be checked easily that equilibrium noise, S "2CR "u", satis"es the #uctuation%% I O dissipation theorem. The resistance R can be extracted from the ac conductance as well. However, O non-equilibrium noise is described by another resistance, R , which is a new quantity. It probes 4 directly the non-diagonal density of states elements N of the charge operator. The non-diagonal *0 density of states elements which describe the charge #uctuations in a conductor in the presence of shot noise can be viewed as the density of states that is associated with a simultaneous current amplitude at contact c and contact d, regardless through which contact the carriers leave the sample. These density of states can be also viewed as blocks of the Wigner}Smith time delay matrix (2pi)\sR ds/dE. For a saddle-point quantum point contact the resistances R and R are evaluated in Ref. [98]. O 4 In the presence of a magnetic "eld R has been calculated for a saddle-point model by one of the 4 authors and Martin [166], and is shown in Fig. 26. For a chaotic cavity connected to two single channel leads both resistances are random quantities, for which the whole distribution function is known [98]. Thus, the resistance R (in units of 2p /e) assumes values between and , O with the average of (orthogonal symmetry) or (unitary symmetry). The resistance R lies in the 4 interval between 0 and , and is on average and for orthogonal and unitary symmetry, respectively.
Fig. 26. R (solid line, in units of 2p /e) and the conductance G (dashed line, in units of e/2p ) as a function of E !< 4 $ for a saddle point QPC with u /u "1 and u /u "4, u is the cyclotron frequency. After Ref. [166]. Copyright 2000 V W V by the American Physical Society.
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4. Shot noise in hybrid normal and superconducting structures The dissipationless current (supercurrent) in superconductors is a property of a ground state, and therefore is noiseless } it is not accompanied by any #uctuations. However, noise appears if the superconductor is in contact with piece(s) of normal metals. This section is devoted to the description of shot noise in these hybrid structures, which exhibit a variety of interesting phenomena. Following the point of view of the previous sections, we present here a description which is based on an extension of the scattering approach to hybrid structures. 4.1. Shot noise of normal-superconductor interfaces 4.1.1. Simple NS interface, scattering theory and general expressions We consider "rst an interface of normal metal and superconductor (NS). If the applied voltage is below the superconducting gap *, the only mechanism of charge transport is Andreev re#ection at the NS interface: an electron with energy E approaching the interface from the normal side is converted into a hole with energy !E. The velocity of the hole is directed back from the interface to a normal metal. The missing charge 2e on the normal side appears as a new Cooper pair on the superconducting side. There is, of course, also a reverse process, when a Cooper pair recombines with a hole in the normal conductor, and creates an electron. At equilibrium, both processes have the same probability, and there is thus no net current. However, if a voltage is applied, a "nite current #ows across the NS interface. The scattering theory which we described in Section 2 has to be extended to take into account the Andreev scattering processes [167]. We give here only a sketch of the derivation; a more detailed description, as well as a comprehensive list of references, may be found in Refs. [72,168]. To set up a scattering problem, we consider the following geometry (Fig. 27): the boundary between normal and superconducting parts is assumed to be sharp, and elastic scattering happens inside the normal metal (shaded region) only. The scattering region is separated from the NS interface by an ideal region 2, which is much longer than the wavelength, and thus we may there use asymptotic expressions for the wave functions. This spatial separation of scattering from the interface is arti"cial. It is not really necessary; our consideration leading to Eq. (159) and (160) does not rely on it. Furthermore, for simplicity, we assume that the number of transverse channels in the normal lead 1 and in the intermediate normal portion 2 is the same. We proceed in much the same way as in Section 2 and de"ne the annihilation operators in region 1 asymptotically far from the scattering area, a( , which annihilate electrons incoming on L
Fig. 27. A simpli"ed model of an NS interface adopted for the scattering description. Scattering is assumed to happen in the shaded area inside the normal metal, which separates the ideal normal parts 1 and 2. Andreev re#ection is happening strictly at the interface separating 2 and the superconductor.
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the sample. These electrons are described by wave functions s (r ) exp(ik z) with unit incident L , $ amplitude, where the coordinate z is directed towards the superconductor (from the left to the right in Fig. 27), the index n labels the transverse channels. Here we neglected the energy dependence of the wave vector, anticipating the fact that only energies close to the Fermi surface will play a role in transport. Similarly, the operator bK annihilates electrons in the outgoing states in the region 1, L s (r ) exp(!ik z). L , $ For holes, we de"ne an annihilation operator in the incoming states in the region 1 as a( , and L the corresponding wave function is s (r ) exp(!ik z). Note that though this wave function is L , $ identical to that for outgoing electrons, it corresponds to the incoming state with energy !E. The velocity of these holes is directed towards the interface. The annihilation operator for holes in the outgoing states, bK , is associated with the wave function s (r ) exp(ik z). Creation operators for L L , $ electrons and holes are de"ned in the same way. Thus, the di!erence with the scattering theory for normal conductors is that we now have an extra index, which assumes values e, h and discriminates between electrons and holes. The electron and hole operators for the outgoing states are related to the electron and hole operators of the incoming states via the scattering matrix,
bK a( s "s , bK a( s
s s
a( , a(
(156)
where the element s gives the outgoing electron current amplitude in response to an incoming electron current amplitude, s gives the outgoing hole current amplitude in response to an incoming electron current amplitude, etc. The generalized current operator (32) for electrons and holes in region 1 is
e dE dE e #\#YR Tr [a( R (E)a( (E )!a( R (E)a( (E ) IK (t)" * 2p
!bK R (E)bK (E )#bK R (E)bK (E )] ,
(157)
or, equivalently,
e IK (t)" dE dE e #\#YR Tr [a( R (E)A (E, E )a( (E )] , (158) * ? ?@ @ 2p
?@ where we have again introduced electron}hole indices a and b, and the trace is taken over channel indices. The matrix A is given by
A(E, E )"K!sR(E)Ks(!E ), K"
1
0
0 !1
,
with the matrix K discriminating between electron and holes. Introducing the distribution functions for electrons f (E)"[exp[(E!e<)/k ¹]#1]\ and holes f (E)"[exp[(E#e<)/ k ¹]#1]\, and acting in a similar way as in Section 2 for normal systems, we obtain from the current operator and the usual quantum statistical assumptions for the averages
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and correlations of the electron and hole operators in the normal reservoir the zero-temperature conductance e G" Tr [sR s ] p
(159)
and the shot noise power 4e"<" Tr [sR s (1!sR s )] S" p
(160)
in the zero-temperature limit up to linear order in the applied voltage. Here we have made use of the unitarity of the scattering matrix, in particular, sR s #sR s "1, and of the particle}hole symmetry. As a consequence, both the conductance and the noise can be expressed in terms of s only. We emphasize that Eqs. (159) and (160) are completely general: In particular, they do not require a clean NS interface and the spatial separation of the scattering region of the normal conductor from the interface. However, without such additional assumptions the evaluation of the scattering matrix can be very di$cult. To gain more insight we now follow Beenakker [72] and assume, as shown in Fig. 27, that a perfect region of normal conductor is inserted between the disordered part of the conductor and the NS-interface. In region 2, incoming states for electrons and outgoing states for holes have wave functions proportional to exp(!ik z), while outgoing states for electrons and incoming states for $ holes contain the factor exp(ik z). We also de"ne annihilation operators a( , a( , bK , bK , and $ L L L L creation operators for this region. The scattering inside the normal lead is described by a 4N ;4N scattering matrix s , , , (N being the number of transverse channels), , bK a( bK a( "s , (161) , a( bK bK a( where operators like bK are vectors, each component denoting an individual transverse channel. The elastic scattering in the normal region does not mix electrons and holes, and therefore in the electron}hole decomposition the matrix s is diagonal, , s (E) 0 r t , s (E)" . s " , 0 sH(!E) t r Here s (E) is the usual 2N ;2N scattering matrix for electrons, which contains re#ection and , , transmission blocks. To leading order in D/E (if both the normal conductors and the superconductor have identical $ Fermi energies) Andreev re#ection at a clean interface is described by a 2N;2N scattering matrix, which is o!-diagonal in the electron}hole decomposition, and is given by
a( 0 exp(i ) "c a( exp(!i ) 0
bK , bK
(162)
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with c,exp[!i arccos(E/D)]. With some algebra we can now "nd expressions for the scattering matrices s , s ,2 in terms of the Andreev re#ection amplitude and the scattering matrix of the normal region [72], s (E)"r (E)#ct (E)rH (!E)[1!cr (E)rH (!E)]\t (E) , s (E)"c exp(i )t (E)[1!crH (!E)r (E)]\tH (!E) , s (E)"c exp(!i )tH (!E)[1!cr (E)rH (!E)]\t (E) ,
(163)
s (E)"rH (!E)#ctH (!E)r (E)[1!crH (!E)r (E)]\tH (!E) . These matrices express amplitudes for an electron (hole) incoming from the left to be eventually re#ected as an electron (hole). The corresponding probability is given by the squared absolute value of the matrix element. In the following, we only consider the case when no magnetic "eld is applied to the structure. Then the matrix s (E) is symmetric. For e"<";D one has c"!i. Taking again the particle}hole symmetry into account, we obtain with the help of Eqs. (159), (162) the conductance ¹ e L , G" p (2!¹ ) L L
(164)
and using Eq. (160), the shot noise 4e"<" 16e"<" ¹(1!¹ ) L , S" Tr [sR s (1!sR s )]" L (2!¹ ) p
p
L L
(165)
where ¹ are eigenvalues of the matrix tR t , i.e. transmission eigenvalues of the normal region L (evaluated at the Fermi surface). As in normal conductors, channels with ¹ "0 and ¹ "1 do not L L contribute to the noise. Note that it is the fact that we have chosen to express the conductance and the noise in terms of the eigenvalues of the normal region which gives rise to the non-linear eigenvalue expressions given by Eqs. (164), (165). In terms of the eigen-channels of s the resulting expression would be formally identical to the conductance and the noise of a normal conductor. Expression (165) was obtained by Khlus [15] using a Keldysh approach for the case when the normal metal and the superconductor are separated by a tunnel barrier. He also investigated the "nite temperature case and derived the Nyquist noise. The results were re-derived within the scattering approach by Muzykantskii and Khmelnitskii [169]. The general case was studied by de Jong and Beenakker [170] in the framework of the scattering approach; we followed their work in the course of the above derivation. Martin [171] obtains the same results using statistical particle counting arguments and investigates the crossover between shot and thermal noise (see below). 4.1.2. Applications If the normal and superconducting electrodes are separated by a tunnel barrier, all the transmission coe$cients ¹ can be taken to be the same, ¹ "¹ (not to be confused with temperature). L L
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Then we obtain ¹ eN , G" , p (2!¹) 16e"<"N ¹(1!¹) , S" , p
(2!¹) with N the number of transverse channels. For the Fano factor this yields [15,170] , S 8(1!¹) F" " . 2eG< (2!¹)
(166)
For low transparency ¹;1 the Fano factor tends to the value of 2. This corresponds to the notion that shot noise in NS junctions is essentially the result of uncorrelated transfer of particles with charge 2e. The shot noise is super-Poissonian (F'1) for ¹(2((2!1)+0.83. For open barriers (¹"1) the shot noise vanishes. Refs. [15,171] have also shown that the crossover between shot and Nyquist noise happens at the temperature k ¹"2e"<", which is one more manifestation of the doubling of the e!ective charge. For the case of a disordered normal metal, Eqs. (164) and (165) have to be averaged over impurity con"gurations. Using the distribution function of the transmission coe$cients in the disordered region (87), we obtain for the Fano factor [170] F". This is twice as high as for a normal disordered wire. Mace( do [85] obtains the weak localization correction and mesoscopic #uctuations of the shot noise power. In particular, he "nds that the mean square of the shot noise power scales as the shot noise power itself, and in this sense the #uctuations are universal (as for normal di!usive conductors). De Jong and Beenakker [170] analyze the case when both a disordered normal metal and tunnel barrier are present, and describe the crossover between the two limiting regimes which are obtained in the absence of a disordered region or in the absence of tunnel barrier. Naidenov and Khlus [172] and Fauche`re et al. [173] analyze the situation when the normal and the superconducting electrodes are separated by a resonant double barrier (in particular, this may correspond to the situation of resonant impurities in the insulating layer separating the two electrodes). Ref. [172] considers the one-channel sample-speci"c case (no averaging) and discusses the resonant structure of the conductance and the noise. Using the distribution function of transmission eigenvalues and assuming that the barrier is symmetric, Ref. [173] "nds for the ensemble average a Fano factor F". This should be contrasted with the result F" for the corresponding normal symmetric resonant double barrier. Schep and Bauer [93] investigate the e!ect of a disordered interface separating the normal metal and the superconductor. The Fano factor also, of course, equals F", which is higher than the value discussed above. 4.1.3. Non-linear regime Khlus [15], and subsequently Anantram and Datta [174] (who used the scattering approach), considered noise in the non-linear regime. Without giving details, we only mention the case of an ideal NS interface coupled to a perfect wire for which all the transmission coe$cients ¹ are equal L
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to one. This ideal contact does not exhibit shot noise for voltages below D/e, as is seen from Eq. (165). The physical reason is that in this case the scattering process is not random: an electron approaching the interface is converted into a hole with probability one and sent back. However, as the voltage increases above D/e, quasiparticle states in the superconductor become available, and electrons can now tunnel into the superconductor without being re#ected as holes (imperfect Andreev reyections). This induces noise even for an ideal interface. For still higher voltages, an even broader range of energies is involved. However, for energies E
, For a non-ideal conductor (0(¹ (1 for at least one channel) the same mechanism leads to the L crossover from Eq. (165) at low voltages to Eq. (57) for high voltages. For barriers of low transparency ¹;1 (for instance, when there is an insulating layer at some distance from the NS interface) another mechanism for non-linear noise takes place, as discussed by Fauche`re et al. [173]. In such a geometry the phases of the scattering matrix are energy sensitive. For e"<";D we obtain, similarly to Eqs. (164) and (165), formulae for non-linear current,
C4 ¹ e L dE , I" ¹#2(1!¹ )(1!cos(a (E)) p
L L L L and noise,
(168)
C 4 ¹(1!¹ )(1!cos(a (E)) 8e L L L dE . (169) S" [¹#2(1!¹ )(1!cos(a (E))] p
L L L L Here we assumed the transmission probabilities ¹ to be energy independent. The phase a is L L a (E)" (E)! (!E)!2 arccos(E/D)"4Ed/ v !2 arccos(E/D) , L L L L where d is the distance between the insulating layer and the NS interface, and v is the velocity in L the channel n. (E) is the phase that an electron with energy E acquires during a round-trip L between the NS interface and the insulating layer. For E"0 we have a "p, and thus in the linear L regime Eqs. (168) and (169) are reduced to Eqs. (164) and (165), respectively. These expressions can be interpreted as follows. The part of the normal metal between the NS interface and the tunnel barrier serves as an Andreev resonant double barrier. The electron entering this region travels to the NS interface, is converted into a hole, then this hole makes a round-trip, and is converted to an electron, which returns to the barrier. The total phase gain during this trip is a (E). The `transmission probabilitya of this process (the integrand in Eq. (168)) shows a proL nounced resonance structure near the energies where the phase a (E) equals 2pm with integer m. L Explicitly, for each channel n, we have a set of resonances (Andreev}Kulik bound states [173]) E "(p v /4d)(2m#1). Thus, the behavior of the transmission probability is similar to that K L describing resonant tunneling in the double-barrier normal system.
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Specializing further to the case of one channel with velocity v and transmission coe$cient $ ¹;1, we write the analog of the Breit}Wigner formula
C4 dE e¹ , I" ¹#(4d/ v )(E!E ) p
$ K K
(170)
and
C 4 (E!E ) dE 4e¹ 4d K . (171) S" [¹#(4d/ v )(E!E )] p v $ K $ K We see that both the current and the noise power show plateaus as a function of applied voltage; sharp transitions between the plateaus take place at resonances, when e"<""E . In particular, K when the voltage e< lies between the resonances (plateau regime), E (e"<"(E , we have + +> I"e¹v /(4d), and S"2eI. Thus, already after the "rst resonance, the Fano factor assumes the $ value F"1, the same as for the normal structure. The explanation is that the transport through Andreev bound states, which dominates in this regime, is not accompanied with the formation of Cooper pairs, and thus the usual classical Schottky value is restored. These considerations should be supplemented by an analysis of the charge and its #uctuations and the role of screening (see Section 5). 4.1.4. Interfaces between normal metals and d-wave superconductors Zhu and Ting [175] considered shot noise of the interface between a normal metal and a superconductor with a d-wave symmetry. For this purpose, they generalized the scattering approach for this situation and subsequently performed numerical studies. Now the shot noise depends on the orientation of the superconducting order parameter at the interface. Zhu and Ting [175] investigated only one particular orientation, when the gaps felt by electrons and holes are of the same magnitude but of di!erent signs. The results they found are drastically di!erent from those for s-wave superconductors. In the tunneling regime, the Fano factor is zero (rather than 2) for low voltages. It grows with voltage and saturates at F"1 for e"<"
e D dE[ f (E, u)#f (E, u)] S(u)" L p
L
# D (1!D ) dE[ f (E, u)#f (E, u)] , L L L
(172)
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with D ,¹(2!¹ )\. Performing the integration, we obtain L L L
u
u#2e< e S(u)" 2 u coth D# ( u#2e<)coth L 2k ¹ 2k ¹ p
L #( u!2e<)coth
u!2e< 2k ¹
D (1!D ) , L L L
(173)
which gives the noise frequency spectrum for arbitrary frequencies, voltages, and temperatures (provided all of them are much below D). For <"0, Eq. (173) agrees with the #uctuationdissipation theorem. At zero temperature, we obtain [176]
2e "u" L DL #2e"<" L DL (1!DL ), "u"(2e"<" , S(u)" p "u" D ,
"u"'2e"<" . L L
(174)
This expression is quite similar to Eq. (129) which describes zero-temperature noise frequency spectrum in the normal contact. One evident di!erence is that the transmission coe$cients ¹ are L replaced by D , due to the modi"cation of scattering by Andreev re#ections. Another observation L is that the electron charge is now doubled. Thus, instead of the singularity at the frequency
u"$e< in a normal metal we have now the singularity at u"$2e<. This is yet one more manifestation of the fact that transport in NS structures is related to the transmission of Cooper pairs. Even more interesting e!ects are expected when the frequency becomes of order D. In this case the total scattering matrix, however, can by no means assumed to be energy independent, and the self-consistent treatment of interactions is needed, as we discussed in Section 3. A step in this direction has been done in Ref. [177], which analyzes noise of a NS interface measured at a capacitively coupled gate and only considers the charge self-consistency. In accordance with the general conclusions of Section 3, the leading order in frequency for this noise is given by S "2C uR e"<", where R is determined by the properties of the interface. %% I 4 4 4.1.6. Multi-terminal devices Consider now a multi-probe hybrid structure, which contains a number of normal and a number of superconducting leads (the superconducting leads are taken at the same chemical potential). The current operator, Eq. (157), can be written for each lead of a multi-terminal structure connected to a superconductor. This leads to a second quantization formulation of the current}current correlations put forth by Anantram and Datta [174]. At each normal contact, labeled a, the current is the sum of an electron current I and a hole current I. In terms of the scattering matrix the resulting ? ? current correlations are
qIqJ dE Tr[A (ak)A (bl)] f H(E)[1!f G(E)] , 1*II*IJ 2" AH BG BG AH A B ? @ p
AH BG By this now we mean self-consistency in both the charge and the superconducting order parameter *.
(175)
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where q"!e and q"e. Here the indices a, b, c, d label the terminals, and i, j, k, l describe the electron}hole decomposition and may assume values e and h. Based on Eq. (175), Anantram and Datta predict that, though correlations at the same contact are always positive, like in the case of normal structures, those at diwerent contacts also may in certain situations become positive. (We remind the reader that cross-correlations are quite generally negative in normal devices, as discussed in Section 2.) Similar results have been obtained by Martin [171] using statistical particle counting arguments. Qualitatively, this conclusion may be understood in the following way. There are two types of processes contributing to noise. First, an electron or hole can be simply re#ected from the interface without being Andreev re#ected. In accordance with the general considerations, this re#ection tends to make the cross-correlation negative. On the other hand, processes involving Andreev re#ection (an electron is converted into a hole or vice versa) provide transport of particles with opposite charges. Due to Eq. (175), these processes are expected to push the cross-correlations towards positive values. This interplay between normal scattering and Andreev reyections determines the total sign of the cross-correlations. This is, indeed, seen from the expressions of Anantram and Datta [174], who decompose current correlations at di!erent contacts into a sum of positive- and negative-de"nite contributions. The interpretation which may be found in the literature, that positive cross-correlations in hybrid structures are due to the bosonic nature of Cooper pairs, does not seem to be plausible. Indeed, the microscopic theory of superconductivity never uses explicitly the Bose statistics of Cooper pairs. In particular, Eq. (175) only contains the (Fermi) distribution functions of electrons and holes, but not the distribution function of Cooper pairs. Quantitative analysis of this e!ect would also require the next step, which is to express the scattering matrix s through the scattering matrices of the normal part of the device (multi-terminal analog of Eq. (163)). The multi-terminal correlations could then be studied for various systems, similarly to the discussion for the normal case (see Section 2). Analytical results are currently only available for systems with an ideal NS interface, where the matrix s is fully determined by Andreev re#ection. Anantram and Datta [174] consider a three-terminal device with two normal contacts and a contact to the superconductor. The superconductor connects to the normal system via two NS-interfaces forming a loop which permits the application of an Aharonov}Bohm #ux. The conductor is a perfect ballistic structure and the NS interfaces are also taken to be ideal. In this system, shot noise is present for arbitrary voltages, since the electron emitted from the normal contact 1, after (several) Andreev re#ections may exit through the normal contact 1 or 2, as an electron or as a hole. Speci"cally, Ref. [174] studies cross-correlations of current at the two normal contacts, and "nds that they may be both positive and negative, depending on the phases of Andreev re#ection which in their geometry can be tuned with the help of an Aharonov}Bohm #ux. Another three-terminal geometry, a wave splitter connected to a superconductor, is discussed by Martin [171] and Torre`s and Martin [178]. The cross-correlation in the normal leads depends on the parameter e which describes the coupling to the superconducting lead (see Eq. (113)), 0(e( [120,121]. For an ideal NS interface, Ref. [178] "nds that the cross-correlations are positive for 0(e((2!1 (weak coupling) and negative for (2!1(e(. TorreH s and Martin [178] also report numerical results for disordered NS interfaces, showing that disorder enhances positive cross-correlations.
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A particularly instructive example has been analyzed in Ref. [179] by Gramespacher and one of the authors of this Review. They investigate the shot noise measurement with a tunneling contact (STM tip) which couples very weakly to a normal conductor which is in turn coupled to a superconductor. If both the normal reservoir and the superconductor are taken at the same potential k , and the tunneling tip at potential k, they "nd from Eq. (175) the following correlations: 1*I *I 2"a4pl "t"l(x , 1 ) , (176) 1*I *I 2"!a4pl "t"l(x , 1 ) , (177) F 1*I *I 2"1*I *I 2"0 , (178) with a"!(e/p )*k, *k"k!k , and "t" the coupling energy of the tip to the sample. Here l(x , 1 ) is the electron density generated at the coupling point x due to injected electrons and l(x , 1 ) is the electron density at the coupling point due to holes injected by the normal reservoir. The total correlation of the currents at contacts 1 and 2 is the sum of all four terms. In the absence of a magnetic "eld, the correlations are proportional to the injected net charge density q(x)" l(x , 1 )!l(x , 1 ), and given by e (179) 1*I *I 2"! *k4pl "t"q(x)"!2G *k q(x)/p(x) , p
where p(x)"l(x , 1 )#l(x , 1 ) is the total particle density of states and G " (e/2p )4pl "t"p(x) is the tip to sample conductance. This result states that if at the point x the electrons injected from contact 1 generate a hole density at x which is larger than the electron density at x, the injected charge becomes negative and the corresponding correlation becomes positive. A more detailed analysis suggest that this e!ect is of order 1/N, where N is the number of channels. Up to now positive correlations in hybrid structures have been theoretically demonstrated only for single-channel conductors. This leaves open the question, on whether or not, ensembleaveraged shot noise spectra can in fact have a positive sign in hybrid structures. 4.1.7. Experiments The only experiment on shot noise in NS structures currently available was performed by Vystavkin and Tarasov [180] long before the current interest on shot noise in mesoscopic systems started. For this reason, they did not study noise systematically, and only concluded that in certain samples it was suppressed below the value 2e1I2. Recently, Jehl et al. [181] experimented with an Nb/Al/Nb structure at temperatures above the critical temperature for Al, but below that for Nb. They estimate that the length of the Al region was longer than the thermal length, which means that the multiple Andreev re#ection processes (see below) are suppressed. Thus, qualitatively their SNS structure acts just as two incoherent NS interfaces, and the expected e!ective charge is 2e (the Fano factor for the di!usive system is ). Indeed, the measurements show that the Fano factor for high temperatures is for all voltages (in accordance with the result for a metallic di!usive wire), while for lower temperatures it grows. The low-temperature behavior is found to be in better agreement with the value F" for an NS
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Fig. 28. Experimental results of Jehl et al. [181]. Solid and dotted lines are theoretical curves corresponding to the e!ective charges 2e and e, respectively. Copyright 1999 by the American Physical Society.
interface, than with the F" prediction for normal systems, though the agreement is far from perfect. The experimental results are shown in Fig. 28. A clear experimental demonstration of the shot noise doubling with clean NS interfaces remains to be performed. 4.2. Noise of Josephson junctions Josephson junctions are contacts which separate two superconducting bulk electrodes by an insulating barrier. We brie#y describe here noise properties for the case when the transmission of this barrier is quite low; other, more interesting, cases are addressed in the next subsection. The transport properties of Josephson junctions can be summarized as follows. First, at zero voltage a Josephson current may #ow across the junction, I"I sin , where is the di!erence of the phases of the superconducting order parameter between the two electrodes. In addition, for "nite voltage tunneling of quasiparticles between the electrodes is possible. For zero temperature this quasiparticle current only exists when the voltage exceeds 2D/e; for "nite temperature an (exponentially small) quasiparticle current #ows at any voltage. The Josephson current is a property of the ground state of the junction, and therefore it does not #uctuate. Hence, shot noise in Josephson junctions is due to the quasiparticle current, and basically coincides with the corresponding shot noise properties of normal tunnel barriers. For zero temperature, there is no shot noise for voltages below 2D/e. Thermal and shot noise in Josephson junctions are analyzed in detail by Rogovin and Scalapino [26], and have been measured by Kanter and Vernon, Jr. [183,184]. As stated by Likharev in his 1979 review [182], `the most important results of all the theories of #uctuations in the Josephson e!ect is that the only intrinsic source of #uctuations is the normal current rather than the supercurrent of the junctiona.
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For completeness, we mention that if a voltage <(t) is applied across the junction, as a consequence of gauge invariance, the Josephson current becomes time-dependent,
2e R I( )"I sin # <(t) dt .
Then, due to any #uctuations of the voltage <(t) (like thermal and shot noise) the Josephson junction starts to radiate. If the voltage < is time-independent on average, the spectral density of this radiation is centered around the resonant frequencies u"2en > \ \ the center of the gap of the superconductor. Note that f "1!f . To investigate the dynamics of \ > this system, in the presence of a bath permitting inelastic transitions, we investigate the relaxation
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of the non-equilibrium distribution o towards the instantaneous equilibrium distribution func! tion with the help of the Debye}Boltzmann-like equations (180) do /dt"!c(o !f! (t)) . ! ! If the system is driven out of equilibrium, the instantaneous distribution function is timedependent. Eq. (180) states that the non-equilibrium distribution tries to follow the instantaneous distribution but can do that at best with a time lag determined by c\. The time-dependent readjustment of the distribution o is achieved with inelastic processes and is thus dissipative. To "nd the resulting noise we investigate the response of the current to a small oscillating phase d (u)e SR superimposed on the dc phase. The current is I"!(e/ )[(de /d )o #(de /d )o ]. > > \ \ The Josephson relation, d /dt"!(2e/ )<, leads to a conductance which in the zero-frequency limit is given by [200]
G( )"
2e c
de df ! d
de
.
(181)
CC> The resulting thermal noise of the Josephson current follows from the #uctuation dissipation theorem S"4k ¹G and, as found by Averin and Imam [201] and MartiD n-Rodero et al. [202], is given by
sin( /2) 2 eD . (182) S" c( ) cosh(e /2k ¹) The peculiar feature of both the conductance (181) and the noise (182) is their divergence as the damping c tends to zero. Furthermore, since (!df/de) in Eq. (181) is proportional to 1/k ¹, the Nyquist noise given by Eq. (182) is not proportional to k ¹, as in open systems. Since in the zero-temperature limit c can be expected to tend rapidly to zero, the Nyquist noise may actually grow as the temperature is lowered. Instead of a small-amplitude ac oscillation of the phase, we can also consider a phase that linearly increases with time, "2e
"$p. To describe this crossing, we extend the range of from !2p to 2p. Ref. [200] "nds that G and G, as given by Eq. (181), are related, ( p de e 1 p > f ( ) d . (183) d G( )" G " ( 4p d > 2pc \p \p Thus, G is inversely proportional to the `e!ective massa (weighted by the equilibrium distribution ( function). We could of course derive this result directly from Eq. (180). In the zero-temperature limit the e!ective mass is 1/mH"2D/p, and the conductance G is "nite and given by ( G "2eD/pc . From the #uctuation dissipation theorem, we obtain an equilibrium noise ( S"4eDk ¹/p c , (184)
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which (unlike Eq. (182)) is proportional to the temperature. Eq. (184) was obtained by Averin and Imam [201] and Cuevas et al. [203]. 4.3.2. Non-equilibrium noise For larger voltages but still e<;D (<'0), the average dc current becomes a non-linear function of voltage. The current peaks for e<& c when dissipation due to the mechanism described above is maximal. We can no longer invoke the #uctuation-dissipation theorem to "nd the noise. Instead, a direct calculation of the current}current correlation function 1IK (t )IK (t )#IK (t )IK (t )2 is needed. Since the current oscillates with frequency 2e
(185)
The voltage dependence of the noise is quite unusual in this case, and exhibits a peak for e<& c. The result (185) has the following interpretation. The mechanism of charge transport in SNS structures for e<;D is multiple Andreev reyections (MARs) [204]. Imagine an electron with energy E, D(E(D#e< (measured from the chemical potential of the right contact), emanating from the right contact. During the motion in the normal region it loses the energy e<, and thus when it arrives at the left superconducting bank it has an energy below D. This electron may only be Andreev re#ected and converted into a hole, which (due to the opposite sign of the charge) loses the energy e< again. The hole is again Andreev re#ected at the right interface, and this process goes on, until the energy of the initial electron falls below !D. The number of these MARs is equal to 2D/e<. In each individual Andreev re#ection the charge 2e is transferred to or from the condensate, and to avoid double counting, we must only take the re#ections happening at the same interface. Therefore, the whole MAR process is accompanied by a transfer of charge 2D/< (for e<;D). In view of this, the noise (185) may be interpreted [201] as `shot noisea of the 2D/e<<1 charge quanta. In this sense this noise is giant: it greatly exceeds the Poisson value 2eI. The general expression for noise can be written explicitly as a sum of contributions of Andreev re#ections of di!erent orders [201,203]. Now we brie#y discuss the case of a non-ideal contact, i.e. when the two electrodes are separated by a barrier of arbitrary transparency. First, for a "nite but small re#ection coe$cient 1!¹ an additional source of noise is given by Landau}Zener transitions between the two subgap states, as
This expression is not explicit in Ref. [201], but can be easily derived in the limit of strictly zero temperature. MAR is destroyed, for instance, if the length of the junction is longer than the inelastic scattering length. In this limit the system acts rather as two independent NS interfaces. Another limitation is e<< c.
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discussed by Averin [205]. The probability of these transitions, which exist even at zero temperature, is l"exp(!p(1!¹)D/e<), and noise is caused by the randomness of these transitions. Naveh and Averin [206] obtained the following result for the noise due to this mechanism: S"(8eD/p <)l(1!l) . They also considered the generalization to the multi-channel case and analyzed a structure with a normal di!usive conductor between the two superconductors. Taking into account the distribution of transmission eigenvalues (87) of a normal conductor yields [206] S"((2D) /(e<) )G((2!1) , where G is the Drude conductance of the normal region. The noise diverges for low voltages as <\ . If the transparency of the barrier is low, we return to the case of a classical Josephson junction. The amplitude of a MAR process containing n Andreev re#ections is proportional to ¹L, where ¹ is the transmission probability of the junction. Thus, for the classical case MARs are totally suppressed. The case of arbitrary transparency was investigated by Cuevas et al. [203], who described the crossover between these two regimes. Bezuglyi et al. [207] considered a tunnel barrier (an insulating layer) inserted in the middle of a long SNS constriction. This geometry is di!erent from the standard Josephson junction problem. Instead, bound Andreev states (similar to what has been discussed before for the NS interface with a barrier) are formed in both parts of the normal region, separated by the insulating layer. An electron in the left part, before being converted to a hole at the left NS interface, is oscillating many times before it tunnels (as an electron or a hole) through the barrier, and starts oscillating again. This picture resembles [207] transport in a di!usive metallic wire, which gives us a hint that shot noise may be suppressed in comparison with its `giant Poissona value, i.e. the value corresponding to the e!ective charge 2D/<. Indeed, Ref. [207] "nds that in the limit of low voltages the e!ective charge is 2D/3<, which surprisingly reminds us of the -suppression of shot noise in metallic di!usive wires. For high voltages e<
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Fig. 29. Experimental results of Dieleman et al. [211]. The Fano factor (plotted here as an e!ective charge, black squares) is compared with the theoretical prediction (solid line). Copyright 1997 by the American Physical Society.
of the Fano factor (which is theoretically predicted to be 2D/e< for e<;D) for voltages up to 2D. For voltages e<&2D/n, n3Z, a step-like structure is observed (Fig. 29). Hoss et al. [212] carried out measurements on Nb/Au/Nb, Al/Au/Al, and Al/Cu/Al junctions, where the Au and Cu layers were essentially di!usive conductors. They observe a well-pronounced peak in the voltage dependence of the shot noise for low voltages (much less than */e). In addition, they also plot the Fano factor. For high temperatures, it turns out to be a linear function of <\ in the whole range of voltages, with the coe$cient which agrees with the theoretical expectations. However, for low temperatures the behavior is not <\, which is currently not understood. In the experiments by Jehl et al. [181] on long Nb/Al/Nb SNS contacts, the two NS interfaces act e!ectively independently, and MAR processes are suppressed. Thus, the physics of this experiment resembles more that of a single NS interface, as we have discussed above.
5. Langevin and master equation approach to noise: double-barrier structures 5.1. Quantum-mechanical versus classical theories of shot noise In this and the next section, we consider classical theories of shot noise in various systems. By doing this, we leave the main road that started from basic quantum mechanics and, through a number of exact transformations and well-justi"ed approximations, lead us to the "nal results for shot noise. In contrast, classical theories are mostly based on the Langevin approach, which has a conceptually much weaker and less transparent foundation. Indeed, the Langevin equation is equivalent to the Fokker}Planck equation under the condition that the random Langevin sources
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are Gaussian distributed (see e.g. Ref. [213]). In turn, the Fokker}Planck equation is derived from the master equation in the di!usion approximation, and this procedure determines the pair correlation function of Langevin sources. In practice, however, such a basic derivation is usually not presented. The correlation function is written based on some ad hoc considerations rather than derived rigorously. For double-barrier structures, which are considered in this section, many results have been derived directly from the master equation, and thus are far better justi"ed than many discussions for other structures. The next section is devoted to the Boltzmann}Langevin approach in disordered conductors, and, to our knowledge, no attempt to obtain the "nal results from the master equation, or to justify microscopically the starting Boltzmann equation with Langevin sources, has ever been performed. The reality, though it may be surprising to some readers, is that in all available cases for which the results of classical calculations of shot noise can be compared to exact quantum results averaged over an ensemble, based on the scattering or Green's functions approaches, they turn out to be identical. It is the fact that for many systems the ensemble-averaged quantities are classical which makes classical Boltzmann}Langevin theories of shot noise in mesoscopic conductors credible even in those situations, where quantum results are not available. The fact that the ensemble-averaged quantum result and the classical result agree is best illustrated by considering for a moment a metallic di!usive wire. A calculation of the conductance can be performed purely quantum-mechanically by "nding the scattering matrix computationally or using random matrix theory. After ensemble averaging the leading-order result for the conductance is just the Drude result for the conductance of the wire which we can "nd by solving a di!usion equation. This situation persists if we consider the noise: the leading order of the noise, the -suppression of shot noise in a metallic di!usive wire can be found by ensemble averaging a quantum-mechanical calculation [74] and/or from a classical consideration [75]. If the two procedures would not agree to leading order, it would imply a gigantic quantum e!ect. Of course, e!ects which are genuinely quantum, like the Aharonov}Bohm e!ect, weak localization, or the quantum Hall e!ect, cannot be described classically. This consideration also indicates the situations where we can expect di!erences between a quantum approach and a classical approach: Whenever the leading-order e!ect is of the same order as the quantum corrections we can obviously not "nd a meaningful classical description. The Langevin approach essentially takes the Poissonian incoming stream of particles and represents it as a random #uctuating force acting even inside the system. In the language of the scattering approach, this would mean that the Poissonian noise of the input stream is converted into the partition noise of the output stream. Another possible classical approach to the shot noise would be to take the Poissonian input stream as a sequence of random events, and to obtain the
Given the results for the distribution of transmitted charge (Appendix A), it is apparent that the Langevin sources are not Gaussian distributed. Possibly, this does not a!ect the shot noise, which is related to the second cumulant of the Langevin sources. Quite recently, Nagaev [214] has shown that the zero-frequency results for shot noise in metallic di!usive wires which are obtained in a quantum-mechanical Green's functions technique, are equivalent to those available from the Boltzmann}Langevin approach, even if the interactions are taken into account. This is a considerable step forward, but it still does not explain why the Boltzmann}Langevin approach works.
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distribution of the carriers in the output stream after the scattering events took place. To our knowledge, this approach has not been realized precisely in this form. Landauer [215] discusses noise in di!usive metallic conductors from a similar point of view, but does not calculate the distribution of outgoing particles. Chen et al. [216] attempt to modify the distribution function in ballistic systems to take into account the Pauli principle, assuming that the time the particle spends inside the system is "nite. Barkai et al. [217] and van Kampen [218] consider the case when the electrons may arrive from two reservoirs, and are transmitted or re#ected with certain probabilities. The Pauli principle forbids two electrons to be in the channel simultaneously. Though there is no doubt that this approach, if realized, would yield the same value of the shot noise as more powerful methods, it would still help to visualize the results and it might allow a simple generalization to interacting systems. Raikh [219] and Imamog lu and Yamamoto [220] have suggested a generalization to the Coulomb blockade regime. Raikh [219] shows how the noise in the Coulomb blockade regime may be expressed if the transformation from the Poisson input stream to the correlated output stream is known for non-interacting electrons. Imamog lu and Yamamoto [220] assume that the Poisson distribution is modi"ed in some particular way by the "nite charging energy, and are able to obtain sub-Poissonian shot noise suppression. We treat shot noise in the Coulomb blockade regime later on (Section 7) by more elaborate methods. 5.2. Suppression of shot noise in double-barrier structures We consider now transport through quantum wells, which were described quantum-mechanically in Section 2. The tunneling rates through the left C and the right C barrier are assumed * 0 to be much lower than all other characteristic energies, including temperature. Introducing the distribution function in the well f (E), we write the charge of the well Q in the form dE dE f (E #E )d(E !EP ) , (186) Q "el A X , X , X L L where the energy E in the well is counted from the band bottom e; in the well (Fig. 8), and the X sum is taken over all the resonant levels. Now we introduce the charges Q (t) and Q (t) which have passed through the left and right * 0 barriers, respectively, from the time t"!R until the time t. At any instant of time the charge of the well is Q (t)"Q (t)!Q (t). The time evolution of the charge Q is determined by the rate * 0 * equation
QQ "e(c !c ) , * * *
(187)
In terms of the quantum-mechanical derivation, this would mean that the transmission coe$cient (75) is replaced by ¹(E)"2pC C C\d(E!EP ). *L 0L L L Note that the notations here and below di!er from those introduced in Section 2: all the energies are measured from the corresponding band bottoms.
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where c and c are transition rates through the left barrier, from the reservoir to the well and * * from the well to the reservoir, respectively. We have
l A c " dE dE C d(E !EP ) f (E #E #e;!e<) * X , *L X L * X ,
C4\3 L ;[1!f (E #E )] , X , l A c " dE dE C d(E !EP ) * X , *L X L
L C4\3 ;[1!f (E #E #e;!e<)] f (E #E ) . (188) * X , X , Here we wrote the distribution function of the left (right) reservoir in such a way that the energy is counted from the chemical potential of the left (right) reservoir. Similarly, QQ "e(c !c ) is 0 0 0 expressed in terms of the transition rates through the right barrier, which are written analogously to Eq. (188). Further progress is easy in two cases: either the tunneling rates C do not depend on n (and * 0L equal C ), or there is only one resonant level of the longitudinal motion in the relevant range of * 0 energies. Taking into account Eq. (186), we obtain QQ "I ! \C (Q !Q ) , * * * * 0 QQ "I # \C (Q !Q ) , 0 0 0 * 0 where we have introduced
(189)
el A dE dE d(E !EP ) f (E #E #e;!e<) , I " C * X , X L * X , *
L C4\3 (190) el A I "! C dE dE d(E !EP ) f (E #E #e;) . 0 * X , X L 0 X ,
L Note that we derived the rate equations (189) without specifying the distribution function f . Thus, within the approximations used here, the rate equations are the same, independently of the relaxation rate in the well (which determines the distribution f ). The current, for instance, through the left barrier, is given by the sum of the particle current I "QQ and the displacement current I "c QQ /(c #c ), where we have introduced the *N * *B * * 0 capacitances c and c of the left and the right barrier, respectively. Using this gives for the total * 0 current I"(c QQ #c QQ )/(c #c ) , (191) * 0 0 * * 0 which is often cited as Ramo}Shockley theorem. The calculation of the current through the right barrier yields the same result, demonstrating that the total current is conserved. Many papers in the "eld are #awed since they use I"(QQ #QQ )/2 and subsequently claim the validity of the results * 0 for arbitrary barriers. Similarly, evaluating Eq. (191) with the help of the free electron results for QQ and QQ leads to * 0 a current-conserving answer but not to a self-consistent result. Compare Eqs. (195) and (201).
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Up to now, we have only discussed average quantities. The idea of the Langevin approach is that the (current) #uctuations can be calculated from the same rate equations (189), if random currents (Langevin sources) are added to their right-hand sides. We write QQ "I ! \C (Q !Q )#m (t) , * * * * 0 *
(192)
QQ "I # \C (Q !Q )#m (t) , 0 0 0 * 0 0 where the Langevin sources m (t) have the following properties. They are zero on average, * 0 1m (t)2"0, a"L, R. Furthermore, they are correlated only for the same barrier, and the correla? tion function describes Poissonian shot noise at each barrier, 1m (t)m (t )2"e1I2d(t!t )d , (193) ? @ ?@ where 1I2 is the average current. To "nd the noise power, we do not need to specify higher cumulants of the Langevin sources. As we mentioned above, the de"nition (193) is intuitive rather than the result of a formal derivation. However, the results we obtain in this way coincide with those found by ensemble averaging the quantum-mechanical results. Eqs. (192) are linear and can be easily solved. The average current is 1I2"(C I #C I )/C , 0 * * 0 where again C"C #C . The "nite frequency shot noise power is found to be * 0 c #c 2 c c 0# C * 0 !C C , S(u)"2e1I2 * * 0 c C#( u) c
(194)
(195)
with the de"nition c,c #c . * 0 For zero frequency, Eq. (195) gives the result (78), F"(C #C )/C. For high frequencies * 0
u
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Beenakker and de Jong [89,90] consider the two-barrier suppression using the conceptually similar Boltzmann}Langevin approach, described in the Section 6. They also investigate the case of n identical barriers in series and obtain for the Fano factor
n(1!¹)(2#¹)!¹ 1 , F" 1# [¹#n(1!¹)] 3 where ¹ is the transmission coe$cient of a barrier. This expression gives F"1!¹ for n"1, F" for nPR (which mimics a di!usive wire), and reduces to F" for ¹;1 for the two-barrier case, n"2. Thus, the crossover of the ensemble-averaged shot noise between a two-barrier and many-barrier (di!usive) system can be described classically. The theory we have presented above has a number of drawbacks, which we discuss now. First, the charges Q and Q are assumed to be continuous, and thus Coulomb blockade e!ects cannot * 0 be treated in this way (see Section 7). Even the charging e!ects which exist if charge quantization can be neglected are not properly taken into account. The discussion thus far has neglected to include the response to the #uctuating electric potential in the well. Thus the Ramo}Shockley theorem, as it has been used here, and is applied in much of the literature, leads to a current conserving, but, as we discuss below, not to a self-consistent result for the frequency dependence of the shot noise. In the next subsection we show that charging e!ects can lead to the enhancement of shot noise above the Poisson value. Furthermore, our consideration is limited to zero temperature, and it is not immediately clear how the Langevin approach should be modi"ed in this case to reproduce correct expressions for the Nyquist noise. Another limitation is that our derivation assumes that the tunneling rates through each of the resonant levels are the same. If this is not the case, the rate equations do not have the simple form (189), but instead start to depend explicitly on the distribution function f . Whereas the above derivation does not require any assumptions on the distribution of the electrons in the well f (i.e. any information on the inelastic processes inside the well), generally this information is required, and it is not a priori clear whether the result on noise suppression depends on the details of the inelastic scattering. The last two issues are relatively easily dealt with in the more general master equation approach. Chen and Ting [134], Chen [227], and independently Davies et al. [54] solved the master equation in the sequential tunneling limit, when the electrons, due to very strong inelastic scattering inside the well, relax to the equilibrium state. They found the results to be identical to those obtained by quantum-mechanical methods (which require quantum coherence, i.e. absence of inelastic scattering), and concluded that inelastic processes do not a!ect shot noise suppression in quantum wells. Later, Iannaccone et al. [228] solved the master equation allowing explicitly for arbitrary inelastic scattering, and found that the noise suppression factor is given by Eq. (78) at zero temperature for arbitrary inelastic scattering provided the reservoirs are ideal. They also studied other cases and temperature e!ects. This picture seems to be consistent with the results obtained quantum-mechanically by attaching voltage probes to the sample (Section 2.7).
To use the kinetic equation formalism, we had to assume k ¹
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We note here, however, that there is no consensus in the literature concerning this issue. First, we discuss the results obtained quantum-mechanically by Davies et al. [229]. They start from the exact expression (74) and average it over the phase , allowing for inelastic scattering (dephasing). Instead of assuming that the phase is a uniformly distributed random variable, they postulated 1exp(i # )2"1exp(i )21exp(i )2. For the Fano factor (at zero frequency) they obtain in this way
C 2C C , F"1! * 0 C#C /2 C where C is the rate of inelastic scattering, proportional to 1!1exp(i )2. Thus, in their model inelastic scattering enhances the Fano factor, driving noise towards the Poisson value. While it is clear that for certain models of inelastic scattering shot noise is a!ected by interactions, we do not see a direct relation of the model of Ref. [229] to the voltage probe models which we consider in Section 2.7 and which yield the result that the Fano factor is interaction insensitive. Furthermore, Isawa et al. [230], using the Green's functions approach, "nd that inelastic processes leading to sequential tunneling a!ect the Fano factor. Their theory, however, is explicitly not current conserving. We mentioned already the result by Lund B+ and Galperin [59], who report suppression of the Fano factor by electron}phonon interactions. Their results clearly contradict the conclusions based on the voltage probe models. A related issue is investigated by Sun and Milburn [223,224], who apply the quantum master equation to the analysis of noise in a double-well (triple-barrier) structure. With this approach, they are able to study the case when the two wells are coupled coherently. Their results show an abundance of regimes depending on the relation between the coupling rates to the reservoirs, elastic rates, and the coupling between the two wells. To conclude this subsection, we address here one more problem. The result (78) predicts that the noise suppression factor may assume values between and 1, depending on the asymmetry of the double barrier. The question is whether interaction e!ects, under some circumstances, may lead to Fano factors above 1 (super-Poissonian noise) or below . Relegating the problem of super Poissonian noise to the next subsection, we only discuss here the possible suppression of shot noise below . Experimentally, noise suppression below was observed in early experiments by Brown [231], and recently by Przadka et al. [68] and by Kuznetsov et al. [232]. Early papers on the subject [233}236] predict shot noise suppression down to zero either with frequency or even at zero frequency. Following Ref. [233], these works treat current #uctuations as a superposition of density and velocity #uctuations, with a self-consistent treatment of interaction e!ects. However, apparently they did not include the partition noise (¹(1!¹)) in their consideration. Since it is precisely the partition noise which produces the minimal suppression of , it is not quite surprising that their theory predicts lower suppression factors. Sugimura [237] and, independently, Carlos Egues et al. [238] propose a model in which the states in the well are inelastically coupled to the degrees of freedom of the reservoirs. This model, indeed, yields noise suppression below in a limited parameter range. The minimal suppression factor given in Ref. [238] is 0.45, which is still above the experimental data. This direction of research looks promising, but certainly requires more e!orts.
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5.3. Interaction ewects and super-Poissonian noise enhancement Prior to the discussion of charging e!ects, we brie#y comment on how quantum wells operate in the strongly non-linear regime (e<&E ), provided charging e!ects are not important. For $ simplicity, we assume that there is only one resonant level of the longitudinal motion E ,E in the relevant range of energies, and all other levels lie too high to be of importance, E <e<, E for L $ n'0. We assume also E 'E , and <'0, then electrons from the right reservoir cannot enter $ the well. Modifying Eq. (79) to take into account that the band bottoms e< in the left reservoir and e; in the quantum well are now "nite, and substituting the transmission coe$cient, ¹(E )" X 2pC C C\d(E !E ), we obtain for the average current * 0 X el A C C * 0 (e
provided E #e;!E (e<(E #e;, and zero otherwise. This dependence, which, of course, $ could also be obtained classically from the rate equations, is shown in Fig. 30a (solid line). The current drops abruptly to zero for e<"E #e;, which corresponds to the passage of the band bottom of the left reservoir through the resonant level in the well. When the smearing of the resonance due to "nite tunneling rates is taken into account (the transmission coe$cient is not replaced by a delta-function), the I}< curve becomes smeared (dashed line in Fig. 30a), and the region of negative diwerential resistance develops around e<&E #e;. This was noted by Tsu and Esaki in their early paper [52], and subsequently observed experimentally in Ref. [53]. Now we consider charging e!ects. The new ingredient is now that the electrostatic potential in the well ; is not an independent parameter any more, but is a function of voltage, which must be calculated self-consistently. Moreover, it has its own dynamics and may #uctuate; we are going to show that the #uctuations of ; may considerably enhance noise. Theoretical papers emphasizing the necessity of charging e!ects for the I}< curve are too numerous to be cited here. For noise, the necessity of a self-consistent treatment in the negative di!erential resistance region was illustrated in the Green's function approach by Levy Yeyati et al. [239], who, however, did not take into account the #uctuations of ;. Iannaccone et al. [240] suggested that these #uctuations may lead to the enhancement of noise above the Poisson value, and provided numerical results supporting this statement. A self-consistent analytical theory of noise in quantum wells including the #uctuations of the band bottom was developed by the authors [241] in the framework of the scattering approach. The classical theory yields the same results [226]; here we give the classical derivation. In the strongly non-linear regime which we discuss here, the energy dependence of the tunneling rates becomes important. To take this into account, we use a simple model and treat each barrier as rectangular. The transmission probability through a rectangular barrier determines the partial decay width of the resonant level, C (E )"a E (E #e;!e<) h(E )h(E #e;!e<) , * X * X X X X C (E )"a E (E #e;) h(E )h(E #e;) , 0 X 0 X X X X The quantity ; serves then as an operator which obeys an operator Poisson equation.
(197)
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Fig. 30. I}< characteristics of the quantum well: (a) charging e!ects are neglected. (b) charging e!ects are taken into account. The values of parameters for (b) are a "a , c "c , E "2E /3, el A"10c . For this case e
where a and a are dimensionless constants (the case of a symmetric well corresponds to a "a , * 0 * 0 not to C "C ). We emphasize that the partial decay widths C are now functions of < and ;. * 0 * 0 Consider "rst the average, stationary quantities. Eqs. (189) are still valid for our case (now I "0, since E 'E ). However, ; is no longer an independent variable, but is related to the 0 $ charge in the well Q !Q . Assuming that the interaction e!ects can be described by a charging * 0 energy only, we write this relation in the form (198) (Q !Q )"c (;!<)#c ; . * 0 * 0 This equation just states that the total charge of the capacitor equals the sum of charges at the left and the right plates. Eqs. (189) and (198) must now be solved together, using the expressions for the partial decay widths (197). Combining them, we obtain a closed equation for ;,
c;Q "el AC (e
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Thus, the new feature due to charging e!ects is the multi-stability of the system in the range of voltages between < and
c
(200)
1 QQ "1I2# [ J#C (c#c )](Q !Q )#m (t) , 0 0 * 0 0
c
where the ;-dependent tunneling rates C are evaluated for ;"; , and the quantities * 0 c "!RQM /R; and J"R(C QM )/ R; (also taken for ;"; ; QM ,el A(e
2 c #c 0# S(u)"2e1I2 * c u#C(c#c ) c c c C ; (c#c ) !C C # * 0 ! J(c#c )(C !C )# J * 0 * 0 c
The quantum-mechanical theory [241] also has been developed in linear approximation.
.
(201)
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The noise power (201) is strongly voltage dependent via the quantities c , J, and C. In particular, for
u&C(c#c )/c, and drops down to zero for <"
(202)
K, J/(c #c #c ) , * 0 which has the form of a dimensionless conductance J/e multiplied by an e!ective charging energy of the well e/(c #c #c ). This quantity contains the relevant information about the charging * 0 e!ects of the well. Eq. (201) is a self-consistent result in contrast to Eq. (195) found by inserting the free electron currents into the Ramo}Shockley formula. We reproduce this result for K"0. We re-emphasize that a calculation of ac conductance or noise using the free-electron results for the currents and the Ramo}Shockley formula is not in general a sound procedure: It assumes that the self-consistent contribution to the currents arising from the internal potential oscillations or #uctuations can be neglected. For
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Fig. 31. Voltage dependence of the Fano factor (202) for the same set of parameters as Fig. 30 (solid line); Fano factor (78) for a charge neutral quantum well (dashed line).
Poissonian. Jahan and Anwar [236], who also found super-Poissonian noise enhancement, included self-consistent e!ect at the level of the stationary transmission probabilities, but also did not take partition noise into account. As we already mentioned, an explanation of the superPoissonian noise in terms of the potential #uctuations was given by Iannaccone et al. [240], and the analytical theory of this enhancement, identifying the relevant energy scales, was proposed in Refs. [241,226]. Experimentally, enhancement of noise in quantum wells, as the voltage approaches the range of negative di!erential resistance, was observed already in the early experiments by Li et al. [64] and by Brown [231]. The super-Poissonian shot noise in the negative di!erential resistance range was observed by Iannaccone et al. [240]. Kuznetsov et al. [232] have presented a detailed investigation of the noise oscillations from sub-Poissonian to super-Poissonian values of a resonant quantum well in the presence of a parallel magnetic "eld. The magnetic "eld leads to multiple voltage ranges of negative di!erential resistance and permits a clear demonstration of the e!ect. Their results are shown in Fig. 32. To conclude this section, we discuss the following issue. To obtain the super-Poissonian noise enhancement, we needed multi-stable behavior of the I}< curve; in turn, the multi-stability in quantum wells is induced by charging e!ects. It is easy to see, however, that the charging (or, generally, interaction) e!ects are not required to cause the multi-stability. Thus, if instead of a voltage-controlled experiment, we discuss a current-controlled experiment, the I}< characteristics for the uncharged quantum well (Fig. 30) are multi-stable for any external current. For the case of an arbitrary load line there typically exists a "nite range of external parameters where multi-stable behavior is developed. Furthermore, the quantum wells are not the only systems with multi-stability; as one well-known example we mention Esaki diodes, where the multi-stability is caused by the structure of the energy bands [243].
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Fig. 32. Experimental results of Kuznetsov et al. [232] which show noise in resonant quantum wells in parallel magnetic "eld. A dotted line represents the Poisson value. It is clearly seen that for certain values of applied bias voltage the noise is super-Poissonian. Copyright 1998 by the American Physical Society.
Usually, such bistable systems are discussed from the point of view of telegraph noise, which is due to spontaneous random transitions between the two states. This is a consideration complementary to the one we developed above. Indeed, in the linear approximation the system does not know that it is multi-stable. The shot noise grows inde"nitely at the instability threshold only because the state around which we have linearized the system becomes unstable rather than metastable. This is a general feature of linear #uctuation theory. Clearly, the divergence of shot noise in the linear approximation must be a general feature of all the systems with multi-stable behavior. Interactions are not the necessary ingredient for this shot noise enhancement. On the other hand, as we have discussed, the transitions between di!erent states, neglected in the linear approximation, will certainly soften the singularity and drive noise to a "nite value at the instability threshold. To describe in this way the interplay between shot noise and random telegraph noise remains an open problem.
6. Boltzmann}Langevin approach to noise: disordered systems 6.1. Fluctuations and the Boltzmann equation In this section we describe the generalization of the Langevin method to disordered systems. As is well known, the evolution of the (average) distribution function fM (r, p, t) is generally described by the Boltzmann equation, (R #* #eERp ) fM (r, p, t)"I[ fM ]#I [ fM ] . R
(203)
Here E is the local electric "eld, I [ fM ] is the inelastic collision integral, due to the electron}electron and electron}phonon scattering (we do not have to specify this integral explicitly at this stage), and
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I[ fM ] is the electron-impurity collision integral. For a d-dimensional disordered system of volume X it is written as I[ fM (r, p, t)]"X
dp [JM ( p , p, r, t)!JM ( p, p , r, t)] , (2p )B
JM ( p, p , r, t),= I ( p, p , r) fM (r, p, t)[1!fM (r, p , t)] ,
(204)
where we have introduced the probability = I of scattering per unit time from the state p to the state p due to the impurity potential ;, = I ( p, p , r)"(2p/ )";pp "d[e( p)!e( p )] . Y Thus, the impurity collision integral can be considered as the sum of particle currents J to/from the state p from/to all the possible "nal states p , taken with appropriate signs. The #uctuations are taken into account via the Boltzmann}Langevin approach, introduced in condensed matter physics by Kogan and Shul'man [244]. This approach assumes that the particle currents between the states p and p #uctuate due to the randomness of the scattering process and partial occupation of the electron states. We write J( p, p , r, t)"JM ( p, p , r, t)#dJ( p, p , r, t) ,
(205)
where the average current JM is given by Eq. (204), and dJ represents the #uctuations. Then the actual distribution function f (r, p, t), which is the sum of the average distribution fM and #uctuating part of the distribution df, f (r, p, t)"fM (r, p, t)#df (r, p, t) ,
(206)
obeys a Boltzmann equation which contains now in addition a #uctuating Langevin source m on the right-hand side, (R #* #eERp ) f (r, p, t)"I[ f ]#I [ f ]#m(r, p, t) , R dp [dJ( p , p, r, t)!dJ( p, p , r, t)] . m(r, p, t)"X (2p )B
(207)
These Langevin sources are zero on average, 1m2"0. To specify the #uctuations, Kogan and Shul'man [244] assumed that the currents J( p, p , r, t) are independent elementary processes. This means that these currents are correlated only when they describe the same process (identical initial and "nal states, space point, and time moment); for the same process, the correlations are taken to be those of a Poisson process. Explicitly, we have (2p )B d( p !p )d( p !p ) 1dJ( p , p , r, t)dJ( p , p , r , t )2" X ;d(r!r )d(t!t )JM ( p , p , r, t) . Eq. (208) then implies the following correlations between the Langevin sources: 1m(r, p, t)m(r , p , t )2"d(r!r )d(t!t )G( p, p , r, t) ,
(208)
(209)
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where
G( p, p , r, t)"X d( p!p ) dp [JM ( p , p, r, t)#JM ( p, p , r, t)]
! [JM ( p, p , r, t)#JM ( p , p, r, t)] .
(210)
Note that the sum rule
dp G( p, p , r, t)" dp G( p, p , r, t)"0
(211)
is ful"lled. This sum rule states that the #uctuations only redistribute the electrons over di!erent states, but do not change the total number of particles. These equations can be further simpli"ed in the important case (which is the main interest in this Section) when all the quantities are sharply peaked around the Fermi energy. Instead of the momentum p, we introduce then the energy E and the direction of the momentum n"p/p. The velocity and the density of states are assumed to be constant and equal to v and l , respectively. $ $ We write X= I ( p, p , r)"l\d(E!E )=(n, n , r) , $ where = is the probability of scattering from the state n to the state n per unit time at the space point r. Furthermore, we will be interested only in the stationary regime, i.e. the averages fM and JM (not the #uctuating parts) are assumed to be time independent. Eliminating the electric "eld E by the substitution EPE!eu(r), with u being the potential, we write the Boltzmann}Langevin equation in the form (R #v n ) f (r, n, E, t)"I[ f ]#I [ f ]#m(r, n, E, t) , R $ where the Langevin sources m are zero on average and are correlated as follows: 1 1m(r, n, E, t)m(r , n , E , t )2" d(r!r )d(t!t )d(E!E )G(n, n , r, E) , l $
(212)
(213)
G(n, n , r, E), dn [d(n!n )!d(n !n )] ;[=(n, n , r) fM (1!fM )#=(n , n, r) fM (1!fM )] .
(214)
We used the notations fM ,fM (r, n, E) and fM ,fM (r, n , E). The expression for the current density,
j(r, t)"ev dn dE nf (r, n, E, t) $
(215)
(with the normalization dn"1), completes the system of equations used in the Boltzmann} Langevin method. We remark that in this formulation the local electric potential does not appear explicitly: for systems with charged carriers such as electric conductors the electric "eld is coupled to the (#uctuating) charge density via the Poisson equation. We will return to this point when we discuss situations in which a treatment of this coupling is essential.
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6.2. Metallic diwusive systems: classical theory of -noise suppression and multi-probe generalization Eqs. (212)}(214) are very general and may be applied to a large variety of systems. Now we turn to the case of metallic di!usive systems, where these equations may be simpli"ed even further and eventually can be solved. A classical theory of noise in metallic di!usive wires was proposed by Nagaev [75], and subsequently by de Jong and Beenakker [89,90]. Sukhorukov and Loss [113,114] gave another derivation of the shot noise suppression for the two-terminal wire and, more importantly, generalized it to treat conductors of arbitrary geometry and with an arbitrary number of contacts. Below we give a sketch of the derivation, following Ref. [114], to which the reader is addressed for further details. The distribution function in metallic di!usive systems is almost isotropic. We then separate it into symmetric and asymmetric parts, f (r, n, E, t)"f (r, E, t)#nf (r, E, t) . (216) For simplicity, we consider here the relaxation time approximation for the electron-impurity collision integral,
n nf ? " dn =(n, n , r)[n !n ] , I[ f ]"! , ? ? q(r) q(r) where q is the average time a carrier travels between collisions with impurities. This approximation is valid when the scattering is isotropic (= depends only on the di!erence "n!n "). The full case is analyzed in Ref. [114]. Integrating Eq. (212) "rst with dn, and then with n dn, we obtain
D ) f "lIM [ f ], IM [ f ], dn I [ f ] ,
(217)
f "!l f #qd nm dn , where we have introduced the mean-free path l"v q and the di!usion coe$cient D"v l/d. We $ $ assume the system to be locally charge neutral. Integrating Eq. (217) with respect to energy, we obtain for the local #uctuation of the current dj#p du"djQ, ) dj"0,
djQ(r, t)"ell n m(r, n, E, t) dn dE . $
(218)
Here p"elD is the conductivity, and u(r) is the electrostatic potential. The currents jQ are correlated as follows: 1djQ(r, t)djQ (r , t )2"2pd d(r!r )d(t!t )P(r) , J K JK
(219)
This is the transport mean-free path; this de"nition di!ers by a numerical factor from that used in Section 2. Following the tradition, we are keeping two di!erent de"nitions for the scattering approach and the kinetic equation approach.
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where the quantity P is expressed through the isotropic part of the average distribution function fM ,
P(r)" dE fM (r, E)[1!fM (r, E)] .
(220)
The distribution fM obeys the equation D fM (r, E)#IM [ fM !ln ) fM ]"0 . (221) The standard boundary conditions for the distribution function are the following. Let ¸ denote L the area of contact n (14n4N), and X the rest of the surface of the sample. At contact n, the non-equilibrium distribution function fM is determined by the equilibrium Fermi distribution function in the reservoir n, fM " L "f (E), whereas away from the contacts the current perpendicular L * to the surface must vanish and thus N ) fM "X "0, where N is the outward normal to the surface. Eqs. (218) and (219) can be used to "nd the current}current #uctuations if the non-equilibrium carrier distribution is known. Thus we proceed "rst to "nd the non-equilibrium distribution function, solving Eq. (221). We follow then Ref. [114] and "nd the characteristic potentials , L which on the ensemble average obey the Poisson equation "0, with the boundary conditions L
" K "d , N )
"X "0 . KL L L* In terms of the characteristic potentials the electrostatic potential is [136] u(r)" (r)< , L L L where < is the voltage applied to the reservoir n. Note that (r)"1 at every space point as L L L a consequence of the invariance of the electrical properties of the conductor under an arbitrary overall voltage shift. With the help of the characteristic potentials, the conductance matrix (which we, as before, de"ne as I "G < , I being the current through ¸ directed into the sample), we K KL L K K obtain
G "p dr
KL K L
(222)
(the integration is carried out over the whole sample). The conductances are independent of the electrical (non-equilibrium) potential inside the conductor. To see this one can rewrite Eq. (222) in terms of a surface integral.
For an arbitrary conductor the electrostatic potential can be expanded as u(r)" (r)< ; Ref. [136] calls the L L coe$cients characteristic potentials. We remark that in the absence of inelastic processes, the average distribution L function can be written as a linear combination of the equilibrium reservoir functions f , fM (r)"l\ l (r) f . Refs. L $ L L [110,111] call the coe$cients l injectivities. In the di!usive metallic conductor of interest here the characteristic L potentials and injectivities are the same functions up to a factor given by the local density of states l . Such an $ equivalence does not hold, for instance, in systems composed of di!erent metallic di!usive conductors, and in general the characteristic potentials and injectivities may have a quite di!erent functional form. Here the use of the characteristic potentials has the advantage that it takes e!ectively the local charge neutrality into account.
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Multiplying Eq. (218) by
and integrating over the whole volume we obtain the #uctuation of L the current through the contact n. The potential #uctuations du(r) actually play no role and are eliminated due to the boundary condition that they vanish at the contacts. At zero temperature, and to linear order in the applied voltage, this is exact: Internal potential #uctuations play a role only in the non-linear voltage dependence of the shot noise and in its temperature dependence. Taking into account the form of the correlation function (219), we "nd the noise power,
S "4p dr
(r)
(r)P(r) . L K KL
(223)
Eq. (223), together with Eq. (221) for the distribution function fM , is the general result for the multi-terminal noise power within the classical approach. At equilibrium P(r)"k ¹, and Eq. (223) reproduces the #uctuation-dissipation theorem. We next apply Eq. (223) to calculate noise suppression in metallic di!usive wires, for the case when the inelastic processes are negligible, IM "0. We consider a wire of length ¸ and width =;¸, situated along the axis x between the point x"0 (reservoir L) and x"¸ (R). The voltage < is applied to the left reservoir. There are only two characteristic potentials,
"1! "1!x/¸ , (224) * 0 which obey the di!usion equation and do not depend on the transverse coordinate. The average distribution function is found as f (x)" (x) f # (x) f , and thus the quantity P for zero * * 0 0 temperature is expressed as P(x)"e< (x)[1! (x)] . * * Subsequently, we "nd the conductance G "p=/¸, and the shot noise ** x 2e1I2 4ep=< * x . (225) S " dx 1! " ** ¸ 3 ¸ ¸ As we mentioned earlier, this expression is due to Nagaev [75]. The Fano factor is , in accordance with the results found using the scattering approach (Section 2). For purely elastic scattering the distribution function fM in an arbitrary geometry quite generally can be written as
fM (r, E)" (r) f (E) . (226) L L L This facilitates the progress for multi-probe geometries. Sukhorukov and Loss [113,114] obtain general expressions for the multi-terminal noise power and use them to study the Hanbury Brown}Twiss e!ect in metallic di!usive conductors. The quantum-mechanical theory of the same e!ect can be found in Ref. [78].
We use two-dimensional notations, d"2.
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6.3. Interaction ewects Interaction e!ects are relatively easy to deal with in the Boltzmann}Langevin approach, in contrast to the di$culties encountered by the scattering theory. 6.3.1. Electron}electron interactions An important feature of electron}electron interactions is that they do not change the total momentum of the electron system. Generally, therefore, electron}electron scattering alone cannot cause transport, and in particular it cannot cause noise. Technically, this is manifested in the fact that the form of current}current #uctuations is given by the same expression (223), as in the non-interacting case. However, electron}electron scattering processes alter the distribution function fM , and thus the value of the shot noise. The inelastic collision integral for electron}electron interactions IM [ f ] (which we denote IM [ f ]) generally has the form
IM [ f (E)]" dE du K(u)+ f (E) f (E )[1!f (E!u)][1!f (E #u)] !f (E!u) f (E #u)[1!f (E)][1!f (E )], ,
(227)
where the kernel K(u) for disordered systems must be found from a microscopic theory. For three-dimensional metallic di!usive systems it was obtained by Schmid [245]; for two- and one-dimensional systems the zero-temperature result may be found in Ref. [246]. For "nite temperatures, a self-consistent treatment is needed [247]. This kernel turns out to be a complicated function of disorder and temperature. In particular, in 1D for zero temperature it diverges as K(u)Ju\ for uP0. The strength of the interaction is characterized by a time q , which we call the electron}electron scattering time. Thus, one needs now to solve Eq. (221) for fM , calculate the quantity P(r), and substitute it into the expression (223) to obtain the noise. Up to now, only two limiting cases have been discussed analytically. First, for weak interactions D/¸
This is only correct if Umklapp processes can be neglected. The in#uence of Umklapp processes on shot noise in mesoscopic systems has not been investigated.
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equilibrium Fermi function with the potential u(r) and a local e!ective temperature ¹(r),
fM (r, E)" exp
E!eu(r) \ #1 . k ¹(r)
(228)
In the literature this distribution function is also referred to as a hot electron distribution. Eq. (221) is now used to "nd the e!ective temperature pro"le ¹(r). Consider the quantity
w(r)" dE E[ fM (E)!h(E!eu(r))] ,
(229)
which up to a coe$cient and an additive constant is the total energy of the system. The substitution of Eq. (228) gives w(r)"p¹(r)/6. Then the application of Eq. (221) yields p [¹(r)]"!e u(r)"!eE(r) , (230) where we have taken into account u"0 to obtain the "nal result. Actually, the term on the right-hand side is proportional to the Joule heating jE, and the equation states that this energy losses are spent to heat the electron gas. In the following, we again specialize to the case of a quasi-one-dimensional metallic di!usive wire between x"0 and ¸. In this case E"
3 e< ¹(x)" ¹ # x(¸!x) . p ¸
(231)
Substituting this into Eq. (223), we "nd the shot noise for hot electrons. In particular, when the bath temperature ¹ equals zero, we "nd for the Fano factor F"((3/4)+0.43 .
(232)
The result (231) is due to Kozub and Rudin [249,250] and Nagaev [248]. The multi-terminal generalization is given by Sukhorukov and Loss [113,114]. Eq. (231) states that shot noise for hot electrons is actually higher than for non-interacting electrons. This agrees with the notion which we obtained considering the scattering approach (Section 2): Electron heating enhances the shot noise. Indeed, enhanced (as compared to the -suppression) shot noise was observed experimentally by Steinbach et al. [80]. Their experimental data are shown in Fig. 33 for a particular sample. Shorter samples in the same experiment [80] exhibit Fano factors which are between and (3/4. The -suppression of shot noise and crossover from the di!usive to the hot-electron regime was very carefully studied by Henny et al. [81], see Section 2.6.4. The hot-electron result (232) is actually independent of the details of electron}electron interaction (independent of the kernel K(u) in Eq. (227)). The cross-over between F" and F"(3/4 does depend on this kernel. Nagaev [143] and Naveh [144] studied this crossover numerically for a particular form of K(u) which assumes that there is no interference between elastic and electron}electron scattering. They suggested that information on the strength of the electron} electron scattering may be extracted from the zero-frequency noise measurements.
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Fig. 33. Shot noise observed at the same sample for three di!erent temperatures by Steinbach et al. [80]. Dashed and solid lines indicate the -suppression and the hot electron result F"(3/4. The temperature is the lowest for the lowest curve. Copyright 1996 by the American Physical Society.
At this point, we summarize the information we obtained from the scattering and Boltzmann} Langevin approaches about the e!ects of the electron}electron interactions on noise. There are two characteristic times, one responsible for dephasing processes, q , and another one due to inelastic ( scattering (electron heating), q . We expect q ;q . Dephasing does not have an e!ect on noise, ( and thus for short enough samples, D/¸
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6.3.2. Electron}phonon interactions In contrast to electron}electron scattering, electron}phonon interactions do change the total momentum of electrons and cause a "nite resistance and noise by themselves. Noise in ballistic, one-channel quantum wires due to electron}phonon interactions within the Boltzmann}Langevin approach was studied by Gurevich and Rudin [253,254], who start directly from Eq. (212) with the electron}phonon collision integral on the right-hand side (no impurity scattering). They consider only the situation of weak interactions, when the electron distribution function is not modi"ed by inelastic scattering. They discover that, for this case, the main e!ect is the absence of noise when the Fermi energy lies below the threshold energy E "2p s, where s is the sound velocity. The $ appearance of this threshold is due to the fact that the maximal wave vector of acoustic phonons which interact with electrons is 2p / . For E 'E shot noise grows. The suppression of shot $ $ noise in long wires, which is a consequence of the equilibration of the electron distribution function due to strong interactions, was beyond the scope of Refs. [253,254]. In disordered systems, one has to take into account three e!ects. First, elastic scattering modi"es the electron}phonon collision integral [255], which assumes di!erent forms depending on the spatial dimension and degree of disorder. Then, it a!ects the distribution function of electrons. Finally, interactions modify the resistance of the sample, and the expression for the shot noise does not have the simple form of Eq. (223). The distribution function of phonons, which enters the electron}phonon collision integral, must, in principle, be found from the Boltzmann equation for phonons, which couples with that for electrons. The standard approximations used to overcome these di$culties and to get reasonable analytical results are as follows. First, for low temperatures, the contribution to the resistance due to electron}phonon collisions is much smaller than that of electron-impurity scattering. Thus, one assumes that Eq. (223) still holds, and electron}phonon collisions only modify the distribution function for electrons and the form of the collision integral. Furthermore, phonons are assumed to be in equilibrium at the lattice temperature. This approach was taken by Nagaev, who calculates the e!ects of electron}phonon scattering on the shot noise of metallic di!usive wires for the case when electron}electron scattering is negligible [75], and subsequently for the case of hot electrons [248]. In addition, careful numerical studies of the role of the electron}phonon interaction in noise in metallic di!usive conductors (starting from the Boltzmann}Langevin approach) are performed by Naveh et al. [256], and Naveh [145]. The results depend on the relation between temperature, applied voltage, and electron}phonon interaction constant, and we refer the reader to Refs. [75,248,256,145] for details. The only feature we want to mention here is that for constant voltage e<
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one "rst goes from the non-interacting regime F" to the hot-electron regime, F"(3/4. For even longer wires electron}phonon collisions play a role, and the Fano factor decreases down to zero. 6.4. Frequency dependence of shot noise While discussing the classical approach to the shot noise suppression, we explicitly assumed that the sample is locally charge neutral: charge pile-up is not allowed in any volume of any size. As a result, we obtained white (frequency independent) shot noise. In reality, however, there is always a "nite (though small) screening radius, which in the case when the system is locally three-dimensional has the form j "(4pel)\ . As Naveh et al. [140] and Turlakov [257] point out, if one of the dimensions of a disordered sample becomes comparable with j , the pile-up of the charge may modify the frequency dependence of noise, though it leaves the zero-frequency noise power unchanged. If all the dimensions of the sample exceed the screening radius (which is typically the case for metallic, and often also for semiconducting) mesoscopic systems, the charge pile-up inside the sample is negligible, and noise stays frequency independent until at least the plasma frequency, which in three-dimensional structures is very high. The situation is di!erent if the sample is capacitively connected to an external gate. As we have seen in the framework of the scattering approach (Section 3), the fact that the sample is now charged, leaves the zero-frequency noise unchanged, but strongly a!ects the frequency dependence of the shot noise. An advantage of the Boltzmann}Langevin approach is that it can treat these e!ects analytically, calculating the potential distribution inside the sample and making use of it to treat the current #uctuations. The general program is as follows. Instead of the charge neutrality condition, one uses the full Poisson equation, relating potential and density #uctuations inside the sample. In their turn, density #uctuations are related to the current #uctuations via the continuity equation. Finally, one expresses the current #uctuations via those of the potential and the Langevin sources. Thus, the system of coupled partial di!erential equations with appropriate boundary conditions needs to be solved. The solution is strongly geometry dependent and has not been written down for an arbitrary geometry. Particular cases, with a simple geometry, were considered by Naveh et al. [140,142], Nagaev [141,143], and Naveh [144,145]. Without even attempting to give derivations, we describe here the main results, referring the reader to these papers for more details. A conductor in proximity to a gate can be charged vis-a`-vis the gate. We can view the conductor and the gate as the two plates of a capacitor. In the limit where the screening length is much larger than the wire radius, only a one-dimensional theory is needed. It is this atypical situation which is considered here. For a wire of cross-section A and a geometrical capacitance c per unit length its low-frequency dynamics is characterized by the electrochemical capacitance c\"c\#(eAl)\ which is the parallel addition of the geometrical capacitance and the I quantum capacitance (eAl)\. Here we have assumed that the potential is uniform both along the wire and more importantly also in the transverse direction of the wire. Any charge accumulated in the wire can dynamically relax via the reservoirs connected to the wire and via the external circuit For a discussion of electrochemical capacitance and ac conductance see Refs. [157,165]; see also Section 3.
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which connects the wire and the gate. For a zero-external impedance circuit this relaxation generates a charge relaxation resistance R (see the discussion in Section 3) which for a metallic O di!usive conductor is of the order of the sample resistance R"¸(pA)\. With these speci"cations we expect that a metallic di!usive wire in proximity of a gate is characterized by a frequency u "1/R C which is given by u "pA/(c¸)#p/(el¸). (Refs. [140,142] express u in 0! O I 0! 0! terms of a generalized di!usion constant D "D#pA/c using the Einstein relation p"elD, such that u " D /¸ has the form of a Thouless energy.) For u;u , noise measured at the 0! 0! contacts to the wire is dominated by the white-noise zero-frequency contribution (the Fano factor equals for independent electrons or (3/4 for hot electrons). For frequencies higher than u the 0! spectrum measured at the contacts of the wire starts to depend on the details of the system, and for in"nite frequency the Fano factor tends to a constant value, which may lie above as well as below the non-interacting value. This is because the zero-temperature quantum noise SJ "u" cannot be obtained by classical means: thus, all the results of these subsections are applicable only outside of the regime when this source of noise is important. In particular, for zero temperature this means u(e"<". The crossover to the quantum noise was recently treated by Nagaev [214] using the Green's functions technique. It is also assumed that the frequency is much below the inverse elastic scattering time, u;q\; outside this regime, the di!usion approximation is not valid. As emphasized in work on chaotic cavities [98] experiments which measure the noise at the gate can also be envisioned: This has the advantage that even for frequencies much smaller than u the noise is frequency dependent and 0! in fact can for metallic systems also be expected to be determined by R and C . O I Nagaev [141,143] considers a circular conductor of length ¸ and radius R, surrounded by a circular gate. For ¸;R (short wire) he "nds that the noise spectrum measured at a contact is frequency independent. However, generally the frequency dependence is quite pronounced. Thus, for the case when the distribution function is described by Eq. (226), and wires are long, ¸
Numerical results for the same model with charge pile-up (only non-thermalized electrons) were previously provided by Nagaev [141].
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non-thermalized electrons, the high-frequency noise (u
) j#eE(r, t)R j"0 #
(233)
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and j"!D(E) o!p(E)E(r, t)R f #dj(r, E, t)"0 . (234) # Here D(E)"2Eq/md and p(E)"el(E)D(E) are the energy-dependent di!usion coe$cient and conductivity, respectively. The electric "eld E is coupled to the charge density via the Poisson equation,
) E(r, t)" dE o(r, E, t) ,
(235)
and the energy-resolved Langevin currents dj are correlated in the following way (cf. Eq. (219)): 1dj (r, E, t)dj (r , E , t )2"2p(E)d d(r!r )d(E!E )d(t!t ) fM (r, E, t) . (236) J K JK Eqs. (233)}(235) describe the response of the system to the #uctuations (236) of the Langevin currents. The main complication as compared to the degenerate case is that the #uctuations of the electric "eld now play an essential role, and cannot be neglected. For this reason, results for the shot noise depend dramatically on the geometry. Further progress can be achieved for the quasi-one-dimensional geometry of a slab 0(x(¸, located between two reservoirs of the same cross-section. One more approximation is the regime of space-charge limiting conduction, corresponding to the boundary condition E(x"0, t)"0. This condition means that the charge in the contacts is well screened, ¸<¸ <¸ , with ¸ and ¸ being the screening length in the contacts and the sample, respectively. Eqs. (233)}(235) are supplemented by the absorbing boundary condition at another contact, o(x"¸, t)"0. Now one can calculate the potential pro"le inside the sample and subsequently the shot noise power. In this way, within the approximation of energy-independent relaxation times, Schomerus et al. [262] obtain the following results:
0.69, d"1 ,
F" 0.44, d"2 ,
(237)
0.31, d"3 .
The numerical values (237) are, indeed, close to 0.7, and , respectively, in accordance with the numerical results by GonzaH lez et al. [258,259], but not precisely equal to them. It is now worthwhile to mention that the results (237), in contrast to the -suppression of shot noise in the degenerate di!usive conductors, are not universal. Whereas the suppression is independent of the geometry of the sample, degree of the disorder, or local dimensionality, the values (237), being geometry independent, do depend on the dimensionality of the sample. Furthermore, they do depend on the disorder, and this dependence enters through their sensitivity to the energy dependence of the relaxation time, as noticed by Nagaev [261]. Schomerus et al. [263] investigated the case q(E)JE?, !4a41, which are the only values of a compatible with the regime of space-charge-limited conduction. In particular, a"! corresponds to scattering on In the case of metallic di!usive wires, this dependence is irrelevant since the relaxation time is evaluated at the Fermi surface.
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short-ranged impurities. They found that the Fano factor in d"3 crosses over monotonically from F"0.38 (a"!) to F"0 (a"1). There is no shot noise in this model for a'1. This example clearly demonstrates that Fermi statistics are not necessary to suppress shot noise, in accordance with general expectations. 6.5.2. Ballistic non-degenerate conductors Bulashenko et al. [264] address the noise in charge-limited ballistic conductors. They consider a two-terminal semiconductor sample with heavily doped contacts. Carriers in the semiconductor exist only due to injection from the contacts which thus determine the potential distribution inside the sample. The self-consistent "eld determines a barrier at which carriers are either completely re#ected or completely transmitted (no tunneling). This system is thus a close analog of the charge-limited shot noise in vacuum tubes. It is easy to adapt the Boltzmann}Langevin formulation to this problem: Since carrier motion inside the conductor is determined by the Vlasov equation (collisionless Boltzmann equation), (238) (R #v R #eE R V ) f (x, p, t)"0 , R V V V N the distribution function f (x, p, t) at any point inside the sample is determined by the distribution function at the surface of the sample. The only source of noise arises from the random injection of carriers at the contacts. Thus the boundary conditions are f (0, p, t)" V "f #df ( p, t) , * * T (239) f (¸, p, t)" V "f #df ( p, t) . 0 0 T The stochastic forces df are zero on average, and their correlation is * 0 1df ( p, t)df ( p , t )2Jd d(p !p )d(t!t ) f (p ) , (240) ? @ ?@ V V ? V where f (p ) is the Maxwell distribution function restricted to p '0 (a"¸) or p (0 (a"R). ? V V V Note that in the degenerate case we would have to write an extra factor 1!f on the right-hand ? side of Eq. (240), thus ensuring that there is no noise at zero temperature. We have checked already in Section 2 that this statement is correct. On the contrary, in the non-degenerate case f ;1, and ? thus (1!f )&1. The results for this regime do not of course allow an extrapolation to k ¹"0. ? The crossover between the shot noise behavior in degenerate and non-degenerate conductors was investigated by Gonza`lez et al. [265] using Monte Carlo simulations. Eq. (238) is coupled to the Poisson equation, !dE /dx"4p f (x, p, t) . V p
(241)
Solving the resulting equations, Bulashenko et al. [264] found that interactions, at least in a certain parameter range, suppress shot noise below the Poisson value. The suppression may be arbitrarily strong in long (but still ballistic) samples. The results are in good agreement with previous numerical studies of shot noise in the same system by GonzaH lez et al. [266,268] and Bulashenko et al. [267]. The crossover between the ballistic and di!usive behavior of the non-degenerate Fermi gas was numerically studied by GonzaH lez et al. [268,259]. Analytical results on this crossover are presently unavailable.
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A similar problem, a ballistic degenerate conductor in the presence of a nearby gate, was posed by Naveh et al. [269]. We conclude this subsection by recalling that the e!ect of interactions on noise arises not only in long ballistic structures but already in samples which are e!ectively zero dimensional, like resonant tunneling diodes [78] or in quantum point contacts [98]. The added complication in extended structures arises from the long-range nature of the Coulomb interaction. 6.6. Boltzmann}Langevin method for shot noise suppression in chaotic cavities with diwusive boundary scattering Now we turn to the classical derivation of the -shot noise suppression in chaotic cavities. In standard cavities, which are regular objects, the chaotic dynamics arises due to the complicated shape of a surface. Thus, scattering at the surface is deterministic and in an individual ensemble member scattering along the surface of the cavity is noiseless. Thus, it is not obvious how to apply the Boltzmann}Langevin equation. However, recently a model of a random billiard } a circular billiard with di!usive boundary scattering } was proposed [270,271] to emulate the behavior of chaotic cavities. It turned out that the model can be relatively easily dealt with, and Refs. [270,271] used it to study spectral and eigenfunction properties of closed systems. Ref. [99] suggests that the same model may be used to study the transport properties of the open chaotic cavities and presents the theory of shot noise based on the Boltzmann}Langevin approach. We consider a circular cavity of radius R connected to the two reservoirs via ideal leads; the angular positions of the leads are h !a /2(h(h #a /2 (left) and !a /2(h(a /2 (right), * * 0 0 see Fig. 34; h is the polar angle. The contacts are assumed to be narrow, a ;1, though the * 0 numbers of the transverse channels, N "p Ra /p , are still assumed to be large compared to * 0 $ * 0 1. Inside the cavity, motion is ballistic, and the average distribution function fM (r, n) obeys the equation n fM (r, n)"0 .
(242)
At the surface (denoted by X) we can choose a di!usive boundary condition: the distribution function of the particles backscattered from the surface is constant (independent of n) and "xed by the condition of current conservation,
(Nn ) fM (r, n ) dn , Nn (0 ,
fM (r, n)"p
(243)
Y Nn
where r3X, N is the outward normal to the surface, and dn"1. Furthermore, we assume that the electrons coming from the leads are described by the equilibrium distribution functions, and are
Earlier, a similar model was numerically implemented to study spectral statistics in closed square billiards [272,273]. This is the simplest possible boundary condition of this kind. For a review, see Ref. [274]. We expect similar results for any other di!usive boundary condition.
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Fig. 34. Geometry of the chaotic cavity with di!usive boundary scattering.
emitted uniformly into all directions. Explicitly, denoting the cross-sections of the left and the right leads by X and X , we have * 0 fM (r, n)"f , r3X , Nn(0 . (244) * 0 * 0 Now we can "nd the average distribution function. Since motion away from the boundary is ballistic, the value of the distribution function, Eq. (242), at a point away from the boundary, is determined by the distribution function at the surface associated with the trajectory that reaches this point after a scattering event at the surface. With the boundary conditions (243) and (244), we can then derive an integral equation for fM (h), 1 fM (h)"X " 4
X
X *
h!h fM (h ) sin dh 2 X 0
(245)
> > subject to the additional conditions fM (h)"X* 0 "f . This exact equation may be considerably * 0 simpli"ed in the limit of narrow leads, a ;1. Integrals of the type X0 F(h) dh can now be replaced * 0 by a F(0). This gives for the distribution function 0 g(0)!g(h ) a a (a !a ) a f #a f 0 ( f !f ) * 0 * 0 0# fM (h)" * * * 0 (a #a ) 4p a #a * 0 * 0 g(h)!g(h!h ) a a * 0 ( f !f ) , # (246) 0 (a #a ) * 4p * 0 with the notation cos lh 1 g(h)" " (3h!6ph#2p), 04h42p . l 12 J The "rst part of the distribution function (246) does not depend on energy and corresponds to the random matrix theory (RMT) results for the transport properties. The second two terms on
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the right-hand side are not universal and generate sample-speci"c corrections to RMT [99], which we do not discuss here. The conductance is easily found to be N N * 0 , (N #N ) * 0 which is identical to the RMT result. The main problem we encounter in attempting to calculate noise via the Boltzmann}Langevin method is that the system is not described by a collision integral of the type (204). Instead, the impurity scattering is hidden inside the boundary condition (243), which, in principle, itself must be derived from the collision integral. This di$culty can, however, be avoided, since we can calculate the probability =(n, n , r) of scattering per unit time from the state n to the state n at the point r (which is, of course, expected to be non-zero only at the di!usive boundary). Indeed, this probability is only "nite for Nn'0 and Nn (0. Under these conditions it does not depend on n , and thus equals =(n, n , r)"2= (n, r), with = being the probability per unit time to scatter out of the state n at the space point r. Consider now the (short) time interval *t. During this time, the particles which are closer to the surface than v nN*t are scattered with probability one, and others $ are not scattered at all. Taking the limit *tP0, we obtain G"(e/2p )
v nN d(R!r), nN'0, n N(0 , =(n, n , r)" $ 0, otherwise .
(247)
Imagine now that we have a collision integral, which is characterized by the scattering probabilities (247). Then, the #uctuation part of the distribution function df obeys Eq. (212) with the Langevin sources correlated according to Eq. (213). A convenient way to proceed was proposed by de Jong and Beenakker [89,90], who showed that quite generally the expression for the noise can be brought into the form
S"2el dE dn dn dr ¹ (n, r)¹ (n , r)G(n, n , r, E) , 0 0 $
(248)
where the function G is given by Eq. (214), and ¹ is the probability that the particle at (n, r) will 0 eventually exit through the right lead. This probability obeys n ¹ "0 with the di!usive 0 boundary conditions at the surface; furthermore, it equals 0 and 1 provided the particle is headed to X and X , respectively. In the leading order ¹ "a /(a #a ); however, this order does not * 0 0 0 * 0 contribute to the noise due to the sum rule (211). The subleading order is that for n pointing out of the left (right) contact, ¹ "0 (1). Substituting the distribution function fM "(a f #a f )/ 0 * * 0 0 (a #a ) into Eq. (248), we obtain for the Fano factor * 0 F"N N /(N #N ) . * 0 * 0
The coe$cient 2, instead of p\, is due to the normalization. We assume that there is no inelastic scattering, I ,0.
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We have thus presented a purely classical derivation of the Fano factor of a chaotic cavity (96). This result was previously derived with the help of the scattering approach and RMT theory (Section 2.6.5). To what extent can the billiard with di!usive boundary scattering also describe cavities which exhibit deterministic surface scattering? As we have stated above, if we consider an ensemble member of a cavity with specular scattering at the surface, such scattering is deterministic and noiseless. Thus, we can de"nitely not expect the model with a di!usive boundary layer to describe an ensemble member. However, to the extent that we are interested in the description only of ensemble-averaged quantities (which we are if we invoke a Boltzmann}Langevin equation) the di!usive boundary layer model can also describe the ensemble-averaged behavior of cavities which are purely deterministic. While in an individual cavity, a particle with an incident direction and velocity generates a de"nite re#ected trajectory, we can, if we consider the ensemble-average, associate with each incident trajectory, a re#ected trajectory of arbitrary direction. In the ensemble, scattering can be considered probabilistic, and the di!usive boundary model can thus also be used to describe cavities with completely deterministic scattering at the surface. This argument is correct, if we can commute ensemble and statistical averages. To investigate this further, we present below another discussion of the deterministic cavity. 6.7. Minimal correlation approach to shot noise in deterministic chaotic cavities In cavities of su$ciently complicated shape, deterministic chaos appears due to specular scattering at the surface. To provide a classical description of shot noise in this type of structures, Ref. [99] designed an approach which it called a `minimal correlationa approach. It is not clear whether it can be applied to a broad class of systems, and therefore, we decided to put it in this section rather than to provide a separate section. In a mesoscopic conductor, in the presence of elastic scattering only, the distribution function is quite generally given by [110,111] l (r) (249) f (r)" L f , l(r) L L where l (r) is the injectivity of contact n (the contribution to the local density of states l(r) of contact L n.) For the ensemble-averaged distribution which we seek, we can replace the actual injectivities and the actual local density of states by their ensemble average. For a cavity with classical contacts (N <1 open quantum channels), the ensemble-averaged injectivities are [138] 1l (r)2" L L l N / N , where l is the ensemble-averaged local density of states. This just states that the $ L L L $ contribution of the nth contact to the local density of states is proportional to its width (number of quantum channels). Thus the ensemble-averaged distribution function, which we denote by f , is !
(250) f " b f , b ,N N . L ! L L L L L L To derive this distribution function classically, we "rst assert that the ensemble-averaged distribution inside the cavity, called f , is a spatially independent constant. This is a consequence of ! the fact that after ensemble averaging at any given point within the cavity there are no preferred
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directions within the cavity if carriers conserve their energy (no inelastic scattering). On the ensemble average the interior of the cavity can be treated as an additional dephasing voltage probe (see Section 2.7). The interior acts as a dephasing probe since we assume that there is no inelastic scattering at the probe. Consequently at such a probe the current in each energy interval is conserved [90]. Thus we must consider the energy-resolved current. Let us denote the current at contact n in an energy interval dE by J (E). The total current at the L contact is I "J dE. In terms of the distribution function of the reservoirs and the cavity the L L energy-resolved current is J (E)"e\G ( f !f ) , (251) L L L ! where G "eN /(2p ) is the (Sharvin) conductance of the nth contact, and N "p = /p is the L L L $ L number of transverse channels. For energy-conserving carrier motion the sum of all currents in each energy interval must vanish (Section 2.7). This requirement immediately gives Eq. (250). Using the distribution function, Eq. (250), gives for the conductance matrix G "(d !b )G . (252) KL KL K L This conductance matrix is symmetric, and for the two-terminal case becomes G " ** (e/2p )(N N /(N #N )), as expected. * 0 * 0 Now we turn to the shot noise. The #uctuation of the current through the contact n is written as ep dI "! $ L 2p
X L
dn dr dE(nN )df (r, n, E, t) , L
(253)
where X and N denote the surface of the contact n and the outward normal to this contact. Here L L df is the #uctuating part of the distribution function, and for further progress we must specify how these #uctuations are correlated. The terms with nN (0 describe #uctuations of the distribution functions of the equilibrium L reservoirs, f (E). These functions #uctuate due to partial occupation of states (equilibrium noise); L the #uctuations of course vanish for k ¹"0. The equal time correlator of these equilibrium #uctuations quite generally is (see e.g. Ref. [275]) 1df (r, n, E, t)df (r , n , E , t)2"l\d(r!r )d(n!n )d(E!E ) fM (r, n, E, t) [1!fM (r, n, E, t)] , (254) $ where in the reservoirs fM "f (E). In particular, the cross-correlations are completely suppressed. L On the other hand, the terms with nN '0 describe #uctuations of the distribution function L inside the cavity. These non-equilibrium #uctuations resemble Eq. (254) very much. Indeed, in the absence of random scattering the only source of noise are the #uctuations of the occupation numbers. Furthermore, in the chaotic cavity the cross-correlations should be suppressed because of multiple random scattering inside the cavity. Thus, we assume that Eq. (254) is valid for #uctuations of the non-equilibrium state of the cavity, where the function f (E) (250) plays the role of ! fM (r, n, E, t). In contrast to the true equilibrium state, these #uctuations persist even for zero temperature, since the average distribution function (250) di!ers from both zero and one. Furthermore, for tOt the correlator obeys the kinetic equation, (R #v n )1df (t)df (t )2"0 R $ [275]. We obtain the following formula: 1df (r, n, E, t)df (r , n , E , t )2"l\d[r!r !v n(t!t )]d(n!n )d(E!E ) f (1!f ) , $ $ ! !
(255)
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which describes strictly ballistic motion and is therefore only valid at the time scales below the time of #ight. An attempt to use Eq. (255) for all times and insert it to Eq. (253) immediately leads to the violation of current conservation. Thus, we must take special care of the #uctuations of the distribution function inside the cavity for the case when t!t is much longer than the dwell time q . In this situation, the electron becomes uniformly distributed and leaves the cavity through the nth contact with the probability b . For times t
S "2G k (¹#¹ ), k ¹ " dE f (1!f ) . KL KL ! ! ! !
(257)
It is easy to check that Eq. (257) actually reproduces all the results we have obtained in Section 2 with the help of the scattering approach. Explicitly, we obtain
e e(< !< ) L K . k ¹ " b b (< !< ) coth ! 2 K L L K 2k ¹ K L At equilibrium ¹ "¹, and the noise power spectra obey the #uctuation-dissipation theorem. For ! zero temperature, in the two-terminal geometry we reproduce the noise suppression factor (96); in the multi-terminal case the Hanbury Brown}Twiss results [115], described in Section 2, also follow from Eq. (257). The perceptive reader notices the close similarity of the discussion given above and the derivation of the classical results from the scattering approach invoking a dephasing voltage probe [90,115] (Section 2.7). The correlations induced by current conservation are built in the discussion of the dephasing voltage probe model of the scattering approach. The discussion which we have given above, is equivalent to this approach, with the only di!erence that #uctuations are at every stage treated with the help of #uctuating distributions. In contrast, the scattering approach with a dephasing voltage probe invokes only stationary, time-averaged distributions. The comparison of the Boltzmann}Langevin method with the dephasing voltage probe approach also serves to indicate the limitations of the above discussion. In general, for partial dephasing (modeled by an additional "ctitious lead) inside the cavity connected via leads with
Minimal means here that the correlations are minimally necessary } the only non-equilibrium type #uctuations considered are those required by current conservation. This kind of #uctuations was discussed by Lax [276]. An approach similar to the one we are discussing was previously applied in Ref. [277] to the shot noise in metallic di!usive wires, where it does not work due to the additional #uctuations induced by the random scattering events.
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a barrier [115], even the averaged distribution function f cannot be determined simply via ! Eq. (251): even on the ensemble average, injectivities are not given simply by conductance ratios.
7. Noise in strongly correlated systems This section is devoted to the shot noise in strongly correlated systems. This is a rather wide subject due to the diversity of the systems considered. There is no unifying approach to treat strongly correlated systems. Typically, the shot noise in interacting systems is described by methods more complicated than the scattering or Langevin approaches. An attempt to present a detailed description of the results and to explain how they are derived would lead us to the necessity to write a separate review (if not a book) for each subject. Our intention is to avoid this, and below we only present some results for particular systems without an attempt to derive them. The discussion is qualitative; for a quantitative description, the reader is referred to the original works. As a consequence, this section has the appearance of a collection of independent results. 7.1. Coulomb blockade The term Coulomb blockade is used to describe phenomena which show a blockage of transport through a system due to the electrostatic e!ects. We recall only some basic facts; the general features of the Coulomb blockade are summarized in the early review article [278]. The most common technique to describe Coulomb blockade e!ects is the master equation approach. 7.1.1. Tunnel barriers The simplest structure for which one might think that the Coulomb blockade is signi"cant is a tunnel junction. The junction is characterized by a capacitance C. From the electrostatic point of view, the system can be regarded as a capacitor where the tunneling between the electrodes is allowed, i.e. the equivalent circuit is the capacitor C connected in parallel with a resistor RJ( ¹ )\. Due to the additional charging energy Q/2C, where Q is the charge of the junction, L L the current through the junction is blocked (i.e. exponentially small for k ¹;e/C) for voltages below < "e/2C (for simplicity, we only consider <'0). The I}< curve is essentially I"R\(
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arrive at the remarkable conclusion that the current conservation for noise is violated for high enough frequencies, (iuC)\;"Z(u)", where Z is the external impedance. Thus, this is an illustration of the statement (Section 3) that one has to take care of the displacement currents, even when the scattering matrices are not energy dependent, for frequencies higher than the inverse collective response time, in this particular situation (R C)\. 7.1.2. Quantum dots An equivalent circuit for a quantum dot with charging (cited in the Coulomb blockade literature as the single-electron transistor, SET) is shown in Fig. 35a. The SET is essentially a two-barrier structure with the capacitances included in parallel to the resistances; in addition, the quantum dot is capacitively coupled to a gate. We assume R
Fig. 35. (a) Equivalent circuit for the single-electron transistor. (b) A sketch of the I}< characteristics (lower curve) and the Fano factor voltage dependence (upper curve) for the very asymmetric case R ;R , C ;C , < "0.
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barriers dominates, see below), and for high voltages is described by a Fano factor F"(R #R )/(R #R ) . (258) The result (258) is genuinely the double-barrier Fano factor (78), since R J¹\ . This high voltage behavior was independently obtained by Hung and Wu [284] using the Green's function technique. Korotkov [281] also studied the frequency dependence of shot noise and found that it is a regular function, which for high frequencies (RC)\;u;e/ C, R\"R\#R\, saturates at an interaction-dependent value. For the general case (Coulomb staircase regime), shot noise was (independently) analyzed by Hersh"eld et al. [283] using a master equation. In the plateau regimes, the transport is via the only state with a "xed number of electrons, and the shot noise is Poissonian (up to exponentially small corrections); in particular, for <(e/2C noise is exponentially small. The situation is, however, di!erent close to the degenerate points (at the center of the step of the staircase), since there are now two charge states available. Hersh"eld et al. [283] "nd that the Fano factor in the vicinity of the degenerate points is F(<)"(C #C )/(C #C ). Here C and C are the (tunneling) rates to add an electron through barrier 1 and remove an electron through barrier 2. Though this result looks similar to the expression for the resonant double-barrier structure (78), the important di!erence is that the rates C are now strongly voltage dependent due to the Coulomb blockade. In particular, at the onset of the step C ;C , and F"1. The Fano factor thus shows dips in the region of the steps. For high voltages the Coulomb blockade is insigni"cant, and the Fano factor assumes the double-barrier suppression value (258). The resulting voltage dependence of the Fano factor is sketched in Fig. 35b, upper curve. Hersh"eld et al. [283] also perform extensive numerical simulations of shot noise and show that the structure in the Fano factor disappears in the symmetric limit R &R . Similar results were subsequently obtained by Galperin et al. [285], and Hanke et al. [286,287], who also investigate the frequency dependence of shot noise. They point out that the quantity RS/Ru is more sensitive to the voltage near the Coulomb blockade steps than the noise spectral power S. Sub-Poissonian shot noise suppression in the Coulomb blockade regime is numerically con"rmed by Anda and LatgeH [288], however, at high bias, they "nd that the average current tends to zero. Another option is to consider transport properties as a function of the gate voltage < . Both current and noise are periodic functions of < with the period of e/C (which corresponds to the period of 1 in N ). As pointed out by Hanke et al. [287], shot noise is then periodically suppressed below the Poisson value. Wang et al. [289] consider shot noise in the Coulomb blockade regime in a semiconductor quantum dot, where the single-particle level spacing is relatively large. In this situation it is not enough to write the interaction potential in the dot in the form Q/2C, and the real space-dependent Coulomb interaction must be taken instead. The current as a function of the gate voltage exhibits a number of well-separated peaks, and, consequently, shot noise is Poissonian everywhere except for the peak positions, where the Fano factor has dips. Surprisingly enough, the numerical results of Wang et al. [289] show that the Fano factor in the dip may be arbitrarily low, certainly below . They explain this as being due to the suppression of the shot noise by Coulomb interactions. We also mention here the papers by Krech et al. [290] and Krech and MuK ller [291], who, based on a master equation, conclude that the Fano factor may be suppressed down to zero in the
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Coulomb blockade regime; in particular, the Fano factor tends to zero for high frequencies. Though we cannot point out an explicit error in these papers, the results seem quite surprising to us. We believe that in the simple model of the Coulomb blockade, when the electrostatic energy is approximated by Q/2C, it is quite unlikely that the shot noise is suppressed below , which is the non-interacting suppression factor for the symmetric double-barrier structure. Clearly, to answer these questions, an analytic investigation, currently unavailable, is required. Experiments on shot noise in the Coulomb blockade regime were performed by Birk et al. [30]. They put a nanoparticle between the STM tip and metallic surface, and were able to change the capacitances C and C and resistances R and R . For R "2R they found the `smootha Coulomb blockade I}< characteristics (no traces of the Coulomb staircase), with the noise crossing over from Poissonian to Eq. (258) with increasing voltage (Fig. 36, left). In contrast, for the very asymmetric case R ;R Birk et al. [30] observe the Coulomb staircase and periodic shot noise suppression below the Poisson value (Fig. 36, right). They compare their experimental data with the theory of Hersh"eld et al. [283] and "nd quantitative agreement. 7.1.3. Arrays of tunnel barriers Consider now a one-dimensional array of metallic grains separated by tunnel barriers. This array also shows Coulomb blockade features: The current is blocked below a certain threshold voltage < (typically higher than the threshold voltage for a single junction), determined by the parameters of the array. Just above < , the transport through the array is determined by a single junction (a bottleneck junction), and corresponds to the transfer of charge e. Thus, the shot noise close to the threshold is Poissonian, the Fano factor equals one. With increasing voltage more and more junctions are opened for transport, and eventually a collective state is established in the array [278]: An addition of an electron to the array causes the polarization of all the grains, such that the e!ective charge transferred throughout the array is e/N, with N<1 being the number of grains. In this regime, the Fano factor is 1/N. The crossover of shot noise from Poissonian to 1/N}suppression was obtained by Matsuoka and Likharev [292], who suggested these qualitative considerations and backed them by numerical calculations. Other interesting opportunities for research open up if the frequency dependence of shot noise is considered. First, in the collective state these arrays exhibit single-electron tunneling oscillations with the frequency u "I/e, I being the average current. These oscillations are a consequence of the discreteness of the charge transfer through the tunnel barrier. The increase of charge on the capacitor is the result of the development of a dipole across the barrier. This is a polarization process for which the charge is not quantized. However, when the polarization charge reaches e/2 the Coulomb blockade is lifted and the capacitance is decharged by the tunneling transfer of a single electron. Both the continuous polarization process and the discrete charge transfer process give rise to currents in the external circuit. It is the di!erence in time scale for the two processes which causes current oscillations. The polarization process is much slower than the decharging process. The discreteness of the charge transfer process is, as we have seen, a key ingredient needed to generate shot noise. However here, in contrast to the Schottky problem, the charge transfer occurs The di!erence of this setup with a SET is that there is no gate. This can be viewed as a transport of solitons with the fractional charge.
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Fig. 36. Experiments by Birk et al. [30]. Left: shot noise in the quantum dot with R "2R (diamonds); the inset shows the I}< characteristics. Right: I}< curve (top) and shot noise (bottom) in a very asymmetric quantum dot. Solid curves show the theory of Hersh"eld et al. [283]. Dashed lines in all cases show the Poisson value 2e1I2 and the value e1I2. Copyright 1995 by the American Physical Society.
at time intervals which are clearly not given by a Poisson distribution. Instead, here, ideally the charge transfer process is clocked and there is no shot noise. However in an array of tunnel junctions, there exists the possibility to observe both current oscillations and shot noise: In an array charge transfer occurs via a soliton (a traveling charge wave). Korotkov et al. [293] and Korotkov [294] "nd that arrays of tunnel barriers exhibit both a current that oscillates (on the average) at a frequency u "I/e, and shot noise which is strongly peaked at the frequency u . Another type of oscillations are Bloch oscillations, which are due to the translational symmetry of the array. Their frequency is proportional to the voltage drop across each contact; if the array is long and <;Ne/C, this frequency is approximately u "e/4p C, u
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which give rise to current slightly above the threshold < "e/C. The dominant process is an electron coming from a reservoir which is Andreev re#ected at the interface, and generates a Cooper pair in the superconducting grain, which is again converted to an electron and a hole pair at the interface. This process results in the transfer of an e!ective charge of 2e; the Fano factor is thus 2. For e/C(e<;D, the Fano factor is expected to be smaller as the voltage is increased and the Coulomb blockade is lifted. For e<&D single-electron tunneling starts to play a role; eventually, for e<
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the asymmetry of the spin-up and spin-down propagation, and the asymmetry of the barrier, respectively. For a given spin projection, the current is proportional to R R /(R #R ), while the shot noise is proportional to R R (R #R )/(R #R ). Here R and R must be taken for each spin projection separately; the total current (shot noise) is then expressed as the sum of the currents (shot noises) of spin-up and spin-down electrons. Evaluating in this way the Fano factor, we obtain F "(1#b)/(1#b) , (259) tt which is precisely the double-barrier Fano factor (78), since the Fano factors for spin-up and spin-down electrons are the same. Similarly, for the anti-parallel (!!N! ) orientation we write R "R , R "aR , t s R "abR , and R "bR . The Fano factor is t s 1#ab a#b 1 \ 1 , (260) # # F " ts (1#ab) (a#b) 1#ab a#b
and depends now on a. In the strongly asymmetric junction, a, b<1, the Fano factor for the parallel orientation (259) becomes 1. At the same time, for the anti-parallel orientation, Eq. (260) is entirely determined by spin-down electrons, and takes the form F "(a#b)/(a#b). Thus, the ts shot noise in the anti-parallel con"guration is suppressed as compared to the parallel case. This situation is, of course, the same in the Coulomb blockade regime, and this is precisely what Bu"ka et al. [300] "nd numerically. In these discussions, it is assumed that both spin channels are independent. In reality, spin relaxation processes which couple the two channels might be important. It is interesting to consider the structure in which one of the reservoirs is ferromagnetic and the another one is normal (FNN junction). Bu"ka et al. [300] only treat the limiting case when, say, spin-down electrons cannot propagate in the ferromagnet at all (R "R). They "nd that the I}< s curve in this case shows negative di!erential resistance, and shot noise may be enhanced above the Poisson value, similarly to what we have discussed in Section 5 for quantum wells. Experimentally, the main problem in ferromagnetic structures is to separate shot noise and 1/f}noise. Nowak et al. [302] measured the low-frequency noise in a tunnel junction with two ferromagnetic electrodes separated by an insulating layer. They succeeded in extracting information on shot noise, and report sub-Poissonian suppression for low voltages, but they did not perform a systematic study of shot noise, and the situation, both theoretically and experimentally, is very far from being clear. 7.1.6. Concluding remarks on the Coulomb blockade A patient reader who followed this Review from the beginning has noticed that we have considerably changed the style. Indeed, in the previous sections we mostly had to deal with results, which are physically appealing, well established, cover the "eld, and in many cases are already experimentally con"rmed. Here, instead, the results are contradictory, in many cases analytically unavailable, and, what is more important, fragmentary } they do not systematically address the "eld. To illustrate this statement, we only give one example; we could have cited dozens of them. Consider a quantum dot under low bias; it is typically Coulomb blocked, unless the gate voltage is tuned to the degenerate state, so that N is half-integer. As we have discussed above, the current dependence on the gate voltage is essentially a set of peaks, separated by the distance of
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approximately e/C. Between the peaks (in the valleys), as we implicitly assumed, the zerotemperature current is due to quantum tunneling, and is exponentially small. We concluded therefore, that the Fano factor is 1 } the shot noise is Poissonian. In reality, however, there are cotunneling processes } virtual transitions via the high-lying state in the dot } which are not exponentially suppressed and thus give the main contribution to the current. Cotunneling is a genuinely quantum phenomenon, and cannot be obtained by means of a classical approach. It is clear that the cotunneling processes may modify the Fano factor in the valleys; moreover, it is a good opportunity to study quantum e!ects in the shot noise. This problem, among many others, remains unaddressed. 7.2. Anderson and Kondo impurities 7.2.1. Anderson impurity model In the context of mesoscopic physics, the Anderson impurity model describes a resonant level with a Hubbard repulsion. It is sometimes taken as a model of a quantum dot. Commonly the entire system is described by the tight-binding model with non-interacting reservoirs and an interaction ;n( n( on the site i"0 (resonant impurity), with n( and n( being the operators of the s t s t number of electrons on this site with spins down and up, respectively. Tunneling into the dot is described, as in the non-interacting case, by partial tunneling widths C and C , which may be * 0 assumed to be energy independent. The on-site repulsion is important (in the linear regime) when ;<E , where E is the energy of the resonant impurity relative to the Fermi level in the reservoirs. In this case, for low temperatures ¹(¹ the spin of electrons traversing the quantum ) dot starts to play a considerable role, and the system shows features essentially similar to the Kondo e!ect [304,305]. Here ¹ is a certain temperature, which is a monotonous function of ;, ) and may be identi"ed with the e!ective Kondo temperature. For ¹(¹ the e!ective transmission ) coe$cient grows as the temperature is decreased, and for ¹"0 reaches the resonant value ¹ "4C C /(C #C ) (see Eq. (75)) for all impurities which are closer to the Fermi surface * 0 * 0 than C #C . Subsequent averaging over the impurities [304] gives rise to the logarithmic * 0 singularity in the conductance for zero temperature. A "nite bias voltage < smears the singularity [304,306]; thus, the di!erential conductance dI/d< as a function of bias shows a narrow peak around <"0 and two broad side peaks for e<"$;. Theoretical results on shot noise are scarce. Hersh"eld [307] performs perturbative analysis in powers of ; of the shot noise, based on the Green's function approach. His results are thus relevant for the high-temperature regime, but do not describe the Kondo physics, which is non-perturbative in ;. He "nds that the noise, apart from the non-interacting contribution (described by the Fano factor (78)) contains also an interacting correction. This correction is a non-trivial function of the applied bias voltage; it is always positive for a symmetric barrier C "C , but may have either sign * 0 for an asymmetric barrier. This interacting correction is zero for zero bias, and peaks around a certain energy E , which is the bare resonant energy E , renormalized by interactions. For higher A more complicated problem } transport through the double quantum dot in the regime when the cotuneling dominates } was investigated by Loss and Sukhorukov [303] with the emphasis on the possibility of probing entanglement (see Appendix B). They report that the Fano factor equals one.
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voltages it falls o! with energy, and in the limit e<<;, E the Fano factor returns to the non-interacting value (78). Thus, in this case the interactions may either enhance or suppress the noise. We also point out the analogy with the Coulomb blockade results: In the symmetric case R "R , the actual Coulomb blockade noise suppression is always stronger, than the non interacting suppression (258); generally, it may be either stronger or weaker. Yamaguchi and Kawamura [308,309] perform a complementary analysis by treating the tunneling Hamiltonian perturbatively. They "nd that the shot noise is strongly suppressed as compared with the Poisson value for voltages e<&E and e<&E #; (the latter resonance corresponds to the addition of the second electron to the resonant state, which is then shifted upwards by ;). For the Kondo regime, we expect that, since the e!ective transmission coe$cient tends to ¹ for zero temperature, the shot noise is a sensitive function of k ¹, which for a symmetric barrier decreases and eventually vanishes as the temperature tends to zero. This regime is investigated by Ding and Ng [310], who complete the Green's functions analysis by numerical simulations. They only plot the results for the symmetric case C "C and only for ¹"¹ ; the * 0 ) shot noise in this regime is, indeed, suppressed below the non-interacting value for any applied bias. Yamaguchi and Kawamura [309], treating the tunneling Hamiltonian perturbatively, report that the Fano factor is suppressed down to zero at zero bias. Results concerning averages over impurities are unavailable. 7.2.3. Kondo model In mesoscopic physics, this is the model of two non-interacting reservoirs which couple to the -spin in the quantum dot via exchange interaction. The interacting part of the Hamiltonian is HK "J?@sH pH, where j"x, y, z and a, b"L, R. Here sH are the matrix elements of the electron H ?@ ?@ spin operator in the basis of the reservoir states, and pH is the spin of the Kondo impurity. In physical systems the coupling J is symmetric; however, to gain some insight and use the exact H solutions, other limits are often considered. To our knowledge, the only results on shot noise in the Kondo model are due to Schiller and Hersh"eld [311], who consider a particular limiting case (Toulouse limit), J?@"J?@, J*0"J0*"0, X W X X and J**"J00. As a function of the bias voltage, the Fano factor is zero at zero bias, and grows X X monotonically. In the high-bias limit the noise is Poissonian rather than suppressed according to Eq. (78). The transport properties of the Kondo model are strongly a!ected by an applied magnetic "eld, which may drive the Fano factor well above the Poisson value. The frequency dependence of the shot noise is sensitive to the spectral function of the Kondo model, and exhibits structure at the inner scales of energy. The studies [311], though quite careful, do not, of course, exhaust the opportunities to investigate shot noise in strongly correlated systems, o!ered by the Kondo model. 7.3. Tomonaga}Luttinger liquids and fractional quantum Hall edge states Many problems concerning (strictly) one-dimensional systems of interacting electrons may be solved exactly by using specially designed techniques. As a result, it turns out that in one dimension, electron}electron interactions are very important. They lead to the formation of a new correlated state of matter, a Tomonaga}Luttinger liquid, which is characterized by the presence of gapless collective excitations, commonly referred as plasmons. In particular, the transport properties of the one-dimensional wires are also quite unusual. We only give here the results which we
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subsequently use for the description of noise; a comprehensive review may be found, e.g. in Ref. [312]. Throughout the whole subsection we assume that the interaction is short-ranged and one-dimensional, <(x!x )"< d(x!x ), and the voltage <'0. For an in"nite homogeneous Luttinger liquid the `conductancea is renormalized by interactions, G"ge/2p , where the dimensionless interaction parameter, g"(1#< /p v )\ , (261) $ will play an important role in what follows. This parameter equals 1 for non-interacting electrons, while g(1 for repulsive interactions. However, if one takes into account the reservoirs, which corresponds to a proper de"nition of conductance, the non-interacting value e/2p is restored, and thus the interaction constant g cannot be probed in this way. If one has an in"nite system (no reservoirs) with a barrier, the situation changes. Even an arbitrarily weak barrier totally suppresses the transmission in the interacting case g(1, and for zero temperature there is no linear dc conductance. The two limiting cases may be treated analytically. For strong barriers (weak tunneling), when the transmission coe$cient is ¹;1, the I}< curve in the leading order in < is 1I2"(e/2p )a¹< E\ ,
(262)
where a is a non-universal (depending on the upper energy cut-o! ) constant. In the opposite case of weak re#ection, 1!¹;1, the interactions renormalize the transmission coe$cient, so that the barrier becomes opaque, and for low voltages we return to the result (262). On the other hand, for high voltages the backscattering may be considered as a small correction, and one obtains 1I2"(ge/2p )
(263)
where b is another non-universal constant. Eq. (263) is only valid when the second term on the right-hand side is small. To emphasize the di!erence, we will refer to the weak and strong tunneling cases (which describe the regimes (262) and (263), respectively), rather than to the cases of transparent and opaque barriers. In particular, whatever the strength of the barrier, for low voltages and temperatures the tunneling is weak. For the transmission through a double-barrier structure resonant tunneling may take place, but the resonances become in"nitesimally narrow in the zero-temperature limit. What is extremely important for the following is that strong tunneling is accompanied by a transfer of charge ge between left- and right-moving particles (we can loosely say that there are quasiparticles with the charge ge which are scattered back from the barrier), while in the case of the weak tunneling the charge transfer across the barrier is e } there is tunneling of real electrons. Whereas the Luttinger liquid state may, in principle, be observed in any one-dimensional system, the most convenient opportunity is o!ered by the fractional quantum Hall e!ect (FQHE) edge states. Indeed, the edge state of a sample in the FQHE regime is a one-dimensional system, and it
This barrier is routinely called `impuritya in the literature. For the FQHE case (see below) this should not cause any confusion: Strong tunneling regime means strong tunneling through the barrier, which is the same as weak backscattering, or weak tunneling of quasiparticles between the edge states. Conversely, the weak tunneling regime means that the edge states are almost not interconnected (Fig. 37).
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may be shown that for the bulk "lling factor l"1/(2m#1), m3Z (Laughlin states), the interaction parameter g takes the same value g"1/(2m#1). The di!erence with the ordinary Luttinger liquid is that the FQHE edge states are chiral: the motion along a certain edge is only possible in one direction. Thus, if we imagine a FQHE strip, the electrons along the upper edge move, say, to the right, and the electrons along the lower edge move to the left. For the transport properties we discuss this plays no role, and expressions (262) and (263) remain valid. In particular, for the FQHE case the quasiparticles with the charge le may be identi"ed with the Laughlin quasiparticles. The distinction between the strong and weak tunneling we described above also gets a clear interpretation (Fig. 37), which is in this form due to Chamon, Freed and Wen [313]. Indeed, consider a FQHE strip with a barrier. If the tunneling is strong (Fig. 37a), the edge states go through the barrier. The backscattering corresponds then to the charge transfer from the upper edge state to the lower one, and this happens via tunneling between the edge states inside the FQHE strip. Thus, in this case, there are Laughlin quasiparticles which tunnel. In principle, the electrons may also be backscattered, but such events have a very low probability (see below). In the opposite regime of weak tunneling, the strip splits into two isolated droplets (Fig. 37b). Now the tunneling through the barrier is again the tunneling between two edge states, but it only may happen outside the FQHE state, where the quasiparticles do not exist. Thus, in this case, one has tunneling of real electrons. 7.3.1. Theory of dc shot noise Shot noise in Luttinger liquids was investigated by Kane and Fisher [314] using the bosonization technique. The conclusion is that for an ideal in"nite one-dimensional system there is no shot noise. Shot noise appears once the barrier is inserted. For the strong tunneling regime the shot noise is S"2ge((ge/2p )
(264)
where the average current 1I2 is given by Eq. (263) in the regime in which the latter is valid. If we introduce the (small) backscattering current I "(e/2p )b¹<E\, the shot noise is written as S"2geI , (265) which physically corresponds to the Poisson backscattering stream of (Laughlin) quasiparticles with the charge ge. Eq. (265) is precisely the analog of the two-terminal expression S"(e
A di!erence for shot noise is that any experiment with the FQHE edge states is always four terminal. The behavior of all correlation functions which we discuss below is the same. Ref. [314] also considers the "nite-temperature case and describes the crossover between thermal and shot noise. The same is, of course, also true for a system between the two reservoirs.
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Fig. 37. The tunneling experiment with the FQHE edge states: (a) strong tunneling; (b) weak tunneling. The shaded areas denote the location of the FQHE droplet(s).
For the case of weak tunneling, the shot noise is Poissonian with the charge e, S"2e1I2: It is determined by the charge of tunneling electrons. Expressions for noise interpolating between this regime and Eq. (264), as well as a numerical evaluation for g", are provided by Fendley et al. [316]; Fendley and Saleur [317] and Weiss [318] generalize them to "nite temperatures. For the resonant tunneling process, at resonance, the shot noise is given by [314] S"4ge((ge/2p )
(266)
which corresponds to the e!ective charge 2eg. This re#ects the fact that at resonance the excitations scatter back in pairs. Sa" [319] argues, however, that the contribution due to the backscattering of single quasiparticles is of the same order; this statement may have implications for the Fano factor, which is then between g and 2g. Sandler et al. [320] consider a situation in the strong tunneling regime with a barrier separating the two FQHE states with di!erent "lling factors l "1/(2m #1) and l "1/(2m #1). One of the states may be in the integer quantum Hall regime, for instance m "0. In particular, the case of l "1 and l " may be solved exactly. They conclude that the noise in this system corresponds to a Poissonian stream (265) of backscattered quasiparticles which are now, however, not the Laughlin quasiparticles of any of the two FQHE states. The charge of these excitations, which is measured by the shot noise, is g e, with g \"(l\#l\)/2"m #m #1. Thus, for l "1 and l " the e!ective charge is e/2. This also implies that the shot noise experiment cannot distinguish certain combinations of "lling factors: the e!ective charge of the } junction is the same as that of the 1} junction. We also note that the above results were obtained for in"nite wires (or FQHE edges). Taking into account the electron reservoirs, as we have mentioned above, changes the conductance of an ideal wire. However, it is not expected to a!ect the Fano factor of a wire with a barrier, which is determined by the scattering processes at the barrier. On the other hand, Ponomarenko and Nagaosa [321] present a calculation which implies that the shot noise in the wire connected to reservoirs is Poissonian with the charge e. In our opinion, this statement is not physically appealing, and we doubt that it is correct. Nevertheless, it deserves a certain attention, and more work is needed in this direction.
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7.3.2. Frequency-dependent noise The frequency dependence of noise was studied by Chamon et al. [322,313], who "rst derived results perturbative in the tunneling strength, and subsequently were able to "nd an exact solution for g". Lesage and Saleur [323,324] and Chamon and Freed [325] developed non-perturbative techniques valid for any g. We brie#y explain the main results, addressing the reader for more details to Refs. [313,323]. The frequency dependence of noise is essentially similar to that for non-interacting electrons in the case when the scattering matrices are energy independent. There is the "u" singularity for zero frequency; the singularity itself is not changed by the interaction, but the coe$cient in front of this singularity is interaction-sensitive. Furthermore, there is a singularity at the frequency u"e
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statistics, which the Laughlin quasiparticles are known to obey, also be probed in the shot noise experiments? This problem was addressed by Isakov et al. [327], who consider the two-terminal experiment for the independent charged particles obeying the exclusion statistics. They obtain that the crossover between zero-temperature shot noise and Nyquist noise is sensitive to the statistics of quasi-particles. We believe that the paper [327] has a number of serious drawbacks. To start with, the statistical particle counting arguments which the paper takes as a departing point, are unable to reproduce the exact results which follow from the scattering matrix approach for bosons, and thus had to add ad hoc certain terms to reproduce these results in the limiting case. Then, the exclusion statistics apply to an ensemble of particles and not to single particles; it is not clear whether the notion of independent particles obeying the exclusion statistics is meaningful. Finally, in the scattering problem one needs to introduce the reservoirs, which are not clearly de"ned in this case. Having said all this, we acknowledge that the question, which Isakov et al. [327] address, is very important. Presumably to attack it one must start with the ensemble of particles; we also note that the e!ects of statistics are best probed in the multi-terminal geometry rather that in the twoterminal case. A demonstration of the HBT-type e!ect with the FQHE edge states would clearly indicate the statistics of the quasiparticles. 7.3.5. Experiments Saminadayar et al. [328] and, independently, de-Picciotto et al. [329] performed measurements on a FQHE strip with the "lling factor l" into which they inserted a quantum point contact. The transmission coe$cient of the point contact can be modi"ed by changing the gate voltages. In particular, Saminadayar et al. [328] obtained the data showing the crossover from strong tunneling to the weak tunneling regime. Their results, shown in Fig. 38, clearly demonstrate that in
Fig. 38. Experimental results of Saminadayar et al. [328] for l" (strong tunneling}weak backscattering regime). Copyright 1997 by the American Physical Society.
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the strong tunneling (weak backscattering) regime the e!ective charge of the carriers is e/3. As the tunneling becomes weak, the e!ective charge crosses over to e, as expected for the strong backscattering regime. They also carefully checked the crossover between the shot noise and the equilibrium noise, and found an excellent agreement with the theory. Though the results of Refs. [328,329] basically coincide, we must point out an important di!erence which is presently not understood. The transmission in the experiment by de-Picciotto et al. [329] is not entirely perfect: they estimate that for the sets of data they plot the transmission coe$cients are ¹"0.82 and 0.73. Taking this into account, they phenomenologically insert the factor ¹ in the expression for the shot noise and "t the data to the curve S"(2e/3)¹I . In this way, they obtain a good agreement between the theory and experiment. On the other hand, Saminadayar et al. [328] "t their data to the curve S"(2e/3)I (without the factor ¹). An attempt to replot the data taking the factor ¹ into account leads to an overestimate of the electron charge. The theory we described above predicts a more complicated dependence than just the factor ¹; therefore it may be important to clarify this detail in order to improve our understanding of the theory of FQHE edge states. Reznikov et al. [330] performed similar measurements in the magnetic "eld corresponding to the "lling factor l". Changing the gate voltage (and thus varying the shape of the quantum point contact) they have observed two plateaus of the conductance, with the heights ()(e/2p ) and (2/5)(e/2p ), respectively. The noise measurements showed that the e!ective charges at these plateaus are e/3 and e/5, respectively. These results are in agreement with the subsequent theory of Imura and Nomura [326], and also with the predictions of the composite fermion model (see below): They assumed that there are two transmission channels which in turn open with the gate voltage. This experiment is important since it clearly shows that what is measured in the FQHE shot noise experiments is not merely a "lling factor (like one could suspect for l"), but really the quasiparticle charge. 7.4. Composite fermions An alternative description of the FQHE systems is achieved in terms of the composite fermions. Starting from the FQHE state with the "lling factor l"p/(2np$1), n, p3Z, one can perform a gauge transformation and attach 2n #ux quanta to each electron. The resulting objects (an electron with the #ux attached) still obey the Fermi statistics and hence are called composite fermions (CF). The initial FQHE state for electrons corresponds in the mean "eld approximation to the "lling factor l"p for CF, i.e. the composite fermions are in the integer quantum Hall regime with the p Landau levels "lled. In particular, the half"lled Landau level corresponds to the CF in zero magnetic "eld. Composite fermions interact electrostatically via their charges, and also via the gauge "elds, which are a measure of the di!erence between the actual #ux quanta attached and the #ux treated in the mean "eld approximation. It is important that, at least in the mean "eld approximation, the composite Fermions do not form a strongly interacting system, and therefore may be regarded as independent particles. One can then proceed by establishing an analogy with the transport of independent or weakly interacting electrons. Von Oppen [331] considers the shot noise of composite fermions at the half-"lled Landau levels, assuming that the sample is disordered. He develops a classical theory based on the
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Boltzmann}Langevin approach (see Section 6), incorporating interactions between them. Though the #uctuations of both electric and magnetic "elds now become important, in the end he obtains the same result as for normal di!usive wires: In the regime of negligible interactions between the CFs, the Fano factor equals and is universal. Likewise, in the regime when the CF distribution function is in local equilibrium (analog of the hot electron regime), the suppression factor is (3/4. Kirczenow [332] considers current #uctuations in the FQHE states, appealing to statistical particle counting arguments. However, he does not take into account any kind of scattering, and only treats equilibrium (Nyquist) noise for which the result is known already. Shot noise in the FQHE strip (l"2/(2p#1)) with a tunnel barrier was discussed by de Picciotto [333], who assumed that the composite Fermions are transmitted through the quantum point contact similarly to the non-interacting electrons. Namely, there are p channels corresponding to the p CF Landau levels. Each channel is characterized by an individual transmission coe$cient. As the gate voltage is changed the channels open (the corresponding transmission coe$cient crosses over from 0 to 1), and the conductance exhibits plateaus with the height (e/2p )l/(2l#1), 1(l(p). The shot noise at the plateau l is then given as the Poisson backscattered current with the e!ective charge e/(2l#1). Though this paper is phenomenological and requires further support from microscopic theory, we note that all the features predicted in Ref. [333] are not only in agreement with the Luttinger liquid approach by Imura and Nomura [326], but also were observed experimentally [330] for l" (see above). In this case there are two channels corresponding to p"1 and 2, which implies conductance plateaus with heights ()(e/2p ) and (for higher gate voltage) ()(e/2p ); the corresponding charges measured in the shot noise experiment are e/3 and e/5, respectively.
8. Concluding remarks, future prospects, and unsolved problems 8.1. General considerations In this section, we try to outline the directions along which the "eld of shot noise in mesoscopic systems has been developing, to point out the unsolved problems which are, in our opinion, important, and to guess how the "eld will further develop. A formal summary can be found at the end of the section. Prior to the development of the theory and the experiments on shot noise in mesoscopic physics, there already existed a considerable amount of knowledge in condensed matter physics, electrical engineering, and especially optics. Both theory and experiments are available in these "elds, and the results are well established. Similarly in mesoscopic physics, there exists a fruitful interaction between theory and experiment. However, presently there are many theoretical predictions concerning shot noise, not only extensions of the existing theories, but which really address new sub-"elds, which are not yet tested experimentally. Below we give a short list of these predictions. To this end, he has to assume implicitly the existence of two CF reservoirs, described by equilibrium Fermi distribution functions.
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Like every list, the choice re#ects very much our taste, and we do not imply that the predictions not included in this list are of minor importance. E -suppression of shot noise in chaotic cavities. E Multi-terminal e!ects probing statistics (exchange Hanbury Brown}Twiss (HBT) e!ect; shot noise at tunnel microscope tips; HBT e!ect with FQHE edge states). E Frequency-dependent noise beyond Nyquist}Johnson (noise measurements which would reveal the inner energy scales of mesoscopic systems); current #uctuations induced into gates or other nearby mesoscopic conductors. E Shot noise of clean NS interfaces; mesoscopic nature of positive cross-correlations in hybrid structures. E Shot noise in high magnetic "elds at the half-"lled Landau level. E Shot noise in hybrid magnetic structures. The theory, in our opinion, is generally well developed for most of the "eld and adequately covers it. However, a number of problems persist: For instance, there is no clear understanding under which conditions the cross-correlations in multi-terminal hybrid structures may be positive. Recent work [179] suggests that it is only a mesoscopic quantum contribution which is positive, but that to leading order the correlations will be negative as in normal conductors. Considerably more work is required on the frequency dependence of shot noise and on strongly correlated systems. The former (Section 3) o!ers the opportunities to probe the inner energy scales and collective relaxation times of the mesoscopic systems; only a few results are presently available. As for the strongly correlated systems (including possibly unconventional superconductors), this may become (and is already becoming) one of the mainstreams of mesoscopic physics; since even the dc shot noise measurements provide valuable information about the charge and statistics of quasiparticles, we expect a lot of theoretical developments in this direction concerning the shot noise. Some of the unsolved problems in this "eld may be found directly in Section 7, one of the most fascinating being the possibility of probing the quasiparticle statistics in multi-terminal noise measurements with FQHE edge states. One more possible development, which we did not mention in the main body of this Review, concerns shot noise far from equilibrium under conditions when the I}< characteristics are non-linear. The situation with non-linear problems resembles very much the frequency dependent ones: Current conservation and gauge invariance are not automatically guaranteed, and interactions must be taken into account to ensure these properties (for a discussion, see e.g. Ref. [153]). Though in the cases which we cited in the Review the non-linear results seem to be credible, it is still desirable to have a gauge-invariant general theory valid for arbitrary non-linear I}< characteristics. It is also desirable to gain insight and develop estimates of the range of applicability of the usual theories. Recently, Wei et al. [334] derived a gauge invariant expression for shot noise in the weakly non-linear regime, expressing it through functional derivatives of the Lindhard function with respect to local potential "elds. They apply the results to the resonant tunneling diode. Wei et al. [334] also discuss the limit in which the tunneling rates may be assumed to be energy independent. Apparently, the theory of Wei et al. does not treat the e!ect of #uctuations of the potential inside the sample, which may be an important source of noise. Furthermore, Green and Das [335}337] proposed a classical theory of shot noise, based on a direct solution of kinetic equations. They discuss the possibility to detect interaction e!ects in the cross-over region from
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thermal to shot noise. It is, however, yet to be shown what results this approach yields in the linear regime and whether it reproduces, for instance, the -suppression of shot noise in metallic di!usive wires. An application of both of these approaches to speci"c systems is highly desirable. What we have mentioned above, concerns the development of a "eld inside mesoscopic physics. We expect, however, that interesting connections will occur across the boundaries of di!erent "elds. An immediate application which can be imagined is the shot noise of photons and phonons. Actually, noise is much better studied in optics than in condensed matter physics (see e.g. the review article [338]), and, as we have just mentioned previously, the theory of shot noise in mesoscopic physics borrowed many ideas from quantum optics. At the same time, mesoscopic physics gained a huge experience in dealing, for instance, with disordered and chaotic systems. Recently a `back-#owa of this experience to quantum optics started. This concerns photonic noise for the transmission through (disordered) waveguides, or due to the radiation of random lasers or cavities of chaotic shape. In particular, the waveguides and cavity may be absorbing or amplifying, which adds new features as compared with condensed matter physics. For references, we cite a recent review by Patra and Beenakker [339]. Possibly, in the future other textbook problems of mesoscopic physics will also "nd their analogies in quantum optics. Phonons are less easy to manipulate with, but, in principle, one can also imagine the same class of problems for them. Generally, shot noise accompanies the propagation of any type of (quasi)-particles; as the last example, we mention plasma waves. 8.2. Summary for a lazy or impatient reader Below is a summary of this Review. Though we encourage the reader to work through the whole text (and then she or he does not need this summary), we understand that certain readers are too lazy or too impatient to do this. For such readers we prepare this summary which permits to acquire some information on shot noise in a very short time span. We only include in this summary the statements which in our opinion are the most important. E Shot noise occurs in a transport state and is due to #uctuations in the occupation number of states caused by (i) thermal random initial #uctuations; (ii) the random nature of quantummechanical transmission/re#ection (partition noise), which, in turn, is a consequence of the discreteness of the charge of the particles. The actual noise is a combination of both of these microscopic sources and typically these sources cannot be separated. E Shot noise provides information about the kinetics of the transport state: In particular, it can be used to obtain information on transmission channels beyond that contained in the conductance. In two-terminal systems, zero-frequency shot noise for non-interacting electrons is suppressed in comparison with the Poisson value S"2e1I2.
One has to take, of course, also into account the di!erences between photon and electron measurements: Apart from the evident Bose versus Fermi and neutral versus charged particles, there are many more. For example, the photon measurement is accompanied by a removal of a photon from the device, while the total electron number is always conserved.
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E For quantum wells the noise suppression is F"S/2e1I2"(C #C )/(C #C ). The sup* 0 * 0 pression is universal for metallic di!usive wires (F") and chaotic cavities (F"). E Far from equilibrium, in the vicinity of instability points the shot noise can exceed the Poisson value. E In the limit of low transmission, the shot noise is Poissonian and measures the charge of transmitting particles. In particular, for normal metal}superconductor interfaces this charge equals 2e, whereas in SNS systems it is greatly enhanced due to multiple Andreev re#ections. In the limit of low re#ection, the shot noise may be understood as Poissonian noise of re#ected particles; in this way, the charge of quasiparticles in the fractional quantum Hall e!ect is measured. E For carriers with Fermi statistics, in multi-terminal systems the zero-frequency correlations of currents at di!erent terminals are always negative. For Bose statistics, they may under certain circumstances become positive. These cross-correlations may be used to probe the statistics of quasiparticles. E The ensemble-averaged shot noise may be described both quantum-mechanically (scattering approach; Green's function technique) and classically (master equation; Langevin and Boltzmann}Langevin approach; minimal correlation approach). Where they can be compared, classical and quantum-mechanical descriptions provide the same results. Classical methods, of course, fail to describe genuinely quantum phenomena like, e.g. the quantum Hall e!ect. E As a function of frequency, the noise crosses over from the shot noise to the equilibrium noise SJ"u". This is only valid when the frequency is low as compared with the inner energy scales of the system and inverse times of the collective response. For higher frequencies, the noise is sensitive to all these scales. However, the current conservation for frequency-dependent noise is not automatically provided, and generally is not achieved in non-interacting systems. E Inelastic scattering may enhance or suppress noise, depending on its nature. In particular, in macroscopic systems shot noise is always suppressed down to zero by inelastic (usually, electron}phonon) scattering. When interactions are strong, shot noise is usually Poissonian, like in the Coulomb blockade plateau regime. E We expect that the future development of the "eld of the shot noise will be mainly along the following directions: Within the "eld of mesoscopic electrical systems: (i) experimental developments; (ii) frequency dependence; and (iii) shot noise in strongly correlated systems; and more generally (iv) shot noise in disordered and chaotic quantum optical systems, shot noise measurements of phonons (and, possibly, of other quasiparticles).
Acknowledgements We have pro"ted from discussions and a number of speci"c comments by Pascal Cedraschi, Thomas Gramespacher, and Andrew M. Martin. Some parts of this Review were written at the Aspen Center for Physics (Y.M.B); at the Max-Planck-Institut fuK r Physik Komplexer Systeme, Dresden (Y.M.B.); at the Centro Stefano Franscini, Ascona (Y.M.B. and M.B.); and at the UniversiteH de Montpellier II (Y.M.B.). We thank these institutions for hospitality and support. This work was supported by the Swiss National Science Foundation via the Nanoscience program.
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Note added in proof Since the submission of this Review, a number of articles on shot noise have come to our attention. We give below a very brief account of them, merely to maintain the completeness of our reference list. Experimentally, the doubling of the e!ective charge in shot noise at interfaces between a di!usive metal and a superconductor was observed by Jehl et al. [357] and Kozhevnikov et al. [358]. Photon-assisted tunneling at such an interface was investigated by Kozhevnikov et al. [359]. Theoretically, the following issues are addressed. Lesovik [360] re-examines the expression for shot noise for non-interacting electrons, and "nds that the contribution of non-parabolicity of the spectrum in the vicinity of the Fermi surface leave the "nite-frequency noise "nite even in the limit of ideal transmission ¹ "1. Gurevich and Muradov [361] study shot noise in the Coulomb drag current, which appears in a ballistic nanowire coupled to a nearby nanowire. Agam et al. [362] investigate shot noise in chaotic cavities not performing an ensemble average, and "nd a suppression below F"1/4 if the electron escape time is "nite. Monte-Carlo simulations of shot noise in non-degenerate double-barrier diodes are performed by Reklaitis and Reggiani [363]. Bulashenko et al. [364] provide an analytical theory of noise in ballistic interacting conductors with an arbitrary distribution function of injected electrons. Turlakov [365] investigates frequency-dependent Nyquist noise in disordered conductors and reports a structure at the Maxwell frequency u"4np, p being the conductivity, which is yet one more collective response frequency, as we discussed in the main body of the Review. Tanaka et al. [366] study shot noise at NS interfaces where the superconductor has a d-symmetry, for di!erent orientations of the interface, and taking into account the spatial dependence of the pair potential. Korotkov and Likharev [367] calculate noise for hopping transport in interacting conductors; the noise is less suppressed than for Coulomb blockade tunnel junctions arrays, allegedly due to redistribution of the electron density. Stopa [368] performs numerical simulations to study #uctuations in arrays of tunnel junctions. Green and Das apply Boltzmann kinetic theory to treat non-linear noise in conductors with screening (emphasizing the conformity with the Fermi liquid) [369] and study non-linear noise in metallic di!usive conductors [370]. Andreev and Kamenev [371] investigate charge counting statistics for the case when the scattering matrix is time-dependent and apply their theory to the counting statistics of adiabatic charge pumping.
Appendix A. Counting statistics and optical analogies The question which naturally originates after consideration of the shot noise is the following: Can we obtain some information about the higher moments of the current? Since, as we have seen, the shot noise at zero frequency contains more information about the transmission channels than the average conductance, the studies of the higher moments may reveal even more information. Also, we have seen that in the classical theories of shot noise the distribution of the Langevin sources (elementary currents) is commonly assumed to be Gaussian, in order to provide the equivalence between the Langevin and Fokker}Planck equations [213]. An independent analysis of the higher moments of the current can reveal whether this equivalence in fact exists, and thus how credible the classical theory is.
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A natural quantity to study is the kth cumulant of the number of particles n(t) which passed through the barrier during the time t (which is assumed to be large). In terms of the current I(t), this cumulant is expressed as
1 R dt 2 dt [I(t )2I(t )\ , (A.1) [nI(t)\" I I eI where [2\ means the cumulant (irreducible part). For the following, we only consider the time-independent problems, i.e. the noise in the presence of a dc voltage. Then the "rst cumulant is [n(t)\"1I2t/e, and the second one is expressed through the zero-temperature shot noise power, [n(t)\"St/2e . In particular, the ratio of [n(t)\ and [n(t)\ gives the Fano factor. The cumulants with k'2 in Eq. (A.1) contain additional information about the statistics of current. Thus, if the distribution of the transmitted charge is Poissonian, all the cumulants have the same value; for the Gaussian distribution all the cumulants with k'2 vanish. The general expression for the cumulants of the number of transmitted particles was obtained by Lee et al. [340], who followed the earlier paper by Levitov and Lesovik [341]. We only give the results for zero temperature. Consider "rst one channel with the transmission probability ¹. The probability that m particles pass through this channel during the time t is given by Bernoulli distribution, as found by Shimizu and Sakaki [343] and Levitov and Lesovik [341], P (t)"CK ¹K(1!¹),\K, m4N , (A.2) K , where N(t) is the `number of attemptsa, on average given by 1N2"e
(A.3)
s(j)" [¹ e H#1!¹ ], , (A.4) H H H where the product is taken over all the transmission channels. The coe$cients in the series expansion of ln s are the cumulants that we are looking for, (ij)I [nI(t)\ , ln s " H k! I
The paper [341] corrects Ref. [342]. In the sense that we can diagonalize the matrix tRt and de"ne the transmission eigenvalues.
(A.5)
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and Lee et al. [340] obtain in this way the explicit expression,
d I\ ¹ . (A.6) [nI(t)\"N ¹(1!¹) d¹ H 22 H We see from Eq. (A.6) that [n(t)\"N ¹ , and [n(t)\"N ¹ (1!¹ ), which are the results for H H H the average current and the zero-frequency shot noise power. However, higher cumulants do not vanish at all. Though they generally cannot be calculated in a closed form, the distributions are studied for many systems, and we give a brief overview below. For the tunnel barrier, when ¹ ;1 for any j, all the cumulants are equal, [nI(t)\"N ¹ , and H H thus the distribution of the transmitted charge is Poissonian. De Jong [344] analyzes the counting statistics for double-barrier structures using the concept of the distribution function for the transmission probabilities. For the symmetric case (C "C "C) * 0 he "nds 1ln s2"2N Ct(exp(ij/2)!1), and explicitly for the cumulants [nI(t)\"N Ct/2I\. , , Here N is the number of transverse channels at the Fermi surface. The cumulants decrease , exponentially with k, and thus the statistics are closer to Gaussian than to Poissonian. De Jong [344] was also able to obtain the same results classically, starting from the master equation. The case of metallic di!usive wires is considered by Lee et al. [340], and subsequently by Nazarov [345]. They "nd that the disorder-averaged logarithm of the characteristic function is 1ln s(j)2"(1I2t/e) arcsinh(e H!1 .
(A.7)
The expressions for the cumulants cannot be found in a closed form. As expected, from Eq. (A.7) we obtain [n\"1I2t/3e, in agreement with the fact that the Fano factor is . For the following cumulants one gets, for instance, [n\"1I2t/15e and [n\"!1I2t/105e. For high k the cumulants [nI\ behave as (k!1)!/((2p)Ik ), i.e. they diverge! Moreover, Lee et al. [340] evaluate the sample-to-sample #uctuations of the cumulants, and "nd that for the high-order cumulants these #uctuations become stronger than the cumulants themselves. Thus, the far tails of the charge distributions are strongly a!ected by disorder. Nazarov [345] generalizes the approach to treat weak localization corrections. For the transmission through a symmetric chaotic cavity, Ref. [347] "nds 41I2t e H #1 ln , 1ln s(j)2" 2 e
(A.8)
with the explicit expression for the cumulants 1I2t 2J!1 [nJ\" B , e 2J\l J
(A.9)
and [nJ>\"0 (l51). Here B are the Bernoulli numbers (B ", B "! ). Indeed, for the I second cumulant we obtain [n\"1I2t/4e, in accordance with the -shot noise suppression in The characteristic function is not self-averaging; the expansion of ln 1s(j)2 yields di!erent expression for the cumulants, as found by Muttalib and Chen [346]. As Lee et al. [340] argue, the correct quantity to average is ln s(j), rather than s(j), since it is linearly related to the cumulants of the transmitted charge.
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symmetric chaotic cavities. Eq. (A.9) can be also obtained classically [347], using the generalization of the minimal correlation approach. Muzykantskii and Khmelnitskii [169] investigate the counting statistics for the NS interface and "nd s(j)" [¹ e H#1!¹ ],, ¹ ,2¹/(2!¹ ) . H H H H H H
(A.10)
It is clearly seen from the comparison with Eq. (A.4) that the particles responsible for transport have e!ective charge 2e. Other important developments include the generalization to the multi-terminal [341] and time-dependent [348}350] problems, and numerical investigation of the counting statistics for the non-degenerate ballistic conductors. Thus, the counting statistics certainly reveal more information about the transport properties of conductors than is contained in either the conductance or the second order shot noise. The drawback is that it is not quite clear how these statistics can be measured. A proposal, due to Levitov et al. [349], is to use the spin- galvanometer, precessing in the magnetic "eld created by the transmission current. The idea is to measure the charge transmitted during a certain time interval through the evolution of the spin precession angle. However, the time-dependent transport is a collective phenomenon (Section 3), and thus the theory of such an e!ect must include electron}electron interactions. In addition, this type of experiments is not easy to realize. On the other hand, measurements of photon numbers are routinely performed in quantum optics. In this "eld concern with counting statistics has already a long history. However, typical mesoscopic aspects } disorder, weak localization, chaotic cavities } and e!ects particular to optics, like absorption and ampli"cation, make the counting statistics of photons a promising tool of research. Some of these aspects (relating to disorder and chaos, where the random matrix theory may be applied), have been recently investigated by Beenakker [351]; however, there are still many unsolved problems.
Appendix B. Spin e4ects and entanglement A notion which mesoscopic physics recently borrowed from quantum optics is entanglement. States are called entangled, if they cannot be written simply as a product of wave functions. For our purpose, we will adopt the following de"nition. Imagine that we have two leads, 1 and 3, which serve as sources of electrons. The entangled states are de"ned as the following two-particle states described in terms of the creation operators, "$2"(1/(2)(a( R (E )a( R (E )$a( R (E )a( R (E ))"02 , s t t s
(B.1)
where a( R (E) is the operator which creates an electron with the energy E and the spin projection ?N p in the source a. The state corresponding to the lower sign in Eq. (B.1) is the spin singlet with the symmetric orbital part of the wave function, while the upper sign describes a triplet (antisymmetric) state.
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Such entangled states are very important in the "eld of quantum computation. Condensed matter systems are full of entangled states: there is hardly a system for which the ground state can be expressed simply in terms of a product of wave functions. The key problem is to "nd ways in which entangled states can be generated and manipulated in a controlled way. Optical experiments on noise have reached a sophisticated stage since there exist optical sources of entangled states (the production of twin photon-pairs through down conversion). It would be highly desirable to have an electronic equivalent of the optical source and to analyze to what extent such experiments can be carried out in electrical conductors [352,108]. An example of such source is a p-n junction which permits the generation of an electron-hole pair and the subsequent separation of the particles. The disadvantage of such an entangled state is that the electron and hole must be kept apart at all times. Similarly, a Cooper pair entering a normal conductor, represents an entangled state. But in the normal conductor it is described as an electron-hole excitation and again we have particles with di!erent charge. To date most proposals in condensed matter related to quantum computation consider entangled states in closed systems. Theoretically, entanglement opens a number of interesting opportunities. One of the questions is: Provided we were able to prepare entangled states, how do we know the states are really entangled? Since we deal with two-particle states, it is clear that entanglement can only be measured in the experiments which are genuinely two-particle. Burkard et al. [353,354] investigate the multiterminal noise. Indeed, add to the structure two more reservoirs (electron detectors) 2 and 4, and imagine that there is no re#ection back to the sources (the geometry of the exchange HBT experiment, Fig. 15b, with the additional `entanglera creating the states (B.1)). The system acts as a three-terminal device, with an input of entangled electrons and measuring the current}current correlation at the two detectors. Since the shot noise is produced by the motion of the electron charge, it is plausible that the noise measurements are in fact sensitive to the symmetry of the orbital part of the wave function, and not to the whole wave function. Thus, the noise power seen at a single contact is expected to be enhanced for the singlet state (symmetric orbital part) and suppressed for the triplet state (anti-symmetric orbital part). It is easy to quantify these considerations by repeating the calculation of Section 2 in the basis of entangled states (B.1). Assuming that the system is of "nite size, so that the set of energies E is discrete, and the incoming stream of entangled electrons is noiseless, Burkard et al. [353] obtained the following result 1I 2"1I 2, S "S "!S , with (B.2) S "2e¹(1!¹)(1Gd ) . # # Thus, indeed, the shot noise is suppressed for the triplet state and enhanced for the singlet state, provided the electrons are taken at (exactly) the same energy. For the singlet state this suppression is an indication of the entanglement, since there are no other singlet states. One can also construct the triplet states which are not entangled, "!!2"a( R (E )a( R (E )"02 , (B.3) t t and an analogous state with spins down. These states, as shown by Burkard et al. [353], produce the same noise as the entangled triplet state. Thus, the noise suppression in this geometry is not a signature of the entanglement. Another proposal, due to Loss and Sukhorukov [303], is that the entangled states prepared in the double quantum dot can be probed by the Aharonov}Bohm transport experiments. The shot
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noise is Poissonian in this setup, and both current and shot noise are sensitive to the symmetry of the orbital wave function. More generally, one can also ask what happens if one can operate with spin-polarized currents separately. (Again, presently no means are known to do this). Burkard et al. [353] considered a transport in a two-terminal conductor where the chemical potentials are di!erent for di!erent spin projections. In particular, if < "!< , the total average current is zero (the spin-polarized t s currents compensate each other). Shot noise, however, exists, and may be used as a means to detect the motion of electrons in this situation.
Appendix C. Noise induced by thermal transport Sukhorukov and Loss [114] consider shot noise in metallic di!usive conductors in the situation when there is no voltage applied between the reservoirs, and the transport is induced by the di!erence of temperatures. To this end, they generalize the Boltzmann}Langevin approach to the case of non-uniform temperature. For the simplest situation of a two-terminal conductor, when one of the reservoirs is kept at zero temperature, and the other at the temperature ¹, their result reads (C.1) S"(1#ln 2)Gk ¹ for the purely elastic scattering, with G being the Drude conductance. This shows, in particular, that the noise induced by thermal transport is also universal } the ratio of the shot noise power to the thermal current does not depend on the details of the sample. This is an experiment that would be interesting to realize. Another prospective problem concerning the noise induced by the non-uniform temperature, is that the applied temperature gradient would cause not only the transport of electrons, but also transport of phonons. Thus, in this kind of experiments one can study shot noise (and, possibly, also counting statistics) of phonons. This really looks very promising, and, to our knowledge, by now has never been discussed.
References [1] M.J.M. de Jong, C.W.J. Beenakker, in: L.L. Sohn, L.P. Kouwenhoven, G. SchoK n (Eds.), Mesoscopic Electron Transport, NATO ASI Series E, Vol. 345, Kluwer Academic Publishing, Dordrecht, 1997, p. 225. [2] Th. Martin, in: D.C. Glattli, M. Sanquer, J. Tra( n Thanh Va( n (Eds.), Coulomb and Interference E!ects in Small Electronic Structures, Editions Frontie`res, Gif-sur-Yvette, 1994, p. 405. [3] Sh. Kogan, Electronic Noise and Fluctuations in Solids, Cambridge University Press, Cambridge, 1996. [4] Y. Imry, Introduction to Mesoscopic Physics, Oxford University Press, Oxford, 1997. [5] W. Schottky, Ann. Phys. (Leipzig) 57 (1918) 541. [6] A. van der Ziel, Noise, Chapman & Hall, London, 1955. [7] R. Hanbury Brown, R.Q. Twiss, Nature 177 (1956) 27. [8] R. Loudon, in: J.A. Blackman, J. Taguena (Eds.), Disorder in Condensed Matter Physics, Clarendon Press, Oxford, 1991, p. 441. [9] M. BuK ttiker, Phys. Rev. B 46 (1992) 12 485. [10] A. Pre( tre, H. Thomas, M. BuK ttiker, Phys. Rev. B 54 (1996) 8130.
Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166 [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57]
159
L.D. Landau, E.M. Lifshits, Statistical Physics, Part 1, Section 118, Butterworth-Heinemann, Oxford, 1980. M. Henny, S. Oberholzer, C. Strunk, T. Heinzel, K. Ensslin, M. Holland, C. SchoK nenberger, Science 284 (1999) 296. G.B. Lesovik, R. Loosen, Z. Phys. B 91 (1993) 531. I.O. Kulik, A.N. Omel'yanchuk, Fiz. Nizk. Temp. 10 (1984) 305 [Sov. J. Low Temp. Phys. 10 (1984) 158]. V.A. Khlus, Zh. ED ksp. Teor. Fiz. 93 (1987) 2179 [Sov. Phys. JETP 66 (1987) 1243]. G.B. Lesovik, Pis'ma Zh. ED ksp. Teor. Fiz. 49 (1989) 513 [JETP Lett. 49 (1989) 592]. B. Yurke, G.P. Kochanski, Phys. Rev. B 41 (1990) 8184. M. BuK ttiker, Phys. Rev. Lett. 65 (1990) 2901. M. BuK ttiker, Physica B 175 (1991) 199. R. Landauer, Th. Martin, Physica B 175 (1991) 167; R. Landauer, Th. Martin, Physica B 182 (1992) 288 (E). Th. Martin, R. Landauer, Phys. Rev. B 45 (1992) 1742. M.H. Pedersen, M. BuK ttiker, unpublished. A.I. Larkin, Yu.N. Ovchinnikov, Zh. ED ksp. Teor. Fiz. 53 (1967) 2159 [Sov. Phys. JETP 26 (1968) 1219]. A.J. Dahm, A. Denenstein, D.N. Langenberg, W.H. Parker, D. Rogovin, D.J. Scalapino, Phys. Rev. Lett. 22 (1969) 1416. D. Rogovin, D.J. Scalapino, Physica 55 (1971) 399. D. Rogovin, D.J. Scalapino, Ann. Phys. (N. Y.) 86 (1974) 1. G. SchoK n, Phys. Rev. B 32 (1985) 4469. H. Lee, L.S. Levitov, Phys. Rev. B 53 (1996) 7383. S.T. Liu, A. van der Ziel, Physica 37 (1967) 241. H. Birk, M.J.M. de Jong, C. SchoK nenberger, Phys. Rev. Lett. 75 (1995) 1610. B.J. van Wees, H. van Houten, C.W.J. Beenakker, J.G. Williamson, L.P. Kouwenhoven, D. van der Marel, C.T. Foxon, Phys. Rev. Lett. 60 (1988) 848. D.A. Wharam, T.J. Thornton, R. Newbury, M. Pepper, H. Ahmed, J.E.F. Frost, D.G. Hasko, D.C. Peacock, D.A. Ritchie, G.A.C. Jones, J. Phys. C 21 (1988) L209. L.I. Glazman, G.B. Lesovik, D.E. Khmel'nitskii, R.I. Shekhter, Pis'ma Zh. ED ksp. Teor. Fiz. 48 (1988) 218 [JETP Lett. 48 (1988) 238]. M. BuK ttiker, Phys. Rev. B 41 (1990) 7906. A.G. Scherbakov, E.N. Bogachek, U. Landman, Phys. Rev. B 57 (1998) 6654. L.Y. Chen, S.C. Ying, Mod. Phys. Lett. B 9 (1995) 573. I.A. Larkin, M. Reznikov, in: M. Sche%er, R. Zimmermann (Eds.), Proceedings of the 23rd International Conference on the Physics of Semiconductors, World Scienti"c, Singapore, 1996, p. 1565. A.I. Akimenko, A.B. Verkin, I.K. Yanson, J. Low Temp. Phys. 54 (1984) 247. Y.P. Li, D.C. Tsui, J.J. Heremans, J.A. Simmons, G.W. Weimann, Appl. Phys. Lett. 57 (1990) 774. C. Dekker, A.J. Scholten, F. Liefrink, R. Eppenga, H. van Houten, C.T. Foxon, Phys. Rev. Lett. 66 (1991) 2148. F. Liefrink, A.J. Scholten, C. Dekker, R. Eppenga, H. van Houten, C.T. Foxon, Physica B 175 (1991) 213. M.I. Reznikov, M. Heiblum, H. Shtrikman, D. Mahalu, Phys. Rev. Lett. 75 (1995) 3340. A. Kumar, L. Saminadayar, D.C. Glattli, Y. Jin, B. Etienne, Phys. Rev. Lett. 76 (1996) 2778. H.E. van den Brom, J.M. van Ruitenbeek, Phys. Rev. Lett. 82 (1999) 1526. J. BuK rki, C.A. Sta!ord, Phys. Rev. Lett. 83 (1999) 3342. A.D. Stone, P.A. Lee, Phys. Rev. Lett. 54 (1985) 1196. M. BuK ttiker, IBM J. Res. Develop. 32 (1988) 63. L.Y. Chen, C.S. Ting, Phys. Rev. B 43 (1991) 4534. A. Reklaitis, L. Reggiani, J. Appl. Phys. 82 (1997) 3161. A. Reklaitis, L. Reggiani, Phys. Stat. Sol. (b) 204 (1997) 459. D.V. Averin, J. Appl. Phys. 73 (1993) 2593. R. Tsu, L. Esaki, Appl. Phys. Lett. 22 (1973) 562. L.L. Chang, L. Esaki, R. Tsu, Appl. Phys. Lett. 24 (1974) 593. J.H. Davies, P. Hyldgaard, S. Hersh"eld, J.W. Wilkins, Phys. Rev. B 46 (1992) 9620. J.A. Melsen, C.W.J. Beenakker, Physica B 203 (1994) 219. V.I. Melnikov, Phys. Lett. A 198 (1995) 459. E. Runge, Phys. Rev. B 47 (1993) 2003.
160 [58] [59] [60] [61] [62] [63] [64] [65] [66] [67] [68] [69] [70] [71] [72] [73] [74] [75] [76] [77] [78] [79] [80] [81] [82] [83] [84] [85] [86] [87] [88] [89] [90] [91] [92] [93] [94] [95] [96] [97] [98] [99] [100] [101] [102] [103] [104] [105] [106]
Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166 ". Lund B+, Yu. Galperin, J. Phys.: Condens. Matter 8 (1996) 3033. ". Lund B+, Yu. Galperin, Phys. Rev. B 55 (1997) 1696. S.-J. Xiong, J. Appl. Phys. 78 (1995) 6079. Yu.V. Nazarov, J.J.R. Struben, Phys. Rev. B 53 (1996) 15 466. A.I. Larkin, K.A. Matveev, Zh. ED ksp. Teor. Fiz. 93 (1987) 1030 [Sov. Phys. JETP 66 (1987) 580]. L.I. Glazman, K.A. Matveev, Pis'ma Zh. ED ksp. Teor. Fiz. 48 (1988) 403 [JETP Lett. 48 (1988) 445]. Y.P. Li, A. Zaslavsky, D.C. Tsui, M. Santos, M. Shayegan, Phys. Rev. B 41 (1990) 8388. T.G. van de Roer, H.C. Heyker, J.J.M. Kwaspen, H.P. Joosten, M. Henini, Electron. Lett. 27 (1991) 2158. P. Ciambrone, M. Macucci, G. Iannaccone, B. Pellegrini, M. Lazzarino, L. Sorba, F. Beltram, Electron. Lett. 31 (1995) 503. H.C. Liu, J. Li, G.C. Aers, C.R. Leavens, M. Buchanan, Z.R. Wasilewski, Phys. Rev. B 51 (1995) 5116. A. Przadka, K.J. Webb, D.B. Janes, H.C. Liu, Z.R. Wasilewski, Appl. Phys. Lett. 71 (1997) 530. S.-T. Yau, H.B. Sun, P.J. Edwards, P. Lynam, Phys. Rev. B 55 (1997) 12 880. O.N. Dorokhov, Solid State Commun. 51 (1984) 381. Y. Imry, Europhys. Lett. 1 (1986) 249. C.W.J. Beenakker, Rev. Mod. Phys. 69 (1997) 731. A.D. Stone, P.A. Mello, K.A. Muttalib, J.-L. Pichard, in: B.L. Altshuler, P.A. Lee, R.A. Webb (Eds.), Mesoscopic Phenomena in Solids, North-Holland, Amsterdam, 1991, p. 369. C.W.J. Beenakker, M. BuK ttiker, Phys. Rev. B 46 (1992) 1889. K.E. Nagaev, Phys. Lett. A 169 (1992) 103. B.L. Altshuler, L.S. Levitov, A.Yu. Yakovets, Pis'ma Zh. ED ksp. Teor. Fiz. 59 (1994) 821 [JETP Lett. 59 (1994) 857]. Yu.V. Nazarov, Phys. Rev. Lett. 73 (1994) 134. Ya.M. Blanter, M. BuK ttiker, Phys. Rev. B 56 (1997) 2127. F. Liefrink, J.I. Dijkhuis, M.J.M. de Jong, L.W. Molenkamp, H. van Houten, Phys. Rev. B 49 (1994) 14 066. A.H. Steinbach, J.M. Martinis, M.H. Devoret, Phys. Rev. Lett. 76 (1996) 3806. M. Henny, S. Oberholzer, C. Strunk, C. SchoK nenberger, Phys. Rev. B 59 (1999) 2871. M.J.M. de Jong, C.W.J. Beenakker, Phys. Rev. B 46 (1992) 13 400. A.M.S. Mace( do, Phys. Rev. B 49 (1994) 1858. A.M.S. Mace( do, J.T. Chalker, Phys. Rev. B 49 (1994) 4695. A.M.S. Mace( do, Phys. Rev. Lett. 79 (1997) 5098. C. Mudry, P.W. Brouwer, A. Furusaki, Phys. Rev. B 59 (1999) 13 221. C. Mudry, P.W. Brouwer, A. Furusaki, unpublished. R. Gade, Nucl. Phys. B 398 (1993) 499. M.J.M. de Jong, C.W.J. Beenakker, Phys. Rev. B 51 (1995) 16 867. M.J.M. de Jong, C.W.J. Beenakker, Physica A 230 (1996) 219. R.C. Liu, P. Eastman, Y. Yamamoto, Solid State Commun. 102 (1997) 785. C.W.J. Beenakker, J.A. Melsen, Phys. Rev. B 50 (1994) 2450. K.M. Schep, G.E.W. Bauer, Phys. Rev. B 56 (1997) 15 860. I.L. Aleiner, L.I. Glazman, Phys. Rev. B 57 (1998) 9608. P.W. Brouwer, I.L. Aleiner, Phys. Rev. Lett. 82 (1999) 390. H.U. Baranger, P. Mello, Phys. Rev. Lett. 73 (1994) 142. R.A. Jalabert, J.-L. Pichard, C.W.J. Beenakker, Europhys. Lett. 27 (1994) 255. M.H. Pedersen, S.A. van Langen, M. BuK ttiker, Phys. Rev. B 57 (1998) 1838. Ya.M. Blanter, E.V. Sukhorukov, Phys. Rev. Lett. 84 (2000) 1280. Yu.V. Nazarov, in: H.A. Cerderia, B. Kramer, G. SchoK n (Eds.), Quantum Dynamics of Submicron Structures, NATO ASI Series E, Vol. 291, Kluwer Academic Publishing, Dordrecht, 1995, p. 687. P.W. Brouwer, C.W.J. Beenakker, J. Math. Phys. 37 (1996) 4904. M. BuK ttiker, Phys. Rev. B 38 (1988) 9375. S. Komiyama, H. Hirai, Phys. Rev. B 54 (1996) 2067. A.J. Kil, R.J.J. Zijlstra, P.M. Koenraad, J.A. Pals, J.P. AndreH , Solid State Commun. 60 (1986) 831. B.E. Kane, D.C. Tsui, G. Weimann, Surf. Sci. 196 (1988) 183. S. Washburn, R.J. Haug, K.Y. Lee, J.M. Hong, Phys. Rev. B 44 (1991) 3875.
Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166 [107] [108] [109] [110] [111] [112] [113] [114] [115] [116] [117] [118] [119] [120] [121] [122] [123] [124] [125] [126] [127] [128] [129] [130] [131] [132] [133] [134] [135] [136] [137] [138] [139] [140] [141] [142] [143] [144] [145] [146] [147] [148] [149] [150] [151] [152] [153] [154] [155]
161
W.D. Oliver, J. Kim, R.C. Liu, Y. Yamamoto, Science 284 (1999) 299. M. BuK ttiker, Science 284 (1999) 275. C7 . Kurdak, C.-J. Chen, D.C. Tsui, J.P. Lu, M. Shayegan, S. Parihar, S.A. Lyon, Surf. Sci. 361/362 (1996) 705. T. Gramespacher, M. BuK ttiker, Phys. Rev. Lett. 81 (1998) 2763. T. Gramespacher, M. BuK ttiker, Phys. Rev. B 60 (1999) 2375. M. BuK ttiker, Phys. Rev. Lett. 68 (1992) 843. E.V. Sukhorukov, D. Loss, Phys. Rev. Lett. 80 (1998) 4959. E.V. Sukhorukov, D. Loss, Phys. Rev. B 59 (1999) 13 054. S.A. van Langen, M. BuK ttiker, Phys. Rev. B 56 (1997) R1680. Ya.M. Blanter, S.A. van Langen, M. BuK ttiker, Usp. Fiz. Nauk 168 (1998) 159 [Physics-Uspekhi 41 (1998) 149]. R.C. Liu, Y. Yamamoto, S. Tarucha, Physica B 249}251 (1998) 152. R.C. Liu, B. Odom, Y. Yamamoto, S. Tarucha, Nature 391 (1998) 263. B.L. Altshuler, A.G. Aronov, B.Z. Spivak, Pis'ma Zh. ED ksp. Teor. Fiz. 33 (1981) 101 [JETP Lett. 33 (1981) 94]. Y. Gefen, Y. Imry, M. Ya. Azbel, Phys. Rev. Lett. 52 (1984) 129. M. BuK ttiker, Y. Imry, M. Ya. Azbel, Phys. Rev. A 30 (1984) 1982. M.A. Davidovich, E.V. Anda, Phys. Rev. B 50 (1994) 15 453. G. Iannaccone, M. Macucci, B. Pellegrini, Phys. Rev. B 56 (1997) 12 104. M. BuK ttiker, in: H.D. Hahlbohm, H. LuK bbig (Eds.), SQUID'85. Superconducting Quantum Interference Devices and their Applications, Walter de Gruyter, Berlin, 1985, p. 529. M. BuK ttiker, Phys. Rev. B 33 (1986) 3020. J.L.D. D'Amato, H.M. Pastawski, Phys. Rev. B 41 (1990) 7411. F. Gagel, K. Maschke, Phys. Rev. B 52 (1995) 2013. G. MuK ller, D. Weiss, S. Koch, K. von Klitzing, H. Nickel, W. Schlapp, R. LoK sch, Phys. Rev. B 42 (1990) 7633. R.C. Liu, Y. Yamamoto, Phys. Rev. B 49 (1994) 10 520. R.C. Liu, Y. Yamamoto, Phys. Rev. B 50 (1994) 17 411. M. BuK ttiker, in: V. Bareikis, R. Katilius (Eds.), Noise in Physical Systems and 1/f Fluctuations, World Scienti"c, Singapore, 1995, p. 35. P.W. Brouwer, C.W.J. Beenakker, Phys. Rev. B 55 (1997) 4695. A. Shimizu, M. Ueda, Phys. Rev. Lett. 69 (1992) 1403. L.Y. Chen, C.S. Ting, Phys. Rev. B 46 (1992) 4714. M. BuK ttiker, in: H.A. Cerdeira, B. Kramer, G. SchoK n (Eds.), Quantum Dynamics of Submicron Structures, NATO ASI Series E, Vol. 291, Kluwer Academic Publishing, Dordrecht, 1995, p. 657. M. BuK ttiker, J. Phys.: Condens. Matter 5 (1993) 9361. I.L. Aleiner, A.I. Larkin, Phys. Rev. B 54 (1996) 14 423. P.W. Brouwer, M. BuK ttiker, Europhys. Lett. 37 (1997) 441. Ya.M. Blanter, M. BuK ttiker, Europhys. Lett. 42 (1998) 535. Y. Naveh, D.V. Averin, K.K. Likharev, Phys. Rev. Lett. 79 (1997) 3482. K.E. Nagaev, Phys. Rev. B 57 (1998) 4628. Y. Naveh, D.V. Averin, K.K. Likharev, Phys. Rev. B 59 (1999) 2848. K.E. Nagaev, Phys. Rev. B 58 (1998) R7513. Y. Naveh, Phys. Rev. B 58 (1998) R13 387. Y. Naveh, cond-mat/9910256. M. BuK ttiker, Phys. Rev. B 45 (1992) 3807. E. Ben-Jacob, E. Mottola, G. SchoK n, Phys. Rev. Lett. 51 (1983) 2064. S.-R.E. Yang, Solid State Commun. 81 (1992) 375. M. Ueda, A. Shimizu, J. Phys. Soc. Jap. 62 (1993) 2994. R.J. Schoelkopf, P.J. Burke, A.A. Kozhevnikov, D.E. Prober, M.J. Rooks, Phys. Rev. Lett. 78 (1997) 3370. D. Zheng, Z.D. Wang, Y. Liu, Phys. Lett. A 208 (1995) 375. G.B. Lesovik, L.S. Levitov, Phys. Rev. Lett. 72 (1994) 538. M.H. Pedersen, M. BuK ttiker, Phys. Rev. B 58 (1998) 12 993. R.J. Schoelkopf, A.A. Kozhevnikov, D.E. Prober, M.J. Rooks, Phys. Rev. Lett. 80 (1998) 2437. Y. Levinson, P. WoK l#e, Phys. Rev. Lett. 83 (1999) 1399.
162
Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166
[156] M. BuK ttiker, in: P.H. Handel, A. Chung (Eds.), Noise in Physical Systems and 1/f Fluctuations, AIP Conf. Proc. No. 285, AIP, New York, 1993, p. 3. [157] M. BuK ttiker, H. Thomas, A. Pre( tre, Phys. Lett. A 180 (1993) 364. [158] T. Kuhn, L. Reggiani, L. Varani, Superlatt. Microstruct. 11 (1992) 205. [159] T. Kuhn, L. Reggiani, L. Varani, Semicond. Sci. Technol. 7 (1992) B495. [160] R. Dashen, S.-k. Ma, H.J. Bernstein, Phys. Rev. 187 (1969) 345. [161] F. Kassubek, C.A. Sta!ord, H. Grabert, Phys. Rev. B 59 (1999) 7560. [162] V.A. Gopar, P.A. Mello, M. BuK ttiker, Phys. Rev. Lett. 77 (1996) 3005. [163] C.W.J. Beenakker, Private communication, 1999. [164] M. BuK ttiker, J. Korean Phys. Soc. 34 (1999) 121, cond-mat/9902054. [165] Ya.M. Blanter, F.W.J. Hekking, M. BuK ttiker, Phys. Rev. Lett. 81 (1998) 1925. [166] M. BuK ttiker, A.M. Martin, Phys. Rev. B 61 (2000) 2737. [167] G.E. Blonder, M. Tinkham, T.M. Klapwijk, Phys. Rev. B 25 (1982) 4515. [168] S. Datta, P.F. Bagwell, M.P. Anantram, Phys. Low-Dim. Struct. 3 (1996) 1. [169] B.A. Muzykantskii, D.E. Khmelnitskii, Phys. Rev. B 50 (1994) 3982. [170] M.J.M. de Jong, C.W.J. Beenakker, Phys. Rev. B 49 (1994) 16 070. [171] Th. Martin, Phys. Lett. A 220 (1996) 137. [172] S.V. Naidenov, V.A. Khlus, Fiz. Nizk. Temp. 21 (1995) 594 [Low Temp. Phys. 21 (1995) 462]. [173] A.L. Fauche`re, G.B. Lesovik, G. Blatter, Phys. Rev. B 58 (1998) 11 177. [174] M.P. Anantram, S. Datta, Phys. Rev. B 53 (1996) 16 390. [175] J.-X. Zhu, C.S. Ting, Phys. Rev. B 59 (1999) R14 165. [176] G.B. Lesovik, Th. Martin, J. Torre`s, Phys. Rev. B 60 (1999) 11 935. [177] A.M. Martin, M. BuK ttiker, unpublished. [178] J. Torre`s, Th. Martin, Eur. Phys. J. B 12 (1999) 319. [179] T. Gramespacher, M. BuK ttiker, Phys. Rev. B 61 (2000) 8125. [180] A.N. Vystavkin, M.A. Tarasov, Pis'ma Zh. Tekh. Phys. 9 (1983) 869 [Sov. Tech. Phys. Lett. 9 (1983) 373]. [181] X. Jehl, P. Payet-Burin, C. Baraduc, R. Calemczuk, M. Sanquer, Phys. Rev. Lett. 83 (1999) 1660. [182] K.K. Likharev, Rev. Mod. Phys. 51 (1979) 101. [183] H. Kanter, F.L. Vernon Jr., Appl. Phys. Lett. 16 (1970) 115. [184] H. Kanter, F.L. Vernon Jr., Phys. Rev. Lett. 25 (1970) 588. [185] M.J. Stephen, Phys. Rev. Lett. 21 (1968) 1629. [186] M.J. Stephen, Phys. Rev. 182 (1969) 531. [187] K.K. Likharev, V.K. Semenov, Pis'ma Zh. ED ksp. Teor. Fiz. 15 (1972) 625 [JETP Lett. 15 (1972) 442]. [188] A.G. Aronov, R. Katilius, Zh. ED ksp. Teor. Fiz. 68 (1975) 2208 [Sov. Phys. JETP 41 (1975) 1106]. [189] P.S. Landa, N.D. Tarankova, Fiz. Nizk. Temp. 3 (1977) 32 [Sov. J. Low Temp. Phys. 3 (1977) 15]. [190] A.B. Zorin, Fiz. Nizk. Temp. 7 (1981) 709 [Sov. J. Low Temp. Phys. 7 (1981) 346]. [191] G.B. Lesovik, Pis'ma Zh. ED ksp. Teor. Fiz. 60 (1994) 806 [JETP Lett. 60 (1994) 820]. [192] W.H. Parker, B.N. Taylor, D.N. Langenberg, Phys. Rev. Lett. 18 (1967) 287. [193] G. Vernet, R. Adde, Appl. Phys. Lett. 19 (1971) 195. [194] R.K. Kirschman, J.E. Mercereau, Phys. Lett. A 35 (1971) 177. [195] M. Tarasov, D. Golubev, V. Shumeiko, Z. Ivanov, E. Stepantsov, O. Harnak, T. Claeson, Pis'ma Zh. ED ksp. Teor. Fiz. 68 (1998) 426 [JETP Lett. 68 (1998) 454]. [196] I.O. Kulik, Zh. ED ksp. Teor. Fiz. 57 (1969) 1745 [Sov. Phys. JETP 30 (1970) 944]. [197] A. Furusaki, M. Tsukada, Physica B 165 & 166 (1990) 967. [198] S.N. Artemenko, A.F. Volkov, A.V. Zaitsev, Zh. ED ksp. Teor. Fiz. 76 (1979) 1816 [Sov. Phys. JETP 49 (1979) 924]. [199] M. BuK ttiker, Phys. Rev. B 32 (1985) 1846. [200] M. BuK ttiker, Ann. N.Y. Acad. Sci. 480 (1986) 194. [201] D. Averin, H.T. Imam, Phys. Rev. Lett. 76 (1996) 3814. [202] A. MartiD n-Rodero, A. Levy Yeyati, F.J. GarciD a-Vidal, Phys. Rev. B 53 (1996) R8891. [203] J.C. Cuevas, A. MartiD n-Rodero, A. Levy Yeyati, Phys. Rev. Lett. 82 (1999) 4086. [204] T.M. Klapwijk, G.E. Blonder, M. Tinkham, Physica B 109 & 110 (1992) 1657. [205] D. Averin, Physica B 227 (1996) 241.
Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166 [206] [207] [208] [209] [210] [211] [212] [213] [214] [215] [216] [217] [218] [219] [220] [221] [222] [223] [224] [225] [226] [227] [228] [229] [230] [231] [232] [233] [234] [235] [236] [237] [238] [239] [240] [241] [242] [243] [244] [245] [246] [247] [248] [249] [250] [251] [252] [253] [254] [255]
163
Y. Naveh, D.V. Averin, Phys. Rev. Lett. 82 (1999) 4090. E.V. Bezuglyi, E.N. Bratus', V.S. Shumeiko, G. Wendin, Phys. Rev. Lett. 83 (1999) 2050. J.P. Hessling, V.S. Shumeiko, Yu.M. Galperin, G. Wendin, Europhys. Lett. 34 (1996) 49. Y. Misaki, A. Saito, K. Hamasaki, Japan J. Appl. Phys. Part 1 35 (1996) 1190. Y. Misaki, A. Saito, S. Anezaki, K. Hamasaki, IEEE Trans. Appl. Supercond. 7 (1997) 2438. P. Dieleman, H.G. Bukkems, T.M. Klapwijk, M. Schicke, K.H. Gundlach, Phys. Rev. Lett. 79 (1997) 3486. T. Hoss, C. Strunk, T. Nussbaumer, R. Huber, U. Staufer, C. SchoK nenberger, cond-mat/9901129. N.G. van Kampen, Stochastic Processes in Physics and Chemistry, North-Holland, Amsterdam, 1992. K.E. Nagaev, cond-mat/9908193. R. Landauer, Phys. Rev. B 47 (1993) 16 427. L.Y. Chen, G. Levine, J. Yang, C.S. Ting, Mod. Phys. Lett. 5 (1991) 315. E. Barkai, R.S. Eisenberg, Z. Schuss, Phys. Rev. E 54 (1996) 1161. N.G. van Kampen, Physica A 244 (1997) 414. M.E. Raikh, Private communication, 1991. A. Imamog lu, Y. Yamamoto, Phys. Rev. Lett. 70 (1993) 3327. L.Y. Chen, Phys. Rev. B 48 (1993) 4914. H.-O. MuK ller, A. HaK dicke, U. Hanke, K.A. Chao, Z. Phys. B 102 (1997) 557. H.B. Sun, G.J. Milburn, Superlatt. Microstruct. 23 (1998) 883. H.B. Sun, G.J. Milburn, Phys. Rev. B 59 (1999) 10 748. G.J. Milburn, cond-mat/9905360. Ya.M. Blanter, M. BuK ttiker, in: D.C. Glattli, M. Sanquer, J. Tra( n Thanh Va( n (Eds.), Quantum Physics at Mesoscopic Scale, Editions Frontie`res, Gif-sur-Yvette, 2000, to be published. L.Y. Chen, Mod. Phys. Lett. 7 (1993) 1677. G. Iannaccone, M. Macucci, B. Pellegrini, Phys. Rev. B 55 (1997) 4539. J.H. Davies, J. Carlos Egues, J.W. Wilkins, Phys. Rev. B 52 (1995) 11 259. Y. Isawa, N. Matsubara, T. Ohuti, Solid-State Electron. 42 (1998) 1515. E.R. Brown, IEEE Trans. Electron Devices 39 (1992) 2686. V.V. Kuznetsov, E.E. Mendez, J.D. Bruno, J.T. Pham, Phys. Rev. B 58 (1998) R10 159. J. Han, F.S. Barnes, IEEE Trans. Electron Devices 38 (1991) 237. M.A. Alam, A.N. Khondker, IEEE Trans. Electron Devices 39 (1992) 2184. H. Sheng, S.-J. Chua, J. Phys. D 27 (1994) 137. M.M. Jahan, A.F.M. Anwar, Solid-State Electron. 38 (1995) 429. A. Sugimura, Solid State Commun. 87 (1993) 461. J. Carlos Egues, S. Hersh"eld, J.W. Wilkins, Phys. Rev. B 49 (1994) 13 517. A. Levy Yeyati, F. Flores, E.V. Anda, Phys. Rev. B 47 (1993) 10 543. G. Iannaccone, G. Lombardi, M. Macucci, B. Pellegrini, Phys. Rev. Lett. 80 (1998) 1054. Ya.M. Blanter, M. BuK ttiker, Phys. Rev. B 59 (1999) 10 217. V.J. Goldman, D.C. Tsui, J.E. Cunningham, Phys. Rev. Lett. 58 (1987) 1256. P. HaK nggi, H. Thomas, Phys. Repts. 88 (1982) 207. Sh.M. Kogan, A.Ya. Shul'man, Zh. ED ksp. Teor. Fiz. 56 (1969) 862 [Sov. Phys. JETP 29 (1969) 467]. A. Schmid, Z. Phys. 271 (1974) 251. B.L. Altshuler, A.G. Aronov, in: A.L. Efros, M. Pollak (Eds.), Electron}electron Interactions in Disordered Systems, North-Holland, Amsterdam, 1985, p. 1. Ya.M. Blanter, Phys. Rev. B 54 (1996) 12 807. K.E. Nagaev, Phys. Rev. B 52 (1995) 4740. V.I. Kozub, A.M. Rudin, Phys. Rev. B 52 (1995) 7853. V.I. Kozub, A.M. Rudin, Pis'ma Zh. ED ksp. Teor. Fiz. 62 (1995) 45 [JETP Lett. 62 (1995) 49]. B.L. Altshuler, A.G. Aronov, D.E. Khmelnitsky, J. Phys. C 15 (1982) 7367. F. von Oppen, A. Stern, Phys. Rev. Lett. 79 (1997) 1114. V.L. Gurevich, A.M. Rudin, Pis'ma Zh. ED ksp. Teor. Fiz. 62 (1995) 13 [JETP Lett. 62 (1995) 12]. V.L. Gurevich, A.M. Rudin, Phys. Rev. B 53 (1996) 10 078. A. Schmid, Z. Phys. B 259 (1973) 421.
164
Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166
[256] Y. Naveh, D.V. Averin, K.K. Likharev, Phys. Rev. B 58 (1998) 15 371. [257] M. Turlakov, cond-mat/9908298. [258] T. GonzaH lez, C. GonzaH lez, J. Mateos, D. Pardo, L. Reggiani, O.M. Bulashenko, J.M. RubiD , Phys. Rev. Lett. 80 (1998) 2901. [259] T. GonzaH lez, J. Mateos, D. Pardo, O.M. Bulashenko, L. Reggiani, Phys. Rev. B 60 (1999) 2670. [260] C.W.J. Beenakker, Phys. Rev. Lett. 82 (1999) 2761. [261] K.E. Nagaev, Phys. Rev. Lett. 83 (1999) 1267. [262] H. Schomerus, E.G. Mishchenko, C.W.J. Beenakker, Phys. Rev. B 60 (1999) 5839. [263] H. Schomerus, E.G. Mishchenko, C.W.J. Beenakker, cond-mat/9907027. [264] O.M. Bulashenko, J.M. RubiD , V.A. Kochelap, Phys. Rev. B 61 (2000) 5511. [265] T. GonzaH lez, J. Mateos, D. Pardo, L. Varani, L. Reggiani, cond-mat/9910125. [266] T. GonzaH lez, O.M. Bulashenko, J. Mateos, D. Pardo, L. Reggiani, Phys. Rev. B 56 (1997) 6424. [267] O.M. Bulashenko, J. Mateos, D. Pardo, T. GonzaH lez, L. Reggiani, J.M. RubiD , Phys. Rev. B 57 (1998) 1366. [268] T. GonzaH lez, O.M. Bulashenko, J. Mateos, D. Pardo, L. Reggiani, J.M. RubiD , Semicond. Sci. Technol. 12 (1997) 1053. [269] Y. Naveh, A.N. Korotkov, K.K. Likharev, Phys. Rev. B 60 (1999) R2169. [270] V. Tripathi, D.E. Khmelnitskii, Phys. Rev. B 58 (1998) 1122. [271] Ya.M. Blanter, A.D. Mirlin, B.A. Muzykantskii, Phys. Rev. Lett. 80 (1998) 4161. [272] E. Cuevas, E. Louis, J.A. VergeH s, Phys. Rev. Lett. 77 (1996) 1970. [273] E. Louis, E. Cuevas, J.A. VergeH s, M. Ortun o, Phys. Rev. B 56 (1997) 2120. [274] V.I. Okulov, V.V. Ustinov, Fiz. Nizk. Temp. 5 (1979) 213 [Sov. J. Low Temp. Phys. 5 (1979) 101]. [275] S.V. Gantsevich, V.L. Gurevich, R. Katilius, Riv. Nuovo Cimento 2 (6) (1979). [276] M. Lax, Rev. Mod. Phys. 32 (1960) 25. [277] C.W.J. Beenakker, H. van Houten, Phys. Rev. B 43 (1991) 12 066. [278] D.V. Averin, K.K. Likharev, in: B.L. Altshuler, P.A. Lee, R.A. Webb (Eds.), Mesoscopic Phenomena in Solids, North-Holland, Amsterdam, 1991, p. 173. [279] M.A. Belogolovskii, L.G. Levin, Pis'ma Zh. Tekh. Phys. 17 (1991) 84 [Sov. Tech. Phys. Lett. 17 (1991) 311]. [280] A.N. Korotkov, D.V. Averin, K.K. Likharev, S.A. Vasenko, in: H. Koch, H. LuK bbig (Eds.), Single-Electron Tunneling and Mesoscopic Devices, Springer Series in Electronics and Photonics, Vol. 31, Springer, Berlin, 1992, p. 45. [281] A.N. Korotkov, Phys. Rev. B 49 (1994) 10 381. [282] A.N. Korotkov, Europhys. Lett. 43 (1998) 343. [283] S. Hersh"eld, J.H. Davies, P. Hyldgaard, C.J. Stanton, J.W. Wilkins, Phys. Rev. B 47 (1993) 1967. [284] K.-M. Hung, G.Y. Wu, Phys. Rev. B 48 (1993) 14 687. [285] Yu.M. Halperin, U. Hanke, K.A. Chao, N. Zou, Mod. Phys. Lett. B 7 (1993) 1159. [286] U. Hanke, Yu.M. Galperin, K.A. Chao, N. Zou, Phys. Rev. B 48 (1993) 17 209. [287] U. Hanke, Yu.M. Galperin, K.A. Chao, Phys. Rev. B 50 (1994) 1595. [288] E.V. Anda, A. LatgeH , Phys. Rev. B 50 (1994) 8559. [289] Z. Wang, M. Iwanaga, T. Miyoshi, Jpn. J. Appl. Phys. Pt. 1 37 (1998) 5894. [290] W. Krech, A. HaK dicke, H.-O. MuK ller, Int. J. Mod. Phys. B 6 (1992) 3555. [291] W. Krech, H.-O. MuK ller, Z. Phys. B 91 (1993) 423. [292] K.A. Matsuoka, K.K. Likharev, Phys. Rev. B 57 (1998) 15 613. [293] A.N. Korotkov, D.V. Averin, K.K. Likharev, Phys. Rev. B 49 (1994) 1915. [294] A.N. Korotkov, Phys. Rev. B 50 (1994) 17 674. [295] D.V. Averin, Yu.V. Nazarov, Phys. Rev. Lett. 69 (1992) 1993. [296] F.W.J. Hekking, L.I. Glazman, K.A. Matveev, R.I. Shekhter, Phys. Rev. Lett. 70 (1993) 4138. [297] W. Krech, H.-O. MuK ller, Mod. Phys. Lett. B 8 (1994) 605. [298] U. Hanke, M. GisselfaK lt, Yu. Galperin, M. Jonson, R.I. Shekhter, K.A. Chao, Phys. Rev. B 50 (1994) 1953. [299] U. Hanke, Yu. Galperin, K.A. Chao, M. GisselfaK lt, M. Jonson, R.I. Shekhter, Phys. Rev. B 51 (1995) 9084. [300] B.R. Bu"ka, J. Martinek, G. Micha"ek, J. BarnasH , Phys. Rev. B 60 (1999) 12 246. [301] Ya.M. Blanter, unpublished. [302] E.R. Nowak, M.B. Weissman, S.S.P. Parkin, Appl. Phys. Lett. 74 (1999) 600. [303] D. Loss, E.V. Sukhorukov, Phys. Rev. Lett. 84 (2000) 1035.
Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166 [304] [305] [306] [307] [308] [309] [310] [311] [312] [313] [314] [315] [316] [317] [318] [319] [320] [321] [322] [323] [324] [325] [326] [327] [328] [329] [330] [331] [332] [333] [334] [335] [336] [337] [338] [339] [340] [341] [342] [343] [344] [345] [346] [347] [348] [349] [350] [351] [352] [353]
165
L.I. Glazman, M.E. Raikh, Pis'ma Zh. ED ksp. Teor. Fiz. 47 (1988) 378 [JETP Lett. 47 (1988) 452]. T.K. Ng, P.A. Lee, Phys. Rev. Lett. 61 (1988) 1768. S. Hersh"eld, J.H. Davies, J.W. Wilkins, Phys. Rev. B 46 (1992) 7046. S. Hersh"eld, Phys. Rev. B 46 (1992) 7061. F. Yamaguchi, K. Kawamura, J. Phys. Soc. Jap. 63 (1994) 1258. F. Yamaguchi, K. Kawamura, Physica B 227 (1996) 116. G.-H. Ding, T.-K. Ng, Phys. Rev. B 56 (1997) R15 521. A. Schiller, S. Hersh"eld, Phys. Rev. B 58 (1998) 14 978. M.P.A. Fisher, L.I. Glazman, in: L.L. Sohn, L.P. Kouwenhoven, G. SchoK n (Eds.), Mesoscopic Electron Transport, NATO ASI Series E, Vol. 345, Kluwer Academic Publishing, Dordrecht, 1997, p. 331. C. de C. Chamon, D.E. Freed, X.G. Wen, Phys. Rev. B 53 (1996) 4033. C.L. Kane, M.P.A. Fisher, Phys. Rev. Lett. 72 (1994) 724. A. Auerbach, Phys. Rev. Lett. 80 (1998) 817. P. Fendley, A.W.W. Ludwig, H. Saleur, Phys. Rev. Lett. 75 (1995) 2196. P. Fendley. H. Saleur, Phys. Rev. B 54 (1996) 10 845. U. Weiss, Solid State Commun. 100 (1996) 281. I. Sa", Phys. Rev. B 56 (1997) R12 691. N.P. Sandler, C. de C. Chamon, E. Fradkin, Phys. Rev. B 59 (1999) 12 521. V.V. Ponomarenko, N. Nagaosa, Solid State Commun. 110 (1999) 321. C. de C. Chamon, D.E. Freed, X.G. Wen, Phys. Rev. B 51 (1995) 2363. F. Lesage, H. Saleur, Nucl. Phys. B 490 (1997) 543. F. Lesage, H. Saleur, Nucl. Phys. B 493 (1997) 613. C. Chamon, D.E. Freed, Phys. Rev. B 60 (1999) 1848. K.-I. Imura, K. Nomura, Europhys. Lett. 47 (1999) 83. S.B. Isakov, Th. Martin, S. Ouvry, Phys. Rev. Lett. 83 (1999) 580. L. Saminadayar, D.C. Glattli, Y. Jin, B. Etienne, Phys. Rev. Lett. 79 (1997) 2526. R. de-Picciotto, M. Reznikov, M. Heiblum, V. Umansky, G. Bunin, D. Mahalu, Nature 389 (1997) 162. M. Reznikov, R. de Picciotto, T.G. Gri$ths, M. Heiblum, V. Umansky, Nature 399 (1999) 238. F. von Oppen, Phys. Rev. B 56 (1997) 9674. G. Kirczenow, Phys. Rev. B 58 (1998) 1457. R. de Picciotto, cond-mat/9802221. Y. Wei, B. Wang, J. Wang, H. Guo, Phys. Rev. B 60 (1999) 16900. F. Green, M.P. Das, cond-mat/9809339. F. Green, M.P. Das, cond-mat/9905086. M.P. Das, F. Green, cond-mat/9910183. C.H. Henry, R.F. Kazarinov, Rev. Mod. Phys. 68 (1996) 801. C.W.J. Beenakker, M. Patra, Mod. Phys. Lett. B 13 (1999) 337. H. Lee, L.S. Levitov, A.Yu. Yakovets, Phys. Rev. B 51 (1995) 4079. L.S. Levitov, G.B. Lesovik, Pis'ma Zh. ED ksp. Teor. Fiz. 58 (1993) 225 [JETP Lett. 58 (1993) 230]. L.S. Levitov, G.B. Lesovik, Pis'ma Zh. ED ksp. Teor. Fiz. 55 (1992) 534 [JETP Lett. 55 (1992) 555]. A. Shimizu, H. Sakaki, Phys. Rev. B 44 (1991) 13 136. M.J.M. de Jong, Phys. Rev. B 54 (1996) 8144. Yu.V. Nazarov, Ann. Phys. (Leipzig) 8 (1999) 507. K.A. Muttalib, Y. Chen, Int. J. Mod. Phys. B 10 (1996) 1999. Ya.M. Blanter, H. Schomerus, C.W.J. Beenakker, cond-mat/9910331; Physica E, to be published. D.A. Ivanov, L.S. Levitov, Pis'ma Zh. ED ksp. Teor. Fiz. 58 (1993) 450 [JETP Lett. 58 (1993) 461]. L.S. Levitov, H. Lee, G.B. Lesovik, J. Math. Phys. 37 (1996) 4845. D.A. Ivanov, H.W. Lee, L.S. Levitov, Phys. Rev. B 56 (1997) 6839. C.W.J. Beenakker, in: J.-P. Fouque (Ed.), Di!use Waves in Complex Media, NATO ASI Series C, Vol. 531, Kluwer Academic Publishing, Dordrecht, 1999, p. 137. D.P. DiVincenzo, D. Loss, J. Magn. Magn. Mater. 200 (1999) 202, cond-mat/9901137. G. Burkard, D. Loss, E.V. Sukhorukov, cond-mat/9906071.
166
Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166
[354] E.V. Sukhorukov, D. Loss, G. Burkard, cond-mat/9909348. [355] S. Oberholzer, M. Henny, C. Strunk, C. SchoK nenberger, T. Heinzel, K. Ensslin, M. Holland, preprint; Physica E (2000), to be published. [356] V.V. Fyodorov, H.-J. Sommers, J. Math. Phys. 38 (1997) 191. [357] X. Jehl, M. Sanquer, R. Calemczuk, D. Mailly, preprint (1999). [358] A.A. Kozhevnikov, R.J. Schoelkopf, L.E. Calvet, M.J. Rooks, D.E. Prober, preprint; J. Low. Temp. Phys. (2000), to be published. [359] A.A. Kozhevnikov, R.J. Schoelkopf, D.E. Prober, preprint (1999). [360] G.B. Lesovik, Pis'ma Zh. ED ksp. Teor. Fiz. 70 (1999) 209 [JETP Lett. 70 (1999) 208]. [361] V.L. Gurevich, M.I. Muradov, cond-mat/0001329. [362] O. Agam, I. Aleiner, A. Larkin, cond-mat/9912086. [363] A. Reklaitis, L. Reggiani, Phys. Rev. B 60 (1999) 11683. [364] O.M. Bulashenko, J.M. RubiD , V.A. Kochelap, cond-mat/9911395. [365] M. Turlakov, cond-mat/0002153. [366] Y. Tanaka, T. Asai, N. Yoshida, J. Inoue, S. Kashiwaya, cond-mat/9912168. [367] A.N. Korotkov, K.K. Likharev, cond-mat/9911318. [368] M. Stopa, preprint (1999). [369] F. Green, M.P. Das, cond-mat/9911251. [370] F. Green, M.P. Das, cond-mat/0001412. [371] A. Andreev, A. Kamenev, cond-mat/0001460.
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POLYMER THEORY: PATH INTEGRALS AND SCALING
T.A. VILGIS Max-Planck-Institut fu( r Polymerforschung, Postfach 3148, 55021 Mainz, Germany Laboratoire EuropeH en AssocieH , Institut Charles Sadron, 6 rue Boussingault, 67083 Strasbourg, France
AMSTERDAM } LAUSANNE } NEW YORK } OXFORD } SHANNON } TOKYO
Physics Reports 336 (2000) 167}254
Polymer theory: path integrals and scaling T.A. Vilgis * Max-Planck-Institut fu( r Polymerforschung, Postfach 3148, 55021 Mainz, Germany Laboratoire Europe& en Associe& , Institut Charles Sadron, 6 rue Boussingault, 67083 Strasbourg, France Received October 1999; editor: M.L. Klein Contents 1. Introduction 2. Polymers and path integrals 2.1. Some general remarks 2.2. Random walks 2.3. Intuitive introduction of the Wiener}Edwards integral 3. Self-avoiding chains 3.1. The problem of self-avoidance 3.2. Polymer magnet analogy 4. Field theories for self-avoiding chains 4.1. Self-avoiding walks and analogy to phase transitions 4.2. Self-avoiding walks and supersymmetry 5. Many-chain systems: melts and screening 5.1. Some general remarks 5.2. Collective variables 5.3. The statistics of tagged chains 5.4. Scaling in semi-dilute polymer solutions 6. Correlations in polymer blends and block copolymer melts 6.1. Some general remarks 6.2. The Edwards Hamiltonian formulation 6.3. Field theoretical formulation of the problem 6.4. Static properties of tagged chains 7. Polymers of larger connectivity: branched polymers and polymeric fractals
170 171 171 172 175 179 179 181 185 185 190 197 197 198 204 207 209 209 210 212 213 215
7.1. Preliminary remarks 7.2. D-dimensionally connected polymers in good solvent 7.3. D-dimensionally connected polymers between two parallel plates in good solvent 7.4. D-dimensionally connected polymers in a cylindrical pore (good solvent) 7.5. Melts of fractals in restricted geometries 7.6. Once more the di!erences 8. Polymers in random potentials: replica analysis 8.1. Frozen disorder and non-Gibbsian statistical mechanics 8.2. A polymer chain in a random potential 8.3. Replica symmetry 8.4. Numerical solution 9. Polymers on disordered surfaces and interfaces 9.1. Scaling argument for fractal spatial disorder 9.2. Variational calculation without replicas 10. Random copolymers 10.1. General remarks 10.2. The model 10.3. The e!ective Hamiltonian 10.4. What can we see?
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* Corresponding author. Laboratoire EuropeH en AssocieH , Institut Charles Sadron, 6 rue Boussingault, 67083 Strasbourg, France. Tel.: #33-3-88-41-4059; fax: #33-3-88-41-4099. E-mail address:
[email protected] (T.A. Vilgis). 0370-1573/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 1 2 2 - 2
T.A. Vilgis / Physics Reports 336 (2000) 167}254 11. Copolymer melts in disordered media 11.1. General remarks 11.2. Replica symmetric solution. The phase diagram 11.3. Stability of the replica symmetric solution
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11.4. Disorder versus ordering in the lamellar phase 12. Final remarks Acknowledgements References
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Abstract In this review paper a number of recent topics on polymer theory are discussed. One of the main issues of the present work is to show that scaling theories together with more sophisticated methods gives answers to the raised questions. The here selected topics are connectivity and interactions. The problem of connectivity is especially important in connection with restricted geometries. Here it is shown that simple scaling theories propose a number of results, which can now checked experimentally. Interactions play a dominant role in polymers blends and copolymers and become especially important when disorder e!ects come into play. 2000 Elsevier Science B.V. All rights reserved. PACS: 05.20.!y; 36.20.!r; 61.25.Hq; 31.15.Kb; 03.50.!z
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1. Introduction Polymer theory has become a well-developed subject since the middle of the 1960s, when Edwards had introduced his famous Hamiltonian for self-avoiding polymer chains [1]. The theoretical success of this new formulation suddenly enabled many theoretical approaches, which had appeared very di$cult before. Edwards formulated the self-avoiding polymer problem in a continuous model, with two terms. The "rst term describes the connectivity which he took as a Wiener process, the second a nonlocal pseudo-potential that describes the repulsive potential between monomers in a very crude manner. This minimum model has been the basis of many di!erent investigations, and progress has been made in many respects. For the "rst time theorists had a kind of uniform method to describe polymers, which is very close to the path integral formulation of quantum mechanics [2] and systematic approximations had been possible. Generalizations to the dynamics of #exible polymers could be made and many predictions have been o!ered [3]. Parallel to this development polymer theory has been in#uenced to a large extent by the theory of phase transitions, and de Gennes [4] revolutionalized the "eld by introducing his celebrated scaling concepts. These have been and are a powerful tool to "nd the essential contents of the physics, which describe at least the basic features of the problem under consideration. Perhaps the most important step in the 1970s was put forward by de Gennes [5], when he showed that the self-avoiding polymer corresponds to a critical phenomenon, when the length of the chain becomes in"nitely long. For the "rst time deep insight was o!ered into the critical exponents and scaling function describing the chain statistics, and very re"ned renormalization methods have been developed [6]. The aim of this review is twofold. First, we review some recent results on the statistical mechanics of polymers and second we present some of the methods, which have not been o!ered in the textbooks so far in a concise manner. Therefore at some places, where I see the necessity to be more formal I will do so. This will be the case especially in Section 4, where we are going to discuss the "eld theoretic formulation of self-avoiding walks. These formulations are the basic theoretical formulations of constructing the corresponding "eld theories and are thus of more general interest, which goes beyond the theory of polymers and soft matter. Indeed, these "eld theories are one possible formal basis for results represented on a less formal level further below in this review (Section 8). This review paper is organized as follows. In a "rst step (Section 2) the basic notation is introduced. Therefore, I introduce the random walks and their formal representation, which yield a heuristic introduction of the path integral. Then, in Section 3 the problem of self-avoidance is discussed on physical grounds. The formal treatment is carried out in more detail in Section 4, where the corresponding "eld theories are constructed, as mentioned earlier. There I distinguish between a bosonic "eld theory, which is that introduced by de Gennes [5] the fermionic representation. The latter goes back to McKane [7] and Parisi [8]. This formulation using Grassmann "elds contains many physical problems. In any case it is an example of avoiding the de Gennes trick of having a zero component "eld theory. The latter is not as well spread as de Gennes nP0 trick, thus I decided to introduce this in more detail. The physics and therefore the methods change completely when many chains are present. This will be discussed in Section 5. Here density functional theories play a dominant role. Many chains
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destroy the self-avoidance correlations the lager the density becomes. Mutual chain interactions yield the screening [9] in concentrated polymer solutions and polymer melts. It is shown, how the dense systems can be described by mean "eld theory. The results obtained here will have immediate application to polymer blends and block copolymers where some results are presented in Section 6. It will be interesting to note, that there the phase separations are described well by mean "eld theory up to very close to the critical point. The distance from the critical point is determined mainly by the chain length itself, where we also discuss the experimental evidence for this behavior. In Section 7 I then move on to a simple description of branched polymers and their behavior in nano pores. In this subject a nice application of scaling theory can be discussed in the context of restricted geometries. The last sections describe some recent results on disorder and polymers. In Section 8 we discuss polymers in disordered media, where the replica method is introduced. In Section 9 I report on polymers at disordered surfaces and discuss the adsorption behavior. There it can be seen that the adsorption is enhanced by the disorder. Then I move to some peculiar problems on random copolymers in Section 10. In Section 11 I will present a more detailed consideration on copolymer melts and disorder. There it will be demonstrated how disorder and microphase separation transitions will interplay with each other. Another point to mention is the richness of polymer and soft matter physics. In this review many di!erent methods will be under consideration. It ranges from "eld theories to scaling methods. The use of such di!erent methods is not done on purpose, but required by the system itself. Dilute solutions need to be described di!erently compared to strong solutions. Di!erent length scales of importance appear in the di!erent regimes. Thus di!erent physical methods and techniques will be needed. This will be visible throughout this review.
2. Polymers and path integrals 2.1. Some general remarks The easiest view of polymer molecules for theorists is to draw lines on a sheet of paper. Of course, this is a highly oversimpli"ed picture, but it helps to formulate the mathematical problem very much. Essentially, two main features must be incorporated into this picture: connectivity and interactions. For this basic introduction we follow the reference of Doi and Edwards [10]. Let us "rst focus on the problem of connectivity and neglect interactions for a moment. To describe the simplest version of a connected line we can build up the polymer by arranging a set of N `bond vectorsa +b ,, which we align in a row. The end-to-end distance, which is a natural length G scale of the polymer can then be counted very simply by adding up all bond vectors. It is given by , , R" b " (R !R ) , (2.1) G G\ G G where R are the position vectors at each bond vector b . The bond vectors are now chosen to be G G random vectors, i.e., they can point in any direction without memory. Thus we have the stochastic
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properties 1b 2"0, G
1b b 2"bd . G H GH
(2.2)
Here d is the usual Kronecker symbol and b the typical size of the bond vector. Indeed, this GH information is enough to compute the typical size of the random chain of bond vectors, which is given by the squared average of the end-to-end distance. A simple one-line calculation gives immediately the result
1R2"
, , " 1b b 2"bN . b G H G G H G
(2.3)
This is the basic relationship for the size of ideal polymers, i.e., polymer molecules which are described by a random walk. The result is coincident with the typical size of a random walk as it must be the case. But in actual fact and despite the simplicity of the input the result contains many more principles of modern theoretical physics than might be obvious. We remark "rst that Eq. (2.3) is a typical example of a scaling law, which de"nes an exponent. The size of the random walk can be casted in the following `scaling equationa [11]: R"bNJ ,
(2.4)
where l". Note that here a sloppy notation has been used, where R denotes the root of 1R2 by using the same symbol. We will do this later on in most cases, when we discuss length scales on a more qualitative basis. There are two interesting aspects concerning the scaling equation. The "rst is that it contains a local length scale b which corresponds to a typical (microscopic) scale in the problem. If we had carried out the random walk on a lattice, b corresponds to the lattice constant. This scale is not universal. On the other hand, the exponent l does not depend on local details. It provides the scaling of the polymer size with the total amount of segments N. It is relatively easy to realize that the value of l does not depend on the lattice type, for example, of the value of b. It seems that l is a universal quantity. Another important observation is that the value of the scaling exponent l does not depend on the space dimension. The argument which gave rise to Eq. (2.3) is completely independent of the dimension d of embedding space. The latter point becomes obvious when we recall that so far we had only studied ideal random walks, i.e., walks without any interaction. Had we studied walks with interactions it would have become obvious that the dimension of space matters. This becomes most obvious in d"1. Here the di!erence between the random walk and a walk, where a segment must not be at the same place is most pronounced: The self-avoiding walk in one dimension has the scaling law R"bN, i.e., it is totally stretched out. 2.2. Random walks To understand the physics we had already found from the very simple remarks above, we have to study the case of random walks in more detail and on a more formal basis [10]. To do so we carry out what we have guessed above. We start from the set of bond vectors +b ,, , which are G G
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statistically independent vectors. The probability to "nd a whole set is given by , P(+b ,, )" p(b ) , (2.5) G G G G where the probability p(b ) is given by G p(b )"(1/4pb)d("b "!b) . (2.6) G G The front factor in Eq. (2.6) comes from the assumption of isotropy and corresponds to the normalization. We are now interested in "nding the distribution of the end-to-end distance to make some statements on the size of the random walk. To do so, we remember the de"nition of the end-to-end distance, Eq. (2.1) and compute its distribution by
, , P(R)" db d R! b P(+b ,, ) . G G G G G The calculation becomes simple if the delta function is parameterized by
(2.7)
, 1 , d R! b " dk exp !ik R! b , (2.8) G G (2p)B where d is the dimension of embedding space. Inserting this into Eq. (2.7) yields the classical Gaussian distribution function P(R)"(d/2pbN)B exp(dR/2bN) .
(2.9)
Another important point to note is that the restriction of "xed length for the bond vectors can be relaxed without problems. Indeed, when the probability equation (2.6) is replaced by an e!ective Gaussian of the form p (b )"(1/4pb)B exp(!b/2b) , (2.10) G G no changes occur in the results. In contrast to Eq. (2.6), where the bond length is constrained to take "xed values, the latter equation "xes only the mean squared distance between the two neighboring bonds. Again, we must note that Eq. (2.9) has a certain scaling function [11]. It contains two important informations. To see this, let us rewrite it in the more convenient form P(R)"(a/mB)F(R/m) .
(2.11)
Here we have introduced the only relevant scale in the problem m"b(N,bNJ (a is just a numerical constant). This scale, which corresponds of course with the size of the ideal polymer, spans out a volume of mB in d space dimensions. There are two important observations. The "rst is that the distribution function depends only on the ratio R/m. The second is that the prefactor must be of the dimension of the volume, because of the requirement of normalization, i.e., dBR P(R)"1. Thus we might expect, that for other polymeric objects we can make use of these facts, even though we cannot compute the analog of P(R) completely. This will happen if we take into account interactions and the chains become self-avoiding.
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Before we proceed in these directions, we have to analyze Eq. (2.9) in more detail. The trivial observation is, as mentioned already, that the distribution function is purely Gaussian. This re#ects once more that we had not taken into account any interactions. The second important point is that the distribution function P(R) is invariant under any rescaling of the chain length, i.e., if N is replaced by NI "N/j, where j is a real number. Naturally, the numerical value of j must be smaller than N itself, so that the rescaled chain behaves still as a random walk. Thus we must require 14j;N. The third point is the mean size of the polymer. The mean end-to-end-distance is computed to
1R2" dBR RP(R)"bN
(2.12)
as it must be the case. However, there is already more physical insight. Eq. (2.12) contains the fractal dimension of the ideal polymer. The fractal dimension is intuitively de"ned by the size}mass relationship. Let us therefore interpret the end-to-end distance by R"bNJM ,
(2.13)
where M is the total mass of the polymer. We have assumed that each monomer has the same mass m such that the total mass is given by M"Nm . The mass M is also called the molecular weight. Eq. (2.13) de"nes again a size mass scaling with the conclusion that the fractal dimension of the random walk or the random walk polymer is given by d "2 , (2.14) which is independent of the space dimension of the Euclidean space in which the polymer is embedded. With just this knowledge it is already easy to realize that the random walk polymer is an unrealistic model for isolated polymers. To see this let us look at the density of the random walk chain or the density correlation function. It has the general form C(R)J1/RB\ .
(2.15)
Obviously, in d"1 the density correlation function and the density grow for large sizes R linearly. Therefore, the density of the one-dimensional walk becomes in"nite 0 C(R, d"1) PR .
(2.16)
This result cannot be true because the density of a real object cannot grow inde"nitely. The physical reason behind this is that the random walk folds very often on itself in one dimension. In d"2 the density of a random walk is constant. This behavior is very similar to that of Euclidean objects. The trajectory of the random walk is therefore space "lling in two dimensions. In three dimensions the random walk behaves as a fractal between the two cuto!s b and the total size R . The density correlation function is given by C(R)J1/R, which resembles the
Ornstein}Zernike law from the mean "eld correlation function in phase transition [12,13]. The Fourier transform of the density correlation function leads to the structure factor S(k) where
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k is the wave vector. It is given by S(k)Jk\B "k\
(2.17)
in the intermediate and large range of the wave vectors. The structure factor de"nes thus an experimental method to measure the fractal dimension, if the scaling behavior extends over a su$cient range of the values for the wave vector. The structure factor of random walk polymers can be computed exactly and yields the Debye function. This will be done shortly in the following. 2.3. Intuitive introduction of the Wiener}Edwards integral The Gaussian chain is a very pedagogical example for the introduction of the path integral description of polymer. The Gaussian chain corresponds to a Feynman}Wiener path integral. Let us therefore present a heuristic argument [10]. The mathematically more interested reader is referred to one of the best introductions to path integrals, i.e., the classic reference of Feynman and Hibbs [14]. We already noticed that Gaussian chains are self-similar [10,15,6]. This point corresponds to the central limit theorem. To understand this we come back to the Gaussian distribution for the mean size of one individual of the bond lengths, p(b ) (d/2pb)B exp+!(d/2b)b, . G G
(2.18)
Of course, the distribution of the set is given by
d B d , exp ! b P(+b ,)" G 2pb 2b G G "
(2.19)
d ,B d , exp ! b . G 2pb 2b G
(2.20)
Now, we recall that each bond vector is given by the di!erence of the spatial vectors of each bond, i.e., b "R !R , and write the total probability as G G G\
d ,B d , P(+b ,)" exp ! (R !R ) . G G G\ 2pb 2b G
(2.21)
Formally, we can associate a `Hamiltoniana with this expression. Indeed, if we write for a moment the distribution as p(+b ,)"N exp+!bH (+R ,), , G G
(2.22)
we can de"ne d , (R !R ) . bH " G G\ 2b G
(2.23)
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Throughout the paper we will call any normalization factor N, which we do not want to determine precisely. From the above, we may recognize a well-known `Hamiltoniana from solid-state physics, which describes a chain of harmonic springs as the simplest example for a one-dimensional solid to describe lattice vibrations [16]. Crudely, we may use the continuum limit (R !R )/1P(RR/Rs) G G>
(2.24)
to arrive at a symbolic notation for the distribution
d , RR ds P(+b ,) N exp ! G 2b Rs
(2.25)
Here N is an appropriate normalization, which we do not specify. Eq. (2.25) has been named the Wiener distribution (or in polymer theory the Wiener}Edwards distribution) for random walk chains. So far, Eq. (2.25) does not contain anything new, except that a more fancy and more useful notation has been introduced. To go from the discrete to the continuous notation we can use the following minimal dictionary: 04s4N 0 14i4N ,
(2.26)
(2.27)
, , ds 0 i"1 . G
Now we have to ask ourselves, what have we gained by reformulating the problem into this language. The advantage will become obvious. Eq. (2.25) is written in a `languagea that allows modern theoretical treatment by using functional integrals, which are well known in theoretical physics, especially in quantum mechanics. Still formally, we can write for the partition function the symbolic expression Z"N
d , RR(s) . exp ! ds Rs 2b RQ
(2.28)
The partition function is now represented as a sum over all possible paths. The physical idea behind this is the following: All possible conformations of the random walk, which is composed of N statistical independent segments contribute to the value of Z. There are of course more probable paths and also less probable paths. As an example for less probable paths we mention a stretched path. The appearance of an almost straight line with RJN is very unlikely from entropic reasons, but nevertheless it contributes to the partition function Z. This mathematical formulation resembles the idea of path integrals in quantum mechanics. Indeed, we are going to build up a simple analogy to the Feynman representation of quantum mechanics [14]. To construct the analogy of `the sum over pathsa we must realize "rst that the random walk polymer satis"es a di!usion equation. This becomes most obvious if we recall that the distribution of the end-to-end distance P(R) satis"es the di!usion equation [10,14] P(R, N)"(d/2pNb)B exp+!(d/2Nb)R,
(2.29)
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which we had derived already. We interpret the equation in the following way: We want to construct all random walks between the space points r "0, where the walk starts and the end point r"R. Additionally, we require that the walker has N steps. It is easy to show that P(R) satis"es the di!usion equation of the form (R/RN!(b/2d) )P(R, N)"0 ∀RO0, N'0 .
(2.30)
We can reformulate this in terms of a Green function for any two points r, r and corresponding contour variables s, s , i.e., (R/Rs!(b/2d) )G(R, R , s, s )"d(R!R )d(s!s ) .
(2.31)
The delta functions on the RHS of Eq. (2.31) are the initial conditions and ensure that the di!usion equation has only physical solutions for s!s 50 and for s!s "0 we have G(r, r , 0)"d(r!r ) The initial conditions therefore are G(R, R, s, s )"0 ∀(s!s )(0 ,
(2.32)
G(R, R , 0, 0)"d(R!R) .
(2.33)
The di!usion equation and the equation for the Green function are formally equivalent to the SchroK dinger equation, apart from constants. Let us therefore write down the SchroK dinger equation for a free quantum particle in the following form: (i R/Rt#( /2m ) )G(r, r , t, t )"!i d(r!r )d(t!t ) . (2.34) The di!usion equation and the SchroK dinger equation can be transformed to each other by the Wick rotation tPis/ , which is an analytic continuation of the time to imaginary values. The Wick rotation has become a standard technique in quantum "eld theory. The role of the masses is played by the step length of the polymer m d/b. It is interesting to realize that the formal analogy holds at di!erent levels. To see this let us compare the Green function of the RW polymer G(R, R, s, s )"(d/2pb(s!s ))B exp+!(d/2b)(R!R )/(s!s ), .
(2.35)
By employing the Wick rotation t (i/ )s this transforms immediately to the Green function of the free quantum particle G(R, R, t, t )"(i/2p (t!t ))B exp+(i/ )(m /2)(R!R )/(t!t ), . (2.36) Indeed, Eq. (2.36) is the Green function of the free quantum particle. Now we are in a position to realize the physical interpretation and the correspondence between the two problems, which we have already mentioned. The physical visualization becomes clear: the propagation of the quantum particle between two points is given by the sum over all possible paths. The corresponding polymer propagator (or the probability) is also the sum over all possible con"gurations. The weight of the di!erent contributions is given by the corresponding measure, i.e., the Lagrange function is the quantum case, and the Wiener distribution in the polymer case [17,18]. It was nice to learn from my friend and colleague Dino Leporini that Giancarlo Wick was born in Torino and that he lectured at the Scuola Normale Superiore in Pisa during the last part of his scienti"c career.
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The tempting idea now is the following question: can the SchroK dinger analogy be used further for interaction of interacting systems? For the free systems the analogy is quite nice, but somehow we have to use it for more complicated problems, such as interacting systems. We show shortly in the following, that this is indeed the case. To begin with, let us study the easiest problem that we can imagine, i.e., a polymer chain in external potential (R(s)), which acts on each monomer along the chain. This problem which at "rst sight seems arti"cial will become very important later. The "rst question we have to ask is what is the total energy ; in the system. This is simple and the answer is ;",ds (R(s)). The next step is to compute the equilibrium conformation of the chain. Therefore, the distribution function of the chain in the external potential. To do so we compute this we need by using the standard Boltzmann factor [10] from
, . (2.37) ds (R(s )) The average 1 2" describes the average over the free chain conformation and can be written as P(+R(s),[;])J1exp+!b;,2" exp !b
(2) exp+!(d/2b), ds(RR(s)/Rs), 122" . exp+!(d/2b), ds(RR/Rs), To simplify the notation let us call the denominator in the following:
(2.38)
d , RR N\" exp ! . (2.39) ds Rs 2b Eq. (2.37) allows us to de"ne a Hamiltonian that takes into account the e!ect of the potential. Therefore, we rewrite Eq. (2.37) in the form P(+R(s),[;])"N exp+!bH(+R(s),), which de"nes the new polymer Hamiltonian as
d , RR(s) , bH" # (R(s)) ds . ds 2b Rs In future, we will also abbreviate the notation by using
(2.40)
(2.41)
N , DR(s) for the functional integrals as sum over paths. It is now easy to realize that the Green function corresponding to the path integral equation (2.37) satis"es the SchroK dinger-type equation +R/Rs!(b/2d) # (r),G(r, r , s, s )"d(r!r )d(s!s ) .
(2.42)
The proof of this statement is very simple and can be directly copied from the standard textbooks [10,17]. The main issue is that indeed problems in polymer theory can be formulated in terms of modern physics and the techniques from quantum mechanics and "eld theory will be useful. To see this in more detail let us now turn to the problem of chains with excluded volume, i.e., the problem of self-avoiding walks (SAW).
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3. Self-avoiding chains 3.1. The problem of self-avoidance So far, the models for the polymer chain we used are too simple. We allowed for random walks and local single-particle interactions in terms of external potentials, only. The most serious drawback of these models is that two chain segments are allowed to have the same coordinates R(s). In more realistic chain models this cannot be the case. We must, however, introduce a repulsive potential [10,6] <(R(s)!R(s )) which prevents two monomers (or chain segments) from being at the same place. To set up a more serious model we use a more plausible Hamiltonian for the self-avoiding walk chain. It is given by
, d , RR 1 , # bH(+R(s),)" ds ds ds <(R(s)!R(s )) . (3.1) Rs 2 2b The potential <(r) is determined by the usual intra-molecular potentials, such as Lennard-Jones potentials, hard core interactions, etc., which are well known from the theory of liquids [19] but we will later on use more simpli"ed pseudo-potentials. It has been realized that a useful pseudopotential approximation is [10,11,6] <(R)"vd(R)Jbd(R) .
(3.2)
This potential is always repulsive as long as the chain segments are at the same place. The strength of the potential is roughly given by the excluded volume between two segments. This is of the order of b. We will see later that the precise value of v is not of signi"cance with respect to the universal properties. First we have to tackle this problem. The "rst di$culty is given by the potential itself. In contrast to the considerations above the excluded volume potential appears as pair interaction. Therefore, we cannot formulate it in terms of the SchroK dinger equation, at least not in the #avor that we have used the terminology above. The "rst serious problem is therefore buried in the nature of the excluded volume: bH of saw does not correspond to a one-particle potential d(R(s)!R(s )). The next serious problem appears if any perturbation theory as an expansion in terms of the excluded volume parameter v is tried, i.e., if we work with an expansion of the form G(R, R , N)"G (R, R , N)#v(2)$v(2) , (3.3) where (2) stands for any expression to be computed. We will immediately realize for this that the perturbation series diverges which means: Red alert! Fixman [20] was the "rst to realize that the perturbation parameter is not a small quantity. The perturbation parameter of relevance is not v itself, but the combination [10,6] v(N, (v(N), (v(N), 2. More generally, in d dimensions the perturbation parameter is vN\B . The result on the chain size is (see, e.g., [3]) 1R2"Nbl [1#(vbN)\B #const. (vbN)\B#2] . (3.4) Thus any perturbation theory in d(4 must break down [6]. This means simply that new physics happens, and we cannot stay within the methods used so far. What will happen can be seen in
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a simple dimensional estimate of the Hamiltonian [21]. To do so we carry out a dimensional argument of bH in the pseudo-potential approximation
, d , RR v , ds # ds ds d(R(s)!R(s )) . bH" Rs 2 sb The steps of the analysis are the following: E E E E
(3.5)
suppose that the size of the polymer has the scaling form R&NJ, estimate connectivity term to &NJ\>, estimate excluded volume &N\BJ, match both terms in the exponents: 2l!1"2!dl and read o! the result l"3/(2#d) .
(3.6)
Here we see that the space dimension enters. Unlike as in the random walk for exponent we can expect a dependence on the space dimension for the size of the chain in the SAW case. Now we have to ask for the quality of the results. The dimensional counting is crude and we cannot expect the results to be of any relevance, but if the estimates and the real values are compared to each other we can be very surprised. Let us "rst summarize the results in Table 1. The only dimension where it goes wrong is d"3. Let us discuss the results in the di!erent dimensions in more detail. The observations are the following. First in d"1 it is exact, since the SAW in one dimension must be a fully stretched chain. d"1 is lower critical dimension since l cannot become larger than 1. Otherwise the chain would be over-stretched. We just mention without proof that the value in d"2 is exact [6]. This has been proven by conformal invariance [22]. In d"3 it is close to the real value of l"0.5892 which has been computed by renormalization group theory. We realize also that v" for d"4. Why is this special? We may not be too surprised, when we see that the perturbation parameter was vN\B . In dimensions larger than four, this parameter becomes really small. To be more precise look at a special Ginzburg argument and let us estimate the energy by
, 1 , N ;" v ds ds d(R(s)!R(s ))J . 2 RB If we put the ideal walk chain size inside we see what happens:
(3.7)
(3.8) ;&v N/RB GHI + vN\B . J( 0 , Thus the SAW interaction is no longer important for d54 for d54 we recover random walk behavior. The case d"4 requires some attention. The exponent l" is exact, but there are, however, logarithmic corrections to the prefactors and scaling functions. This can be seen intuitively, since the scaling estimate of the interaction potential is ;JN, which in most cases monitors the existence of logarithmic corrections. These have been worked out in detail [6] and more recently by SchaK fer and coworkers [23]. The latter reference is an extended review on these topics and we refer the reader to the corresponding issue.
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Table 1 Flory-type estimates for the critical exponents l d
l $
Comment
1 2 3 4
1
Exact Exact Wrong Exact
At the present level, we are not able to compute the exponents more accurately. This requires more work, which we will outline in the next section. We can, however, use the scaling forms to "nd the asymptotic form of the distribution function for example. In the case of the random walk we had found that the probability distribution was of a scaling form, see Eq. (2.11). We might assume that the SAW is also a self-similar object and we can use the same argumentation. In doing so we might immediately guess the form [11]. P (R, N)J(1/N)JB(R/bNJ)A\ J exp+!(R/bNJ) \J, , (3.9) 1 5 where c is another exponent, which is c"1 for the random walk. For many more lucid discussions on these issues see the brilliant book by des Cloizeaux and Jannink [6]. Another point of interest is the structure factor and the density}density correlation function. These can be also estimated by the scaling idea. The density}density correlation is simply given by S(r)J1/rB\B "1/rB\ J
(3.10)
which is S(r)Jr\ in three dimensions. The corresponding structure factor scales of course as, S(k)Jk\ J. 3.2. Polymer magnet analogy As we have seen above there is a natural connection between phase transition and self-avoiding walks. So far, this connection is intuitive and must be shown on a more rigorous basis. A natural way is to think about a way to represent the SAWs by an Ising model. Obviously, the spin model cannot be the classical Ising model, where the spin takes only two values S "$1, which has only G a phase transition at zero temperature with very peculiar exponents, which do not resemble any of the walk statistics known so far. The standard Ising model is described by the Hamiltonian H"!J S S , (3.11) G H G HZL L where J is the coupling constant, i.e., the interaction between two neighboring spins (see Fig. 1). The sum over the spins runs over the nearest neighbors. The properties of this model are well known and can be found in the standard textbooks of statistical physics (see for example [24]). S "$1. G
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Fig. 1. A simple visualization of the Ising model. The spins are sitting on a lattice. Their nearest-neighbor interaction is described by the exchange constant J.
The model can be generalized for vector spins in the form (3.12) SPS"+S , S ,2, S , , L where the number of components corresponds to di!erent physical systems. Thus, the case n"1 describes the classical Ising model, n"2 the two-dimensional case and n"3 the Heisenberg model. If the spin variable is de"ned as complex, then super conductivity can be described, and "nally the limit nPR corresponds to the spherical model, which is exactly solvable [21]. We will see later that even negative values, e.g., n"!2 describe reasonable physics. The partition function can be written as
dS k(S ) exp bJ S ) S . (3.13) G G G H G G HZL @ The partition function must be evaluated under the constraint that the length of the spins is "xed. For the n-vector model the integration therefore must be carried out under Z"
k(S)"d(S!n) ,
(3.14)
which restricts the magnitude of the spins to an n-dimensional sphere. The next step is the evaluation of the partition function for any value of the components of the spins n. One possibility is to carry out a high-temperature expansion which becomes valid for bJ;1, i.e., small b or high temperatures,
1 exp bJ S ) S "1#bJ S ) S # (bJ) S ) S #O((bJ)) . G H G H 2 G H G HZL L G HZL L G HZL L I JZL L
(3.15)
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To calculate the appearing spin correlation functions from Eq. (3.15) we use the measure equation (3.14) and "nd for the approximate partition function a series which contains the terms
: 1S S 2S K 2 " G I G G
dS k(S ) S S 2S K . G G G G G G To be more precise it is useful to de"ne a characteristic function f (p) f ( p) " : 1e pS2
(3.16)
(3.17)
in the usual form, which generates all correlations by the corresponding derivative with respect to the generating variable p which is also an n-component vector. Mean values are therefore de"ned as Rf (p)/Rp""i1S2 .
(3.18)
How can the characteristic function be computed? To do this, it is useful to consider the second derivative and try to "nd a di!erential equation which determines f (p). The second derivative is given by S Rf/Rp"!1GHI e pS2 . L
(3.19)
Now we use the fact that S"n and "nd a simple di!erential equation for the generating function (D#n) f (p)"0 .
(3.20)
The latter equation is interesting, because it shows for isotropic reasons that the generating function depends only on p""p". Thus we may use for the isotropic case the n-dimensional Laplace-operator in the form D"R/Rp#[(n!1)/p]R/Rp
(3.21)
and "nd for Eq. (3.20) the form (R/Rp#((n!1)/p)R/Rp#n) f (k)"0 .
(3.22)
Now let us consider at "rst sight strange limit nP0, for which we "nd the simple version f (p)!( f (p)/p)"0 ,
(3.23)
which can be solved very easily. To do so we impose the initial conditions f (0)"1 und f (0)"0 and "nd for Eq. (3.17) immediately f (p)"1!p , (3.24) this very formal result showing that for the case n"0 only the quadratic correlation function survives and all higher correlations and moments vanish, since the generating function does not
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Fig. 2. Representation of a self-avoiding walk (with closed ends). One bond can have only two neighbors, which corresponds to the generating function of order p. Higher orders, i.e., joining four lattice sites are not possible in the limit n"0. Fig. 3. A simple sketch of a self-avoiding chain. Di!erent chain segments repel each other and swell the chain. The interactions can be of any order, i.e., binary, ternary 2 contacts.
contain higher orders as O(p). Therefore, the conclusion is that correlation functions always have the form
1 for m"2, 0 , 1S S 2S K 2" G G G 0 otherwise .
(3.25)
In the case n"0 Eq. (3.15) reduces greatly, since only terms of the order S remain and many simpli"cations can be expected. What has this to do with polymers and self-avoiding walks? The answer can be given by viewing the results graphically. The remaining terms can be visualized as lines on a lattice, see Fig. 2. The "gure is constructed as follows: the products between two neighboring spin variables are graphically represented as connected lines between consecutive lattice sites. Therefore, only terms of order 2 appear, i.e., only pairs of spin variables survive in the limit n"0. Such diagrams can be understood as self-avoiding walks on a lattice. It is interesting to note that the SAW problem is characterized by a partition function Z"1, a result which can be easily seen from the computations above (the case p"0). The propagator of the SAW can also be visualized in this representation. To do so we introduce in Eq. (3.13) additional spin variables for the starting point and the end point and "nd that G(r, r )"1S(r)S(r )2"1S(r)S(r )2 /Z . (3.26) I Taking into account that we always have Z"1 we see that only those diagrams contribute to Eq. (3.26) which start at r and end at r .
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This is now a starting point for more re"ned theories. Indeed these more heuristic arguments show a deep connection between phase transitions and self-avoiding walks, which represent a minimal model for polymers. In the next sections we are going to follow these ideas and construct the corresponding "eld theories.
4. Field theories for self-avoiding chains In this section we are going to drive the "eld theories for the self-avoiding chain. It is important to note at this early stage, that the points below apply only for single chains, i.e., for very dilute solutions, such that di!erent chains do not interact mutually. In fact, for higher concentrations the methods introduced below do not make too much sense, since the physics will change greatly. In Section 3.2 we have already shown that self-avoiding walks and phase transition have many interrelations. Here we are going to show them in more detail. The following two chapters are presented in a very formal way. Nevertheless, it appears useful since more general methods are introduced, which can be useful in di!erent parts of statistical physics and "eld theory. The main di$culty on a physical basis is to describe all self-avoiding correlations of the chain in a consistent manner. Intuitively it is clear what happens already from Fig. 3. We can see, that only contacts of arbitrary orders are possible. Thus we expect in a diagrammatic language, that no loops appear. They simply do not belong to the single chain problem. Therefore this point must be recovered under construction of the corresponding "eld theories. It is shown below, that two simple possibilities exist. First, a bosonic "eld theory, in the framework of Landau}Ginzburg-type theories and a fermionic version, which uses Grassmann-type "elds. Since both techniques are probably not well known to a broad readership we spell them out in more detail. 4.1. Self-avoiding walks and analogy to phase transitions In the previous section, we saw that conventional methods applied to the problem of calculating the critical exponent l of the (non-Gaussian) SAW failed. To solve this problem we may employ "eld theoretic methods. The starting point is that the random walk corresponds to the Gaussian model in "eld theory. Therefore, the question is, what is the corresponding "eld theory for the SAW. To construct this "eld theory we start from the Green's function for the self-avoiding walk [10]
,r , d , Rr 1 , ds Dr(s) exp ! exp ! v ds ds d(r(s)!r(s )) . (4.1) Rs 2b 2 r 0 As we have been realizing so far, it would be useful to rewrite this problem such that it can be mapped to a SchroK dinger equation. At "rst sight this does not seem possible since the interaction term is a nonlocal pair interaction. Fortunately, these pair interactions can be decoupled by using a Hubbard}Stratonovich transformation in the form [10,25] r
G(r, 0; N, 0)"
, , 1 , 1 exp ! v ds ds d(r(s)!r(s )) ,N D (x) exp ! dBx (x)!i ds (r(s)) . (4.2) 2 2v
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This transformation introduces an auxiliary real-scalar stochastic "eld (r) with a Gaussian distribution
1 P[ ]Jexp ! dBx (x) 2v
(4.3)
and moments 1 (x)2 "0 , ( 1 (x) (0)2 "vd(x) . ( This allows to rewrite the Green's function in a form without pair interactions as
,r
r
(4.4) (4.5)
Dr D exp[!bH [r, ]] , (4.6) 0 , d , Rr 1 #i ds (r(s))! dBx (x) , (4.7) bH (+r(s),, + ,) " : ds Rs 2b 2v but includes an additional averaging over the Gaussian distribution over the random "eld (x). As a result, the pair interactions are replaced exactly by a stochastic "eld (4.3) acting on each point of the walk. The promising feature is that this auxiliary "eld is Gaussian so that we can have some hope to do better on the SAW than before. Viewing the -integration in (4.6) as a statistical average, we rewrite the Green's function as G(r, 0; N, 0)"N
r
1 G(r, 0; N, 0)"N D (x) exp ! dBx (x) G(r, 0; N, 0; [ ]) 2v
(4.8)
":1G(r, 0; N, 0; [ ])2 , (4.9) ( where a new Green's function (4.9) depending on a given realization [ ] of the stochastic "eld has to be averaged 1 2 over all possible realizations of this "eld. It is well known that expression (4.9) ( corresponds to di!usion of a particle in an imaginary random external potential [10,12,26] [R/Rs!(b/2d) x #i (x)]G(r, 0; N, 0; [ ])"d(r)d(N)
(4.10)
which corresponds to Eq. (2.42). The next step is to solve it formally by the inverse operator, i.e.,
(4.11)
": Gd(x!x )d(s!s ) ,
(4.12)
R \ l d(x!x )d(s!s ) ! x #i (x) Rs 2d
G(x, x : s, s ; [ ])"
note that up to this end everything was based on identities. To proceed further with the formal (exact) solution (4.12) we introduce the Laplace transform
G(x, x ; m; [ ])"
ds exp[!ms]G(x, s; x , s ; [ ])
(4.13)
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for the Green's function
\ b d(x!x ) . G(x, x ; m; [ ])" m! x #i (x) 2d
(4.14)
It is now simple to represent the Laplace-transformed Green's function or resolvent as
1 G(x, x ; m; [ ]), Du(x)u(x)u(x ) Z
1 ;exp ! dBx dBx u(x)G\(x, x ; m; [ ])u(x ) 2 ": 1u(x)u(x )2 ,
(4.15)
a functional average over the correlation function of an auxiliary "eld u(x). However, G\ has the convenient form (4.14) so that we can immediately rewrite all this to
1 G(x, x ; m; [ ])" Duu(x)u(x ) Z
1 b ;exp ! dBx u(x) m! x #i (x) u(x) . 2 2d
(4.16)
The average 1 2 has to be performed, over all realizations of the random "eld . This appears to ( be a nontrivial problem, since we introduced in (4.16) a denominator which also depends on this "eld. One possible solution to this problem is the conventional trick, often applied in theoretical physics in di!erent contexts [12,26}28] 1/Z,lim ZL\ . L
(4.17)
This identity complicates again the expression, but will lead "nally to the desired results. G(x, x ; m; [ ]) "lim L
1 b D u(x) exp ! dBx u(x) m! x #i (x) u(x) 2 2d
1 b ; Duu(x)u(x ) exp ! dBx u(x) m! x #i (x) u(x) 2 2d L "&&lim L ?
Du u (x)u (x ) exp[!bH(+u ,, )]'' , ? ?
L\
(4.18)
(4.19)
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since now an O(n)-symmetric "eld theory has been derived with a Hamiltonian as
b 1 L dBx u (x) m! x #iu(x) u (x) (4.20) bH(+u ,, ) " : ? ? ? 2d 2 ? and the "eld u"(u ,2,u ), and the limit lim . By analytic continuation n3- Pn31, we L L > have "nally obtained an exact formulation of the SAW problem as the zero-component limit of an n-dimensional bosonic "eld theory. Since we have still the Green's function G(x, x ; m; [ ]) for one realization of the -"elds, the "nal Gaussian average
1 dBx (x) lim DLu(x)u (x)u (x ) G(x, x ; m)"N D (x) exp ! 2v L 1 ;exp ! dBx u(x) [m! #i (x)]u(x) 2
(4.21)
has to be carried out. Thus, we get
1 G(x, x ; m)"lim DLu(x)u (x)u (x ) exp ! dBx u(x)[m! ]u(x) 2 L 1 1 ;N D (x) exp ! dBx (x)! dBx u(x)i (x)u(x) 2v 2
"lim DLu(x)u (x)u (x ) L 1 ;exp ! dBx u(x) [m! ]u(x)#v(u(x)u(x)) 2
(4.22)
(4.23)
after performing the Gaussian -integration. To get more physical insight we rewrite (4.23) in the conventional form
[u]] G(x, x ; m)" DLu(x)u (x)u (x ) exp[!bH %*5
(4.24)
with
L L [" u (x)"#mu]#v uu (4.25) ? ? ? @ ? ? @ and we "nally have the well-known result that the SAW corresponds to standard "eld theoretical formulations, i.e. the u-formulation with its Ginzburg}Landau}Wilson functional [13,26,29]. The physical picture for the nP0 limit can be seen easily in the diagrammatic expansion. The perturbation series expansion for an n-component u-theory contains all known diagrams, i.e. 1 bH [u]" dBx %*5 2
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Fig. 4. Two typical Feynman diagrams. The upper one corresponds to a polymer single-chain diagram. Two polymer segments of the polymer (thick straght line) interact with each other. The tadpole diagram does not contribute to the SAW. These types are removed by taking the limit n"0.
loops, etc. Especially, the loops do not correspond to the SAW. Such tadpole diagrams do not correspond to con"gurations of one walker since they are constituted by two or more interacting free propagators. If one actually carries out the calculation of the perturbation expansion, one "nds that all these loop graphs are proportional to n. Thus, in the language of Feynman diagrams the nP0 trick is nothing else but ignoring diagrams which do not contribute to true walk con"gurations (see Fig. 4). A beautiful connection of walks, polymers and "eld theories has been found by de Gennes in 1972 [5]. What we gain with this analysis is now the correct value for the SAW critical exponent l. Knowing that it is represented by the nP0 limit of an n-component u-"eld theory, one looks up the e"4!d expansion from the renormalization group treatment, e.g. [26,29,30] l "[1#((n#2)/(n#8))e#O(e)] and take the limit n"0
(4.26)
lim l "[1#e#O(e)] (4.27) L at the end which is correct to "rst order in e. This can now be compared to the variational SAW exponent l "[1#e#O(e)] , Flory's estimate
(4.28)
l "[1#e#O(e)] . (4.29) $ Thus we see that the Flory estimate is wrong already at "rst order in the e-expansion. Nevertheless, for practical purposes the Flory value of l" is still useful.
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4.2. Self-avoiding walks and supersymmetry 4.2.1. Grassmann variables and their Gaussian integrals Before we can actually introduce this idea, we have to present a short introduction to the Grassmann algebra and the most important consequences for our purpose; a more detailed discussion can be found in Ryders text book [31] or many other books in quantum "eld theory. Beautiful discussions are provided by Refs. [26,32]. Up to now, we used commuting variables [u , u ] " : u u !u u "0 G H G H H G
(4.30)
throughout, but one may explore also the properties of [c , c ] " : c c #c c "0 G H > G H H G
(4.31)
anticommuting variables +c , which constitute the Grassmann algebra. G In the following, let us just explore some properties of Grassmann variables. From de"nition (4.31) it follows for i"j that the square of any Grassmann variable c"0
(4.32)
vanishes. This has the nice consequence f (c)"f #f c#f c#f c#2 GFFHFFI
(4.33)
,
,f #f c
(4.34)
that the "rst-order Taylor expansion is already the exact representation of f. Equivalently, one sees from (4.34) that dLf (c)/dcL,0 for all n'1 ,
(4.35)
all derivatives of higher order than one vanish. The integration operation has to be de"ned only for integrations over constants and linear functions
dc"0 ,
(4.36)
dc c"1 ,
(4.37)
such that all the properties of usual integrals are satis"ed; the normalization of the integrals may di!er depending on the de"nition. If we now integrate an arbitray function
dcf (c)" dc( f #f c)"f ,
(4.38)
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we "nd the surprising result that
df , dc f (c) , dc
(4.39)
integration and di!erentiation yield the same result. These are already all the properties we need in order to use Grassmann "elds later on. The most important feature which is the key for Grassmann "elds are their Gaussian integrals
I" dc
dc exp[!c jc ]" dc dc (1!c jc )
"!j dc dc c c "j dc dc c c "j, GHI GFHFI
(4.40)
note that II "dc exp[!jc] gives II "0. Using c " : c and c> " : c (as the adjoint of c) one can also write
dc dc> exp[!cjc>]"j
(4.41)
which should be compared to
1 1 dz dz夹 exp[!zjz夹]" j p
(4.42)
for usual complex variables z, z夹3C.
L 1 dz dz夹 exp ! z M z夹 "[det ,]\ +z , complex , G G G GH H G p G H G
(4.43)
L G
(4.44)
dc dc> exp ! c M c> "[det ,]> +c , Grassmann G G G GH G G G H
which are di!erent only in the exponents of the determinant. The generalization of this for functional integrals over Grassmann "elds t(r) and t>(r) is straightforward and yields
1 Dt Dt> exp ! dBx dBx t(x)M(x, x )t>(x ) "det , . 2
(4.45)
With this knowledge, we are now in a position to explore the use of these results in the framework of statistical "elds theories. For a quick sketch, recall again the propagator of the
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self-avoiding walk [12]
u u
1 1 dBx (x) DLu DLu夹u (r)u夹(r ) G(r, r ; m)" D exp ! Z 2v 1 dBx (x)[m! #i ] ;exp ! 2v
1 " D exp ! dBx (x) 2v
r
夹
(x)
(4.46)
1 r m! #i
(4.47)
and compare it to
u u u
1 G(r, r ; m)" D exp ! dBx (x) det[m! #i ] 2v ; DL
DL
夹
u (r)u夹(r )
1 dBx (x)[m! #i ]u夹(x) ;exp ! 2v
(4.48)
another expression. The idea is to rewrite the determinant in the numerator of (4.48) as a Gaussian}Grassmann integral, see (4.45), thus
1 dBx (x) G(r, r ; m)" D exp ! 2v
u
1 ; Dt Dt> exp ! dBx t(x)[m! #i ]t>(x) 2 ; DL
1 DLu夹u (r)u夹(r ) exp ! dBx u(x)[m! #i ]u夹(x) , 2v
(4.49)
we obtain an expression where the determinant has vanished. This means that in the present case Grassmann "elds can overcome the di$culty of taking the n"0 limit. For a "rst guess expression (4.49) can be rewritten as
G(r, r ; m)" D Dt Dt> DLu DLu夹 exp[!bH[u, u夹, t, t>, ]]
(4.50)
with the Hamiltonian
1 bH[u, u夹, t, t>, ]" dBx u(x)[m! ]u夹(x)t(x)[m! ]t>(r) 2 #i (x)[u(x)u夹(x)#t(x)t>(x)]# (x)
1
(x) 2v
(4.51)
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containing bosonic as well as fermionic "elds. The equation is set in such a way that the di!erent contributions appear in one line. The "rst new term is a free fermionic "eld theory, whereas the third term is nothing else but a coupling of the bosonic and fermionic degrees of freedom via the Hubbard}Stratonovich "eld. Now what can we learn from all this? 4.2.2. Scalar Grassmann xeld theories and n"0 limit In this section, we go back to the original problem and work out more carefully and systematically the e!ects of auxiliary Grassmann integrations; we follow the very pedagogical presentation in Ref. [33]. To begin with, we look "rst at an integral of the generic u-type
1 v dLu exp ! u! (u) , 2 4!
(4.52)
where u"(u ,2,u ) is a 2n-dimensional vector. We introduce new (n#1)-dimensional vectors L U" : (U ,2,U , c) , L
(4.53)
U> " : (U>,2,U>, c>) L
(4.54)
with the components U "(1/(2)(u #iu ) , H H\ H
(4.55)
U>"(1/(2)(u> #iu> ) H\ H H
(4.56)
and [c, c>] "0. We further generalize the form of the Ginzburg}Landau integral (4.52) >
u
v I " dL>U dL>U>exp !U>U# (U>U) 3! , dL
(4.57)
1 v v dc dc> exp ! u# (u)#c>c# uc>c , 2 4! 3!
(4.58)
where the de"nitions of the U-"elds have been used. The fourth-order terms can be again decoupled as usual by a Hubbard}Stratonovich transformation
1 da exp[!a#2iax] exp[!x]" (p
(4.59)
with the identi"cation x"v(u)/4!#vuc>c/3! or x"(u#2c>c)(v/4!. Furthermore, the Grassmann properties (c>c)"c>cc>c"!c>ccc>"0 have been used. Thus, we arrive at
v v 1 1 dLu dc> dc da exp ! u#c>c#a#2ia u#4ia c>c I " (p 4! 4! 2
(4.60)
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an expression where an additional integration is present. The Gaussian u, c, and c> integrations can be carried out
1 1 L 1 L v I " 2p 1#4ia da exp[!a] (p 2p 4! 1#4ia(v/4!
"N da exp[!a]
1
1#4ia(v/4!
L\
.
(4.61) (4.62)
Let us look for another integral of the type
1 v I " dKu exp ! u ! (u ) 2 4!
(4.63)
with p-dimensional vectors u "(u ,2, u ) and treat the fourth-order term by another Hubbard} N Stratonovich transformation:
v 1 u !a I "N dKu da exp ! u #2ia 4! 2
(4.64)
(4.65)
"N da exp[!a]
1
1#4ia(v/4!
N
NL\ , I .
(4.66)
This is the same answer as for integral (4.57) if the variable dimension p is chosen appropriately such that p/2"n!1! The generalization for functional integrals is obtained via the lattice formulation uPu which in turn is carried over to the continuum limit u(r). Correspondingly, the G Hubbard}Stratonovich variable a becomes the auxiliary "eld a(r) and dLuP [dLu ]PDLu. G G Thus, we showed that a theory formulated in 2n bosonic "elds and 2 additional fermionic Grassmann "elds is equivalent to a theory with 2n!2 bosonic degrees of freedom. We will exploit this property in the next section in order to get rid of the determinant in the denominator with a proper choice of the number of Grassmann "elds relative to the bosonic degrees of freedom. 4.2.3. Fermionic theory of the self-avoiding walk Let us start for the reader's convenience by compiling again the basic relations for the self-avoiding walk in the pure bosonic language
G(r, r ; m)"lim DLuu (r)u (r ) exp[!bH[u]] L with a standard Laplace-transformed Ginzburg}Landau-type Hamiltonian
v bH[u]" dBx u(x)[m! ]u(x)# (u(x)u(x)) 4!
(4.67)
(4.68)
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de"ned in n-dimensional (commuting) "elds u"(u ,2, u ). Using the Hubbard}Stratonovich L transformation, which then allows to carry out the Gaussian u-integration, yields
G(r, r ; m)"lim Da exp ! dBx a(x) L
v \ a(x) r 4!
v a(x) 4!
;det\L m! #2i " Da exp ! dBx a(x)
r m! #2i
(4.69)
r m! #2i
v \ a(x) r 4!
(4.70)
an expression where the nP0 limit can be carried out formally. This is what we already know about the self-avoiding walk. Stimulated by the results of the zero-dimensional theory from above and in the last section (see (4.66)) for mixed Hamiltonians we now introduce as an Ansatz a more complicated Hamiltonian than (4.68)
bH [u, t, t>, a]" dBx u(x)[m! ]u(x)w(x)[m! ]w>(r) # 2i
v a(x)[u(x)u(x)#w>(x)w(x)]#a(x) 4!
(4.71)
formulated in both a real n-component commuting vector "eld u"(u ,2, u ) and its p-compon L ent anticommuting counterpart w"(t ,2, t ) where w> is again the adjoint of w, which is the N result of Parisi and Sourlas [34]. This supersymmetric Hamiltonian contains again free bosonic and fermionic parts as well as the Hubbard}Stratonovich "eld and the coupling of the di!erent types of "elds via a.
G (r, r ; m)" Da DLu DKw DKw>u (r)u (r ) exp[!bH [u, t, t>, a]]
(4.72)
becomes identical to the one obtained from purely commuting "elds. By use of the Gaussian Grassmann integrals in Section 4.2.1, one can integrate out all the "elds except the a and obtain an expression
G (r, r ; m)" Da exp ! dBx a(x)
;detK m! #2i NL , G(r, r ; m) ,
r m! #2i
v a(x) 4!
v \ a(x) r 4!
(4.73)
(4.74)
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which is identical to the usual formulation of the Green's function for the SAW (4.70) if the Grassmann "elds w and w> have each p"n/2 components. In other words, the sometimes mathematically suspicious nP0 trick can actually be avoided in case of the SAW if the problem is formulated based on a supersymmetric Hamiltonian (4.71). Having shown the equivalence of these two "eld theoretical formulations of the SAW, we now want to give a heuristic argument why the nP0 limit can be avoided with Grassmann "elds. Recall the diagrammatic expansion of the SAW Green's function formulated in the u-"elds from Section 4: the nP0 trick removed all the loop graphs since they have a weight at least proportional to n. The equivalent expansion for the supersymmetric Hamiltonian contains all, for the SAW unphysical diagrams from the u-"elds as for the bosonic Hamiltonian. But the essential feature is that the fermionic graphs of the fourth line just cancel with the unphysical loop diagrams of the third line due to the fermionic negative sign. This is now the precise correspondence between the n"0 bosonic formulation and the theory containing Grassmann "elds. What is meant by the term supersymmetry? This notion comes from quantum "eld theories and is a symmetry of the Hamiltonian if the coupling part (4.71) &u(x)u(x)#w>(x)w(x)
(4.75)
stays invariant under the so-called superrotation
u , , u ? A P , (4.76) w w , , B @ where the matrices , and , have commuting elements whereas the matrices , and , are ? @ A B built up with anticommuting elements. This can be shown if a super"eld
W(x, H, H>) " : u(x)#H>w(x)#w>(x)H#HH>u(x)
(4.77)
is introduced; H and H> are scalar Grassmann variables and u denotes the increment of the rotation. This allows to rewrite the supersymmetric Hamiltonian as
1 bH [W]" d(x, H, H>)W(x, H, H>) W(x, H, H>)#<[W] 2
(4.78)
with
" : #(R/RHRH>) (4.79) in isomorphic to any standard "eld theory with a nonlinear potential <. Before leaving the supersymmetric formulation of the SAW, we look at the heuristic n"!2 limit often used in the literature [29,30,35]. From the discussion of the general n dependence discussed above, this limit was motivated by the RG value for the n-vector model (4.80) l "[1#((n#2)/(n#8))e#O(e)] , yielding the exponent for the Gaussian model lim l " which is the MF value. In this limit, L \ the supersymmetric Hamiltonian reduces exactly
bH [w, w>]" dBx[w>[m! ]w#(w>w)] $
(4.81)
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to a purely fermionic w-theory. Is this really a fourth-order form? The answer is no, as the property (4.32) of Grassmann variables shows, and it can be seen immediately that the Hamiltonian is actually that of a free theory, whence the index F. Thus, it does not surprise us that even the strange limit n"!2 has actually a physical meaning if formulated in the proper language. There are some remarks which are necessary. The fermionic formulation seems to be work out nicely. The removal of the unphysical diagrams, i.e., loops and tadpoles, which cannot contribute to a single SAW chain, by their corresponding diagrams with negative sign is proper. There are, however, a number of di$culties to be mentioned. The critical point in the polymer problem (NPR) corresponds to spontaneous symmetry breaking. In the bosonic "eld theory there appears no di$culty and can be visualized in a Landau theory. Here, in the fermionic version it is not possible to "nd a nonzero expectation value and it is di$cult to interpret the symmetry breaking in terms of the fermionic "eld theory.
5. Many-chain systems: melts and screening 5.1. Some general remarks So far we had studied a single chain in good solvents, which corresponds to the case of the self-avoiding walk. The most important result was the case of the swollen chain with the scaling law R&N . This introduced a new fractal dimension d 5/3
(5.1)
for the chain. In the following, we are going to study the problem of polymer melts or correspondingly concentrated polymer solutions. In other words, we want to study the physical behavior of many-chain systems. What can we expect? To see this from a pictorial point let us imagine a snap shot of a concentrated polymer solution. Excluded volume correlations are now not only taking place within one single chain, but an increasing number of contact points from other chains at increasing polymer concentration yield additional excluded volume. On the other hand, the correlations within one chain become more and more destroyed. To some extent, less correlations rule the statistical behavior of individual chains in the concentrated solution or the polymer melt. We will show below that these additional contacts will have severe e!ects on the statistical behavior of the individual chains. This situation is cartooned in Fig. 5. The sketch in Fig. 5 suggests the following guess for highly concentrated systems. We must distinguish between (at least) two di!erent length scales. One regime will be given by r4m. At these scales the chain piece experiences only correlations from itself, i.e., we may expect the classical self-avoiding behavior. For larger scales r5m the self-avoiding correlations do not play a signi"cant role and we can expect chain statistics close to a Gaussian chain. From this naive picture we must conclude that m must be a function of the concentration. At this intuitive level we can already "gure out one signi"cant concentration CH, which characterizes the overlap between the chains. If the polymers do just overlap one chain is occupied in its own volume. Thus we have [11] CH"N/RB"N/NBJ"N\JB .
(5.2)
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Fig. 5. The single-chain versus a labeled chain in a melt. The single-chain experiences only contacts with itself, whereas the melt chain has contacts with all neighboring chains. Thus the self-contacts, depicted by the circles become irrelevant as the concentration increases.
This is an important result and we have to note that for large chain lengths N the overlap concentration CH is very small. Therefore the underlying physics becomes completely changed. The question is, what are the appropriate methods to describe the concentrated polymer solutions. In fact, below we will see that the methods introduced in the previous sections are no longer the methods of choice. The SAW correlations are changed and thus the partition function is no longer ruled by all the SAW paths, but by the remaining density #uctuations which are determined by the overall concentration of the polymer solution. In the limit of very large concentrations, imagined by an almost completely "lled lattice, the density #uctuations become less and less important. It is thus useful to introduce collective variables, such as collective densities and construct free energy functionals which contain the collective properties. 5.2. Collective variables To some extent we would like to formulate the problem more in terms of the chain model and the Edwards Hamiltonian. To begin with, let us generalize the Edwards formulation to many chains. This is very simple and all that has to be done is to take into account the interaction between all the chain segments. This is re#ected in the Hamiltonian [10,25,27]
1 L ,? ,@ d L ,? RR ? ds# v ds ds d(R (s)!R (s)) , (5.3) bH" ? @ Rs 2 2b ? ? @ where d is the space dimension, n the number of polymer chains present and all other symbols have the same meaning as before. The principal task is to compute the partition function
L (5.4) Z" DR (s) exp(!bH([R (s)])) . ? ? ? Of course, this is generally not simple and the partition function cannot be computed exactly. Therefore a number of simpli"cations are necessary. The "rst one is to assume monodispersity which means that all chains have the same length. Mathematically, this corresponds to N "N ∀a, b. The next problem is that the partition function contains too many degrees of ? @
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freedom. The number of chains n involved can be very large, however, every chain itself has internal degrees of freedom, since they are assumed to be totally #exible. For this reason it is convenient to introduce collective variables which are in this case the polymer segment densities de"ned as
1 L , ds d(x!R (s)) . (5.5) o(x)" ? < ? In fact, o(x) can be viewed as microscopic density operator whose value de"nes the density at an arbitrary point x. It is therefore desirable to transform Edward's Hamiltonian which is a function of the real chain variables to an e!ective one, which depends only on the collective density variables. Let us therefore try a transformation, which is written formally as H(+R (s),)P ?
H(+o(x),)
GHI
.
(5.6)
& The resulting Hamiltonian is called `e!ectivea here since it does not contain all the initial information. It can be imagined that the transformation cannot be carried out exactly. In the following, we will show more of the transformation, since it has become an important tool in polymer physics. The technical strategy is quite simple. It corresponds to the simple mathematical change of variables. The only di!erence is that it has to be done under the functional aspect. The result which we will aim for corresponds to the so-called random phase approximation (RPA) which has been frequently used in solid-state physics. In the following, we will not present the computation in detail but we outline the important steps of computation. Some of the details can be found in [10,27]. 1. Transformation to k-space: The "rst step is to use a formulation in reciprocal space. The advantage of this is to "nd a reasonable simpli"cation in notation. To start with, let us transform the density variable into k-space. This is very simple, and the result can be immediately written down as
, 1 L o(x)" ds e\ kx\R? Q" e\ kxok .
(5.7)
1 L , ds e k R? Q . (5.8) ok , < ? One technical problem is how to treat the sum over all wave vectors k. The exact enumeration can be carried out on a lattice, but it is useful to handle the sum over the wave vectors in its continuum version
< dBk . " (2p)B k
(5.9)
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The sum over the k-vectors appears very complicated, but is much simpler if we note that the density must be a real number. Thus we make use of o(x)31N(ok )H"o k \
(5.10)
in the following and realize that only a certain number of values will contribute to the sum, i.e., only k'0 are independent. In Eq. (5.10) the H corresponds to complex conjugate. 2. Transformation of variables: The second step is the most technical point. Here we have to transform the Hamiltonian from the chain variables R (s) to the collective variables ok . The ? computation is very involved and we are not going to write all details here, but concentrate on the main issues. Formally, we may write the transformation as H(+R(s),)PH(+ok ,)
(5.11)
and it becomes clear that it cannot be carried out exactly. Moreover, we will see later on that we can only go in the direction of the arrow in Eq. (5.11). Thus the transformation cannot be inverted. The "rst formal step is to use the identity for the partition function
GFFHFFI
Z" DR (s) dok d(ok !o( k ) e\@&R? Q . ? k ?
(5.12)
,
Here we have just inserted a 1 which is expressed as a complicated functional integral over density variables. The hated density corresponds to the de"nition of ok , i.e., to Eq. (5.8). This expression contains all monomer positions R (s), which we want to get rid of. A common way to proceed is to ? use a functional Fourier representation of the delta function d(o!o( ) in the form
, 1 dok d ok ! ds e kR? Q < k ?
L , o( k ok (5.13) ds e kR? Q , k k k ? where we have introduced an auxiliary "eld k for each value of the (formally) discrete wave vector. This auxiliary "eld parameterizes each of the terms in the product of delta functions in Eq. (5.13). 3. Put together and exchange integration: We are now in a position to compute the partition function. To do so, we put the parameterized delta functions into the partition function and interchange the order of the integration. Thus we write the partition function in the following order: , dok d k exp !i
\k
Z" dok d k exp !i k ok \ k k k
L ; DR (s) exp !bH(+R (s),)#i k o( k . ? ? k ?
(5.14)
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The main advantage of the expression, Eq. (5.14) is that the term in the second line depends only on the auxiliary "eld variables k after the integrals over the chain variables are carried out. The latter point cannot be done exactly as already mentioned earlier, but this is the main issue. Mathematically, the second term contains the Jacobian of the variable transformation and physically it corresponds to a Legendre transformation of the original partition function. Another essential issue is that the exponential in the "rst line can be written as
1 , o( k " k ds e k R? Q \
k
Z" dok exp(!H (ok )) . k
(5.16)
This de"nes itself, via the partition function of the desired Hamiltonian. Nevertheless, the strategy has now made clear that we wish to go one step further and study the structure of both `Hamiltoniansa H(ok , k ) and H (ok ) in more detail. The reason is the following. Below we will show that the auxiliary variable k has indeed a physical meaning, although it has been introduced just to parameterize the delta function during the change of the variables from R (s)Pok . ? To begin with let us apply the procedure to the problem we would like to study. The transformation of the many-chain Hamiltonian is the "rst step. The part of the interactions, i.e., the mutual self-avoidance between all chains is very simple. We write down again the starting point for the many-chain problem
RR (s) v , , 3 L , ? ds # ds ds d(R (s)!R (s )) bH(+R (s),)" ? @ ? Rs 2 2b ? ?@ GFFFFFHFFFFFI
k
(5.17)
Mk M\k
and realize that the Hamiltonian has always the general structure v H (ok )"H (ok )# ok o k , \ 2k
(5.18)
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as long as only two-body interactions are of importance. The main task in the following will be the determination of the contribution H . 4. Determination of H : To do this we have to consider the following integral for A : L , RR , 3 ? #i k ds e kR? Q . (5.19) A " DR (s) exp ! ? \ Rs 2b k ? ? ? Formally, A corresponds to the partition function of a set of n polymers in a random "eld
(R (s)). This is a well-posed problem which we had already dealt with. We see the real advantage ? of this procedure: The problem is now diagonal in all monomer indices, i.e., no couplings between di!erent monomers s, s and a, b appear. Next we carry out the R (s)-integration. We will then be ? left with an expression which depends only on the auxiliary variable. Of course, this is in general only possible by approximations. The most important one is the assumption of small #uctuations in variables ok and k . This turns out to be consistent with the assumption of dense systems. In fact, the larger the polymer density the smaller the #uctuations, and hence, the better the assumptions of small #uctuations. Intuitively it can be easily imagined. In dense melts the density #uctuations are much less pronounced, compared to a dilute solution, just because of the space "lling fraction of polymers. In low-concentration solutions the spatial #uctuations are given by the single-chain conformations, whereas in melts the scales of the chain sizes do not play a major role, and #uctuations of the density are less pronounced. The above assumption allows cumulant expansion of the integral. To simplify notation, we use the operator form of the collective density, i.e., Eq. (4.51). Eq. (5.19) takes then the more convenient form
L 3 , RR #i k o( k . A " DR (s) exp ! ? \ Rs 2b k ? To proceed we try to approximate Eq. (5.20) in the following form:
(5.20)
1
C(k , k , k ) k k k k $etc. . A Kexp ! k C kk k ! Y Y k k k k k \ \ \ 2 kk \ Y Indeed this can be done and it turns out that the lowest order function C is given by C kk "1o( k o( k 2 . Y Y The average here is de"ned as
(5.21)
(5.22)
L 3 , , RR ? , (5.23) 122 "N DR (2) exp ! ? Rs 2b ? ? where N is an appropriate normalization. At this point the advantage of the expansion becomes visible. All terms of the contributions to bH can be expressed as averages of the ideal chain measure, i.e., the noninteracting random walk. Thus the terms can be evaluated very simply. Obviously, the lowest order contribution corresponds to the structure factor of the ideal chain d(k!k ) . C S(k) kk , Y GHI
(5.24)
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Let us stick to the Gaussian order, which becomes reasonable for very strong polymer solutions and polymer melts, where the #uctuations are very small. In this limit everything can be calculated on a simple level. The e!ective Hamiltonians H[ok , k ] become of Gaussian order H[o, ]"v k
v 1 "ok "# S(k)" k " # i k o k #higher order in k . \ 2 2 k
(5.25)
The latter expression appears not as a real Hamiltonian, because it contains complex contributions at "rst sight. This is a somewhat confusing notation, but the problem can be resolved immediately, if the conjugated part is added.Then, as before the term containing the imaginary unit i must be replaced by
k ok P( k ok # k o k ) , \ \ \ then in summary will be well de"ned and the exponent will be real. Only then all averages are well de"ned. 5. Gaussian model: Now we carry out the "nal -integration at Gaussian level, that is, that all higher orders of the expansion in terms of the auxiliary "eld are neglected. Thus we start from the expression for the e!ective Hamiltonian
1 1 #v ok o k #O(o, o,2) H [ok , k ]" \ 2 k S(k)
(5.26)
and compute the corresponding averages. The most important one is the structure factor of the interacting system. The computation is trivial and starts from the de"nition of the structure factor as density}density correlation function, S(k)"1ok o k 2 . \ Again, we recall the de"nition of the averages in terms of the collective variables, i.e.,
(5.27)
1 122" dok ok o k exp(!H ) , (5.28) \ Z which yields simply the celebrated equation for the structure factor of the strong polymer solution 1/S(k)"1/S(k)#v .
(5.29)
The latter is often called the standard RPA results for polymer melts [10,11]. It is remarkable, that the RPA result works very well despite the crude approximations and Eq. (5.29) has been con"rmed by experiments with dense polymer solutions. In fact, we will come back later to this equation in the context of polymer blends and copolymers. At this moment we could stop, because we have achieved what we wanted. The transformation of the Hamiltonian to collective variables is complete, and the structure factor is computed, at least in lowest order. On the other hand, we have still some unsolved questions, even on this level of the approximations. The next questions we want to look at are: does the auxiliary "eld have a physical meaning? And what about the chain conformations in melts and concentrated solutions? We have already understood that the single-chain correlations become obviously destroyed as the polymer concentration increases. Can we then expect e!ects on the size of a labeled, or tagged chain?
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5.3. The statistics of tagged chains So far, we discussed the behavior of the dense polymer solution in terms of collective properties, such as the structure factor and the scattering properties of the system. One question which we have not addressed yet is the behavior of the chains in the melt. In the introductory remarks we had thought about the statistic of chains in the solution and the melt. We had guessed, that the size of the chain cannot be ruled by excluded volume forces alone, as in the case of isolated chains, since additional correlations from other chains play also an important role. This question is fortunately connected to another formal one. We had introduced an auxiliary "eld to represent the delta function, when we changed the variables. This was a very formal point but the legal question is: does the auxiliary "eld k have a physical meaning? The answer is of course, yes. To see this let us compute the correlator 1 k k 2 \ in the lowest Gaussian order. Again the calculation is trivial, but very instructive, and the result is given by 1 k k 2"v/(1#vS(k)),; . \
(5.30)
This result was given for the "rst time by Edwards [27] in a di!erent form and by a di!erent calculation. A simple dimensional analysis shows that ; must be a renormalized interaction. This can be seen by its units, it must be the same as the `barea excluded volume parameter v. It is instructive to bring Eq. (5.30) into the more appropriate form 1/;"1/v#S(k) .
(5.31)
This result yields that the original excluded volume interaction becomes renormalized by the presence of the other chains and, the renormalized interaction in the melt is smaller compared to the bare interaction. The interaction becomes, however, screened out. Edwards [27] used a di!erent route. He derived the same result for the potential by using collective variables and integrated over all chains (in terms of collective variables), except one. Then he was left with an e!ective chain in the melt. The form he derived was (5.32)
;(k)"v!v/(1/S(k)#v)
which agrees with the two equations given above. This form of the e!ective interaction is very instructive: the bare excluded volume v becomes reduced by a term stemming from the collective properties. We need now a physical picture and an interpretation for this point. To do so, we are going to have a look at e!ective ok -Hamiltonian, i.e., when we have integrated out the k -"elds in Eq. (5.25). By some simple manipulations we see that
1 v #v ok o k H [ok ]" \ 2 k S(k) v " 2 . & k
GHI
kb v 6 1 # #v ok o k (k#m)ok o k \ \ 12c 2 k 12c NC
GHI
(5.33)
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appropriate forms for the physical interpretation. The steps of the small computation involve "rst, a PadeH approximation for the structure factor, and second, to neglect the small 1/N-term. In the last line we introduced a screening length m"(b/12cv) .
(5.34)
Why did we call the length scale a screening length? To see this let us again compute the structure factor in terms of the length m. It is in this notation given by S(k)K(12c/vb)1/(k#m\)
(5.35)
and can be immediately transformed back in three-dimensional real space. The structure factor in real space is then S(r)"(3c/pb)(1/r) e\P K
(5.36)
which is of the Yukawa type. We see that m plays the role of a characteristic length scale. At scales below m the structure factor shows strong 1/r correlations. At larger scales r'm the correlations fall o! exponentially, i.e., they are screened out. The screening length depends strongly on the concentration. The screening length becomes smaller with increasing concentration. This means that the strong correlations are destroyed faster at smaller length scales. In other words, we can say that the interactions become screened by increasing concentration. On the other hand, we can already see that something is missing in the computation. For scales r(m the density}density correlation function is S(r)J1/r instead of J1/r as suggested by Eq. (3.10). This point must be clari"ed later. To con"rm this idea of screening interactions it is useful to calculate Hamiltonian H(R (s)) for 2 test chain in the strong solution. To do so, we represent the n chain Hamiltonian by the collective variables described above. Into this medium we can insert an additional chain of the same chain length N which we call R (s). Of course, this test chain interacts with itself and the medium. The 2 interaction term with the medium represents a coupling between the test chain and the medium. Thus, we may represent the Hamiltonian for the test chain and the medium simply by
3 , RR (s) v , , 2 # ds ds ds d(R (s)!R (s )) H(R (s))" 2 2 2 Rs 2 2b #v
, 1 1 ds o(R (s))# #v ok o k #O(ok ) . 2 \ 2 k S(k)
(5.37)
Here the "rst line of the equation represents the bare test chain Hamiltonian, the second term the coupling to the medium, i.e., the melt composed of the same chains, and the third term represents the polymer melt (medium) itself. Again, if higher orders in ok are neglected the collective variables can be integrated out and we are left with an e!ective Hamiltonian for the test chain. It is given by
3 , RR 1 , , 2 # H[R (s)]" ds ds ds ;(R (s)!R (s )) , 2 2 2 2b Rs 2
(5.38)
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where the potential ;(r) is called the `screened potentiala. It has the explicit form in k-space ;(k)"v k/(k#(1/m))"v/(1#S(k)v),1 k 2 . I \
(5.39)
The agreement with the correlation function of the k "elds is obvious and shows that the previous guess is correct. It is instructive to calculate its Fourier transform to real space, which represents itself as
;(R)"v
d(R)
(1/4pm) e\0 K/R !GFFHFFI GHI
.
(5.40)
The latter equation shows explicitly why the melt potential is called `screened potentiala. The bare excluded volume interaction v becomes screened out by the presence of the other chains. The range of interaction is mainly given by the screening length m. Moreover, at large values of r, or equivalently at values k"0 the interaction is zero. This yields the conclusions that the chains in dense polymer solution or in polymer melts will have somehow Gaussian statistics, i.e., their mean size R will obey the Gaussian scaling, RJ(N. This point can already be seen by considering a simple perturbation analysis for the size of the chain, where the perturbation is the e!ective potential ;(r). This calculation is straightforward and we only quote the result. The size of the chain is given by 1R2 Nb[1#(12/p)v/bm] .
(5.41)
Eq. (5.41) shows, that the chain behaves like a Gaussian with respect to the scaling with the chain length, but with a renormalized prefactor. The prefactor contains the screening length m and is thus concentration dependent. It is also easy to show that all higher order terms in the perturbation analysis are of less importance. The next step is to develop a physical picture of this formal result. Obviously, the screening can be understood by introducing a blob size which comes from the screening length m. From Eq. (5.41) we see explicitly that the exponent in melts is l" and the chain is no longer self-avoiding. Higher order perturbations are of course smaller and not N-dependent as in the SAW case. This allows to con"rm the `blob picturea that we already speculated about. Indeed the chain can be replaced by an e!ective chain of blobs. These blobs have a diameter of the order of m. Inside these blobs the SAW correlations dominate the physical behavior, i.e., the chain is expected to behave SAW-like. Outside the blobs, the many-chain correlations destroy the SAW character, and we have just shown in Eq. (5.41) that the chain becomes Gaussian (see Fig. 6). We have, however, made a mistake somewhere. It becomes obvious, if we have a look at Eq. (5.36). We have just required that inside the blob we should have excluded volume correlations, thus we must expect a structure factor S(r)Jr\ in three dimensions, since in reciprocal space it scales as S(k)Jk\ in d"3. Eq. (5.36) however shows only a scaling with the inverse of the distance for scales r(m. The mistake becomes immediately clear, since we had worked out these results only in Gaussian approximation and we neglected all higher order terms in k or ok . On the other hand, it is possible to recover the correct scaling by higher order expansions together with the renormalization theory [6]. The technical details of these computations are beyond the scope of
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Fig. 6. The blob picture of chains in concentrated polymer solutions. The chain can be divided into blobs of size m. Inside the blobs the chain behaves unperturbed. Outside the blobs the SAW correlations are destroyed. The chain becomes Gaussian.
the present article, but we will come back to this problem by the scaling theory. Shortly, we will put forward physical arguments, which yield the desired results, without doing too much technical work. This is the great advantage of scaling theory: simple physical pictures together with physical intuition yields fast "rst-order results. Before progressing in this direction let us stay with the Gaussian approximation and compute the free energy of the solution bF"log Z. The technical part of the calculation is skipped here, because it corresponds exactly to the Gaussian model for phase transitions in statistical physics [13,30] and is quite simple. The main issue of this calculation is that it will provide the osmotic pressure of the polymer solution, simply by the derivative P"!RF/Rv. This calculation is indeed straightforward and yields
(5.42) P"k ¹ C/N#vc ((3/p)(Cv) /6 . GFHFI !GFFHFFI Again we see that obviously something must have gone wrong, because at small concentrations C the osmotic pressure becomes negative. This mistake also can be drawn back to the problem of staying with the Gaussian approximation. The problem appears in the osmotic pressure at small concentrations C, which correspond to large screening lengths. Then the SAW correlations become most important at longer length scales. In e!ect, below the overlap concentration the chain size is determined by the SAW behavior, then the Gaussian approximation must break down, because the #uctuations dominate the physics. 5.4. Scaling in semi-dilute polymer solutions Let us now come back to a scaling argument for this behavior. The only input we need is the information from above, i.e., the calculations and the results of the Gaussian approximation, the obvious reason why it breaks down and physical intuition. Let us therefore use the following physical arguments [11]: 1. At large concentrations C the osmotic pressure P must not depend on N: The physical origin of this assumption is quite natural. In a snapshot of a strong polymer solution where the chains overlap
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3.
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quite severely the chain length itself and the degree of polymerization N cannot be of any importance. The only size, or length scale which rules the physical picture is the correlation length m, i.e., roughly the mesh size of the network given by the snapshot. Choose therefore an ansatz P"(C/N)k ¹f (C/CH): This scaling form of the osmotic pressure is given by two things. First, the prefactor C/N stems from the classical dependence of P at very small concentration, i.e., the van't Ho! law. Since the osmotic pressure must be concentration dependent we assume that it scales with a function f (C/CH). The latter form comes from the idea that only one length is determining the physical behavior and one concentration, which is the overlap concentration CH. In fact, we must use the fact that the formula for the osmotic pressure is able to interpolate between the small and large concentration regimes. The scaling function f (C) is unknown so far. Remember that CHJN\BJ: The overlap concentration is known and we use the precise scaling. Here we see that the number of monomers comes into play. Since CH depends on the chain length we can learn something about the unknown scaling function f. Guess a scaling form for f: This step is now to use only what is known so far, to make a simple guess for the leading behavior of f. The simplest assumption is that f scales as a power law. Indeed this must be the case (at least to leading order), since somehow we must get rid of the chain length dependence. Thus we require PK(C/N)k ¹(C/CH)V"(C/N)k ¹CHNVBJ\&N. Result: Together with the requirements and the physical picture, this yields immediately the scaling of the osmotic pressure to PJ(Ck ¹/N)(C/CH) BJ\ which is in three dimension d"3,lK the des Cloizeaux result PJC only by the requirement of the independence of P from the chain length at larger concentration compared to the overlap-CH.
We do not really have to mention that this law has been con"rmed by experiments and renormalization calculations. The results of the Gaussian model are thus not su$cient in the values of the exponents. Nevertheless, it was useful to take up these considerations before. The model sets up the physical idea of strong polymer solutions. Now, we may get the idea, that the screening length m is also not correct from its dependence on the concentration. Indeed here also the mean "eld computation of m is not su$cient, recall that we had truncated all series and cumulant expansion at the Gaussian level. To make progress we again put forward a scaling argument for the screening length and follow the same ideas. We assume therefore a dependence
m"NJbf (C/CH)"
NJb
for CP0 ,
Nb for C
(5.43)
To put this forward we try out the ansatz f (C/CH) (C/CH)W and we use again the previous result cH"N\BJ. Then we "nd y"!l/(ld!1)
(5.44)
and the screening length mJCJ JB\JC .
(5.45)
Here the concentration dependence is di!erent from the corresponding one in the mean "eld theory mJC\. Again we see that the answer is di!erent from the one gained by the Gaussian model.
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As one of the most important consequences we "nd for polymer melts and large C polymer solutions in three dimensions d"3 that the chains are almost ideal, i.e., their size scales as a random walk. To understand this point better and to gain more physical insight, we can again put forward a physical argument. First let us remember, that the upper critical dimension for the SAW chain was d "4. For melts we must therefore expect a di!erent upper critical dimension. We have now all means to do this quite e!ectively. Let us recall the e!ective Hamiltonian for the chain in the melt.
, , dk 3 , RR 2 # ds ds ds ;(k) exp(ik(R (s))!R (s )) , bH[R ]" 2 2 2 Rs (2p) 2b
(5.46)
where ;(k) is the screened potential. Instead of doing elaborate computations with the Hamiltonian of the melt chain let us try a simple Flory estimate, which can always be of use for getting some "rst insight into the physics. The above Hamiltonian corresponds to a free energy of the form, bF+R/bN# [N/CN/(1/CN#N)]N/RB
GFFFFHFFFFI .
(5.47)
0
k
Let us have a closer look at the new interaction potential. ; +(1/CN)N/RB&O(N\B ) .
(5.48)
This e!ective potential is now of the order of N\B and suggests an upper critical dimension d of d +2
(5.49)
which is not quite correct as has been shown by conformal invariance. Indeed the upper critical dimension is slightly larger than two [36]. Nevertheless, this simple argument allows us to understand de Gennes headline in his book [11] `molten chains are ideala in another way.
6. Correlations in polymer blends and block copolymer melts 6.1. Some general remarks In this Section we will extend what we have learned about dense polymer systems and polymer melts. Here we apply these ideas to polymer blend, i.e., the mixture of two di!erent types of polymers and to block copolymers. In these systems we expect phase transitions, because the di!erent components tend to demix at certain temperatures. We have seen that we can achieve the random phase approximation by using Edward's method. We will also show that we can make statements about the collective properties and the single-chain statistics close to any phase transition. We will use the same method that we had proposed when we studied the problem of strong polymer solutions.
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As in the case of polymer melts we study from the Edwards formulation of the random-phase approximation (RPA), but take into account that we have now two species. To begin with, we start by looking at the collective properties and introduce densities of the well-known form o " , ds exp+ikR(s), but now for each species A and B, say. Thus the collective densities can be ? I got. If we follow the steps from Section 5 we can already guess as a natural generalization of Eq. (4.51) the structure factor of the polymer blend [11,37] 1/S(k)"1/S (k)#1/S (k)!2s , $
(6.1)
where we have two components A and B and a natural interaction parameter will appear. This is the so-called Flory s-parameter which is de"ned as 2s "2< !(< #< ), which describes the $ energetic situation of the blend. The appearance of the unperturbed structure factors play the same role as in the case of strong solutions and are therefore nothing else but S(k)"F(kR), a function E of the unperturbed chain dimension. Thus, the RPA does not account for chain dimensions of a tagged chain in the blend or in the copolymer melt, at least at "rst sight. Thus one problem we have to solve is the question, how is the size of the chain behaving close to an eventual phase separation. In the case of the strong solution we have seen that it depends on the correlation length (blob size). Can we expect here that it depends, for example, on the distance from the critical point?. Obviously, the critical point can be de"ned as such, when the structure factor S(k) diverges, since this corresponds to large concentration #uctuations, and corresponds further to the critical opalescence. These questions can be solved within the RPA by studying the e!ective potentials and by employing the common formalism introduced earlier. To see how all this emerges let us follow the main steps again, without detailed computations.
6.2. The Edwards Hamiltonian formulation To begin with we start from the Edwards Hamiltonian for n chains of the di!erent types. Again we assume for simplicity monodisperse chains, i.e., all chains are composed of N steps. The Hamiltonian for the two-component blend system is then
1 , , ds ds < (R (s)!R (s )) H"H # 2
, , 1 , , #H # ds ds < (R (s)!R (s ))# ds ds < (R (s)!R (s )) . 2
(6.2)
Here we have abbreviated the Gaussian}Hamiltonian as
3 , RR (s) N HN " 2b Rs for all species as usual. To describe the blend we have to specify all interactions V by a matrix, < , where p, p "A, B. These are the mutual self-avoiding interactions between all chains. This N NY
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Hamiltonian formulation enables calculation of properties of solution of the blend consisting totally of n polymers via the partition function
L Z" DR (s) e\@&+R? Q, ? ?
(6.3)
as usual. From now on we follow the scheme presented in Section 4. First, we carry out the transformation to collective coordinates in the scheme H(+R (s), R (s ),)PH(+o k , ok ,) .
(6.4)
For a "rst trial we work to the Gaussian approximation, i.e., only to order "o ". According to our I earlier results we get H(+o k , o k ,)" \ k
1 1 #< o #< k ok # S S (k)
#higher orders ooo#oooo#etc .
ok o
\k
#2< o k o k \
(6.5)
The calculation of scattering functions is trivial. Let us therefore give as an example the structure factor S(k) of an incompressible binary blend. Incompressibility means that all #uctuations in the o-variables add up to zero, i.e.,
o k #ok "
0
for kO0 ,
o
for k"0 ,
(6.6)
where o is the overall density of the sample. Thus we arrive immediately at the structure factor 1/S(k)"1/S (k)#1/S (k)#< #< !2< , GFFHFFI
(6.7)
\Q$ an equation that we could almost guess. This is de Gennes celebrated result, which we recovered here. On the other hand, we can get the same result here by another possibility. Therefore, let us forget about incompressibility "rst and rewrite the Hamiltonian H(+o k , ok ,), see Eq. (6.5) in matrix form (up to the Gaussian contribution). Thus, we write it in terms of 1 bH" qk +(S(k))\#V,q2k . 2k
(6.8)
Here we have introduced a vector notation, i.e., qk "(o k , ok ), and the bare structure factor matrix
S(k)"
S 0
0 S
(6.9)
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and the matrix of the bare interactions
< < . (6.10) < < In this compact notation the generalization to block copolymers is now simple. The only di!erence is that in the bare structure factor matrix additional AB correlations appear, since the chains are bound together. Thus we can immediately write for the bare structure factor matrix V(k)"
S S , (6.11) S S whereas the interaction matrix remains unaltered. This can be proven rigorously with the Edwards Hamiltonian formulation as introduced above, which is skipped here. In the incompressible limit, which can be achieved either by condition (6.6) or by the limit < "<#e together with
S #S #2S 1 !2s o o . H" $ k k det S 2k
(6.12)
At this stage, we recover Leibler's result in Gaussian approximation for block copolymers [38]. 6.3. Field theoretical formulation of the problem Let us brie#y mention the more general theory which we had developed in Section 5. Again, the main aim is to treat the collective properties and single-chain properties within the same formalism? Of course, the answer is yes. This is nothing but a simple generalizations of the results from the concentrated solutions. The procedure follows the scheme that we had employed before. 1. Write down partition function
1 DR (s) exp ! HN ! <(R (s)!R (s )) , (6.13) N 2 N O N N O where p labels the di!erent chemical components, and HN has the same meaning as before. 2. Use collective coordinates oNk " e kRN Q and rewrite the partition function as Z"
1 Z" DR (s)DoNk d oNk ! e kRN Q exp ! HN ! oNk < oO k . N 2 N O \ k N k N N O N
(6.14)
3. Parameterize the d-function for each species by a "eld tNk
, 3 RR N #itN(RN(s)) Z" DR (s)DoMk DtNk exp ! N 2b Rs N 1 ! oNk < oO k #i tNk oN k . NO \ \ 2 k N O N k
(6.15)
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Indeed the last equation contains all the information we need (at least on the Gaussian level) for the moment. Now, we can use this for studying the collective properties and the tagged chain properties. We start from the generalized Edwards transformation chains in (complex) random "elds and use the Gaussian approximation of Z
1 1 Z" DwDq exp ! wk S(k)w k ! qk Vq k #i qk w!k , \ \ 2k 2k k
(6.16)
where S(k) is the matrix of the unperturbed structure factors. This reformulation of the problem contains "rst all structures factors and we recover the RPA equation for blends and copolymers, i.e., (6.17) S (k)"1oNk oO k 2"((S)\#V )\ . \ NO NO Furthermore, it contains all e!ective potentials by computing the correlations of the auxiliary "elds w in the form ; (k)"1tNk tO k 2"(S#<\)\ . \ NO NO If we turn to the incompressible limit by setting
(6.18)
< "< #e (6.19) NO NO and letting < PR we recover all that we had found before. So far, the method works and we are now going to employ it to answer several question concerning blends and copolymers. The main issue we want to raise is the behavior of the tagged chain in the blend, or block copolymer melt close to the phase transition. In the following, we will stay for simplicity in the mean "eld limit. 6.4. Static properties of tagged chains So far we have only recovered well-known results from the RPA. The x#next step is now to investigate consequences on tagged chains. The mathematical procedure is now clear. We have to follow the routes introduced in the case of strong solutions and introduce the appropriate auxiliary "elds. This procedure leads to e!ective Hamiltonian of a tagged chain (for example we study here an A chain. This choice is without loss of generality). The e!ective Hamiltonian has the form , , 3 , RR # ds ds ds ;(R (s)!r (s)) , (6.20) H" Rs 2b where the e!ective potential is given by the corresponding matrix element of the interaction matrix equation (6.18), i.e., p"q"A:
;k " dr exp+ikr,;(r)
(6.21)
and the Fourier transform ;k "(1/S (k))(1/S (k)!2s )/(1/S (k)#1/S (k)!2s ) . $ $
(6.22)
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Fig. 7. The radius of gyration as a function of temperature in an AB-blend as obtained in simulations. Fig. 8. The radius of gyration as a function of temperature in an AB-blend. The chain size decreases on approaching the critical temperature. The symbols are light scattering experiments, whereas the line is given by a "t of Eq. (6.23). The mean "eld approximation breaks down at a certain temperature, which is marked in the "gure. This special temperature is sometimes called Ginzburg-temperature.
The most important observation is: if s increases ; changes sign before phase separation takes $ I place. Thus at certain values of the Flory interaction parameter the e!ective potential becomes negative, hence attractive. The important temperature ranges are the following: 0(x (1/S repulsive, s "(1/S (k));k "0 , $ $ 1/S (s (1/S #1/S attractive . $ Since the potential changes sign we can expect an e!ect on the size of tagged polymer. Let us "rst look at simulations of blends (see [39], for an exhaustive review). The "rst observation is that the tagged chain of A-molecules shrinks. Let us therefore compute the size of a single chain of type A in an AB-blend by perturbation theory of "rst order. The result is then
< 1! 2s N < #c $ , (6.23) N (s !s )
$ where a, c are numerical prefactors of less important interest. This is an interesting result and we can look at it in the following way. The "rst term corresponds to the unperturbed size of the chain. The second term is the classical Edward's screening term, which we had already derived in the previous chapter. The third term is the `blenda e!ect. Of course it depends on the (mean "eld) correlation length m"(s !s )\ . Thus, we see that before the phase separation $ takes place the end-to-end distance (or the size of the chains), 1R2, decreases with increasing interaction which is con"rmed by numerical simulation [39], see Fig. 7 and by experiments [40], see Figs. 8 and 9. We can employ a similar calculation for block copolymers. The formulae are complicated and lengthy [41] and we just mention the main results. The copolymer chain becomes stretched due to the strong repulsion between the A and B blocks. The individual arms become contracted for the same reasons as given in blends. This has also been seen in simulations from us [42]. In addition, 1R2"N b 1#a
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Fig. 9. When the e!ective propagator for the blend at the critical temperature is used, i.e., S(k)"Jk\\E, the curves can be "tted also close to the phase separation. Fig. 10. The radius of gyration of a copolymer chain in a melt of block copolymers. The overall size of the chain increases. The di!erent cases belong to di!erent situations, see Ref. [44] for details.
we have shown that these create #uctuation-induced orientational correlations [43], which we do not treat here in detail. The simulations have given the following result, reproduced here in Fig. 10.
7. Polymers of larger connectivity: branched polymers and polymeric fractals 7.1. Preliminary remarks So far, we have discussed the behavior of linear polymer chains. The origin of linear chains comes naturally from linear polymerization, corresponding from a chemical reaction, which connects bifunctional (linear) monomers at their end which form a linear chain. We can associate a (internal) dimension of the linear chains by stretching it out completely and by coarse graining. Then we will have a one-dimensional object, i.e., the chain is of one-dimensional connectivity D"1. This new dimension is connected with the connectivity and is called spectral dimension. If higher functional monomers are used for the polymerization process, branched polymers are generated, see Fig. 11 for a visualization. The challenging question is, how to describe them by methods similar to what we used so far. Let us therefore try a simple generalization for the connectivity part, where we make use of the idea of higher dimensional connectivity. Thus we try the more general Edwards Hamiltonian
1 d"x( x R(x)) . bH" 2b
(7.1)
This is at "rst sight a strange object, because it contains the internal and the spatial dimensions. The variables x are the internal variables of dimensionality dim x"D. The vector in bold face characters R describes the external variables with the dimension dim R"d, i.e., the dimension of embedding space. Naturally, we must require D4d. Let us see in the following, if this analytical
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Fig. 11. Linear versus branched polymerization.
Fig. 12. Di!erent examples for the spectral dimension.
continuation makes sense. For simple visualization we show some examples for the analytic continuation for the spectral dimension (see Fig. 12). 7.2. D-dimensionally connected polymers in good solvent The arguments from above can be made more familiar if a common generalization of the Edwards Hamiltonian for linear polymers [10] is introduced for polymeric fractals [45] and D-dimensional manifolds in the following standard way:
H" d"x( x R)!v d"x d"x dB(R(x)!R(x )) .
(7.2)
D is the spectral dimension as noted earlier. Well-known special examples are D"1, that are linear polymers and D"2 that are random tethered surfaces. The analytic continuation of D to
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noninteger values, i.e., 1(D(2 correspond to any polymeric fractal of arbitrary connectivity. x is a D-dimensional vector of the manifold embedded in d space dimensions described by the vector R, and N is the linear dimension in one arbitrary direction. The "rst term is the Gaussian connectivity and the second term the usual excluded volume interaction. In this paper only objects with D(2 are considered for convenience. Hamiltonian (7.2) does not make sense for fractional values of D, without de"ning fractional di!erentials and integrals properly. In the scaling limit it can be used without problems. We restrict ourselves to this latter case. Although we do not use the full Hamiltonian in this paper, its introduction is helpful to derive its scaling properties [21,46], especially for readers who are not familiar with the notation used below. The standard estimation uses the replacement R(x)PR and "x"PN. All integrations yield trivially a factor of N". It is remembered again that N is not the entire number of monomers but only the number of monomers in one given direction in spectral vector space. Then Hamiltonians such as that given by Eq. (7.2) can be transformed easily into a Flory free energy by dimensional analysis [46] F"R/bN\"#bN"/RB .
(7.3)
By the substitution M"N" it is easy to show that Eq. (7.3) transforms into the well-known Flory form of polymeric fractals [45,47,48] with the Gaussian fractal dimension d "2D/(2!D), which recovers cases of linear polymers (D"1), randomly branched polymers (D"), and tethered membranes (D"2). The standard result is obtained by minimization of Eq. (7.3), that yields the usual Flory exponent for the size of the polymer in the swollen (crumbled) state which is found to be [49] l"(D#2)/(d#2) .
(7.4)
To avoid misunderstanding at this early stage it has to be mentioned that this exponent accounts for the linear (chemical) size N and not for the total mass M. By simple arguments it is easily found that the corresponding fractal dimension is given by d "D/l. In the following, we restrict ourselves to the three dimensional (d"3) embedding space. 7.3. D-dimensionally connected polymers between two parallel plates in good solvent As a "rst example the case of a D-dimensional polymeric manifold in good solvent between two plates is studied. The situation is depicted in Fig. 13. The "rst attempt to solve the problem is to use Flory's theory. This is very simple and the Flory free energy (see Eq. (7.3)) for this problem is given by F"R /bN\"#bN"/(HR ). H is the distance between the two parallel plates and , , R measures the size of the polymer parallel to the plates. Minimizing the free energy with respect , to the size R yields the desired result , R "b(b/H) N>" . (7.5) , Note that the N dependence in Eq. (7.5) corresponds to a two-dimensional polymer with spectral dimension D. For D"1 (and N"M) the correct exponent l" is recovered [11,6]. For polymer sheets, such as #exibly polymerized membranes (D"2) the reasonable exponent l"1 is obtained. The latter corresponds to the case where the tethered membrane is #at between two very narrow
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Fig. 13. A Linear polymer chain berween two parallel plates. The polymer chain can immediately be replaced by a chain of any connectivity. For simplicity only the case D"1 is shown.
parallel plates with undulation #uctuations of the order of the distance between the two parallel plates. Although reasonable limits are predicted by this Flory model it is di$cult to be sure about the validity of this result if it is not derived by a di!erent method, such as, e.g. the scaling theory. This will be provided in the next paragraph. The scaling analysis can be done in close analogy to the case of linear chains. The radius for the chain between two plates can be written as R "R f (R /H) , (7.6) , $ $ where R is again the extension of the chain parallel to the plate, and H the distance between the , two plates. R is the geometrically unconstrained Flory radius in good solvent and is de"ned by the $ Flory exponent given above. The scaling function f (x) has two limits. The "rst is when x"R /H D tends to zero, i.e. when the plates are separated very far from each other, f (x)P1. The opposite limit, when the parallel plates are placed very narrow the two-dimensional con"guration appears, which determines in the usual manner the exponent x of the scaling function. This corresponds to the two-dimensional con"guration of the D-polymer which is calculated by the Flory model above. The usual argumentation provides the same answer as derived in Eq. (7.5). A more appropriate form is given by R "H(b/H) N>" . (7.7) , It is tempting to generalize de Gennes' blob picture to such D-dimensional polymers. This has not been done yet, although it is simple. To do this, assume that the D-polymer between two plates behaves as a fractal made out of blobs of size H. Thus, it is reasonable to assume that the size of the object is given by R "Hn>" " where n is the total number of blobs. Note that the fractal , dimension of the e!ectively two-dimensional object d "(2#D)/4D has been used to account for the mass in the fractal. The number of blobs n can be calculated by determination of the mass m (number of monomers) inside the blob. Inside the blob the branched structure shows good solvent behavior that yields: m"(H/b)" >". Therefore, the number of such blobs is given by n"M/m, where M is the total mass M"N" as before. Following de Gennes' argumentation the result R "H(b/H) M\" " (7.8) , is obtained, which is identical to those obtained by the Flory theory and the scaling approach. This almost trivial example shows that the blob picture can be used to construct the same physically
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reasonable results for branched chains similar to the case of linear chains. The point is to use the information also from the scaling in terms of the number of monomers in the chemical path N additionally to the mass scaling in the blobs. The cross check of all results is to consider the "lling fraction f"bN"/(HR) [11]. It turns out later that this quantity is useful in more respects. For the polymeric manifold (or polymeric fractal) in the slap f is given by f"(b/H) n"\ . This result makes sense physically, and for D"1 the classical polymer behavior is recovered [11,50]. For D'2 the "lling fraction becomes unphysically large. Trivially, a polymeric membrane can be pressed completely between two very narrow plates, i.e., H"O(b). In this special case the "lling fraction becomes independent of the molecular weight, as it is intuitively clear (lower critical dimension). The example of the arbitrarily self-similarly branched polymer between two plates has been explicitly discussed in more detail to demonstrate how the blob picture and the scaling arguments can be generalized to branched polymers or arbitrary higher connected polymeric objects. 7.4. D-dimensionally connected polymers in a cylindrical pore (good solvent) Severe problems occur when such self-similarly branched, nonlinear objects are forced into cylindrical pores when in other words the space available for the polymer is further restricted, see Fig. 14. The simple dimensional analysis from above has to be modi"ed in the usual sense, that the d-dimensional Dirac function becomes anisotropic and the lateral dimensions are given by the pore size. Thus, we estimate the relevant excluded volume from Eqs. (7.2), (7.3) to be d(R(x)!R(x ))J(1/H)1/R where H is now the pore diameter and R is again the chain size , , parallel to the pore. Now begin with the consideration of the Flory free energy for the manifold in the pore F"R /N\"#vN"/(HR ) . , , Minimization yields immediately the result for the parallel exponent
(7.9)
l "D"2/3 (7.10) , which agrees for D"1 with the standard exponent [11], i.e., l "1 corresponding to the , stretching of the linear chain along the pore. Obviously, this exponent is ill-de"ned whenever D'1, that is whenever the polymers are of higher connectivity as the linear ones. The linear dimension through the fractal or the membrane must not be larger than N itself that corresponds to the entirely stretched limit. It is now easy to understand, that scaling and a blob model as it was presented for the slap cannot work anymore for the case of the cylindrical pore (Fig. 15). For example, if a scaling argument is considered which assumes a good solvent behavior for large cylindrical pores and a linear (entirely stretched) branched polymer for narrow pores, contradictions will show up such that the "lling fraction is unphysically large for all values for the spectral dimension D'1. One way out of this di$culty is the postulation of a minimum pore size through which the branched polymer is able to pass through. Thus the minimum pore size can be found
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Fig. 14. A linear chain in a cylindrical pore. The parallel size of the chain is stretched. Inside the blob the chain remains unperturbed by the pore size. Fig. 15. A branched structure in a cylindrical pore. The connectivity is another limitation and in contrast to the linear chain the pore diameter cannot be of the order of the monomer size.
such that for the parallel direction the value R JN is taken. It is given by , H JN"\ ,M"\ " . (7.11)
This result makes sense physically. The minimum pore size for linear polymers is independent of the molecular weight. Thus, linear polymers (D"1) "nd their way even through a very small pore, if H is of the order of the Kuhn length, but with an extremely low probability and in a very long
time. In branched polymers as D'1 another limitation is important: the connectivity. The larger the connectivity, the larger is the minimum pore size. It should therefore be possible to construct a porous medium that is able to separate a mixture of branched and linear molecules on the basis of their connectivity. This can be done by an appropriate minimum pore size through which linear polymers can pass, whereas branched polymers cannot. For the construction of such a chromatographer dynamical aspects have to be taken into account, since the other selection constraint is the "nite time to pass through a pore, as mentioned in Eq. (7.11). Such aspects have been studied later on in detail by Gay [51]. The essential point to be made is that the minimum pore size is entirely de"ned by the spectral dimension and the molecular weight. Therefore, the pore is able to select objects with respect to their connectivity, i.e., their spectral dimension. This possibility has been called spectral chromatography in the previous paper } to distinguish from classical chromatography, which selects only with respect to molecular weight. When, for example, a membrane is put into a small enough pore it cannot #atten out in the remaining space but has either to crumble in a speci"c direction, if the pore is large enough, or to saturate (collapse) in smaller pores in the lateral direction. Whether one "nds the crumbled or collapsed case is determined by the value of the minimum pore size H .
It could be guessed that the geometric restriction of the pore becomes less important if a theta (H) solvent is used for transporting the polymer through the pore [52]. For a linear polymer the H-exponent is l" that is less than the swollen exponent [11]. Thus the total size of the chain in H-solvent is smaller, compared to the good solvent case. For polymeric manifolds of larger connectivity this is also the case and it could be concluded that the limitation de"ned by the pore is less severe. This is indeed not the case. It can be immediately estimated by using a Flory argument of a similar type as above but including the third viral coe$cient, whereas the excluded volume is e!ectively zero [11]. It can be shown that the minimum pore size is given by the same value
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H JN"\ as above. We will not elaborate in more detail on this since it is well known that
the Flory argumentation yields very bad values for the exponents l in two dimensions, but exact values at the lower critical dimension for three-body interactions d"1 and the upper critical dimension d"3 [6]. Little is known for the validity of Flory values of branched structures and manifolds under H-conditions, and we do not discuss it further. The value for H under
H-conditions is, however, the indication that the minimum pore size is determined by the connectivity only and not by thermodynamic conditions. It has been mentioned earlier that an interesting check about the consistency of the results is to calculate the internal concentration, or "lling factor, f"N"/HR . The "lling fraction becomes , independent of the molecular weight M"N" of the manifold at the point when the pore takes its minimum size. This indicated that the pore size is still a natural scale for the D-dimensional polymers. 7.5. Melts of fractals in restricted geometries The case of melts of fractals and branched polymers is also of interest. The case of melts of linear chains in small cylindrical pores has been studied by Brochard and de Gennes [53]. Again, this case of D"1-linear polymer melts in restricted geometries is simple. The generalization to branched polymers and polymers with higher connectivity is not as trivial and simple as it might seem at "rst sight. With our previous considerations for polymer melts consisting of linear chains it is easy to see [48] that melts of branched polymers with a spectral dimension D'1 must be divided into two classes. Whenever the connectivity is larger than a threshold value D D'D "2d/(d#2) the fractals do not interpenetrate in melts as do linear chains. In such systems the connectivity and space "lling are too high. Instead of interpenetration the polymers saturate and form separated balls of their natural density, i.e., RJN" B. The short argument follows the same argumentation as before: take a melt of branched polymers and integrate the collective variables out to be left with one test fractal in the melt. The corresponding Hamiltonian is then given by
v 1 d"x( x R(x))# d"x d"x exp+ik(R(x)!R (x )), , bH[R (x)]" 2b k 1#vS(k)
(7.12)
where the second term is the well-known e!ective potential. The Flory estimate just uses again the zero wave vector term and counts the structure factor as S&CN"(1#0(k)), where C is again the concentration. This suggests the melt value of the excluded volume parameter to be of the order vJN\" [12]. The Flory free energy for melts is then F"R /N\"#bN"/RB ,
(7.13)
which leads formally to the D-independent melt exponent l"2/(d#2). This result is of no physical sense since it yields a fractal dimension larger than the space dimension, i.e., d "D(d#2)/2, which is larger than the space dimension d whenever the spectral dimension D is larger than D "2d/(2#d). For smaller connectivities the situation is very di!erent and the result is that the polymers take their unperturbed size, that is given by RJN\B . This happens since for all cases the system is above the upper critical dimension. The upper critical dimension in the melt is
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given by d "2D/(2!D)
(7.14)
which is of factor 2 smaller than in the case of good solvent. The latter point is due to excluded volume screening. The two di!erent cases are now discussed in detail. For simplicity only the case of the cylindrical pore is considered. The case of the slap is very easy and can be done in a straightforward manner. The following subsections consider the two di!erent cases, when the spectral dimension is larger or smaller compared to the critical value D . 7.5.1. D'D In this regime the melt of fractals is saturated. That means that, unlike linear polymers the polymers of higher connectivity do not interpenetrate each other since their connectivity is so large that this cannot happen. One limitation for such polymers in the pore is that they form a row of balls each of size R "bN" . The "lling fraction for this situation is easily calculated and it is given by f"bN"/HR "(b/H)N"\" .
(7.15)
The "lling fraction cannot be larger than one and, therefore, the pore diameter is limited to the value HH"bN" , which is the saturation radius of the polymer itself. This situation is given in Fig. 16. When the pore is smaller each of the individual saturated fractals can become compressed further as their saturated radius of gyration can be elongated to form ellipses. Again, the maximum parallel radius of the polymer is given by R "bN, and for this case the "lling fraction is given by f"bN"/HR "(b/H)N"\
(7.16)
and the limiting pore size is given by H JN"\ which is naturally identical to that of good
solutions and theta-solutions. Again, we "nd that the pore size does not depend on the thermodynamic state of the manifold. H is only determined by the molecular weight and the connectivity.
To pass through this minimum pore size each of the saturated fractals has to be compressed by a factor of j"N\"\ , which is for membranes j"N and for randomly branched polymers j"N as both cases belong to the class D'D . A visualization of this situation is schematically shown in Fig. 17. 7.5.2. D(D In this case the physical picture is very similar to the case of linear polymers. As long as the connectivity is less than the critical D " in three dimensions, the fractal takes its ideal Gaussian dimensions, i.e. R "bN\" . This is because the upper critical dimension for the melt is always less than the dimension itself, i.e. in the present case d"3. Therefore excluded volume forces are screened completely and the manifold behaves ideally. In addition, this means that the manifolds can interpenetrate each other as the connectivity is very small. There exist again two basic length scales for the pore size. The "rst one is given by the limitation where the manifolds pass through
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Fig. 16. A melt of highly connected polymers or manifolds in a small cylindrical pore. The connectivity is larger than D". The manifolds cannot penetrate each other. They form separated balls. The pore diameter is identical to the radius of gyration in the saturated case. Fig. 17. Same as above but with smaller pore diameter. The individual fractals can be further compressed until they are stretched to their maximum radius of gyration R"JN.
without changing their shape. This can be read o! again from the "lling fraction. f"bN"/HR "(b/H)N"\ .
(7.17)
The limiting value for the pore size is given by D "bN\" . For larger pore size the polymers in the melt are able to pass through without changing their shape. Smaller pore sizes are possible, when the manifolds stretch out. The maximum stretching is given by R "bN, and the limiting
"lling fraction predicts D "aN"\ , which is again identical to the same limiting pore size
as in all other cases. 7.6. Once more the diwerences In this paper a scaling theory for arbitrarily connected polymeric manifold in a simple restricted form has been presented. The main point is that larger connectivities give rise to severe restrictions for the conformation and the behavior of such molecules in restricted geometries such as parallel plates or pores. The radius of gyration can be calculated for the case of two parallel plates using Flory theory, blob model, and scaling theory. The blob model has been constructed using arbitrary spectral and fractal dimensions. This has not been done yet. To our knowledge, the blob model has only been used for regularly branched molecules, such as, for example, star branched polymers [54]. These treatments can break down when the D-dimensional manifold is studied in a small cylindrical pore. Indeed, the study of the behavior of the manifold in the cylindrical pore yields the most serious restriction. An observation that is in contrast to linear polymers. Manifolds with larger spectral dimension D"1 do not pass through small pores without problems. A minimum pore size has to be assumed to "nd consistent results. Another main result is that this minimum pore size does not depend on the thermodynamic conditions of the manifold, such as good solvent, theta solvent, or melt conditions. The minimum pore size is determined by the connectivity, i.e. the spectral dimension only.
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8. Polymers in random potentials: replica analysis In this part of the paper we will discuss the problem of frozen disorder and polymers. This will bring us a number of new methods for treating these systems. We will mainly review here the replica method and a variational technique. To do so we will study two di!erent problems, i.e., the polymer chain in random potential (see Fig. 18). There we will "nd localization properties of the polymers in disordered potentials (or random media) under di!erent variational methods. The second main problem we study is a dense system of statistical copolymers, their phase separation, their freezing, and their collective behavior. In the latter point we will again come back to the techniques from Section 5, but in a much more complicated form. 8.1. Frozen disorder and non-Gibbsian statistical mechanics Let us introduce the concept by an example which we will study later on. This will be the problem of a polymer chain in porous medium. The problem which arises here naturally is that we have to cope with two types of variables. The "rst set of variables are the quenched variables which describe all coordinates of the medium. Quenched means here that these variables do not change in time. In other words, these variables are not #uctuating in the theromdynamical sense. On the other hand, there are #uctuating thermo-statistical variables. In the present problems these are the chain variables R(s) which describe the chain. To set up a simple model let us make life simple and use for the frozen variables a potential which represents all obstacles in the disordered medium. These obstacles are special points in the sample, which are forbidden for the polymer. Therefore, the polymer has to cope with conformations that are dominated by the frozen potential [;], i.e., the con"guration of the pores. Therefore these systems need a di!erent treatment than the usual Gibbsian thermodynamics. In the following, we will introduce this formalism, which has become a standard tool in theoretical physics, especially in spin glasses [28]. The main problem which arises is in connection with quenched variables when the free energy or the partition function is studied. To see this let us study the partition function of an arbitrary system in the form
Z[;]" DR(s) exp+!bH[R, ;], .
(8.1)
Fig. 18. A polymer in random media. The polymer di!uses through the media and thus samples all the disorder, nevertheless it localizes in a favorable region in the medium, which is entropically preferred.
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It is important to realize that Z itself depends on the quenched variables, here represented by the potential ;. This is so far an unusual situation, because, obviously, the free energy corresponding to the partition function depends also on the quenched variables. The experimentally observable free energy, on the other hand, must not depend on the quenched variables, at least it should not, because in the thermodynamics limit is should not, and also because the precise con"guration of the porous medium is not known exactly. Therefore, we wish to average over the quenched variables in the following sense. We must average the free energy as F "1F[;]2 "!k ¹1ln Z[;]2 (8.2) 3 3
which is the so-called quenched average. We see that in actual fact the logarithm of the partition function Z[;] must be averaged over a (given) distribution for the random potential ; itself. Mathematically this is a very di$cult task, and the much simpler average which we have used so far is the annealed average, which is de"ned as F "!k ¹ ln1Z[;]2 . (8.3) 3
Roughly speaking, in the annealed average the partition function is averaged directly. This is of course only allowed if the variable ; is fast enough. This mathematical identity is the basis for the replica trick. It is interesting to note that the replica trick has been used for the problem of rubber elasticity [55}57] before it has been applied to spin glasses [58]. The main problem in coping with the quenched average is purely mathematical, i.e., to compute the average over the logarithm. To do so it was convenient to introduce the replica method. The calculation of the quenched average is carried out by the identity 1ln Z[;]2 "lim R1Z[;]L2 /Rn . (8.4) 3
3
L Of course this equation is somehow ill de"ned, because it makes only real sense in the limit n3-. Then the average can be computed without problems and we may formally write
L L (8.5) ZL[;]" DR (s) exp !b H[R , ;] . ? ? ? ? This expression shows, why the method has been called replica trick: the procedure reproduces the original Hamiltonian n times. The next step is to analytically continue to n31 and hopefully the limit exists. We have to recall that this procedure is really di!erent compared to that of the "eld theory of the SAW. In the latter problem no disorder has been present. The main advantage of this method is now, that the average on the quenched potential can be performed. When the average has been performed the result will be an e!ective Hamiltonian H [R ] without disorder, but with coupling of di!erent replicas. Then a number of questions arise: what is the symmetry based on permutations of the replicas? Is there a spontaneous breaking of this symmetry? There is so far only one problem in theoretical physics for which these questions have been studied in detail. This was done mainly on the problem of spin glasses in connection with the ergodicity-nonergodicity transition, which corresponds to a freezing of the dynamics of the thermodynamic variables. In the next section we are going to illustrate some of these issues, when we study the chain in a random potential.
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8.2. A polymer chain in a random potential To de"ne the problem in more detail we start from the Edwards Hamiltonian of a Gaussian chain in random potential [59]
RR , d , ds #b ds <(R(s)) . bH" Rs 2b
(8.6)
We have now speci"ed the arbitrary potential ; to a special case <(R)s, i.e., a potential that acts on each monomer of the polymer with an arbitrary value. For simplicity, we assume that <(R) has a Gaussian distribution and is thus d-correlated in the sense
1 dBr <(r) . P[<]"N D<(r) exp ! 2D
(8.7)
A corresponding situation is given in Fig. 19 below. The disorder average is now simple to perform and the result is the replica Hamiltonian with coupled replicas
d L L , RR 1 , , ? ! bD bH " ds ds ds d(R (s)!R (s )) . 2b ? @ Rs 2 ? @ ?
(8.8)
This is an interesting result. The disorder average yields a coupling between the di!erent replicas. Moreover, the disorder average produces an attractive potential of strength D. The e!ective attractive potential resembles an attraction between an arti"cial generated n-chain model. Thus all replicas attract each other with an `excluded volumea pseudo-potential. The problem becomes in fact `unstablea because of the attraction and we cannot expect to solve the problem by the methods used so far. Instead we will use in the following a variation method with variation Hamiltonian h based on Feynman's inequality [14,60,61] but in the form appropriate for replicas [62] F 41H !h2 #F . F F
(8.9)
The most important trick on employing the variational technique is the choice of the variational Hamiltonian h. The requirement on h is of course, that its corresponding partition function is exactly known. Therefore a tempting ansatz for harmonic variational Hamiltonian is a generalized harmonic oscillator of the form
d L , RR 1 L , , ? ! bh" ds ds ds p (s!s )R (s)R (s ) , ?@ ? @ 2b Rs 2 ? ? @
(8.10)
where the strength of the harmonic potential is the variational parameter. Here the variational `parametera is a whole function which we want to determine. This ansatz makes sense physically, because the overall potential is attractive and we may guess that the chain is somehow localized in the random potential. The value and the shape of the function p(s!s ) will provide a deeper insight into the localization of the chain.
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Fig. 19. Sketch of a Gaussian random potential, which has been generated on a computer by Gaussian statistics.
To proceed we may employ a Fourier transformation of the form
, p (s!s )Pp (k)" ds p (s )e\ IQ ,
K
(8.11)
where the double underlining is a symbolic notation for the matrix in terms of the replicas a, b. Of course the values of the variational function p are determined by "nding the minimum. To start with let us "rst assume replica symmetry, i.e., the matrix p is symmetric with respect to the replica indices. 8.3. Replica symmetry Let us "rst try a self-consistent solution with the extra diagonal elements of the form p(k)"p d(k). The solution for the variational parameter is then given by
dkI 1 \B \ . p "!*I N 2d 2p ((d/b)kI #k!p(kI ))
(8.12)
The diagonal elements are then given by p(k)"p #DI
,
ds(1!e IQ) 2d
dkI (1!e II Q) \B \ . 2p (d/b)kI #k!p(kI )
(8.13)
At this point we cannot proceed further with analytical computations since the self-consistent equation has no closed analytical solution. What we can see is at the simplest level the localization
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Fig. 20. Numerical solution for the chain size. The localization transition becomes visible for large values of s. The parameters are N"4096, D"0.001. Fig. 21. Bifurcation of the solutions. The solutions are no longer unique and the result cannot be interpreted in a simple way.
of the chain. When neglecting the k-dependent contribution to the above equations we get a localized solution as it has been given earlier by Edwards and Muthukumar [59] when they "rst analyzed the problem. 1(R(s)!R(s ))2"bl (1!exp+!b(s!s )/l ,) , (8.14) This result tells us the following. First we realize that a new length scale l appears, which we have to interpret. We see that at small values of s!s the size of the chain is independent of the new length scale and we recover the classical random walk behavior. At large values of s!s we can neglect the exponential term and we "nd that the size of the chain is independent of the chain length. Thus the chain is localized to a certain size. The value of l is given by l "((l/d)p )\ JD\ \B . (8.15) This means that the length is determined by the disorder strength D and has the correct scaling with the disorder parameter. The exponent !2/(4!d) was con"rmed by di!erent computations of this localization transition [59,63,64]. Localization is here meant in the sense that the chain size no longer depends on the chain length, but only on the strength of the disorder D. 8.4. Numerical solution In spite of the problem that we cannot go further by analyzing the self-consistent equations with analytical methods we proceed with the numerical examination of the complete system of equations. To do so, we use a discretization of the chain Hamiltonian and introduce periodic boundary condition. The solution is then obtained by iteration. For moderate values of the disorder strength D the localization transition is con"rmed. At large values of the contour variable the size of the chain becomes independent of s. At small values we "nd roughly the random walk behavior, despite deviations (Fig. 20).
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Now, what happens at larger disorder strength? To see this we plot in Fig. 21 the same as in Fig. 20 for chain length N"4096 and increasing disorder strength bD. It is seen that the solutions of the equation system becomes unstable, resulting in a bifurcation. With increasing disorder D the curves split up more and more. The "nal question is, do we have here replica symmetry breaking? We do not answer this question, but instead use a di!erent method to discuss the problem physically.
9. Polymers on disordered surfaces and interfaces Instead of aswering this di$cult question we go back to the physical problem of a chain at a random potential. Indeed, the related physical question is to consider a chain at a disordered surface. In addition, we would like to study this problem without using the replica method. Therefore let us "rst recall the `easya methods and problems we know. The problem of polymers on #at and homogeneous (attractive) surfaces has been studied many times [65,66], but here we put the emphasis on disordered surfaces [67}75]. The problem seems to be very di$cult, but we had made so far always some progress by using simple physical arguments containing the statement that upon increasing the surface irregularity, the number of polymer} surface interactions is strongly enhanced relative to the idealized planar surface (see Fig. 22). This is a consequence of a larger probability of polymer}surface intersection with increasing roughness. To understand what is going on we perform again a simple scaling argument based on ideas put forward by de Gennes. 9.1. Scaling argument for fractal spatial disorder 9.1.1. Flat surface First, we brie#y review a simple scaling treatment of an ideal chain adsorbed on a #at surface, as introduced by de Gennes [11]. Let R and R KR KbN be the mean size of an ideal polymer , , (with N monomers and e!ective monomer length b) perpendicular and parallel to the surface, respectively. The monomer density is assumed to be constant in a region of size R R . Then the , , number N of monomers bounded to the surface is estimated as N"bR N/R R "bN/R . , , , , Consequently, the free energy can be written as
(9.1)
bF+R /R !bwN"bN/R !bw bN/R , (9.2) , , , the "rst term being the con"nement energy, the second one due to contact interactions with the surface, !w denotes the e!ective monomer attraction. Minimization of the free energy RF/RR "0 gives an expression for the polymer thickness R perpendicular to the surface, , , R Kb/bw . (9.3) , Thus, the thickness of the polymer reduces with growing attractive interaction strength, as expected. The independence of the chain length N indicates that the polymer is in the localized regime [64]. Again, as before, the chain size does not depend on the number of monomers N.
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Fig. 22. Sketch of the main mechanism of adsorption enhancement by surface roughness: the number of possible binding sites increases without being balanced by a loss in con"gurational entropy.
9.1.2. Adsorption on fractal surfaces, scaling A fractal surface may be characterized by its fractal dimension d , 24d 43, a value d "2 1 1 1 corresponding to a #at surface. The limit d P3 produces an extremely rough, space-"lling surface, 1 Brownian surfaces [76] are characterized by d "2.5. 1 Now the number of bounded monomers is written as N"b\B1 RB1 N/R . , , Running through the same procedure as above yields
(9.4)
(9.5) R Kb/(bw) B1 \ , , so that result (9.3) for the case of a #at surface d "2 is recovered. 1 From (9.3) we have bw(1 because of b;R for weak adsorption, where no complete collapse , on the surface takes place. In fact for most materials values about bw&0.0120.1 are found [77]. Therefore the polymer adsorption on rough surfaces, with d '2 generally is enhanced compared 1 to the case of a #at surface, i.e. R(R . , , Although this is a crude argument, it gives an insight into the main aspects of adsorption enhancement: the crucial point is the competition between the gain in potential energy obtained by binding to the surface and the loss in chain entropy associated with the con"nement of chains in comparison to free chains. Therefore, the dominating mechanism in our consideration above is the increasing number of binding sites at a rough surface without paying an entropy penalty, which means that a chain has to lose less con"gurational entropy to adsorb on a rough surface. This is in agreement with results of much more expensive previous calculations by Douglas et al. [74] and Hone et al. [75]. A similar argument holds for the case of energetical heterogeneity [71] with a distribution of the interaction strength on the surface, the chain can select the strong binding points without changing its con"guration too much, thus seeing a larger e!ective interaction strength. 9.2. Variational calculation without replicas For a systematic study of R in the case of spatial and energetical heterogeneity, the free energy , is calculated via a variational procedure, where the disorder is treated as a quenched randomness. The replica method is avoided by the introduction of an additional variational parameter, see the next section. We consider an ideal chain at an in"nite, penetrable, well-de"ned surface with low pro"le. Furthermore, we assume an attractive contact (i.e., extremely short range) interaction between chain and surface that can be mimicked by a delta potential.
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Now this system is represented by its Edwards Hamiltonian, which reads
3 , RR(s) , bH" ds #b ds<(R(s)) , 2b Rs
(9.6)
R(s) being the chain segment vector position. The potential contains the polymer}surface coupling,
<(R(s))"! dx K[h(x)]bw(x)d(R(s)!h(x))
(9.7)
with h(x)"(x, h(x)), where x"(x , x ) is an internal surface vector. The surface disorder is by w(x ) for energetical disorder, described i.e. an interaction strength varying on the surface, and by h(x) for spatial disorder, i.e. a rough surface pro"le. The factor K[h(x)]"(1#" h(x)") takes account of the local de#ection of the surface in cartesian coordinates. In order to approximate the free energy we make use of a Feynman variational procedure of the same form as before, i.e., 1e\@&\& 2 5e\@6&\& 7& . &
(9.8)
To avoid confusion with earlier considerations we use a di!erent notation for the di!erent Hamiltonians. Again, 122 denotes the average with respect to a trial Hamiltonian H . This & gives an upper bound to the free energy bF4bFH"bF #b1H!H 2 &
(9.9)
with the abbreviation
bF "log
DR exp+!bH , .
(9.10)
bFH has to be minimized to give the best estimate for the true free energy bF. 9.2.1. Trial Hamiltonian of Garel and Orland The appropriate choice of the trial Hamiltonian is most signi"cant for the utility of the variational procedure. Here we take an extension of a form suggested by Garel and Orland [78],
1 , , bH " ds ds (R (s)!B )g\("s!s ")(R (s )!B ) . 2 H H H H H H
(9.11)
Of course we had used the same main features for the trial Hamiltonian, (a) it is quadratic in R(s), so that an exact calculation of bF is possible; (b) the coupling of chain segments is mediated by the variational parameters g ("s!s "), one for each direction of space: the indices 1 and 2 are identi"ed H with the coordinates x and x of the surface parameterization, index 3 corresponds to the z coordinate parallel to the average surface normal; (c) there is an additional variational parameter B, equivalent to a translation of the center of mass of the chain. It should be mentioned that this
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type of variational principle was originally designed to avoid replica theory in random systems [78]. This is another reason why the Garel}Orland method is chosen here. If the polymer is assumed to stick permanently at some place along the disordered surface, the problem falls into classes dealing with `quenched disordera and di$culties with replica symmetry breaking arise. In the following, we will show that the Garel}Orland method is indeed useful to treat the problem of polymer adsorption on disordered surfaces as it yields physically sensible results. Assuming cyclic boundary conditions R(N),R(0), the variational free energy (9.9) can now be calculated to give g (n) g (n) bFH"! log H # Nu H #bW(B, G) L b b L H L H
(9.12)
with u "2np/N. Here the interaction energy W(B, G) is the only term depending on the L interaction potential,
!Nb (B !h (x)) G W(B, G)" dx K[h(x)]w(x) exp ! G , 2G (2p) (G G G ) G G
(9.13)
with h (x) being the components of h(x), i.e. h (x),h(x). The parameters G are de"ned by G H 2 , G "2 g (n)" ds cos(u s)g (s), j"1, 2, 3 . (9.14) H H L H N L L
G can be identi"ed with the mean square radius of the polymer parallel (G ) or perpendicular H (G , and G ) to the surface normal. 9.2.2. Minimization of the free energy Following the lines of Garel and Orland, the minimization of bFH with respect to g (n) and H B leads to r 0
B W(B, G)"
(9.15)
b r g (n)" , H Nu#bb RW(B, G)/RB L H
(9.16)
and
because RW(B, G)/(Rg (n))"RW(B, G)/(RB). As already discussed by Garel and Orland [78], one H H does, in general, expect the variational equations to have several solutions. This especially applies to (9.15), since we consider an in"nite surface, e.g. leading to an in"nite number of solutions in the case of a periodic surface heterogeneity. All these solutions have equal free energy. Introducing the abbreviation a "((Nb/4(2p) ) b"w"/G ) , H H H
(9.17)
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the optimized parameter G is calculated from (9.16) as H
Nb coth(a )!a\ H H for w50 , H 4 a H (9.18) G" H Nb cot(a )!a\ H H ! for w(0 . H 4 a H The e!ective interaction strength w contains all relevant surface and polymer properties and is H given by
(2p) G RW(B,G) H w" . H Nb RB 0 BW B G
H In two special cases results can be obtained very easily:
(9.19)
(i) if there is no interacting surface at all, i.e. w(x),0, then we immediately have a "0 and H therefore G "Nb/12,R /2, the chain conformation is purely Gaussian in all directions, as H expected; (ii) for an ideal surface, which means w(x),w and h(x),h , the e!ective interactions strengths are calculated as w "0 and w"w . So de"nition (9.19) of w guarantees correct results H for this case. The explicit form of w for various special sorts of surface heterogeneity is calculated in the next H section. The discussion of (9.18) is complicated by the fact that the e!ective interaction strength itself is a function of the polymer extensions in di!erent directions. But, in general, (9.18) can for w50 be H expanded in the limits of small and large a . This yields the mean polymer extension into the H di!erent directions of space, RM being parallel to the surface normal, RM and RM perpendicular to it, if 1h(x)2"0 is assumed,
R +1!cN bw, H RM K(G K b p H H 1! N(bw) bw H H
for bw;N\ , H for bw
(9.20)
where, as above, R denotes the radius of gyration of the corresponding Gaussian chain. Thus, in the limit of small e!ective interaction strength the chain has Gaussian conformation (see Fig. 23), whereas for high bw the chain is localized at the surface, leading to a mean polymer size which in lowest order shows the same characteristics as the result of the scaling argument (9.3). From the conditions for the limiting cases, a localization criterion bw +N\ (9.21) H can be found, which means that bw +0 for long chains. Therefore, very long chains always are H adsorbed, i.e. localized, at an attractive surface. This of course is a consequence of our assumption of a penetrable surface, since in the opposite case of impenetrable surfaces adsorption takes place only from a "nite interaction strength [65], i.e. bw '0. H
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Fig. 23. Numerical result for the optimized variational parameter G KRM as a function of interaction strength (bw) , and chain length N. The localization transition can be identi"ed: for small values of w, G grows linearly with N, whereas G is independent of N for large values of w.
For negative e!ective interaction strength w(0, only the case b"w";N\ is important for H H us, as we are mainly interested in the adsorption behavior. Expansion of (9.18) in this case yields RM K(G KR +1#cN bw, . H H H
(9.22)
9.2.3. Ewective interactions The full general form of the e!ective interaction strength is
(B !h (x)) (B !h (x)) G H G H dx K[h(x)]w(x) 1! H exp ! G w" H G 2G 2p(G G G ) H G G
, (9.23)
where the translational parameter B has to be chosen such that B W(B, G)"0. In the following, the surface is assumed to be symmetrical with respect to the coordinates x and x . Then we r 0 and B " r 0 as a solution of the minimization equation (9.15). For immediately have B " simplicity, we additionally assume the surface heterogeneity to depend only on one space direction x , which means that w(x),w(x ) and h(x),h(x ). Hence, w"0 and the polymer extension into the direction of x equals that of a Gaussian chain. In this case the expressions for w and w reduce to
x x (B !h(x)) 1 dx K[h(x)]w(x) 1! exp ! ! , w" G 2G 2G (2pG )
(9.24)
(9.25)
1 x (B !h(x)) w" dx K[h(x)]w(x) exp ! ! . (2pG ) 2G 2G Now a straightforward calculation for various types of surface heterogeneity is possible.
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(1) For a #at surface with energetical heterogeneity, h(x)"h , the minimization condition (9.15) r 0, so that the center of mass of the chain is located on the surface. Inserting results in B " w(x)" dq exp+iqx, w (q) leads to \ G q G , (9.26) w" dq qw (q) exp ! 2 G \
w"
\
G q . dq w (q) exp ! 2
(9.27)
As can be seen from the notation w (q)"w d(q)#w H(q), the e!ective interaction strength parallel to the surface is independent of the mean interaction strength w . E For a periodic interaction strength w (q)"w d(q)#(A /2)+d(q!f )#d(q#f ), with ampli U tude A and wave number f, we have U G w" A f e\% D , G U
(9.28)
w"w #A e\% D . U
(9.29)
Thus w takes on its maximum w #A , if the wavelength of the heterogeneity exceeds the U polymer size parallel to the surface, jKf \
(9.30)
w"w #cD /(G #m) . U
(9.31)
The magnitude of the heterogeneity is determined by both m and D : the smaller the U correlation width and larger the variance, the stronger are the #uctuations, which leads to an increase of the e!ective interaction strength. The limiting cases perpendicular to the surface are
w #cD m\ for m
(9.32)
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Fig. 24. Sketch of the optimized translational shift B of the center of mass of the chain for a sinusoidal surface pro"le.
(2) In the case of a heterogeneous surface pro"le (whereas the interaction strength w(x)"w is assumed constant) the disorder has to be weak in order to make the x integration feasible. Therefore, we only investigate the case "h(x)";1 and " h(x)";1, where 1h(x)2"0, and restrict the calculation to "rst order in the #uctuation of h(x). With h(x)" dq cos(qx)hI (q), minimiz ation (9.15) yields B + dq hI (q) exp+!G q/2,. This means that the center of mass of the chain to some extent follows the surface pro"le. For an example see Fig. 24, where B is sketched for a periodic surface pro"le. Now the de#ection factor can be approximated by
1 K[h(x)]"(1#" h(x)") +1# dq dq hI (q)hI (q )qq sin(qx) sin(q x) . 2
(9.33)
If additionally the part of the exponent in (9.24) and (9.25) which depends on h(x) is expanded, we obtain in lowest order of hI (q)
G (q#q ) G w+w dq dq hI (q) hI (q ) exp ! G 2
(q#q ) q# (G qq sinh(G qq )!cosh(G qq )) , 2
1 G (q#q ) w+w 1# dq dq hI (q)hI (q ) exp ! 2G 2
(9.34)
[3#G qq sinh(G qq )!cosh(G qq )], .
(9.35)
E A periodic surface geometry hI (q)"A d(q!f ) leads to F w+w (G /G ) (A f /2)+G f (1!e\% D)!(1!e\% D), , F
(9.36)
w+w +1#(A f /4)(1!e\% D)!(3A/4G )(1!e\% D), . F F
(9.37)
For a #at surface, the polymer extension RM parallel to the mean surface normal (i.e. in z direction) is identical to the size perpendicular to the surface. This is di!erent for a rough
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surface pro"le, which now turns out to be important for the interpretation of (9.36) and (9.37). If, for example, G is small compared with the squared wavelength jKf , then the polymer sticks to the surface, following the de#ections. Therefore, the result for RM K(G exceeds the polymer size perpendicular to the surface by the amount of the de#ection, the e!ective interaction strength w accordingly is smaller than w . Adsorption enhancement therefore only can be obtained in the opposite case RM
(9.38)
For a very large correlation width, which in the limit mPR corresponds to a #at surface, we again have the e!ect of a reduction of the e!ective interaction strength compared with the #at surface, w(w . Therefore, the result which is relevant for adsorption enhancement here is obtained in the case D 4m;G 4G , where the e!ective interaction strength has its F maximum value w +w +1#(cD/m(G ), . F
(9.39)
An analytic expression for w is not available, but the main features of the result can be estimated to strongly resemble those of w for a periodic surface pro"le discussed above. 9.2.4. Advantage of the method In this special example we have seen that we could achieve more compared to the replica method. For example we succeeded in computing the e!ective interaction strength. This is not to say that the replica method is less useful, but rather it is to say that for some questions alternatives are useful. Let us discuss the main features again. The variational calculation presented here is valid for weak spatial disorder only (therefore it does not reproduce the scaling behavior for fractal surfaces). Nevertheless, the mechanism of adsorption enhancement is well reproduced, we "nd agreement of the results in all special cases which were already investigated in the literature. A special feature of the variational method employed here is the possibility of quantifying the localization transition, i.e., the transition from a slightly deformed Gaussian coil to a localized
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conformation, where the polymer size perpendicular to the surface no longer depends on the chain length. According to result (9.21) the localization can be obtained by increase either of the e!ective interaction strength or of the chain length. This helps to compare the strength of adsorption enhancement for the two sorts of disorder considered here: as can be seen from the maximum values of (9.29) and (9.37) or from a comparison of (9.31) and (9.39), the localization transition is only slightly a!ected by a rough surface pro"le, whereas energetical heterogeneity can induce the transition even at vanishing mean interaction strength. Therefore, we conclude that the disorderinduced enhancement of polymer adsorption is much more signi"cant for a heterogeneous interaction strength than for spatial roughness [68,79]. Our "ndings concerning the localization behavior are a!ected qualitatively by the assumption of surface penetrability: for in"nitely long chains at a #at and homogeneous impenetrable surface, the localization transition in contrast to (9.21) only occurs for some nonzero value of the attractive potential [65]. Nevertheless, we expect our main statement on the signi"cance of adsorption enhancement to hold also for impenetrable surfaces, since the comparison of transparent and opaque surfaces in simple soluble cases by Hone et al. [75] shows that they should not be a!ected di!erently by weak surface heterogeneities. The polymer size parallel to the surface does not directly depend on the mean interaction strength, but only through the extension perpendicular to the surface. Thus, because it is less a!ected by heterogeneity, the former always exceeds the latter, except for one special case: for a #at, neutral surface with a periodic interaction strength, the polymer size parallel to the surface is smaller than that perpendicular to it, if the period "ts the polymer size such that it is concentrated to a maximum of w(x) and even restricted by the neighboring repulsive regions.
10. Random copolymers 10.1. General remarks In the following, we study polymers with spin glass disorder [28] themselves. As examples we investigate random copolymers, which play a certain role in the phase behavior of mixtures. We will present a very simple model study on random copolymers, where we will restrict ourselves to the quenched randomness. From the chemists point of view this might be unrealistic, since in the reactor not only one type of random copolymer with a certain disorder will be synthesized, but a whole ensemble of di!erent random copolymers. In this case the quenched randomness does not play a signi"cant role [80,81]. The phase diagram has been computed and it does contain unexpected macrophase separations where chains of certain architecture in the randomness phase separate on macroscopic scales. Of course, microphase separation transitions are possible, sampling the disorder. We use a di!erent system, where we assume that only one architecture on randomness exists. This might not be very useful at "rst sight, but we will see that this system has unexpected features in two space dimensions. The main point is to study the competition between microphase separation and frozen states. This question has been addressed in several papers [82,83]. We show shortly below, that indeed in two dimensions glassy phases can exist. The physical reason is simple: two dimensions are close to the upper critical dimension in polymer melts.
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Fig. 25. The random copolymers are modeled by a chemical sequence of two monomers AB. The excluded volume interaction is no longer uniform along the chain. The probability of "nding a sequence is given by a conditional probability P(A"B).
Another area where theoretical models of random copolymers are useful are biological systems. There are di!erent models for protein folding which treat the disordered quenched } but not really random } sequence of the amino acids of a protein by an ensemble of random copolymer species. There were former calculations by Fredrickson et al. [81,84] that showed for the many-chain melt system a demixing transition of di!erent chain species. On the other hand, there were calculations by Sfatos et al. [82] that show a spin glass-like freezing transition in these systems. It is this discrepancy that we want to resolve by our calculations. 10.2. The model We describe the sequence of di!erent monomers of a chain a by a quenched disorder variable p (s), which gives the chemical species of the monomers. s"02N parameterizes the contour of ? the polymer chain. A simple picture of such a chain is cartooned in Fig. 25. We use a Gaussian distribution with chemical correlation length N and a mole fraction f of @ A-monomers to describe the correlated sequence of monomers on the chain, so that we obtain for the cumulants of the disorder variable p (s) ? 1p (s)2 "2f!1 , (10.1) ? ) (10.2) 1p (s)p (s )2 "d 4f (1!f )e\ Q\QY ,@ . ? @ ) ?@ All higher cumulants we set to zero in this Gaussian approximation. This distribution de"nes an average `blockinessa b"N/N of the copolymers [85,84]. We describe the system by our @ well-known Edwards Hamiltonian. We include the entropy of the chains in the form of a Wiener term
LN d , RR (s) ? ds . bH" 2b Rs ? Besides the overall excluded volume interaction of all monomers
, , 1 LN ds ds d(R (s)!R (s )) , bH " v ? @ 2 ? @
(10.3)
(10.4)
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where so far nothing new has appeared. In addition to these two contributions we must account for the random interactions. Here we assume a repulsive interaction of unlike monomers,
, , 1 LN LN 1 ds ds (1!p (s) ) p (s ))d(R (s)!R (s )) . bH " o\s ? @ ? @ 2 2 ? @
(10.5)
s is the interaction parameter between unlike monomers. Here other interactions between A}A and B}B monomers can be included but we do not do this here for the sake of simplicity. For the quenched average over the disorder p we use the replica method [28], 1log Z2 "lim (1ZL2 !1)/n . N
N
L
(10.6)
We average the replica partition function
L L ZL[p]" DR (s) exp !b (H [R ]#H [R ]#H [R , p]) ? ? ? ? ? ?
for integer n. Having done this we obtain a replica Edwards Hamiltonian which contains couplings that can be expressed in density variables
, o( (R)" ds d(R!R? (s)) . ? ? ?
(10.7)
These are the same density variables as introduced in Section 5, but they depend on each of the replicas. Therefore they describe the density in each replica. Second, we have spin glass-like overlap variables [28] of the form
, , ds ds e\ Q\QY ,@ d(R!R? (s))d(R !R@ (s )) . QK (R, R )" ? ? ?@ ?
(10.8)
Using these variables the averaged replica partition function can be written as
, RR? (s) d L LN L LN ? 1Z[p]L2" DR? (s) exp ! ds ? Rs 2b ? ? ? ?
!
1 L 1 1 V# o\s dBr o( (r)! Tr ln[1!2o\sQK (r, r )] ? 2 2 2 ?
L s dBr dBr o( (r)(1!2o\sQK )\o( (r ) #o\ (2f!1) ? ?@ @ 2 ? @
(10.9)
with s"f (1!f )s. The last two terms in Eq. (10.9) can be expanded in powers of QK . For the Gaussian approximation we keep only the terms up to second order.
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10.3. The ewective Hamiltonian The two collective variables o( and QK (R, R ) are the natural choice as collective variables of the ? ?@ problem and we want to perform a transformation to these "elds. To achieve this, we introduce into the partition function two identities for these two collective "elds. For the density variables we introduce
, Do (k)d o (k)! ds e\ k R?? Q ? ? k ? ? GFFHFFI M? (
k
L BkU k k ( k " Do (k)DU (k)e ? ? \ M? \M? ,1 ? ? ? and for the overlap variables we introduce also
(10.10)
L DQ (k, k )d(Q (k, k )!QK (k, k ))" DQ (k, k ) DW (k, k ) ?@ ?@ ?@ ?@ ?@ kk ? @ Y ?@
;exp i dBk dBk W (!k,!k )(Q (k, k )!QK (k, k )) ,1 . (10.11) ?@ ?@ ?@ ? @ To integrate out the chain degrees of freedom we make a cumulant expansion in the collective "elds o( (k) and QK (k, k ). We expand up to second order in the auxiliary "elds W and U . After the ? ?@ ?@ ? conformational average using the Wiener term of the Hamiltonian and integration over the auxiliary "elds we obtain an e!ective Gaussian}Hamiltonian for the collective "elds o (k) and ? Q (k, k ), ?@
1 L bHL" dBk(S\(k)#V#2s)o?(k)o?(!k) M 2 ? 1 L ! s(2f!1)f (1!f ) dBk dBk o?(!k)Q (k, k )o@(!k ) ?@ 2 ?@ 1 L # dBk(SI \!o\s)(Q (k,!k)!SI (k, s)) / ?? / 2 ? 1 L # dBk dBk (S\!2o\s)Q (k, k )Q (!k,!k ) . (10.12) / ?@ ?@ 2 ?$@ S, S and SI are correlation functions, originating from the RPA transformation, calculated for M / / a system of free Gaussian chains,
S"1o( (k)o( (!k)2 M ? ? &
(10.13)
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for the density variables, SI "1QK (k,!k)QK (k,!k)2 / ?? ?? & for the diagonal elements and
(10.14)
S "1QK (k,!k)QK (k,!k)2 / ?@ ?@ &
(10.15)
for the o!-diagonal elements of Q . These give the entropic contribution to the e!ective Hamiltonian due to the chain degrees offreedom, whereas the energetic contribution, proportional to s, was obtained from the disorder average. SI is in this replica formulation the structure factor of the / composition #uctuations in the homogeneous state of the system SI "1Q (k,!k)2"1"o (k)!o (k)"2 . (10.16) / ?? We emphasize here that the Q-"eld is nonlocal due to the chemical correlations present for the sequence of the chains. For what follows, we will assume incompressibility of the system so that all o(k)"0 for kO0. Using the Gaussian}Hamiltonian we can determine the limits of absolute stability of the homogeneous phase (spinodal). There are two possible instabilities originating from the last two terms in (10.12), one related to the diagonal Q and the other related to the ?? o!-diagonal Q . ?@ The bare two-point correlation functions (10.14) and (10.15), which give the entropic contribution, and the energetic contribution have a di!erent scaling with the number n of polymer chains. Hence one recognizes that for the thermodynamic limit of a melt n PR,
(10.17)
g(b)"(2/b)(e\@!1#b) .
(10.18)
This instability of the o!-diagonal elements of the overlap matrix Q leads to a spin glass-like ?@ freezing of the composition #uctuations. Here the system freezes to a disordered state, where the macroscopic #uctuations (k"0) are the "rst modes to freeze.
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Fig. 26. The structure factor S as a function of the scaled wave vector. With increasing incompatibility between the two components the system becomes unstable. Fig. 27. The critical lines for the diagonal and o!-diagonal elements.
As we have calculated here only the Gaussian part of the Hamiltonian we cannot decide if replica symmetry breaking is relevant for this problem or if the freezing transition of the o!-diagonal elements is of Mattis-type. The last is the case in high dimensions where the chain constraint is of no importance. For low dimensions instead there could be entropic contributions of higher order that break the replica symmetry. Comparing the two obtained transitions shows that for all blockiness parameters b the spin glass-like freezing transition is the "rst to set in as shown in Fig. 27. Thus no microphase-like transition due to the instability of the diagonal elements Q should be ?? observed. With our choice of the order parameters (10.7) and (10.8) we are not able to detect for a system of many chains (melt) any phase separation of di!erent monomer species like Fredrickson [84]. This is due to the assumption of no correlations between the monomers of di!erent chains. The spin glass-like transition that was obtained by Sfatos et al. [82], on the other hand, exists only for single-chain systems. There the system is fully frustrated as the single-chain conformation cannot accommodate the disordered demixing pattern of the di!erent monomers. For the melt the situation is di!erent as a chain may explore the whole system to "nd a convenient pattern without dragging the whole rest of the chains. This shows up in our calculations by the dominance of the entropy term in the e!ective Hamiltonian. 10.4. What can we see? We considered di!erent dense systems of random copolymers with correlated chemical sequence. By application of an RPA-like method to the averaged replica Hamiltonian of the system we were able to show that for many-chain systems like melts there are no phase transitions to be expected. This result was obtained due to a di!erent scaling with the number of polymer chains of the interaction energy and the conformational entropy of the system.
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For single-chain systems there show up to possible transitions in the diagonal and o!-diagonal elements of the overlap matrix. The diagonal elements allow the calculation of composition #uctuations characterized by the structure factor. Here shows up a possible microphase separation at "nite length scale, but we "nd out that the instability of the o!-diagonal elements sets in "rst for all chemical correlation lengths (or blockiness parameters b). This transition of the o!-diagonal elements corresponds to a freezing transition of the dense globule, as known from spin glass theory, which sets in before the microphase separation. We emphasize again that this is the case only for dense globular-like single chains and not for melts, where the conformational entropy dominates.
11. Copolymer melts in disordered media 11.1. General remarks In this section we will emphasize on the interacting copolymer melt in disordered systems. Here we are going to analyze the interplay between disorder and phase separation. As before, the copolymers can be characterized by the overall degree of polymerization N, the composition f"N /N, and the interaction energies < (r)"< d(r) between the monomers. In the case when ?@ ?@ < '< , < the contacts between A and B monomers are unfavorable, which drives the system to segregate. Quantitatively, the tendency to segregation can be expressed by the Flory}Huggins parameter s"2< !< !< . Since the entropic and enthalpic contributions to the free energy scale as N\ and s, respectively, block copolymer phase state is dictated by the product sN. For many materials the Flory}Huggins parameter has the temperature dependence s+a/¹#b, where a'0 and b is a constant, so the tendency for segregation will be increased with decreasing the temperature ¹. The system without disorder undergoes a microphase separation, as mentioned in Section 6. The domain structure appears as a result of competition between short-range monomer}monomer interaction that seeks to decrease the number of unfavorable contacts between A and B monomers and long-range correlation due to chemical bonds between those parts of the chains that tend to segregate into domains. The "rst stage of the domain structure formation, when the amplitude of the local composition #uctuations is small in comparison with its average value, can be described in terms of Landau}Brazovskii e!ective Hamiltonian [38,86]. This Hamiltonian describes the phase transitions in systems such as weakly anisotropic antiferromagnets [87], the isotropic-cholesteric and nematic-smectic C transition in liquid crystals [86], and even pion condensation in neutron stars [88]. The homogeneous state of these systems is unstable with respect to #uctuations of the "nite wave number q that below phase transition point results in formation of the modulated structure with period ¸"2p/q (for recent excellent reviews see [80,89]). An interesting question is, as we have seen brie#y before, can disorder destroy the structure of the microphase separation, and is it even possible that a glassy phase exists? The description of the copolymer melt is based on the introduction instead of the individual monomeric coordinates r (s ) (the index i"1,2, n , counts the copolymer chains, s gives the G G A G position of the segment along the chain) the densities of polymer segments (collective coordinates) o (r)" LA D,ds d(r!r (s)) and o (r)" LA , ds d(r!r (s)). For the incompressible (o (r)#oG (r)"o , o is the average G D,density of the melt) the order parameter U(r) copolymerG melt
is the deviation of the local density of A monomers o (r) from its average concentration fo . There
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is one characteristic length scale of the system under consideration, the gyration radius of the copolymer chain R "bN /6 , where l is the bond size. It is convenient to normalize the length scales on the gyration radius and introduce dimensionless variables q"qR , U(r)"U(r)/(o NR), j"u N/(o R), q"2(s !s)N .
E
(11.1)
s is the value of Flory}Huggins parameter on the spinodal computed by Leibler [38] (s N"10.495 at f"), q is the e!ective temperature, and the parameter u depends on the copolymer composition. Near the phase transition point the free energy of the system can be expanded in the power series of the order parameter U(r) which corresponds to the di!erence of the density variables [87,90] (see also Section 6). The e!ective Hamiltonian can then be written in terms of these variables as 1 H(U)" 2
O
#
(("q"!q )#q)U(q)U(!q)
(No ) j
U(q )d q # h(!q)U(q) , G G R 4! G O G O G
(11.2)
where q ("1.94 at f") is the peak position of the scattering factor and U(q) is the Fourier transform of the order parameter. The last term in (11.2) describes the interaction of the copolymers with an external "eld; h(r)"; (r)!; (r) with ; (r) being the interaction energy of the ath ? monomer. For an AB copolymer melt that is immersed in a gel matrix the random "eld ; (r) ? describes the interaction of monomers of ath type with the gel monomers. The "eld h models the disorder which can be seen as a random "eld interaction with all monomers in the polymers. In this case the random "eld ; (r) is ? ; (r)"!ve U (r) , ? ?
(11.3)
where e is the adsorption energy of ath monomer in the units of the thermal energy k¹, v is the ? excluded volume of the gel}copolymer interaction, which is assumed to be the same for all types of interactions, U (r) is the local gel concentration. The gel structure can be characterized by the "rst two correlation functions [91] 1U (r)2"o "N /R ,
1U (r)U (r )2"G(r!r ) ,
(11.4)
where N is the number of monomers between gel crosslinks and R is mesh size distance being for
rigid network proportional to N and for Gaussian one &N . Taking into account relations (11.4) we can write the "rst two moments of the random "elds ; (r) as follows: ? 1; (r)2"ve o , ? ?
1; (r); (r)2"ve e NR\d(r!r ) . ? @ ? @
(11.5)
It is interesting to note that the last equality on the r.h.s. of Eq. (11.5) is valid as long as the characteristic length scale of the copolymer melt ¸ is larger than R . The average over the random
"eld h(q) can be carried out by using the replica trick. As a result the multi-replica e!ective
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Hamiltonian is obtained as
L j L U (q)(d G\(q)!D)U (!q)# U (q )d q , (11.6) G ? ?@ @ ? G 4! O ? @ ? G OG G where n is the number of replicas, and D"No v(e !e )N/R . To calculate the free energy
F we use a variational principle based on the second Legendre transformation [92]. In the L framework of this approach the n-replica free energy of the system under consideration is 1 H (+U,)" L 2
F "min = (+U (q),, +G (q),) L L ? ?@
(11.7)
with
1 L 1 (d G\(q)!D)G (!q) = "! Sp ln G (q)# ?@ ?@ ?@ L 2 2 O ? @ O 1 L 1U (!q)2(d G\(q)!D)1U (q)2 # ? ?@ @ 2 ? @ O j L L j 1U (!q)2 G (q ) 1U (q)2# dr1U (r)2 # ? ?? ? ? 4! 4 OY ? O ? j L j L G (q) ! drG (r) , (11.8) # ?? ?@ 48 8 O ? ? @ where the minimum of the functional = is to be sought with respect to functions 1U (q)2 and the L ? renormalized Green's function G (q) that are considered as independent variables at the "xed ?@ parameters j, q, D. The functional = contains only 2-irreducible diagrams that cannot be cut into L two independent parts by removing any two lines between vertices j.
11.2. Replica symmetric solution. The phase diagram In this subsection we will calculate the phase diagram of the copolymer melt in the gel taking into account only one-loop diagrams in the functional = . In this case there is only a replica L symmetric solution. In the replica symmetric case the e!ective propagator is expected to have the following form: G (q)"g (q)d #*g (q) , (11.9) ?@ ?@ with g (q)"1/(("q"!q )#r), where r is the renormalized temperature. In the ordered phase there is a nonzero average value of the order parameter 1U(q)2, which describes the appearance of the domain structure. The Fourier transform of the order parameter has the form 1U (q)2"A(d(q !q )#d(q #q )) . (11.10) ? X X Here, we will consider only the ordered phase with the lamellar type of symmetry of the microphase structure that has the lowest free energy for the e!ective Hamiltonian (11.6). Substituting the trial
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functions (11.9)}(11.10) into the expression of the free energy we can write the free energy as a functional of A and g (q). In order to "nd the values A and g (q) corresponding to the extremum of the functional = we have to take variational derivatives of = with respect to these functions. L L The analysis of the extremal equations can be simpli"ed if one introduces the new reduced variables t"q/(jq /2p) , d"D/(jq /2p) , z"r/(jq /2p) . (11.11) In the homogeneous state the extremal equation for the renormalized temperature r reads (11.12) z"t#z\ (1#(d/2z)) . Repeating the same calculations for the ordered phase for which A"2r/j, one can obtain !t"z#z\ (1#(d/2z)) . (11.13) To calculate the phase diagram of the system under consideration the energies of the homogeneous and ordered phases have to be compared. Substituting trial functions given by Eqs. (11.9) into the expression for n-replica free energy F and taking limit 1F2 "lim F /n one can write the L L L expressions for the free energy of homogeneous phase as 1F2 "(q /2p) j ((z!d/(z#z!t) and for the ordered one as
(11.14)
(11.15) 1F2 "(q /2p) j ((z!(d/(z)!z!t) . The phase diagram of the copolymer system in random media is sketched in Fig. 28 in the plane (d"D/((jq /2p) ), t"q/((jq /2p) )). In the limit of weak random "eld the "rst-order phase transition occurs at q K!1.3(jq /2p) . At the strong random "eld the temperature of the phase transition is proportional to D . 11.3. Stability of the replica symmetric solution In this subsection we consider the stability of the replica symmetric solution (11.9) with respect to replica symmetry breaking. With this purpose let us represent the renormalized two-replica correlation function G (q) in the following form: ?@ G (q)"g (q)d #*g (q)e2e #dQ (q) , (11.16) ?@ ?@ ? @ ?@ where e"(1,2, 1) and dQ (q) is a function that equals zero for a"b. Substituting the function ?@ G (q) into the r.h.s. of the n-replica free energy (11.8) and expanding Sp ln(G ) in the power of the ?@ ?@ function dQ (q) one "nds ?@ d 2D 1 @A ! e2e dQ (q)dQ (q) *F " ?@ A? L 4 g (q) g (q) @ A O j dQ (k)dQ (p) f (q) f (q!k!p) , ! (11.17) ?@ ?@ 2 O I N
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Fig. 28. The phase diagram of the copolymer melt in a random "eld environment obtained within the one-loop replica symmetric solution. The continuous curve is the coexistence curve between the ordered (lamellar) and disordered states. It is computed by setting equal Eqs. (11.14)}(11.15). The dashed line is the stability line of the replica symmetric solution. It is derived by solving Eqs. (11.21), (11.12).
where in the r.h.s. of Eq. (11.17) the summation over all repeat indices is assumed. The analysis of the Dyson equation for the o!-diagonal part of the two-replica correlation function including the second-order diagram in the power of vertex j shows that the #uctuations with "q""q give the main contribution to the renormalization of the bare characteristic of the system, so that we can choose dQ (q) in the form dQ (q)"Q g (q), where Q is an n;n matrix. Introducing the Parisi ?@ ?@ ?@ ?@ function q(x) [8] de"ned in the interval [0, 1] and connected to Q by ?@ 1 QI ∀, k , (11.18) qI(x) dx"lim ?@ n(n!1) ? @ L the quadratic part of the free energy in power of Q is obtained as ?@ 1 ((I !I )d(x!y)#I )q(x)q(y) dx dy , *F"! (11.19) 4 where
I "(q /4p)r\ , I "(1/512p)jDq r\, I "(3/8p)q *r\ . (11.20) The replica symmetric solution is unstable in the region of parameters where the matrix (I !I )d(x!y)#I has a negative eigenvalue j "I #I !I 10 (or in the reduced variables \ (11.11)) j &(1#d/z!c d/z )40 , (11.21) \ where c"((2p) /64)(j/q ) (1. Eq. (11.21) determines the spinodal line of the replica symmetric solution. The boundary of the RS solution, which is computed by using Eqs. (11.21), (11.12) for the value c"0.5, is also plotted in Fig. 28. It follows that RSB occurs already in the disordered phase. 11.4. Disorder versus ordering in the lamellar phase In the weak segregation limit for which the local composition #uctuations are small in comparison with its average value we can use Hamiltonian (11.2) to describe the system below microphase
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separation transition. Keeping only gradient and random "eld terms the e!ective Hamiltonian is
1 (No )
H(+W(r),)" (( #q )U(r))! h(r)U(r) . 8q R P P E
(11.22)
For periodic structure in the z-direction the average value of the order parameter 1U(r)2 is 1U(r)2"2A cos(q (z#u(r))) ,
(11.23)
where scalar function u(r) describes the deformation of the layers in the z-direction. Substituting Eq. (11.23) into the r.h.s. of Eq. (11.22), one obtains after averaging over all possible con"gurations of the random "eld h(r) the e!ective Hamiltonian for function u(r) as
L A H (+u (r),)" ? ((* u (r))#4q ( u (r))) V W ? X ? L ? 4 P ? L (11.24) ! *A A cos (q (u (r)!u (r))) , ? @ ? @ P ? @ where the "rst term in the r.h.s. of Eq. (11.24) describes the deformation of the lamellar layers and the second one couples this deformation in di!erent replicas. It is interesting to note that the cosine-like coupling term between #uctuations of the order parameter in two di!erent replicas appears in disordered physical systems such as an array of #ux-line in type II superconducting "lm in magnetic "lm [93], a crystalline surface with a disordered substrate [94], random "eld X> model [95]. In all these systems the cosine term results in the breaking of the long-range order and spontaneous replica symmetry breaking [96]. In the framework of the Gaussian variational principle [62] the contribution to the free energy of the system due to #uctuations of the displacement u (r) in di!erent replicas is ?
1 F "! Sp ln(G (q))#1H(+u (q),)!H 2 , ?@ ? 2 O
(11.25)
where we introduced
1 L G\(q)u (q)u (!q) (11.26) H " ?@ ? @ 2 ? @ O and G\(q) is the two-replica trial function whose form has to be de"ned self-consistently and the ?@ brackets 122 denote the thermal averaging with the weight exp(!H ). Analyzing the last equation one can conclude that the function G\(q) does not depend on the momentum q for aOb. ?@ So, we can de"ne G\(q)"!p (aOb). In the case of the one-step replica symmetry breaking for ?@ ?@ which the elements of the matrix p are assumed to have two di!erent values p and p depending ?@ on whether the two indices a and b belong to the same blocks of the length m or not one can rewrite the extremal equations as p "> exp(g ln t), p "> exp(g ln t!(1/m)(2g ln ¸q #g ln t)) ,
(11.27)
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where ¸ is the linear size of the system and we introduced the following parameters assuming A to ? be the same in the all replicas and equal A: g"q /16pA, t"2m(p !p )/Aq , >"2*q A . (11.28) In the thermodynamic limit ¸PR Eq. (11.27) gives p "0. Substitution of the solution for the trial function G (q) into variational free energy yields [90] ?@
(1!m)tE , (11.29) f "lim (F (t)!F (0))" 4 m n L where > "2>/(Aq )"4D/q is introduced. The equilibrium values of the parameters m and t can be found from the system of the equations
(1/m)gt!> tE"0, (1!(1/m))#> (1!m)tE\"0 .
(11.30)
For g'1 this system has only trivial solution m"1 and t"0 that corresponds to the replica symmetric solution with all o!-diagonal elements of the matrix G (q) equal to zero. The nontrivial ?@ solution appears for g41, that reads m"g, t"(> g) \E .
(11.31)
In other words, at g"1, the system undergoes a phase transition for which the correlation function G (q) changes the form. We can rewrite the condition g"1 in terms of the parameters of the ?@ system. In the mean-"eld approximation A"2"q"/j that gives the e!ective temperature of the phase transition "q""q j/32p. Comparing this temperature with the temperature of the "rst-order phase transition "q"+(q jD) one can see that for D5q j there is a "rst-order phase transition between the disordered state (A"0) and the ordered phase (AO0) with one-step replica symmetry breaking for the correlation function 1u (q)u (!q)2. The form of the correlation ? @ function, which was computed in [90], shows that there are two di!erent regions. Inside the domains of size "x !x "( m and "z!z "(m , where m q "t\ and m q "t\ are the , , V X X V correlations lengths in the z and x, y directions, the system behaves like skmectic A [97]. On the larger length scales the #uctuations of the layer displacement u(r) wash out the long-range modulated order and result in formation of the highly anisotropic translational incoherent domains with m /m &D\ \E. Therefore, we may conclude that the symmetric AB copolymers X V in the gel matrix with a weak preferential adsorption of A monomers is an example of a random"eld system. By using the method of the 2nd Legendre transform we have shown that two-replica correlation function is unstable with respect to replica symmetry breaking at "nite values of the random "eld. We interpret this phase as glassy state. The stability of the lamellar phase in the weak segregation limit was considered by mapping the copolymer Hamiltonian onto the Hamiltonian of the random "eld XY model. In the ordered phase (AO0) very weak random "eld D&¸\\E destroys the long-range modulated order, resulting in formation of the highly anisotropic translational incoherent domains. 12. Final remarks Several remarks should be made at this point. In this review we considered some aspects of the statistical mechanics of #exible polymers. An important aspect which was ignored completely are
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entanglements and knots. Obviously, entanglements and knots have important properties for #exible polymers and rule more or less the dynamics of long chains in concentrated systems such as strong solutions and melts [10]. There is also progress on knots and topological problems in polymer physics [98}100] in the methodological framework which has been discussed here in this review. Especially in [100] we presented results on entangled DNA rings. There we computed the size distribution as a function of the degree of knotting between two DNA rings, which is in good agreement with experimental observations [101,102]. Most of the recent developments are summarized in a review paper [103]. A "eld theory for entangled polymers has been formulated by Ferrari [104]. Numerical simulations yield also deep insight for this problem [105]. A new developing subject especially important for biological physics comprises problems concerning semi-#exible polymers [106,107]. Polymers in reality are never #exible at short distances. In biological polymers e!ects stemming from these properties are of importance, much progress has been made in this direction, see e.g. [108,109] and references therein. These biological molecules, such as actin, form networks at certain time scales and new processes appear. There are several new results [110,111]. This review is far from being complete and the author apologizes for that. The choice of the subjects has been subjective and guided by the authors own work and his coworkers in the recent years.
Acknowledgements The author acknowledges the excellent collaboration with his coworkers and friends, who made this possible over the past years, namely Andrea Weyersberg, Jochen Eckert, Matthias Otto, Gregor Huber, Michael Solf, Peter Haronska, Mabrouk Benhamou, Michael Brereton, Mustapha Benmouna, Semjon Stepanow and many others. Without their help, work and friendship these lectures could never have been held. Special thanks to Mrs. Charlotte Muench and Mrs Irene Nanz. Without them this paper would never have been "nished. The author would like to express his special thanks also to Dino Leporini from Pisa. Parts of this review have been the subject of a summer school for condensed matter physics in Torino in Italy. Apart from his great help in all aspects concerning the scuolo, he opened his eyes and papillas for Roero, Neive, Barbaresco, Barolo and related pleasure. Mille grazie, Dino! Parts of this paper have been written at the Institut Charles Sadron in Strasbourg, France. The authors thanks Jean-Franc7 ois Joanny, Albert Johner and Henri Benoi) t for many discussions, their friendship, and moral support. L'auteur aimerait bien dire merci aussi a` Jacques, qui lui a montreH un chemin nouveau.
References [1] [2] [3] [4] [5]
S.F. Edwards, Proc. Phys. Soc. 85 (1965) 613. R.P. Feynman, A.R. Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill, New York, 1965. M. Doi, S.F. Edwards, The Theory of Polymer Dynamics, Clarendon Press, Oxford, 1986. P.-G. de Gennes, Scaling Concepts in Polymer Physics, Cornell University Press, Ithaca, 1979. P.G. de Gennes, Phys. Lett. A 38 (1972) 339.
252 [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51]
T.A. Vilgis / Physics Reports 336 (2000) 167}254 J. des Cloizeaux, G. Jannik, Polymers in Solution; Their Modelling and Structure, Clarendon Press, Oxford, 1990. A.J. McKane, Phys. Lett. A 76 (1980) 22. G. Parisi, N. Sourlas, J. Phys. Lett. 41 (1980) L403. S.F. Edwards, J. Phys. A 8 (1975) 1670. M. Doi, S.F. Edwards, The Theory of Polymer Dynamics, Clarendon Press, Oxford, 1986. P.G. de Gennes, Scaling Concepts in Polymer Physics, Cornell University Press, Ithaca, 1979. G. Parisi, Statistical Field Theory, Addison-Wesley, Redwood City, 1978. S.K. Ma, Modern Theory of Critical Phenomena, Benjamin, Reading, MA, 1976. R.P. Feynman, A.R. Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill, New York, 1965. R. Zallen, The Physics of Amorphous Solids, Wiley, New York, 1983, 0-471-01968-2. C. Kittel, Introduction to Solid Physics, 7th Edition, Wiley, New York, 1996. F.W. Wiegel, Introduction to Path-Integral Methods in Physics and Polymer Science, World Scienti"c, Singapore, 1986. H. Kleinert, Path Integrals in Quantum Mechanics, Statistics and Polymer Physics, World Scienti"c, Singapore, 1990. J.P. Hansen, I. MacDonald, Theory of Simple Liquids, Academic Press, New York, 1986. M. Fixman, J. Chem. Phys. 23 (1955) 1656. J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, 2nd Edition, Clarendon Press, Oxford, 1993. J. Cardy, Scaling and Renormalization in Statistical Physics, Cambridge University Press, Cambridge, MA, 1996. L. SchaK fer, Phys. Rep. 301 (1998) 205. L.E. Reichl, A Modern Course in Statistical Physics, Edward Arnold Publishers, London, 1987. S.F. Edwards, Proc. Phys. Soc. 85 (1965) 613. C. Itzykson, J.M. Drou!e, Statistical Field Theory, Cambridge University Press, Cambridge, MA, 1989. S.F. Edwards, P.W. Anderson, J. Phys. F 74 (1975) 965. M. MeH zard, G. Parisi, M.V. Virasoro, Spin Glasses and Beyond, World Scienti"c, Singapore, 1987. D.J. Amit, Field Theory, the Renormalization Group and Critical Phenomena, World Scienti"c, Singapore, 1984. P. Pfeuty, G. Toulouse, Introduction to the Renormalization Group and the Critical Phenomena, Wiley, London, 1977. L.H. Ryder, Quantum Field Theory, Cambridge University Press, Cambridge, MA, 1985. B.F. Hat"eld, Quantum Field Theory of Point Particles and Strings, Addison-Wesley, Redwood City, 1989. A. McKane, Phys. Lett. A 76 (1980) 22. G. Parisi, N. Sourlas, J. Phys. Lett. 41 (1980) L403. P. Pfeuty, G. Toulouse, Introduction to the Renormalization Group and to Critical Phenomena, Wiley, London, 1977. B. Duplantier, H. Saleur, Phys. Rev. Lett. 60 (1988) 2343. T.A. Vilgis, M. Benmouna, H. Benoit, Macromol. 24 (1991) 4481. L. Leibler, Macromolecules 13 (1980) 1602. K. Binder, Adv. Polym. Sci. 112 (1994) 181. W. Theobald, A. Sans-Penninckx, G. Meier, T. Vilgis, Phys. Rev. E 55 (1997) 5723. T.A. Vilgis, R. Borsali, Macromolecules 23 (1990) 3172. A. Weyersberg, T.A. Vilgis, Phys. Rev. E 48 (1993) 277. T.A. Vilgis, A. Weyersberg, M.G. Brereton, Phys. Rev. E 49 (1994) 3031. H. Fried, K. Binder, J. Chem. Phys. 94 (1991) 8349. M. Cates, J. Phys. France 46 (1985) 1059. T.A. Vilgis, J. Phys. II France 2 (1992) 1961. T.A. Vilgis, J. Phys. II France 2 (1992) 2097. T.A. Vilgis, Physica A 153 (1988) 341. D. Nelson, T. Piran, S. Weinberg, Statistical mechanics of membranes and surfaces, World Scienti"c, Singapore, 1989. M. Daoud, P.G. de Gennes, J. Phys. 38 (1977) 85. C. Gay, P.G. de Gennes, E. RaphaeK l, F. Brochard-Wyart, Macromolecules 29 (1996) 8379.
T.A. Vilgis / Physics Reports 336 (2000) 167}254 [52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66] [67] [68] [69] [70] [71] [72] [73] [74] [75] [76] [77] [78] [79] [80] [81] [82] [83] [84] [85] [86] [87] [88] [89] [90] [91] [92] [93] [94] [95] [96] [97] [98] [99] [100] [101]
253
E. Raphael, P. Pincus, J. Phys. II 2 (1992) 1341. F. Brochard, P.G. de Gennes, J. Phys. (France) 40 (1979) L399. M. Daoud, J. Cotton, J. Phys. France 40 (1982) 531. S.F. Edwards, in: A. Chrompf, S. Newman (Eds.), Polymer Networks, Plenum Press, New York, 1971. R.T. Deam, S.F. Edwards, Proc. Trans. Roy. Soc. London A 280 (1976) 317. S.F. Edwards, T.A. Vilgis, Rep. Prog. Phys. 52 (1988) 247. G. Parisi, Phys. Rev. Lett. 43 (1979) 1754. S.F. Edwards, M. Muthukumar, J. Chem. Phys 89 (1988) 2435. R.P. Feynman, Statistical Mechanics, Addison-Wesley, Reading, MA, 1972. R.P. Feynman, Statistical Mechanics: a Set of Lectures, Benjamin, Reading, MA, 1992. M. MeH zard, G. Parisi, J. Phys. I (France) 1 (1991) 809. J. Machta, T. Kirkpatrick, Phys. Rev. A 40 (1990) 5345. M. Cates, R.C. Ball, J. Phys. (France) 49 (1988) 2009. E. Eisenriegler, Polymers Near Surfaces, World Scienti"c, Singapore, 1993. G.J. Fleer et al., Polymers at Interfaces, Chapman & Hall, London, 1993. G. Heinrich, T. Vilgis, Rubber Chem. Technol. 68 (1995) 26. T.A. Vilgis, G. Heinrich, Macromolecules 27 (1994) 6743. C. van der Linden, B. van Lent, F. Leermakers, G. Fleer, Macromolecules 27 (1994) 1915. D. Andelman, J. Joanny, J. Phys. II 3 (1993) 121. A. BaumgaK rtner, M. Muthukumar, J. Chem. Phys. 94 (1991) 4062. M. Blunt, W. Barford, R. Ball, Macromolecules 22 (1989) 1458. R. Ball, M. Blunt, W. Barford, J. Phys. A (Math. Gen.) 22 (1989) 2587. M. Douglas, Macromolecules 22 (1989) 3707. D. Hone, H. Ji, P. Pincus, Macromolecules 20 (1987) 2543. J. Feder, Fractals, Plenum Press, New York, 1988. J. Charvolin, J. Joanny, J. Zinn-Justin, Liquids at Interfaces, Les Houches 1988 Summer School XLVIII, North-Holland, Amsterdam, 1988. T. Garel, H. Orland, Phys. Rev. B 55 (1997) 226. G. Huber, T.A. Vilgis, Eur. Phys. J. B 3 (1998) 217. F.S. Bates, G.H. Fredrickson, Ann. Rev. Phys. Chem. 41 (1990) 525. G.H. Fredrickson, S.T. Milner, Phys. Rev. Lett. 67 (1991) 835. C.D. Sfatos, A.M. Gutin, E.I. Shakhnovich, Phys. Rev. E 48 (1993) 465. E.I. Shakhnovich, A.M. Gutin, Nature 346 (1990) 773. G.H. Fredrickson, S.T. Milner, L. Leibler, Macromolecules 25 (1992) 6341. G.H. Fredrickson, L. Leibler, Macromolecules 23 (1990) 531. S. Brazovskii, I. Dzaloshinskii, A. Muratov, Sov. Phys. JETP 66 (1975) 625. S.G. Brazovskii, Sov. Phys. JETP 41 (1975) 85. R. Sawyer, Phys. Rev. Lett. 29 (1972) 382. K. Binder, Adv. Pol. Sci. 112 (1994) 181. S. Stepanow, A. Dobrynin, T.A. Vilgis, K. Binder, J. Phys. I (France) 6 (1998) 837. P.G. de Gennes, J. Chem. Phys. 88 (1987) 6469. J. Cornwall, R. Jackiw, E. Tomboulis, Phys. Rev. D 10 (1974) 2428. J. Cardy, S. Ostlund, Phys. Rev. B 25 (1983) 6899. Y.-C. Tsai, Y. Shapir, Phys. Rev. Lett. 69 (1992) 1773. M. Fisher, Phys. Rev. Lett. 62 (1989) 1415. P.L. Doussal, T. Giamarchi, Phys. Rev. Lett. 74 (1995) 606. P. de Gennes, J. Prost, The Physics of Liquid Crystals, Clarendon Press, Oxford, UK, 1993. M.G. Brereton, T.A. Vilgis, J. Phys. A 28 (1995) 1149. M. Otto, T.A. Vilgis, J. Phys. A 29 (1996) 1451. M. Otto, T.A. Vilgis, Phys. Rev. Lett. 80 (1998) 881. C.L. Adams, The Knot Book, Freeman, New York, 1994.
254 [102] [103] [104] [105] [106] [107] [108] [109] [110] [111]
T.A. Vilgis / Physics Reports 336 (2000) 167}254 S.D. Levene, C. Donahue, T.C. Boles, N.R. Cozzarelli, Biophys. J. 69 (1995) 1036. A.L. Kholodenko, T.A. Vilgis, Phys. Rep. 298 (1998) 251. F. Ferrari, I. Lazzizzera, hep-th/9903084, preprint, 1998. R. Everaers, K. Kremer, Phys. Rev. E 53 (1996) R37. J. des Cloizeaux, Macromolecules 6 (1973) 403. M. Otto, J. Eckert, T.A. Vilgis, Macromol. Theory Simulation 3 (1994) 543. A.L. Kholodenko, T.A. Vilgis, Phys. Rev. E 59 (1994) 1257. D.K. Klimov, D. Thirumalai, Proc. Natl. Acad. Sci. USA 96 (1999) 6166. E. Frey, K. Kroy, J. Wilhelm, Polymer Networ Group Series 2, preprint, 1999. K. Kroy, E. Frey, Scattering in polymeric and colloidal systems, preprint, 1999.
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RESONANT INTERACTIONS OF DIATOMIC MOLECULES WITH INTENSE LASER FIELDS: TIME-INDEPENDENT MULTI-CHANNEL GREEN FUNCTION THEORY AND APPLICATION TO EXPERIMENT
Alexander I. PEGARKOV Physics Faculty, Voronezh State University, 1 University Square, Voronezh 394693, Russia
AMSTERDAM } LAUSANNE } NEW YORK } OXFORD } SHANNON } TOKYO
Physics Reports 336 (2000) 255}411
Resonant interactions of diatomic molecules with intense laser "elds: time-independent multi-channel Green function theory and application to experiment Alexander I. Pegarkov Physics Faculty, Voronezh State University, 1 University Square, Voronezh 394693, Russia Received March 2000; editor: S. Peyerimho!
Contents 1. Introduction 2. Diatomic molecule in intense resonant laser "eld. Quantum-electrodynamic picture 2.1. Molecular Hamiltonian 2.2. Laser "eld Hamiltonian 2.3. Hamiltonian of laser}molecule interaction 2.4. Quantum-electrodynamic states of diatomic molecule in laser "eld 2.5. Radial, rotation and laser-induced couplings. Applicability of multi-channel quantum-electrodynamic approach 3. Multi-channel Green's functions 3.1. Matrix form for the solutions of multichannel SchroK dinger equation 3.2. Time-independent Green's function of multi-channel SchroK dinger equation 3.3 Spectral expansion for multi-channel Green's function 3.4 Expression of multi-channel Green's function through the one-matrix solution 4. Quasi-classical approximation for multichannel systems 4.1. Quasi-classical propagation of multichannel solutions through singular points
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4.2. Regular solutions of multi-channel SchroK dinger equation 4.3. Restrictions of the multi-channel quasi-classical approximation 5. Delay of direct photodissociation and orientation of molecules in intense resonant "elds 5.1. Direct molecular photodissociation 5.2. Direct photodissociation in multi-channel quasi-classical approximation 5.3. High-order perturbation theory limit of the non-perturbative quasi-classical photodissociation cross-section 5.4. Over-excited molecule and laser-induced quasi-bound nuclear states 5.5. Non-linear photodissociation of noble gas dimers. Numerical results 5.6. Stable and unstable quasi-bound states. Softening and hardening of bonds in an over-excited molecule 5.7. Molecular orientation in resonant laser "elds 6. Non-adiabatic channel of resonant photoabsorption in intense laser "elds
E-mail address: [email protected] (A.I. Pegarkov). 0370-1573/99/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 0 0 ) 0 0 0 5 3 - 3
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A.I. Pegarkov / Physics Reports 336 (2000) 255}411 6.1. Non-linear photoabsorption in intense resonant "eld 6.2. Energy spectrum of the CO molecule in the F -excimer laser with j"158 nm 6.3. Resonant photoabsorption by the CO molecule in the j"158 nm F -laser. Numerical results 6.4. Mechanism of resonant one-photon absorption in resonant laser "eld 6.5. Experimental observation of nonlinear laser absorption in vapours and solutions 7. Laser-induced resonances in two-photon dissociation and three-photon ionization 7.1. Resonant two-photon dissociation in intense "elds 7.2. Laser-induced resonances in the Na dissociation 7.3. Resonant three-photon ionization in intense "elds 7.4. Resonant three-photon dissociative ionization in intense "elds 7.5. Laser-power dependence of resonant ionization yield 7.6. Laser-induced resonance in threephoton ionization of Na 8. Dynamic polarizability of diatomics beyond adiabatic approximation 8.1. High-order perturbation theory method to include the non-adiabatic couplings to polarizability calculations 8.2. Multi-channel Green's functions method to include the non-adiabatic couplings to polarizability calculations 8.3. Non-adiabatic calculations of dynamic polarizabilities 9. E!ect of dynamic and laser-induced nonadiabatic interactions upon resonance Raman scattering 9.1. Non-adiabatic theory of resonance Raman scattering of intense laser radiation from rotating diatomic molecules 9.2. Non-adiabatic calculations of resonance Raman scattering from IBr and Ar>
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9.3. The RRS ampli"cation due to laserinduced quasi-bound states 10. New spectral lines of VUV absorption in presence of powerful IR radiation 10.1. One-photon molecular absorption in two-colour laser "eld 10.2. Two-photon molecular absorption in two-colour laser "eld 10.3. The VUV photoabsorption spectrum of N in "eld of intense Nd:YAG-laser with j"1.064 lm 11. Trends in experiment Acknowledgements Appendix A. Wave functions of some multichannel systems in quasi-classical approximation A.1. Multi-channel solutions A.2. Two-channel wave function for resonant photoabsorption A.3. Two-channel wave function for two-colour absorption Appendix B. Some multi-channel Green's functions in quasi-classical approximation B.1. Two-channel Green's functions B.2. Three-channel Green's functions for term system &&attractive#attractive#attractive'' B.3. Quasi-classical multi-channel Green's functions in the limits of weak and strong non-adiabatic couplings Appendix C. Stationary phase estimation for nuclear integrals C.1. Stationary phase method C.2. Application to nuclear integrals Appendix D. Analytic approximation for molecular potentials D.1. Morse-type potential D.2. Exponent-type potential Appendix E. Exact wave functions of the system &&(quasi-) molecular ion#photoelectron'' E.1. System &&molecular ion#photoelectron'' E.2. System &&quasi-molecular ion#photoelectron'' Appendix F. Bipolar compound harmonics Appendix G. Exact analytic equations for highorder tensors of resonance Raman scattering from rotating diatomics References
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381 382 386 386 387 388 390
391 394 394 394 396 396 398 398 398 400 400
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Abstract The resonant interactions of diatomic molecules with UV and VUV intense laser radiation are investigated within an original non-perturbative approach of multi-channel nuclear wave functions and Green's functions. The interactions are considered in a quantum-electrodynamic picture where the laser "eld and molecule are treated as quantum objects. The presented theoretical approach is formulated as a multiquantum technique to calculate analytically the resonant laser}molecule interactions and molecular photoprocesses. The method permits to include the non-linear impact of a resonant laser upon intra-molecular motion beyond an adiabatic approximation and may be applied for laser intensities until 10 W cm\. The theory of wave functions and Green's functions of multi-channel SchroK dinger equation is elaborated without perturbative restrictions on inter-channel couplings. The multi-channel quasi-classical approximation is developed to integrate the coupled equations for nuclear motion in simple matrix form. The theoretical approach proposed is used to study the intense-"eld photodissociation and photoabsorption, multi-photon resonant ionization and dissociation, dynamic polarizability and resonance Raman scattering from rotating diatomic molecules. The well-known laser-induced hardening and softening of intra-molecular bonds are reformulated. The new resonances in the molecular absorption, ionization and dissociation, arisen in resonant laser "elds, are examined too. The phenomena of resonant laser "eld selective orientation and alignment of non-dipole diatomic molecules, laser-induced molecular dipole moment, temperature-dependent polarization of molecular gases, non-perturbative ampli"cation of scattered laser radiation in resonant medium are analysed in detail. The known experimental observations of laser-induced non-linear e!ects and circumstances for new strong-"eld experiments are discussed. 2000 Elsevier Science B.V. All rights reserved. PACS: 42.50 Hr; 42.65.Vh; 33.20.Lg; 33.80.Rv Keywords: Resonant interactions; Non-linear phenomena; Molecules; Lasers; Green's functions
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1. Introduction Laser}molecule interaction impacts upon full quantum states of the system `molecule#laser xeld a. The interaction changes the electronic}rotation}vibration states of the molecule and laser "eld. The most interesting laser}molecule processes are those where the laser radiation is in resonance with quantum transitions in the molecules. In this case the di!erence of total energies of the compound system in its initial and "nal states may be much less than the energy of a state-to-state transition in free molecule and, as a result, the radiation interaction, needed to induce a non-linear phenomenon, may be much weaker than that in a non-resonant "eld. This fact implies important peculiarities of the resonant laser}molecule interaction and brings considerable di!erences between the dynamics of resonant molecular photoprocesses and that of non-resonant processes. The radiation of either ultraviolet (UV) or vacuum-ultraviolet (VUV) frequencies is always in resonance with the transitions between electronic molecular terms. Therefore, the physics of the molecular photoprocesses in the UV, VUV laser "elds di!ers from the physics of the non-resonant processes in intense laser "elds of infra-red or optical diapasons. The molecular photoprocesses in o!-resonant laser "elds have multi-photon or one-step nature, where few non-resonant photons are absorbed (emitted) by the molecule at the same time and the molecular quantum states are non-perturbed by the "eld. Here the laser}molecule interaction makes no impact upon intra-molecular motion. The non-resonant photoprocesses in the diatomic molecules are analogous with the multi-photon processes in atoms and have been studied carefully in [1}5]. The photoprocesses in resonant laser "elds have a non-adiabatic or a step-wise nature where one or few resonant photons are absorbed (emitted) by the molecule whose quantum states are perturbed by the laser and the laser}molecule interaction impacts on the intra-molecular motion signi"cantly. The resonant laser radiation induces non-adiabatic interactions of resonant molecular terms, which depend on the laser frequency, intensity and polarization and invokes non-linear phenomena in molecular photoprocesses. Because the probabilities of the resonant photoprocesses are higher than those in the o!-resonant "elds, weaker laser intensities are needed in order to observe the non-linear e!ects in the resonant "elds than to do it in the o!-resonant "elds. Moreover, the resonant "elds create the picture of laser}molecule interaction being quite di!erent from the non-resonant interaction due to a laser-induced non-adiabatic coupling of resonant molecular terms. Firstly, the quantitative theory of molecular photoprocesses was developed within the Born}Oppenheimer approximation where the rotation}vibration, rotation-electronic, vibrationelectronic, and laser}molecule non-adiabatic couplings were omitted [1}7]. This approach is valid as for the multiphoton non-resonant processes in the laser "elds of intensity I410 W cm\ as for the resonant processes in the weak laser xelds I410 W cm\. The Born}Oppenheimer approximation is violated either if the non-adiabatic couplings are intense or if the laser radiation is in resonance with intra-molecular energy transitions. The non-adiabatic rotation}vibration interaction is considerable in resonant laser "eld of IR diapason [8]. The rotation-electronic and vibration-electronic couplings are important in optical and weak near-UV radiation [9]. The laser}molecule non-adiabatic coupling has to be taken into account for the diatomic molecules in the UV or VUV "elds. There the laser radiation is in resonance with electronic
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transitions and the nuclear motion in the resonant electronic terms is mixed non-perturbatively by the "elds of intensity I510 W cm\. As a result, the resonant photoprocesses may not be considered more within the picture of isolated non-perturbed molecular terms. The molecular nuclei move through compound electronic potential formed from the resonant electronic terms dressed by the laser xeld. The compound potentials involve the laser}molecule interaction via the radiation non-adiabatic coupling. The coupling is a function of laser intensity, frequency and polarization and modi"es the nuclear motion. The dynamics of nuclear motion in the new complicated electronic potential depends on laser power, that permits to use the powerful lasers as a tool to control and to operate on the photochemical processes in the diatomic molecules [11,12]. Due to the presence of the laser-induced non-adiabatic coupling, the adiabatic Born}Oppenheimer approach is inapplicable to consider the resonant photo-processes in diatomic molecules in intense laser "elds. If the "eld is strong as 10 W cm\;I;10 W cm\, the resonant photoprocesses may be considered within the Fedorov's strong-xeld adiabatic approximation [13]. The Fedorov's strong-"eld approximation is inapplicable to study the processes in laser "elds of intensity 10 W cm\4I410 W cm\ (the so-called "elds of moderate intensities), for which the perturbation theory approach of Bunkin and Tugov [1}5] is inapplicable too. Therefore, it is interesting to develop a theoretical approach which may be valid to consider the resonant laser}molecule interaction for all the laser intensities as in weak, moderate, and strong "elds like I410 W cm\. The following theoretical methods have been elaborated originally to study the resonant laser}molecule interactions in a time-independent picture: E E E E
methods of laser-induced non-adiabatic transitions [10,14,15], complex quasi-vibration energy formalism [16,17], coupled equations approach [18], method of non-adiabatic wave functions and Green's functions [19,20].
The scope of this review is restricted to the photoprocesses in picosecond and microsecond laser pulses where the time-independent picture is well applicable to describe the laser}molecule interactions. The molecular photoprocesses in laser pulses of femtosecond duration are far from the subject of the review and all the time-dependent approaches including the modern achievements [21}41] are not discussed here. The developed time-independent approaches [10,14}20] have permitted to analyse theoretically few important non-linear resonant photoprocesses before they have been observed experimentally.
The depicted threshold of violation of perturbation theory approach depends on the irradiated molecule and its activated resonant electronic transition. So, the breakdown threshold is equal to: 1.2;10 W cm\ for the AR>!XR> transition in the Na molecule, 1;10 W cm\ for the LiH (AR>!XR>) transition, and 2.3;10 W cm\ for the XeF(III 1/2!X 1/2) one [10]. The lower and upper bounds of this interval of intensity depend on the molecule and dipole moment of its resonant electronic transition: 1.2;10 W cm\4I45;10 W cm\ stands for the Na (AR>!XR>) transition, 1.0;10 W cm\4I41.2;10 W cm\ is for LiH(AR>!XR>), and 2.3;10 W cm\4I43.9;10 W cm\ is for XeF(III 1/2!X 1/2). For the examples given above the intensity threshold is exactly as follows: I42;10 W cm\ for Na (AR>! XR>), I42;10 W cm\ for LiH(AR>!XR>), and I43;10 W cm\ for XeF(III 1/2!X /12). For details see subsection 2.5.
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In the "rst papers [10,15] the resonant photodissociation of diatomic molecules was considered as a non-adiabatic transition induced by resonant laser xeld within semi-classical non-adiabatic picture. The principal ideas of semi-classical theory of non-adiabatic transitions under slow atomic collisions [42,43], were developed there and the equations for collisional probabilities, obtained earlier, were generalized upon the bound}free laser-induced transitions in the molecules. The complex quasi-vibration energy formalism [16,17] studied one-photon and multi-photon dissociation and ionization of molecules within the quasi-energy approach proposed "rstly by Shirley [51] and the complex coordinates method of Reinhardt [52]. This approach permits to make the high-precise computer calculations to receive the transition probabilities only in the case where one of the molecular states is unbound. Therefore, there is the important restriction of the complex coordinates method: the complex quasi-vibration energy formalism may be applied to study either ionization or dissociation processes but not the processes involving only the bound molecular states. We emphasize that the quasi-energy approach itself [51] does not have such limitations and may be generalized upon intense-"eld bound}bound multi-photon and non-linear optical processes of di!erent types [54]. The coupled equations approach [18] makes it possible to integrate the non-diagonal system of second-order di!erential equations for nuclear motion, coupled non-adiabatically, and it extracts the non-perturbative transition probabilities of the multi-photon processes from it. The coupled equations method is correct for the weak, moderate and strong laser "elds [55]. This approach may be applied to bound}bound and bound}free transitions but needs to introduce artixcial channels in order to use an S-matrix computation algorithm. The arti"cial channels were introduced and justi"ed "rstly by Shapiro for the direct photodissociation [56]. The method of non-adiabatic nuclear wave functions and Green's functions [19,20] was developed in order to obtain the transition probability equations in closed analytic form without a numerical integration of the nuclear Schro( dinger equation. It was possible to do so due to elaboration of an original theoretical method to solve the multi-channel system of second-order di!erential equations and to receive its Green function analytically in a simple matrix representation. The method of non-adiabatic wave functions and Green's functions considers the full quantum non-adiabatic states of the `molecule (or molecular pair,quasi-molecule)#laser xeld a system too, but, in contrast to the previous approaches [10,14}18], calculates the time-independent transition probabilities of various resonant photoprocesses without matrix diagonalization or arti"cial channels. The non-linear e!ects induced by the intense resonant laser "eld in molecular photoprocesses were discussed "rstly theoretically in [10,13}16]. The appearence of new vibration states induced by the resonant laser}molecule interaction had been predicted in 1975 by Fedorov [13]. In 1978, 1981 new radiation non-linear e!ects such as laser-induced avoidance of dressed molecular terms crossing, particle trapping by laser-induced potential well, laser-induced impact upon molecular orientation and transparence of molecular medium were considered theoretically in papers [10,15]. In 1987, 1988 a possibility to observe experimentally a delay of direct molecular photodissociation in
This semi-classical non-adiabatic method, been very e!ective in theory of radiative-collision phenomena [44}48], has been generalized also to obtain the non-adiabatic probabilities for radiation and non-radiation bound}bound transitions [49,50]. For an excellent review on the complex coordinates method see [53].
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the intense resonant laser xeld had been analysed quantitatively in [57}59]. In [57,58] the e!ect has been attributed to quasi-bound vibration states of the molecule, created by the resonant "eld in molecular continuum. For the "rst time, this non-linear e!ect had been observed by Zavriyev et al. in 1990 for hydrogen molecules [60] and named as suppression of photodissociation or molecular stabilization in intense laser "elds [61}63]. New interesting results for the intense-"eld photodissociation have been obtained recently by Wunderlich et al. in experiments with Ar> molecules [64,65]. In 1985 the generation of new resonances and complicated non-linear dependence of spectral line intensities on laser power had been observed "rstly in multi-photon intense-"eld ionization of sodium vapours [66,67]. The multi-channel nuclear Green's functions have been applied in order to explain these e!ects and to develop their quantitative non-adiabatic theory [19,68}70]. Our theoretical consideration has deduced that the resonant intense laser radiation can induce new resonant peaks in the Na ionization spectrum and alter non-linearly the intensities of ionization spectral resonances. In 1992 the experiments of Baumert et al. [71,72] showed the new laserinduced peaks in the sodium ionization spectra and con"rmed our earlier theoretical predictions. In 1987 a non-adiabatic theory of two-photon resonant dissociation of sodium vapour has been developed within the multi-channel nuclear Green's function approach in our paper [57]. In that case the laser radiation was in resonance with electronic transition from ground term to one of the excited attractive molecular terms and from it to a high-excited dissociation term. The couplings of the attractive terms modi"ed the spectrum of the two-photon dissociation. It was obtained that the frequencies of spectrum resonances were shifted and new dissociation resonance arose as the laser power increased. As it was emphasized in [57], the powerful resonance radiation of a second laser may be used in order to observe the laser-induced e!ects in the photodissociation from the "rst laser. The Na molecule has been chosen for calculation in [57] in order to display the laser-induced non-linear e!ects because the Na electronic dipole transition moment was high enough (about 10 Deb). The magnitude of the moment made it possible to observe the non-linear laser-induced e!ects in the Na photoprocesses in laser "elds of intensity I410 W cm\. This is the moderate laser intensity which has been used later to observe the new laser-induced resonances from Na in experiments [71,72]. In 1986, as the non-adiabatic approach of multi-channel nuclear wave functions and Green's function was beginning to be worked out, no experiments could con"rm the non-linear e!ects predicted. The "rst experiment that con"rmed qualitatively our multi-channel theory [57,58] was that of Zavriyev et al. for hydrogen [60]. The experiment showed that the strong laser "eld I&10 W cm\ modi"ed the molecular electronic structure, the nuclei moved through the new laser-induced adiabatic terms formed from the attractive and dissociative diabatic terms, that impacted non-linearly on molecular dissociation and ionization. The second experiment that con"rmed our earlier multi-channel Green's functions theory [68}70] was that of Baumert et al. for sodium [71]. These authors observed a new intense resonant peak in the spectrum of three-photon Due to its high electronic transition moment, the Na molecule has become nowadays a perfect object for interesting experiments and calculations on two-colour laser control in weak "eld [12,73]. The same experimental Na scheme realized for the case of moderate and strong lasers can show new laser-induced peculiarities in the two-colour laser control.
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resonant ionization as the laser intensity reached I&10 W cm\. The peak was created by the coupling of vibration motion in the ground electronic term to the motion in intermediate molecular terms. As the laser intensity increased from 10 W cm\ until 10 W cm\, one could see the considerable changes in the Na ionization spectra: the relative intensities and frequencies of the existing peaks were altered and also new resonances arose. This experiment together with the observations of Wu et al. [66] and Burkhardt et al. [67] con"rmed, that under the intense laser "eld the vibration motion in all resonant electronic terms are excited and coupled coherently and the nuclear motion passes over a new multi-term electronic state formed by the laser radiation from dressed resonant molecular terms. The frequencies of the new resonant peaks are unlike the frequencies of an unperturbed molecule. An aim of this review is to present the results and advances of the time-independent multichannel wave functions and Green's functions approach obtained from its application to intensexeld resonant photoprocesses in diatomics. The approach is very e!ective and simple to use. It may apply to the intense-xeld resonant photoprocesses on colliding atoms [74}77] and polyatomics [78,79]. During the last two decades the strong-"eld photoionization and photodissociation of diatomics have been studied very well in di!erent theoretical and experimental investigations. In the present review the laser-induced non-linear e!ects are considered not only in the photoionization and photodissociation but also in other resonant molecular photoprocesses which have not been studied enough yet. It would be important to attract the attention of experimentalists to investigate the "eld}molecule interactions under the circumstances depicted below. Such a study can bring to light new facts regarding physics of laser}molecule resonant interactions. In 1992 the resonant intense-"eld photoabsorption by the CO molecules has been considered within the multi-channel nuclear wave functions and Green's functions method [20]. The VUV laser radiation couples the CO(XR>) and CO(AP) resonant attractive terms and modi"es the vibration states of the "nal molecular term whose structure forms the photoabsorption spectrum. The results obtained display that the one-photon absorption in the intense UV laser "eld di!ers from that in a weak "eld. The photoabsorption occurs to the quite new vibration}rotation states of the excited molecular term, which are formed by the two resonant molecular terms and do not coincide with the states of pure diabatic "eld-free or adiabatic strong-"eld potentials. In the moderate laser "eld I&10 W cm\ there are two channels of the resonant photoabsorption: an adiabatic channel and a non-adiabatic channel. The di!erence of their full energies is small and depends on the laser power. The contributions of the two photoabsorption channels to total cross-section di!er from each other and have quite di!erent dependence on laser intensity. The contribution of the non-adiabatic channel is determined by concentration of the diatomics stronger than that of adiabatic channel. Under the high molecular concentration only the adiabatic channel brings a principal contribution to the total photoabsorption and the impact of the non-adiabatic channel is weak. But if the molecular concentration is low, the non-adiabatic channel plays a leading role and brings an observable non-linear contribution to the photoabsorption spectrum. The quantitative calculations showed [20] that for the example of the CO gas irradiated by the F excimer laser (j"158 nm) few non-linear laser-induced e!ects could be registered under laser intensities I510 W cm\ and molecular concentration n "10 m\.
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The spectrum of molecular photoabsorption may be modi"ed by the powerful "eld of an external laser being in resonance with a transition between two excited electronic terms. This is the so-called two-colour resonant photoabsorption. Experimentally, this photoabsorption scheme can be realized as follows: two lasers } weak and strong } irradiate together a gas of diatomic molecules. The photoabsorption spectrum from the weak laser is measured. The strong laser couples a pair of excited molecular terms and its power may vary form 10}10 W cm\ until 10}10 W cm\. Such an external intense "eld induces new lines in the photoabsorption spectrum and alters intensities and frequencies of existing spectral lines. As the external laser "eld increases, the laser-induced non-linear peculiarities arise in the weak-laser photoabsorption spectrum. The increase of the external laser power suppresses the existing photoabsorption lines and invokes new lines whose frequencies are unusual for the unperturbed molecule. The model calculations for the N molecules displayed that the suppression and generation of new photoabsorption lines can be observed under the intensity of the external laser as I "10}10 W cm\ [80]. The multi-channel nuclear wave functions and Green's functions approach has been applied to investigate the dynamic polarizability of diatomics and resonance Raman scattering from rotating molecules [19,78,79,81}84]. The method permitted, for the "rst time, to calculate in a simple analytic form the dynamic polarizabilities of multi-electron diatomic molecules under nonadiabatic conditions and to obtain, that the resonant UV radiation can induce an extra electronic dipole moment of the molecular due to the laser "eld modi"cation of its electronic shell [83]. For the laser-"eld resonance Raman scattering, in contrast to the numerical method of Atabek et al. and Bandrauk et al. [18,55,85}87], the multi-channel Green's functions approach presents an analytic technique to obtain equations for resonance Raman cross-section in closed form [19,84]. This analytic technique permits to study the non-radiative and laser-induced non-adiabatic peculiarities in the RRS spectra from a common point of view. The multi-channel nuclear wave functions and Green's functions analytic approach revealed theoretically an unexpected impact of the laser-induced quasi-bound vibration states on the resonance Raman scattering [19,84]. Within the approach it was deduced that the anti-Stokes RRS via nuclear continuum is modi"ed from continuum RRS to the discrete RRS as the laser intensity increases. The discrete-like form of the excitation pro"le of the strong-"eld anti-Stokes continuum RRS occurs due to the quasi-bound nuclear states and their transformation to the pure bound states of nuclear motion in attractive laser-induced adiabatic molecular term. The laser-induced attractive term is formed by the radiation non-adiabatic interaction of the ground attractive and excited dissociative terms of unperturbed molecule and traps vibrationally the nuclei into the bound states. Therefore, the RRS via continuum becomes like the RRS via an attractive electronic term. Increasing the laser intensity, one can control the process of formation of the attractive laser-induced term and operate for intensity and form of the scattered electromagnetic signal. The quantitative calculations for the Ar> molecules displayed that the laser-induced quasi-bound nuclear states may be e!ective for the RRS already under laser intensity I'10 W cm\ [19,78,84]. The method of non-adiabatic wave functions and Green's functions is a time-independent approach and may be applied to calculate the various resonant laser}molecule interactions until the laser pulse intensity 10 W cm\ and pulse duration 10\ s for light molecules and 10\ s for heavy molecules, which can be simply reached in a modern experimental setup. Therefore, the new experiments on the intense-"eld photoabsorption, two-photon resonant dissociation, and
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resonance Raman scattering could bring new information about strong-"eld dynamics of the resonant laser}molecule interactions. This paper presents a new approach of multi-channel nuclear wave functions and Green's functions in the quantum-mechanical theory of resonant laser}molecule interactions. This approach is developed within only the time-independent picture and does not expand here on femtosecond diapason of laser pulse duration. In the present review the laser}molecule interactions are studied for the case of long laser pulse of nano- or micro-second durations in terms of the traditional laser spectroscopy of molecules [9]. A laser-induced interactions is considered as a non-adiabatic one which brings a non-perturbative non-linear e!ect upon the molecular spectra and may not be included as a weak perturbation. This article reviews the referred author's papers [19,20,57,58,68}70,78}84] as well as a progress in the discussed topics, reached by means of the new approach. The review is organized as follows. In Section 2 the non-adiabatic Hamiltonian of a diatomic molecule in quantized laser "eld is considered. The restrictions of the developed approach are depicted. In Section 3 a general theory of Green's functions of multi-channel SchroK dinger equation is elaborated. In Section 4 the analytic solutions of the multi-channel equation are obtained and the multi-channel nuclear wave functions are considered. In Section 5 the direct molecular photodissociation in intense "elds is studied, the delay of the photodissociation and its physical nature are discussed in detail. In Section 6 the non-adiabatic resonant photoabsorption and impact of the non-adiabatic channel upon it are investigated. In Section 7 the multi-channel Green's function approach is used to elaborate the theory of resonant multi-photon dissociation and ionization and to explain the laser-induced non-linear e!ects in their spectra. In Section 8 the new method is applied to study the dynamic polarizability of diatomics under non-adiabatic conditions. Section 9 presents the non-adiabatic theory of radiation and non-radiation e!ects in the spectra of resonance Raman scattering and discusses the impact of quasi-bound nuclear states upon the spectra. In Section 10 the resonance absorption of the vacuum-ultraviolet radiation by diatomics in the "eld of a powerful infrared laser is treated and the new laser-induced absorption lines are analyzed. In Section 10 the new experiments, which can be carried out, and the laser-induced non-linear e!ects, which may be observed, are discussed.
2. Diatomic molecule in intense resonant laser 5eld. Quantum-electrodynamic picture The states of a diatomic molecule irradiated by quantized electromagnetic "eld have to be considered as the quantum states of a compound quantum system molecule#xeld. In resonant laser "eld the molecule}laser interaction is comparable with the energy di!erences between the states of the compound system and, in principle, may not be included perturbatively. Quantum electrodynamics gives the following form for the full molecule#laser "eld Hamiltonian H"H #H , (1) where H is the unperturbed Hamiltonian, being a sum of the Born}Oppenheimer Hamiltonian of free molecule and Hamiltonian of free electromagnetic "eld H "H - #H .
(2)
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The operator of the total non-adiabatic interaction H contains the operator of non-adiabatic interaction in the free molecule < and operator of radiation interaction < : H "< #< . (3) 2.1. Molecular Hamiltonian In order to write out the full Hamiltonian H "H - #< (4)
and wave functions of the free diatomic molecule within a representation, one has to de"ne "rstly the type of electronic coordinates. The space and spin coordinates of molecular electrons may be de"ned either in a molecular reference frame (MF) or in a laboratory reference frame (LF). The corresponding complete sets of the molecular generalized coordinates contain the following variables: m "+X, a, bI , c"0, R, y , p , , (5) ? G G m "+X, a, bI , c"0, R, x , p , , (6) @ G G where y and p (x and p ) are the sets of the space and spin variables of all molecular electrons in G G G G MF +x, y, z, p , (or in LF +x , y , z , p ,), i"1, 2,2, n (n stands for the full number of the molecular G G electrons); p and p are proportional to the electronic spin projections onto the axis oz and G G oz : p ,p "$1. The MF's origin coincides with the centre of mass of molecular nuclei, whose G G radius-vector is denoted as X, R is the internuclear distance, a, bI , c are the Euler angles of the molecular axis onto LF. For the diatomics whose nuclear charges are high, Z'10, the spin-orbit interaction becomes comparable with the Coulomb interaction of molecular electrons. In this case the total molecular spin is not conserved and both electronic momentum L and total electronic momentum S form the total electronic angular momentum J ? J "L#S . ? In this case the generalized coordinates m Eq. (6) have to be used. The form of the free molecular ? Hamiltonian, Eq. (4), in the m -coordinate set, Eq. (6), is well known [9,88], but its form in the @ m representation, Eq. (5), is non-trivial and can lose some important terms. ? The form of the total molecular Hamiltonian in the m representation, Eq. (5) ? R R JK !JK
?X , R ! (7) H (m )"HC#H #H !
? 2kR RR RR
has been corrected in [89] to include all the important terms. HC in Eq. (7) is the electronic Hamiltonian, being a sum of the electronic kinetic energy operator ¹C, the operator of Coulomb electronic-nuclear interaction ; and the operator of electronic spin}orbit interaction ; (the nuclear spins are equal to zero): HC"¹C#; #; ,
(8)
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the H term allows for correction induced by nuclear motion 1 1 [PC]# (JK !JK ) , (9) H " ?X 2(M #M ) 2kR ? M and M are the masses of both nuclei, k stands for the molecular reduced mass, P stands for the total electronic kinetic momentum
H
L P"!i yG , G is the operator of rotation interaction J>J\#J\J> ? , H " ? 2kR
where A> and A\ designate the raising and lowering operators A!"A $iA , V W JK is the operator of total molecular angular momentum squared:
JK "!
1 R R R 1 2i cos b R JK sin b # ! JK ! ?X ?X Ra sin b Rb Rb sin b Ra
.
(10)
The operators of projections of the momenta JK and JK onto MF have the forms ? R R i R #cot bJK , JK "!i , JK "!i "JK JK " ?X W X ?X V sin b Ra Rb Rc and "t to the following commutation relations: [JK , JK ]"!i e JK , [JK , JK ]"!i e JK , G H GHI I ?G ?H GHI ?I I I where i, j, k"x, y, z and e is the third rank unit antisymmetric tensor. GHI Within the generalized coordinates m , Eq. (5), the operators of electronic Hamiltonian, Eq. (8), ? acquire the forms
L yG , ¹C(m )"! ? 2m C G Z e L Z Z e ; (m )" ! ? [y #y #(y #RM /(M #M ))] R GV GW GX G Z e # [y #y #(y !RM /(M #M ))] GV GW GX e # , [(y !y )#(y !y )#(y !y )] GV HV GW HW GX HX GH a L Z 1 Z (l ) s )# (l ) s )! ; (m )" (l ) (s #2s )) , ? H 2 m e "y " G G "y " GH G "y " G G G C G H$G GH G (11)
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where a is the "ne-structure constant, l "[ y ;p ], l"1, 2, j, y "y !y , p "!i
yG , GJ GJ G GH G H G y "y $M R/(M #M ), R"(0, 0, R) . G G It is easy to prove that the operator HC, Eqs. (8) and (11), commutes with the operators J, J ? ?XY and J . ?X Note that the spin}orbit interaction ; , Eq. (11), is non-equal to zero even if the total electronic spin of the molecule is equal to zero (S"0). This fact can be seen from Eq. (11) if we put there, e.g. n"2 and s "!s . Thus, the singlet electronic states of heavy molecules also fall in Hund 's case c [89]. Let us compose a complete set of the orthonormalized functions 1m "pJMX2 as ? 2J#1 (12) D( X (a, bI , 0)UCX (R, y , p ) , 1m "pJMX2" N G G + ? 4p
where the quantum numbers M and X are the J's projections onto the oz and oz axes, p denotes the other quantum numbers (excluding the vibration one), D( X (a, bI , 0) is the Wigner function describ+ ing the molecular rotations, UCX (R, y , p ) is the relativistic electronic wave function. G G N The functions, Eq. (12), form a basis over the space spanned by the functions of the generalized molecular coordinates m , Eq. (5). For any "xed value of R the functions, Eq. (12), obey the ? conditions of orthonormality and completeness: , 11p J M X "pJMX22"d d d d NNY ((Y ++Y XXY
(13)
L 1m "pJMX21pJMX"m 2"d(RK !RK ) d(y !y )d G G , ? ? G G NN G N(+X where the following designation is used:
(14)
L p L dy 11p J M X "pJMX22, da dbI sin(bI )1p J M X "m 21m "pJMX2 . G ? ? N G +y By B2ByL , G The operators J!, J! transfer the electronic-rotation functions Eq. (12) to each other as ? J!"pJMX2" [J(J#1)!X(XG1)] "pJMXG12 , J!"pJMX2" [J (J #1)!X(X$1)] "pJMX$12 . ? ? ? Within the "pJMX2 representation the total molecular Hamiltonian, Eqs. (4) and (8), takes the form of a non-diagonal matrix H (R)"""1p J M X "H - #< "pJMX2""
]d d "" , """[H - (R)d dXX #<X (R)dXX #< X Xd NYN Y Y NNY ((Y ++Y
NNY Y where H - (R) is the Born}Oppenheimer vibration Hamiltonian of the free molecule:
R R H - (R)"! R #;(X (R) ,
N 2kR RR RR
(15)
(16)
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;(X (R) stands for the e!ective electronic potential N
[J(J#1)!X] , ;(X (R)"; X (R)# N N 2kR
(17)
; X (R) is the total adiabatic electronic potential, being an eigenvalue of the operator sum N HC#H (18) (HC#H )UCX (R, y , p )"; X (R)UCX (R, y , p ) , G G N N G G N <X (R) is the operator of radial non-adiabatic coupling NYN
RUCX (R, y , p ) N G G G<X (R)"! UCX (R, y , p ) NYN NY G G 2k RR
#2
UCX (R, y , p ) NY G G
RUCX (R, y , p ) N G G RR
d R G # , R RR
i"0, 1 ,
(19)
< X X is the operator of rotation non-adiabatic coupling Y
#j\ ,, +j> < X X (R)"! X (J, J )dX X X (J, J )dX X Y ? Y > ? Y \ 2kR
(20) j! X (J, J )"[(J (J #1)!X(X$1))(J(J#1)!X(X$1))] . ? ? ? One can deduce from Eq. (15) that in a free molecule there is a conservation of the total molecular momentum J and its space-"xed projection M as well as non-conservation of its molecule-"xed projection X and the other quantum numbers p. The non-adiabatic couplings, Eqs. (19) and (20), mix the states of nuclear motion in di!erent unperturbed e!ective electronic terms, Eq. (17), and induce the non-adiabatic e!ects in free diatomics. The wave function of the free diatomic molecule, "tted to the total molecular Hamiltonian, Eq. (7), may be expanded over the basis Eq. (12) as W (m )" f X s(+ (R)1m "pJMX2 , (21) (+ ? N NX ? NX (R) is the wave function of nuclear motion allowing for the where f X is a phase coe$cient, s(+ NX N non-adiabatic couplings, Eqs. (19) and (20), and satisfying the nuclear di!erential equation with matrix Hamiltonian, Eq. (15), as H (R)v(+(R)"Ev(+(R) ,
here E is the energy of free diatomic, v(+(R) is the nuclear vector function
(22)
(R)"" . v(+(R)"""s(+ NX The wave functions, Eq. (21), describe the states of free diatomic molecules with radial and rotation non-adiabatic intra-molecular couplings, Eqs. (19) and (20). Eq. (21) presents the full non-adiabatic wave function of the free molecule. The careful numerical allowing for the corrections, Eqs. (9), (10), (19) and (20), permits to calculate the ab initio electronic potentials and rotation}vibration non-adiabatic spectra [90,91].
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If the non-adiabatic couplings are weak, the molecular wave function may be written as (m )"(2!dX )\ (1m "pJMX2#P1m "pJM!X2)X (R) , UQ ? ? TN( NT(+ X ? P"(!1)(>X\R>1(1!dX ) , 1 L (23) R" p , G 2 G where X (R) is the unperturbed wave function of nuclear motion, which "ts the diabatic TN( SchroK dinger equation with the Born}Oppenheimer Hamiltonian, Eq. (16), as H - (R)X (R)"E X X (R) , (24)
TN( NT( TN( v denotes the vibration molecular state, s denotes the sign of molecular state (s"0 represents a positive sign and s"1 a negative sign). The diabatic wave function, Eq. (23), describes the electronic}rotation}vibration state of free diatomic molecule without intra-molecular nonadiabatic couplings. 2.2. Laser xeld Hamiltonian In a general case, the free laser "eld can be like a polychromatic one. Its Hamiltonian may be presented in second quantization form as (25) H " u (a>a # ) , H H H H where a> and a are the operators of creation and annihilation of the laser photon j with H H polarization e , wave vector k "nu /c, and frequency u . H H H H The wave functions of the free "eld Hamiltonian, Eq. (25) are represented here through the photon occupation numbers as "N , N ,2, N ,22. The free laser "eld wave functions describe H the states of the free polychromatic laser "eld including N photons (u , e ), N photons (u , e ),2, N photons (u , e ),2 and "t to the following equation: H H H (26) H "N , N ,2, N ,22" u (N #)"N , N ,2, N ,22 . HY H HY HY HY The creation and annihilation operators act on the laser "eld wave functions "N , N ,2, N ,22 H as follows: a>"N , N ,2, N ,22"(N #1"N , N ,2, N #1,22 , H H H H a "N , N ,2, N ,22"(N "N , N ,2, N !1,22 . H H H H The description of the "eld Hamiltonian in the form of Eqs. (25) and (26) is the so-called Fock quantum representation in terms of photon occupation numbers [92]. The "eld equations (25) and (26) present a full quantum description of laser "eld and di!er from the well-known semi-classical representation in terms of quasi-energies [51] used by other authors frequently (see for example [8,13,14,16,17,54,62]). The general equations for molecular transition probabilities in resonant laser "elds, obtained within the Fock quantum representation, are the
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same as those obtained within the quasi-energy approach. Moreover, one can deduce formally that both representations are equivalent and one may use either the "rst or the second one without any restriction [92,93]. There is only one di!erence between the quantum and semi-classical representations } this is the methodological point of view. The di!erences of laser}molecule interaction pictures within both approaches are the same as for laser}atom interactions that can be obtained from a comparison of both theoretical formalisms in books [93,94]. In this review, following the "rst papers of Atabek et al. [85] and Bandrauk et al. [15,86,87], the Fock quantization form is used to describe the laser "eld and laser}molecule interaction. Preference has been given to the Fock occupation numbers representation because the resonant laser}molecule interactions may be described by methods of time-independent quantum mechanics. This is due to the fact that the duration of laser pulse *q considered in this paper is much longer than temporal characteristics of all intra-molecular motions *q
"e " e G ( p A )# G A !Coulomb gauge ,
(28)
<#%"! l E !electric ,eld gauge , (29) G G G where e , m and p are the charge, mass and momentum of the ith electron, c is the speed of light, G G G A is the quantized vector potential of the "eld G 2p c +e a e krG #eHa>e\ krG , , A " H H H H G Vk H V is the "eld quantization volume, r is the radius-vector of the electron within the space-"xed reference frame. The vector l in Eq. (29) is the electric dipole moment of the ith electron, G l "!"e "r and the vector E is the operator of the electric component of the laser "eld G G G G 2p u E "!i +e a e krG !eHa>e\ krG , . (30) H H G H H V H
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It has been deduced elsewhere [95] that for the general case of a multielectron molecule the laser}molecule interaction operators, Eqs. (28) and (29), are related to each other by a canonical transformation and the matrix elements of both operators, calculated on the energy shell, are equivalent. Therefore, the matrix elements of a photoprocess, calculated using either the Coulomb or electric "eld gauge, must be the same. However, for the computations of real molecules, where a restricted electronic basis is used, the electric "eld form, Eq. (29), is preferable [96]. The transformation between the gauges, Eqs. (28) and (29), in the laser}molecule theory is a non-trivial problem. For a detailed discussion of the relations between the Coulomb gauge, the electric "eld gauge and their connection with a diabatic, adiabatic, and acceleration frame representations see papers [97}100]. The dipole approximation is traditionally used in order to calculate the photoprocesses in molecules. It consists of the following simpli"cation of Eq. (30) "e krG "+1 , that is true if "kr ";1 . (31) G Then, the laser}molecule quantum-electrodynamic Hamiltonian may be written in the well-known form
2p u H V "iD +e a !eHa>, , (32) H H H H V H where D is the full dipole moment of the molecule, D " L l . G G There are no questions to use the dipole approximation for the UV and long-wave lasers [101] } one may use the approximation to calculate the interactions of such lasers with low-excited molecules. However, one must justify the dipole approximation if the interactions of molecules with the VUV or shorter electromagnetic waves are studied [78,79]. The short-wave lasers excite the high-lying Rydberg states of the molecule and couple them radiatively with the ground one as well as with the low-lying electronic terms. Within a oneelectron approximation, the averaged size of a Rydberg molecule may be estimated as follows: 3 a (33) 1r 2+ n , G 2 0Z where n is the principal quantum number of the Rydberg state, Z is the charge of molecular core 0 and a is the Bohr radius. From Eqs. (31) and (33) it is easy to show that j<10a n /Z . 0 Therefore, the dipole approximation is applicable to study the short-wave laser}molecule interaction if the radiation wavelength satis"es the following condition: j550a n /Z . (34) 0 Eq. (34) gives a restriction for the radiation wavelength and molecules which may be considered within the approach presented in this review.
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Table 1 Regions of ultraviolet radiation Type
j (As ) j (nm)
Energy (eV)
UV VUV XUV Soft X-ray
4000}1800 400}180 1800}1050 180}105 1050}100 105}10 100}1 10}0.1
(Ry)
(cm\)
3.10}6.89
0.229}0.506
2.5;10}5.556;10
6.89}11.81
0.506}0.868
5.556;10}9.524;10
0.868}9.114
9.524;10}10
9.114}911.4
10}10
11.81}124 124}1.24;10
The dipole approximation may be applied to calculate the short-wave photoprocess involving: (a) low-lying Rydberg states (n "1}3) of multi-charged molecular ions (Z 510) in all the UV 0 diapasons and in the soft X-ray one (see Table 1), (b) the "rst Rydberg state of neutral molecules (Z "1, n "1) in all the UV diapasons 0 including the soft X-ray one being not shorter than 25 As , (c) low-Rydberg molecules (Z "1, n "2}5) in all the UV diapasons including the XUV one, 0 (d) high-Rydberg low-charged molecular ions (Z "2}5, n 510) in the VUV and longer 0 radiation, (e) high-Rydberg molecules (Z "1, n 510) in the "elds of radiation not as short as UV, and 0 (f) super Rydberg molecules (Z "1, n 530) in the near-UV and longer radiation "elds. 0 For other situations (e.g., the super-Rydberg molecules in the VUV or X-ray "elds) the dipole approximation is not applicable and a non-dipole approach must be used [102]. 2.4. Quantum-electrodynamic states of diatomic molecule in laser xeld A molecule irradiated by an electromagnetic "eld may not be considered more as a free one. A spontaneous emission of "eld photons occurs even in a weak laser "eld and one must consider the quantum interactive system `molecule#"elda in the weak "eld too. The "eld}molecule interaction impacts upon both parts of the system, changes their quantum states and permits, in principle, for all their quantum states to be occupied. The presence of the laser}molecule interaction brings the crucial modi"cation to dynamics of the molecule and laser "eld, that must be taken into account in theoretical study. The interaction changes the inner dynamics of the laser "eld too because it makes possible the emission and absorption of the "eld photons. The "eld-induced reconstruction of intra-molecular dynamics is more complicated than that for photon "eld because the "eld}molecule coupling impacts not only on molecular electrons and nuclei but also on their reciprocal electromagnetic interactions. Therefore, the "eld and molecule, interacting with each other, are not more as free objects and, in general, one may not consider the laser}molecule radiation transitions to occur between their free quantum states.
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The exact quantum states of the compound system `molecule#"elda generate a space of states E. The eigenvectors of the Hamiltonian of free laser "eld H Eq. (25) and the sum of the following Hamiltonians of free molecule HC#H # J/2kR, Eqs. (8)}(10), form sub-space E of the space E, which is a tensor product of the space of free "eld states E and that of electronic-rotation states of the free molecule, Eq. (12), E U E "E E . (35) U Vectors of space, Eq. (35), de"ne the laser}molecule states, in which the occupation numbers of the laser "eld N , N ,2, N ,2, the e!ective molecular term Eq. (17), the total molecular momentum H J and its projections M and X are constants for each "xed internuclear distance R during the laser}molecule interaction. The product of the free molecular electronic-rotation states and free "eld ones, being the eigenstates of the unperturbed laser}molecule Hamiltonian, Eq. (2), form the full orthonormal function set +1m "pJMX2"N , N ,2, N ,22, ? H 1N , N ,2, N ,2"11p J M X "pJMX22"N , N ,2, N ,22 H H "d d d d d H H , ,, NNY ((Y ++Y XXY H 1m "pJMX2"N , N ,2, N ,221N , N ,2, N ,2"1pJMX"m 2 ? H H ? X , , , N(+ L "d(RK !RK ) d( y !y )d G G , G G NN G which may be used as a representation basis (or projection sub-space) to consider the laser}molecule interactions. In reality, the radiation transitions in the system `molecule#"elda involve its perturbed states which can be expanded over the unperturbed vectors of space, Eq. (35). In the non-perturbative picture, discussed in this review, the unperturbed vectors of the space, Eq. (35), are the non-existent states during the laser}molecule interaction. A perturbed laser}molecule state evolves to a set of such unperturbed states after termination of the laser}molecule interaction. Within the long laser pulse approximation, Eq. (27), the quantum wave function of the interactive system `molecule#"elda "ts to the following time-independent SchroK dinger equation with Hamiltonian, Eq. (1), as 2 H2
H(m )W (m )"EW (m ) . (36) ? # ? # ? The full quantum-electrodynamic laser}molecule wave function W (m) may be expanded over the # basis +1m "pJMX2"N , N ,2, N ,22, as ? H W (m )" f +,H ,X s(+ (R; +N ,)R\1m "pJMX2"N , N ,2, N ,22 , N(+ NX # ? H ? H + , X ,H N(+
(37)
The vectors of the space, Eq. (35), may be involved to the free molecule wave functions, Eqs. (21) and (23), in order to calculate the radiation transitions in the weak laser "eld only.
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where f is the phase factor, dependent on all the summation quantum numbers +N ,, p, J, M, X, and H +N , designates the set of the "eld occupation numbers H +N ,,N , N ,2, N ,2 , H H (R; +N ,) depends on inter-nuclear distance and is the full quantum-electrodynamic wave s(+ H NX function of the nuclear motion in the set of intersecting electronic-rotation molecular terms dressed by the laser "eld. (R; +N ,) satisfy the following coupled equations: The nuclear wave functions s(+ H NX (R; +N ,)# [ <X (R)dXXI #< (R; +N ,) [H (R)!E]s(+ XXI ]s(+ NN H N XI H NX XI N (I +I *' (I +I (38) # [<>*' I I XI (R; e )s XI (R; +N #1,)#< I I XI (R; eH)s XI (R; +N !1,)]"0 . N H H N H H ( + N ( + N N (I +I XI H The operator H (R) is the unperturbed laser}molecule Hamiltonian, Eq. (2), presented here as
d H (R)"! #;(X (R; +N ,) , N H 2k dR where ;(X (R; +N ,) is the molecular electronic-rotation term, Eq. (17), dressed by the polychroH N matic laser "eld, Eq. (25), as ;(X (R; +N ,)";(X (R)# u (N #) . (39) N H N HY HY HY X I are the operators of radial and rotation non-adiabatic couplings, The operators < (R) and <XX NN Eqs. (19) and (20), allowing for dynamic non-linear e!ects in a free molecule. The operator <X (R) NN di!ers from another one <X (R), Eq. (19), due to the expansion form with R\s(R) admitted to NN (R; e ) and <*' (R; eH) in Eq. (38) are the Eq. (37) unlike Eq. (21). The other operators <>*' N (I +I XI H N (I +I XI H quantum operators of the non-adiabatic couplings in the molecule, induced by the (u e )-mode of H H the polychromatic laser "eld, written within the Fock representation as X <>L*' (R; eH)" F>LD (R)AN(+ (eH) , N (I +I XI N (I +I XI H H H NN 2JI #1 X AN(+ C(+I I C(X I XI nJ (eH)k , I I XI (eH)," I(+ J( NN H N(+ H 2J#1 IJ 8p u (N #m) H H , F>K" H l
(40) (41) (42)
(eH) , n are the cyclic components of the laser photon polarization vector and unit vector H I NN n ,DK (R) along the molecular dipole transition moment D (R) calculated in MF: NN NN NN L D (R)" dy [UCX (R, y , p )]HDUC (R, y , p ) . (43) N XI NN G N G G G G +y y 2 y , L B B B G NG
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The symbol +N $1, in Eq. (38) designates the following set of photon occupation numbers H +N $1,,N , N ,2, N $1,2 . H H (R; +N ,) of the expansion, Eq. (37), As it follows from the coupled equations (38), the coe$cient s(+ H NX is the wave function of nuclear motion in the set +;(X (R; +N ,), of interacting and intersecting H N electronic-rotation molecular terms dressed by the laser "eld, Eq. (39). The nuclear wave functions s(+ (R; +N ,) allow for the radiation and non-radiation non-adiabatic couplings and describe the NX H quantum states of molecular nuclei under laser}molecule resonant interaction exactly. The nuclear wave functions ful"l the boundary and normalization conditions depending on the type of laser}molecule interaction. In the spectroscopic transitions the initial and "nal states of the laser}molecule system are bound and its full quantum-electrodynamic wave function is normalized to unity: . 1W (m )"W (m )2 ? "d ##Y #Y ? # ? K Eq. (44) gives the following orthonormalization equation for the nuclear wave functions
(44)
(R; +N ,)"s(+# (R; +N ,)2 "d . (45) 1s(+#Y NX H NX H 0 ##Y X+ , N(+ ,H For the laser}molecule interactions like the photochemical reactions the full states are unbound (or scattering-like) and normalized as (46) 1W (m )"W (m )2 ? "d(E!E )d(q( !q( ) , #Y ? # ? K where q( is the direction of the relative kinetic momentum of reactive particles q at in"nity. The normalization equation for the nuclear wave functions has the form of (R; +N ,)"s(+C (R; +N ,)2 "d(e!e ) , 1s(+CY NX H NX H 0 NX+,H , where e is the e!ective kinetic energy of the scattering particles e" q/2k .
(47)
(48)
The boundary condition for all the nuclear wave functions in zero is (RP0; +N ,)P0 . (49) s(+ H NX The boundary condition at in"nity depends on the type of the laser}molecule interaction process: For the spectroscopic transitions: s(+ (RPR; +N ,)P0 . NX H For the photochemical transitions:
2k cos(qR! ) , (RPR; +N ,)P s(+ H NX p q
is a phase.
For details on the wave functions of photochemical reactions see, for example, papers [74}77].
(50)
(51)
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The laser-induced non-diagonal couplings, Eq. (40) in Eq. (38), allow for the quantum interaction between the molecule and quantized laser "eld and give di!erent contributions to the nuclear wave functions because N #1ON . In intense laser "eld the occupation number of its strong mode is H H much more than unity N <1, N #1+N H H H
(52)
and the occupation number N is connected with the laser "eld strength F and intensity I of the H H H strong mode as
2p u (N #1) 2p u N F 2pI H H H H H + + H" . V V 2 c
(53)
Therefore, in a polychromatic "eld its strong mode may be considered in the classical limit, Eq. (53), where F>+F, Eq. (42), but its weak modes, for which the approximation, Eqs. (52) and (53), is H H inapplicable (F>OF), must be done beyond the classical limit. Eq. (53) is the so-called HY HY semi-classical limit of the Fock quantization for laser "elds [92]. In order to "nd the quantum functions of nuclear motion one must solve the rigorous quantum-electrodynamic coupled equations (38) supplemented with the boundary conditions for all the channels participating, Eqs. (49) and (50) or Eqs. (49) and (51) and normalization equation, Eq. (45) or Eq. (47). In dependence on the total laser}molecule energy E and form of the molecular electronic terms involved in the photointeraction, the nuclear motion in each of the interaction channels can be either "nite } this is the vibration motion of the nuclei, or in"nite } this is the scattering motion of the nuclei. Therefore, it is more convenient to speak about wave functions of nuclear motion instead of the vibration functions. The quantum wave functions of nuclear motion, calculated from Eqs. (38), (45), (49) and (50) or Eqs. (38), (47), (49) and (51), describe the real motion of molecular nuclei a!ected by resonant impact of intense laser "eld on the molecule beyond a perturbation theory framework. These functions bring full quantum-mechanical information about the dynamics of intra-molecular motion in resonant laser "elds of intensity up to 10 W cm\. Today more than twenty six numerical methods are known for computer integration of the sets of second-order di!erential equations like Eq. (38) [103,104]. 2.5. Radial, rotation and laser-induced couplings. Applicability of multi-channel quantum-electrodynamic approach The non-diagonal parts, Eqs. (19), (20) and (40), of the coupled equations (38) play quite di!erent roles in the resonant laser}molecule interaction. So, the radial and rotation interactions, Eqs. (19) and (20), couple the molecular terms, Eq. (39), dressed by the same numbers of photons. The
Sometimes one describes the strong "eld mode classically and the weak one quantum mechanically. This way complicates the physics picture of the interaction since it does not allow for quantum absorption or emission of photons from the strong laser mode. Within the Fock quantization, although the approximation (52) is correct, the laser "eld states with di!erent occupation numbers are quite di!erent, "N 2O"N #12. H H
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rotation coupling, Eq. (20), can be estimated as I (R)"& (¹ /k) /RH;¹ , "<XX I I where ¹ is the kinetic energy of relative motion of nuclei whose magnitude is I ¹ &(1}10) u , I
u &10 cm\ is the molecular vibration quantum, RH&10 As is the e!ective diameter of the molecule. The radial coupling, Eq. (19), depends on the structure of electronic molecular terms, relative velocity of nuclei v as (R)" "G<X (R)"&E /g, g"*;RH/ v, *;""; X (R)!; NN I N N$N XI $X and is signi"cantly high if the Messiah parameter g is small. Therefore, for a pair of interacting terms the rotation coupling is less than the radial coupling of the terms and may be omitted in comparison with that. The laser-induced non-adiabatic interaction, Eq. (40), couples the dressed molecular terms, Eq. (39), whose "eld occupation numbers di!er from each other by unity only for the one laser mode and are equal for the other modes. The laser-induced coupling is signi"cant in the resonant laser "elds, where two dressed terms, Eq. (39), are intersected. The laser-induced interaction is proportional to the laser "eld strength (see Eqs. (40) and (53)) (R; eH)"&F "D (R)" "<*' H H NN N (I +I XI and can intensively couple the vibration states in the resonant molecular terms as the laser strength increases. So, the resonant "eld becomes a tool to control and to force upon the dynamics of nuclear motion. In the resonant laser mode (u , e ), whose frequency is much higher than the vibration frequency H H u and is comparable with the frequency of an electronic transition
u ; u &¹ H C (¹ is the energy of the electronic transition, ¹ &10}10 cm\), the laser-induced non-adiabatic C C coupling of the dressed molecular terms dominates over their radial and rotation non-adiabatic couplings. Therefore, all the non-adiabatic interactions of a term pair in a diatomic molecule irradiated by the resonant UV laser "eld may be estimated as I (R)";"<X (R)";"<*' (R; eH)";¹ . (54) "<XX N (I +I XI H C NN Eq. (54) gives the following general condition to apply the presented non-perturbative quantumelectrodynamic approach to non-adiabatic study of a resonant laser}molecule interaction: X *' H)",;D , J , max+"< XX (R)", "< (R)", "< I I XI (R, e N(+ H Y NN
(55)
The laser-induced coupling of vibration states of each molecular term (p "p, XI "X) is very weak here. This vibration coupling is intense in the strong IR laser "elds only [8].
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where D and J are the dissociation and ionization potentials of the molecule. It needs to be emphasized that, contrary to Eq. (54), the quantum numbers p, J, M, X, +N , in Eq. (55) may H correspond to di!erent pairs of coupled molecular terms. The condition of applicability of our quantum-electrodynamic non-perturbative non-adiabatic approach has only the upper limit for the non-adiabatic couplings, Eq. (55), that permits to study the resonant laser}molecule interactions by means of common theoretical technique in the laser "elds of intensity up to 10 W cm\. The approach elaborated is as valid for the perturbative limit as for the strong xeld limit. Moreover, our approach can describe the laser}molecule interaction in the "elds, for which both perturbative and strong "eld approximations are invalid. The perturbative approach, proposed "rstly by Bunkin and Tugov, may be applied if the laserinduced non-adiabatic coupling is restricted as [1}5,81,82] (56) "1v"<*' X (R; eH)"v 2";"E X $ u !E XY " , H NYTY(Y NT( H NY(Y+Y Y where "v 2 and E X are the eigenvector and eigenvalue of the Born}Oppenheimer Hamiltonian of NT( free molecule, Eq. (24). Eq. (56) gives the following limit for the laser "eld intensity in the perturbation theory approach c du , I ; H 2peRq C TTY R is the molecular equilibrium distance, q is the Franck}Condon factor: C TTY q , [sM (R)]HsM (R) dR , TTY NT( NYTY( du is the resonance detuning:
(57)
(58)
du"""E X !E XY "/ !u " . (59) NT( NYTY(Y H Thus, the perturbation approach is totally inapplicable either in the exact resonance du"0 or in the strong "elds of intensity c du . I * H 2peRq C TTY The Fedorov's strong xeld approximation [13] is valid if "G<X (R)";"<*' (R; eH)";¹ , N (I +I XI NN H C or, through the mode intensity, c u c u ;I ; H . H 8peR 32peR C C Therefore, there is the following gap in the laser intensity: c du c u , 4I 4 H 32peR 2peRq C TTY C
(60)
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for which both perturbation and strong "eld approximations are incorrect. The multi-channel quantum-electrodynamic approach developed in this review may be successfully applied to investigate the resonant laser}molecule interactions within this gap as well as under the weak and strong intensities, Eqs. (57) and (60), c u H . I ; H 8neR C
(61)
3. Multi-channel Green's functions The Green function formalism is applied often in physics due to its possibility to sum over all the virtual intermediate states of a quantum system in an analytic form [105]. In theory of multiphoton processes the Green functions permit to sum over all the spectrum and one does not need to "nd the wave functions of each intermediate state that participated in multiphoton interactions [106]. To "nd the Green function one needs to solve the SchroK dinger equation for the given quantum system and obtain only two of its linearly independent solutions which construct the Green function in a simple analytic form. A one-electron Green's function is applied to develop analytic methods in calculations of multi-photon processes in atoms [106]. This is the Green's function of the one electron placed in Coulomb "eld of atomic core. It does not allow for multi-electron correlations in atoms and is useful to study the laser}atom interactions within the perturbation theory framework. The Green functions of model electronic potentials are a powerful analytic tool to investigate the multi-photon processes in diatomic molecules. The functions had been obtained "rstly by Bunkin and Tugov for Morse-type and hard ionic core-type molecular terms and applied to calculate perturbatively the multi-photon processes in homo- and hetero-polar diatomics [1}4]. A multi-channel Green's function appears if the coupling of few channels (either of few electrons or few molecular potentials) must be taken into account. The multi-channel Green's functions (or Green's functions in matrix representation) have been considered earlier in papers [107}118] in connection with the di!erent problems of physics and chemical physics. Here the approach of multi-channel Green's functions developed in my papers [19,20] is presented. In [19,20] the method, proposed by Ignat'ev and Polikanov within a perturbative picture [107], has been generalized beyond the perturbative framework. In papers [19,20] under the non-perturbation conditions the spectral expansion, the matrix representation through regular and non-regular solutions of multi-channel SchroK dinger equation as well as the representation
One can see from Eqs. (57)}(60) that sometimes the perturbation approach and strong "eld approximation may be applied for the same "eld intensities because the Frank}Condon factor q can be very small and the detuning is big, TTY du&u . Therefore, the strong "eld approximation [13] may not be considered as opposed to the perturbative one [1}5]. For some cases, the perturbative calculations give correct results for the "elds admitted to the strong "eld approximation, although both techniques are quite di!erent. Nevertheless, our approach can be applied commonly for all the intensities, Eq. (61), without any trouble. Sometimes only one solution of the SchroK dinger equation may be enough to write out the Green function [107].
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through only one regular matrix solution have been "rstly obtained for the multi-channel Green's function for various symmetries of the matrix Hamiltonian and applied to study resonant interactions of intense laser "elds with diatomic molecules and quasi-molecules [57,68}70,74}81,83,84]. 3.1. Matrix form for the solutions of multi-channel SchroK dinger equation Beyond an adiabatic approximation the coupled equations (38) may be written in a simple matrix form as +IE!H(R),U(R)"0 , I"""d "", 0"""0 ) d "", i, j"1, 2,2, k , GH GH where the Hamiltonian H(R) is presented in the matrix form too:
(62)
H(R)"H(R)#V(R) ,
(63)
d #U(R) , H(R)"""H(R)d """!I G GH 2k dR
(64)
(65) U(R)"diag+; (R), ; (R),2, ; (R), , I U(R) is the diagonal matrix containing the potentials of unperturbed channels ; (R) on its G diagonal, ; (R) is the potential of the ith channel, V(R)"""< (R)"" is the non-diagonal matrix of G GH channel couplings, k is the number of all channels considered. For the case of resonant laser}molecule interactions, ; (R) is the e!ective molecular term dressed by laser "eld, Eq. (39), G < is the total operator of non-adiabatic couplings, Eqs. (19), (20) and (40). GH The solution U(R) of matrix equation (62) is presented here in the form of a (k;k) matrix U(R)""" (R)"" (66) GH which contains all the partial solution vectors of the coupled equations (38). The representation of the solution of the multi-channel di!erential equation (62) in the matrix form, Eq. (66), is analogous with that used by Mies in a multi-channel quantum defect theory [119]. The important problem of the multi-channel non-adiabatic theory is: How must the channel solutions of Eq. (62) be placed into the matrix Eq. (66) } by rows or by columns? This thing is the principal one because their false place in the matrix, Eq. (66), causes incorrect form of the multi-channel Green's function and its boundary limits. In order to answer this question the non-diagonal matrix H(R), Eq. (63), has to be considered. If the channel coupling vanishes, the non-diagonal matrix Hamiltonian, Eq. (63), converges to the unperturbed diagonal Hamiltonian H(R), Eq. (64), as (67) lim H(R)PH(R) . 0 0 The non-diagonal matrix solution of the multi-channel equation (62) must converge to the diagonal solution of the SchroK dinger equation with unperturbed Hamiltonian, Eq. (64): V
lim U(R)PU(R) , 0 0 U(R),"" (R)d "" . G GH V
(68)
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Let us rewrite Eq. (62) as I (E!H(R)) (R)" < (R) (R) . (69) G GH GK KH K The wave function (R) "ts to the inhomogeneous SchroK dinger equation for the ith channel. It is GH easy to see from Eqs. (66)}(69) that each of the wave functions (R), j"1, 2,2, k, is a perturbed GH partial wave in the ith channel. Therefore, we may say, that the row [ (R), (R),2, (R)] G G GI contains all the partial solutions for the ith channel. So, the correct answer for the question above is as follows: E all the channel solutions of Eq. (62) must be placed into the matrix function, Eq. (66), by rows, E the function (R) is a partial solution in the ith channel, GH E the partial wave (R) is excited by the other channel j interacting with the ith one. GH In an approximation of pair interaction of channels in the "xed point X: < (R)"< (X )d , GH GH GH HK
(R) is the wave transited from the jth channel to the ith one after its passage through their GH interaction point X . GH We have to emphasize, that only the multi-channel solutions in matrix representation, Eq. (66), may be used to construct the Green function of multi-channel SchroK dinger equation. It is simple to see below, that the use of fundamental solutions in vector representation gives an incorrect equation for the multi-channel Green's function. Following Eqs. (67) and (68) the limit of the multi-channel Green's function, which is searched for, is (70) lim G(R, R ; E)PG(R, R ; E) , 0 0 G(R, R ; E)"""G(R, R ; E)d "" , (71) G GH where a coupling-free diagonal matrix Green's function, Eq. (71), "ts the diagonal Hamiltonian, Eq. (64), as follows: V
+IE!H(R),G(R, R ; E)"Id(R!R ) and each of its components "ts the one-channel unperturbed SchroK dinger equation +E!H(R),G(R, R ; E)"d(R!R ) . G G
(72)
3.2. Time-independent Green's function of multi-channel SchroK dinger equation The Green function of the multi-channel Hamiltonian, Eq. (63), ful"ls the following second-order di!erential inhomogeneous equation +IE!H(R),G(R, R ; E)"Id(R!R )
(73)
and is a matrix function of two variables R and R , depending on the energy E as on a parameter G(R, R ; E)"""GGH(R, R ; E)"" .
(74)
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The Green function de"ned by Eqs. (73) and (74) is continuous at a point R"R G(R#e, R; E)"G(R!e, R; E), eP0 ,
(75)
and has a "rst kind discontinuity there
d ! G(R, R ; E) dR
d G(R, R ; E) dR
2k " I.
(76)
0 0Y>C 0 0Y\C Let us search for a solution of the inhomogeneous di!erential equation (73) in the form
2k U (R)A(R ), R'R , G(R, R ; E)" (77)
U (R)B(R ), R(R , where U (R) and U (R) are the two linearly independent solutions of Eq. (62), which "tted the standard boundary conditions at zero and at in"nity. A(R) and B(R) are the matrices to be found. The multi-channel functions U (R) and U (R) have non-diagonal matrix form of Eq. (66). Substituting Eq. (77) into Eq. (76) we obtain U A#U B"I . Premultiplying Eq. (78) by [U ]\ on the left and allowing for Eq. (75) yields A"[U !U U\U ]\ , if, of course,
(78)
(79)
det[U !U U\U ]O0 . Like Eq. (79), we obtain that
(80)
B"![U !U U\U ]\ , det[U !U U\U ]O0 . Now we show that if the matrices of potentials U(R) and V(R) are symmetric:
(81)
V(R)"V2(R), U(R)"U2(R)
(82)
(83)
or Hermitian: V(R)"V>(R), U(R)"U>(R) , then Eqs. (79) and (81) can be expressed through the regular solutions U and U . Substituting Eqs. (63) and (64) into Eq. (62), premultiplying it by function U or U and using the fact that UU\"U\U"I , The superscript 2 means transpose, the > denotes Hermitian conjugation, A\ is an inverse matrix to A, AA\" A\A"I.
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one gets the equation 2k [U (R)U\ (R)] !U (R)[U\ (R)] " [U(R)#V(R)!EI ] .
(84)
In order to "nd the functions [U\ ] we use the following property of derivative of a matrix [120] (p is an integer) d dA(t) [A(t)]\N"!A\N(t) A\N(t) . dt dt Then, [U\ (R)] "!U\ (R)U (R)U\ (R) . (85) Substituting Eq. (85) into Eq. (84) yields the matrix Riccati equation in the form like [107]: 2k f (R)#[ f (R)]" [U(R)#V(R)!EI ] ,
(86)
where (R),U (R)U\ (R) . Under the condition of Eq. (83) the functions f 2 (R) "t the Riccati equation (86) too. Therefore, if the equation f
(R )"f 2 (R ) is true in a "xed point R then it is true for any R f (R)"f 2 (R) . Making some transformations with Eqs. (79) and (81) and substituting Eq. (85) into them permit to deduce "nally the following equation for the multi-channel Green's function (74) of the symmetric SchroK dinger Hamiltonian f
H(R)"H2(R) ,
(87)
R'R , 2k U (R)W\U2 (R ), G(R, R ; E)"
U (R)[W2]\U2 (R ), R(R , where W is an invertible R-independent matrix W"U2 (R)[U (R)] ![U2 (R)] U (R)"constO0 . The multi-channel Green's function, Eq. (88), obeys the following symmetry properties: G(R, R ; E)"G2(R , R; E) , G>(R, R ; E)"GH(R , R; E) . If the Green's function (88) is a real one (of, in general, complex energy, EOEH), then G>(R, R ; E)"G(R , R; EH) .
(88)
(89)
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Starting from Eq. (86), proposing the potential matrices to be Hermitian and the energy to be real one can deduce the following equations for the multi-channel Green's function of the Hermitian SchroK dinger Hamiltonian H(R)"H>(R) ,
(90)
(R ), R'R , 2k U (R)W\U> G(R, R ; E)"
U (R)[W>]\U>(R ), R(R , W"U>(R)[U (R)] ![U>(R)] U (R)"constO0 , G(R, R ; E)"G>(R , R; E) .
(91) (92)
If the Hamiltonian, Eq. (63), has no symmetry
H(R)O
H2(R)
, EOEH H>(R)
(93)
the multi-channel Green's function, Eq. (74), may be written in its general form as
(R )U (R )]\, R'R , 2k U (R)[U (R )!U (R )U\ G(R, R ; E)" (94)
U (R)[U (R )U\(R )U (R )!U (R )]\, R(R . Let us consider now the multi-channel Green's function Eq. (88). Using the determination of inverse matrix and properties of invertible matrices [120] gives the multi-channel Green's function components as 2k 1 I (R) (R )A , GGH(R, R ; E; R'R )" GJ HK KJ
"W" J K 2k 1 I GGH(R, R ; E; R(R )" I (R) (R )A I , GJ HK JK
"W" I J K I I "W",det W" =( J , J )A , ? K ? J KJ J ?J J J A "(!1)K>J (!1)R.J =( P , P ) , KJ ?P ?P P ?P .J P "+f , f , f ,2, f , , J I
(95) (96) (97)
where P is a permutation +f , f , f ,2, f , of k!1 following numbers 1, 2, 3,2, l!1, l#1,2, k J I written in any sequence, t(P ) is the number of transpositions which bring the +f , f , f ,2, f , J I permutation to the normally graduated one +1, 2, 3,2, l!1, l#1,2, k, (the P permutation takes J all the (k!1)! possible permutations), =(u, v)"uv !vu . Applying Eqs. (95)}(97) one can formulate a simple rule to write out the GGH(R, R ; E; R(R )components of the multi-channel Green's function, Eq. (88) or Eq. (91), using its GGH(R, R ; E; R'R )-ones: in order to get the GGH(R, R ; E; R(R )-component of Eq. (88) one needs
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to replace the variables R and R with each other in the GHG(R, R ; E; R'R )-component GGH(R, R ; E; R(R )"GHG(R, R ; E; R'R )" ; 0@0Y for the components of Eq. (91) the rule is: GGH(R, R ; E; R(R )"[GHG(R, R ; E; R'R )" ]H . 0@0Y
(98)
(99)
3.3. Spectral expansion for multi-channel Green's function A spectral expansion of the multi-channel Green's function can be obtained via eigenfunctions of multi-channel Hamiltonian, Eq. (63). So, the eigenfunctions W(R) of the symmetric Hamiltonian, Eq. (87), H(R)W (R)"E W (R) K K K satisfy the following conditions of orthonormalization and completeness:
dR W (R)W2 (R)"Id , K KY KKY
(100)
(101)
W (R)W2 (R )"Id(R!R ) . (102) K K K Substituting Eq. (102) into Eq. (73) and its simple transformations yield the following spectral expansion for the multi-channel Green's function, Eq. (88): W (R)W2 (R ) K . G(R, R ; E)" K E!E #i0 K K The eigenfunctions of the Hermitian Hamiltonian, Eq. (90), "t the equations
dR W (R)W> (R)"Id , K KY KKY
W (R)W>(R )"Id(R!R ) K K K and the spectral expansion for its Green's function, Eq. (91), is
(103)
(104) (105)
W (R)W>(R ) K . (106) G(R, R ; E)" K E!E #i0 K K The spectral expansion for the multi-channel Green's function of the Hamiltonian without symmetry, Eq. (93), cannot be obtained because the set of its eigenfunctions does not form a basis.
Another multi-channel Green's function has been used in papers [121}123] too, but the results obtained in [121}123] contradict those of papers [107}117] and our Eqs. (70), (77), (91) and (101)}(106).
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3.4. Expression of multi-channel Green's function through the one-matrix solution Here the method of paper [107] is generalized upon the non-perturbative channel interactions. Let us express the multi-channel Green's function of the symmetric Hamiltonian, Eq. (88), through the regular in zero solution U (R). From Eqs. (88) and (89) one gets 0 U (R)"U (R) [U2 (X)U (X)]\W dX , (107) ?7 where "a2 is the vector limit, that
U\(R)U (R)" "0 . 07 ?7 Substituting Eq. (107) into Eq. (88) yields the expression of the multi-channel Green's function Eq. (88) through only the regular in zero multi-channel solution 2k G(R, R ; E)" U (R)J(R )U2 (R ) ,
0
[U2 (X)U (X)]\W dX, R "max+R, R , . ?7 The multi-channel Green's function of the Hermitian Hamiltonian, Eq. (91), is expressed through the regular in zero solution as J(R)"
2k G(R, R ; E)" U (R)J(R )U>(R ) ,
J(R)"
0
[U>(X)U (X)]\W dX .
?7 Therefore, the multi-channel Green's function may be expressed in the analytic form using Eqs. (88), (91) or (94). The form to write out the Green function depends on symmetry of the laser}molecular Hamiltonian and can be various for di!erent laser molecule multi-photon interaction. If we consider the spectroscopic bound}bound transitions (the dissociative and ionizing channels are absent), then the multi-channel Green's function is presented by Eqs. (91) and (106). If the bound}free transitions involved in the dissociative channels (scattering-like processes) are studied, then the Green function is given by Eqs. (88) and (103). For the non-adiabatic interactions inducing the non-perturbative coupling with ionization channels the general expression of the multi-channel Green's function, Eq. (94), should be used.
4. Quasi-classical approximation for multi-channel systems To "nd the multi-channel wave functions which are solutions of coupled equations (62) is a complex problem. The multi-channel wave functions have been obtained either by means of numerical integration of Eq. (62) [103,104,114] or from the integral Lipmann}Schwinger equation [108]. In this review a quasi-classical method to integrate the coupled equations (62) is presented.
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Fig. 1. Intersecting potentials in general multi-channel case in dependence on inter-nuclear distance R. R (R ) is the left G G (right) turning point in unperturbed potential ; (R), X is the branch point (the point of crossing of the ; (R) and ; (R) G GH G H potentials, X "X ). GH HG
Within the method the multi-channel SchroK dinger equation is considered in the matrix representation, Eq. (62), where the potential consists of a system of k attractive and/or dissociative potentials coupled with each other in few intersection points (Fig. 1). The matrix SchroK dinger equation is integrated in a simple analytic form without perturbative restrictions upon inter-channel couplings. Within the quasi-classical approach the multi-channel solutions of the coupled equations (62) are obtained in the analytic form for all the points of R-variable except for a few singular points. The multi-channel solution U(R) may be written in the quasi-classical approximation as [19,20]: U(R)"E>(R , R)a>#E\(R , R)a\ , (108) G G where E!(R , R ) are the diagonal matrices of quasi-classical waves, a! are the non-diagonal R-independent matrices of wave amplitudes: E!(R , R )"""j\ (R )F!(R , R )d "" , K K KJ a!"""a! "", m, l"1, 2,2, k . KJ In a classical permitted point R the wave in the m th channel F!(R , R ) has the form K F!(R , R )"exp+$i¸ (R , R ), , K K and in a classical forbidden one
(109) (110)
(111)
F!(R , R )"exp+s"¸ (R , R )", , (112) K K the sign s can be equal to 1 or !1 in dependence on the boundary condition for the m th channel,
j (R) is the classical kinetic momentum in the m th channel ¸ (R , R ) is the reduced action along K K
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the channel potential ; (R) K
j (R)" \[2k(E!; (R))] , ¸ (R , R )" K K K
289
0
j (R) dR . (113) K 0 The quasi-classical representation for the multi-channel solution, Eqs. (108)}(113), is true if the condition
dj\(R) K ;1 dR
(114)
holds for all the channel potentials. 4.1. Quasi-classical propagation of multi-channel solutions through singular points The multi-channel solution, Eq. (108), propagates through the isolated turning point as follows: if the (R) component in a classical forbidden R-region has the following view: GH a (115)
(R)""j (R)"\ GH exp+!"¸ (R , R)", , G G GH G 2 then its form in a classical permitted R-region is
(R)"j (R)\ a cos("¸ (R , R)"!p/4) ; GH G GH G G if the wave component in the classical permitted region is
(116)
(R)"j (R)\ +a> exp+i["¸ (R , R)"#p/4],#a\ exp+!i["¸ (R , R)"#p/4],, , GH G GH G G GH G G then in the classical forbidden region it is
(117)
(R)""j (R)"\ (a>#a\) exp+"¸ (R , R)", . (118) GH G GH GH G G Eqs. (115)}(118) are a simple multi-channel representation of the well-known one-channel formalism of FroK man and FroK man [124]. The propagation through a branch point (the point of crossing of diabatic potentials and that of quasi-crossing of adiabatic ones) changes the wave amplitudes, Eq. (110). Here we consider the crossing picture, for which the following conditions are true: ¸ <1, "p "<1 , (119) G GH ¸ ,¸ (R , R ) p ,¸ (R , X )!¸ (R , X ) . (120) G G G G GH G G GH H H GH In a point R between two neighbouring branch points X ,X ,X (R(X (Fig. 1) the GH\ GH GH\ GH multi-channel solution, Eq. (108), may be written as a sum of incoming and outgoing matrix waves as U(R)"E>(X , R)a>#E\(X , R)a\ . (121) GH GH For the R-point between the next branch points X , X , X (R(X the multi-channel GH GH> GH GH> solution may be written as U(R)"E>(X , R)a>#E\(X , R)a\ . GH GH
(122)
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The amplitudes a! and a! in Eqs. (121) and (122) are connected by the k;k matrix N : GH a>"N a> , GH a\"NH a\ . (123) GH The N matrix has the following structure: GH its components N "d for all m, lOi, j , KJ KJ its components N , N , N , N form the 2;2 matrix , GG GH HG HH N N GH . (124) " GG N N HG HH The -matrix connects the wave amplitudes of the two interaction channels i and j on the left and right sides from the non-adiabatic point X and is expressed in di!erent analytic forms within GH either a diabatic basis or an adiabatic basis for the total energy E lying either above or under the crossing point energy ; (X ). G GH The 2;2 matrix of non-adiabatic transitions has been studied earlier very widely in atomic collisions theory to obtain the S matrix of reactive scattering [125}129]. Here we generalize the main results of papers [125}129] upon the multi-term wave function problem. For the abovebarrier case the matrix may be expressed as (the branch point lies in the classical permitted region, the diabatic and adiabatic curves are marked as those in Fig. 2): In the diabatic basis
(P exp[it ] GH GH (1!P exp[!i ] GH GH In the adiabatic basis "
!(1!P exp[i ] GH GH . !(P exp[!it ] GH GH
(125)
(1!P exp[i ] (P exp[it ] GH GH GH GH . (126) !(P exp[!it ] (1!P exp[!i ] GH GH GH GH The non-adiabatic parameters P , t , describe the interaction of the intersecting channel GH GH GH potentials ; (R) and ; (R). They depend on velocity of e!ective particle in the X point, on the G H GH form of the potential curves and their reciprocal position, and on the non-adiabatic coupling strength "< (R)". The functional form of the non-adiabatic parameters is determined by a model of GH non-adiabatic transitions. The model used may be any one of the Landau}Zener, Nikitin, Demkov}Osherov, Nakamura models and their last generalizations [42,126,130}134]. The choice of the model is subjected to the curves picture to be appropriate for the transition process. The model admitted gives only the analytic equations for the non-adiabatic parameters but the general form of the multi-channel transition matrix N, Eqs. (124)}(126), connecting the quasi-classical wave amplitudes, Eq. (123), is the same for all the models. "
E(; (X ) is the so-called under-barrier non-adiabatic transition, E'; (X ) is the so-called above-barrier G GH G GH non-adiabatic transition.
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One can see from Eqs. (121)}(126) that the multi-channel approach, developed here, may be applied with the same success either for the diabatic or adiabatic basis. A basis from the two ones is only a representation (a picture) to describe the two-curve crossing problem. So, one may use the diabatic basis, say about the crossing diabatic potentials ;(R) and do the calculations with the G diabatic 2;2 matrix Eq. (125). On the other hand, one may use the adiabatic basis, say about the quasi-crossing adiabatic potentials ; (R) and calculate with the adiabatic matrix , Eq. (126), G (Fig. 2). This choice of the basis brings no formal changes to the construction of the non-adiabatic multi-channel wave solutions, Eq. (108). Therefore, the non-adiabatic approach presented in this review is a universal one to solve the multi-channel SchroK dinger equation independent of the non-adiabatic transition model and the type of basis used and may be applied to study other multi-potential problems of quantum mechanics. 4.2. Regular solutions of multi-channel Schro( dinger equation Using the rules formulated above permits to construct the regular in zero and regular at in"nity solutions of the multi-channel SchroK dinger equation (62). In the R-point lying between two branch points X and X , X (R(X both solutions are: GH GH> GH GH> U (R)"E>(X , R)Aa>#E\(X , R)AHa\ , GH * GH *
(127)
U (R)"E>(X , R)Ba>#E\(X , R)BHa\ , GH 0 GH 0
(128)
Fig. 2. Branches of diabatic (solid lines) and adiabatic (dashed lines) potentials ; (R) and ; (R) form diabatic and G H G H adiabatic bases. The subscripts of diabatic and adiabatic curves are the same if R(X . GH
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where A, B, a!, a! are the following R-independent matrices: * 0 A"N F>(X , X )N N F>(X , X )N , (129) GH GH\ GH GH\ 2 B"F>(X , X )N> N> F>(X ,X )N> , (130) GH> GH GH> 2 LL\ LL\ LL\ LL\ F!(X , X )"""F!(X , X )d "", [F!]2"F! , K KJ a! """(a! ) d "", [a! ]2"a! . (131) * 0 * 0 K KJ * 0 * 0 Here we consider the general case, where EHOE and, therefore, ¸H O¸ . In the opposite case, K K Eqs. (131) are supplemented by the following: [F!]>"F8, [a! ]>"a8 . The channel left-side * 0 * 0 amplitudes are (both for open and closed m-channels): In the classical permitted region
p a (a!) " K exp $i ¸ (R , X )! K K * K 4 2
,
In the classical forbidden region a (a!) " K exp+!"¸ (R , X )", ; * K K K 2 the channel right-side amplitude are: (i) for the case of the closed m-channel: In the classical permitted region
b p (a!) " K exp $i ¸ (R , X )# 0 K K K LL\ 2 4
,
,
In the classical forbidden region b (a!) " K exp+!"¸ (R , X )", , 0 K K K LL\ 2 (ii) for the case of the open m-channel:
p )# (a!) "b! exp $i ¸ (R , X K K LL\ 0 K K 4
where a , b , b! are constants. K K K The matrix amplitudes of regular solutions, Eqs. (127) and (128), are connected with each other by means of T-matrix as Aa>"T(ij)AHa\ , (132) * * Ba>"T(ij)BHa\ . (133) 0 0 Including Eqs. (127) and (128) into the matrix Wronskian, Eqs. (89) and (92) deduces its independence on the R-variable. In the quasi-classical approximation, Eqs. (89) and (92) are formally equal
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to each other and have the form W"2i+a\Sa>!a>SHa\, , * 0 * 0
(134)
S"A>B"N> F\(X , X )N> 2N> F\(X ,X )N> . LL\ LL\ LL\ LL\ The matrices A and B, Eqs. (129) and (130), are the quasi-classical multi-channel propagators evolving the regular channel waves along the multi-channel passage from their origins to the R point: A propagates the multi-channel wave from zero and B propagates it from in"nity. Both propagators allow for only the inter-channel interactions to exist between the wave origin and the R point. So, the A-propagator involves the interactions lying between zero and R and not those lying between in"nity and R; the B-propagator involves the interactions lying between in"nity and R and not those lying between zero and R. For a quantum state in such multi-curve potential the regular in zero solution, Eq. (127), must be equal to the regular at in"nity one, Eq. (128), in an R-point as well as their "rst derivatives must be equal to each other in the same point. The multi-channel regular solutions, Eqs. (127) and (128), may be obtained quasi-classically for di!erent resonant photoprocesses in molecules. In dependence on type of the resonant phototransition the solutions are substituted into a multi-channel Green's function from those obtained above, Eqs. (88), (91) and (94), that gives the multi-channel Green's function of the nuclear motion through intersecting molecular curves without any restriction upon the form of the terms and strength of non-adiabatic couplings. The various quasi-classical multi-channel wave functions and Green's functions are presented in Appendices A and B. 4.3. Restrictions of the multi-channel quasi-classical approximation The multi-channel quasi-classical approach presented here does not consider the potential barriers. A method to take into account the barrier e!ects has been elaborated in papers [129,135] and can be simply generalized based upon the multi-channel propagators. The potential barriers are important only either for the resonant multi-photon processes involving high-excited rotation}vibration states of molecules (the so-called near threshold processes) or for the radiative collisions of atoms with very low relative kinetic energies (the so-called cold atomic collisions, Fig. 3). The barriers do not a!ect signi"cantly the dynamics of resonant laser}molecule interactions in ordinary experimental circumstances and are omitted in our formulas. The quasi-classical approach developed in this review permits to obtain the multi-channel wave functions and Green's functions in the analytic form, which is useful to consider the principal picture of the laser}molecule interactions and to receive simple equations for cross sections of resonant intense-"eld photoprocesses. Our multi-channel method permits to get the equations in the form to be appropriate for a simple physics analysis without routine numerical computations. The quasi-classical functions obtained may be used, if the conditions, Eqs. (114) and (119), are true. In papers [136,137] it has been proved that the analytic equations for amplitudes of the resonant phototransitions, received with the quasi-classical functions, are applicable to calculate the laser}molecule transition probabilities even beyond the quasi-classical conditions, Eqs. (114) and (119).
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Fig. 3. Cases of strong impact of barrier e!ects. (a), (b) Forms and positions of electronic terms for the processes involving high-excited molecular rotation}vibration states: (a) the molecular energy lies near the dissociation threshold of the molecule, E X &D , (b) the energy of dressed molecule is near the dissociation threshold of excited molecule, NT( E X # u&D . (c), (d) Radiative collision processes impacted by barrier e!ects: (c) cold collisions, ¹C#e&¹C, (d) laser NT( cooling of colliding atoms ¹C #e#u&¹C .
5. Delay of direct photodissociation and orientation of molecules in intense resonant 5elds The general theoretical approach elaborated above is applied to study a few photoprocesses during laser}molecule resonant interactions. The principal aspects are that the laser radiation is in resonance with an electronic transition in the diatomics and the increase of laser intensity invokes new phenomena which can be unregistered in the weak resonant "elds. Some of the photoprocesses described below have been studied also by means of other theoretical methods both before and after our multi-channel consideration. Here the advances of our non-perturbative approach of non-adiabatic nuclear wave functions and Green's functions applied to the resonant laser}molecule interactions are presented. In a few cases this method permitted to predict surprising strong "eld e!ects observed and calculated later in other works. The "rst process during the resonant laser molecule interaction discussed here is a direct photodissociation of diatomic molecules in intense "eld. This photoprocess was maybe the "rst one that revealed its surprising behaviour in strong laser "elds [60]. 5.1. Direct molecular photodissociation The photodissociation is a physical-chemical process, in which a molecule AB, which absorbed the laser photon u, dissociates into atoms A and B [138,139] (Fig. 4): AB#(N#1) uNA#B#N u .
(135)
There was a comprehensive consideration of the photodissociation within a perturbation theory, where the laser}molecule interaction was approximated to be small and its matrix element was calculated as the "rst-order one over the laser}molecule perturbation [3,4,7]. Within the adiabatic
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Fig. 4. Direct molecular photodissociation, Eq. (135), in two-term approximation.
approximation the electronic and nuclear motions in the molecule may be separated in its initial and "nal states and the problem is reduced to the determination of the initial and "nal nuclear wave functions and dipole moment of the resonant transition between the electronic terms that participated in the process. Firstly, the nuclear wave functions of attractive and dissociative molecular terms were obtained by means of numerical integration of adiabatic SchroK dinger equation and the photodissociation cross section was calculated numerically (see the pioneering paper [7] as well as papers [140}142]). A "t of the initial and "nal terms to the Morse-type or hard ionic core-type potentials and a canonical expansion of the electronic dipole transition moment permitted "rstly to receive the analytic equations for the photodissociation cross-section [2}6]. The equations became simpler if the adiabatic nuclear wave functions were written in quasi-classical approximation and the electronic transition moment was assumed to be an R-independent constant [143,144]. In the intense resonant laser "eld, which couples the quantum states of nuclear motions in the attractive and dissociative terms, the picture of dissociation is qualitatively changed. A delay of photodissociation takes place and some peculiarities occur in angular distribution of the photodissociation fragments [10,15,16,145}147]. In this review the non-perturbative theory of photodissociation based upon our quasi-classical multi-channel wave functions is presented. In accordance with a time-dependent scattering theory [148,149] the probability (per time unit) of process, Eq. (135), may be written as 2p dP " "1W\"< "U 2"d(E !E) do ,
(136)
where "W 2, "U 2 and E , E are the eigenvectors and eigenvalues of the full and unperturbed laser}molecule Hamiltonians, Eqs. (1) and (2), H"W 2"E "W 2 , H "U 2"E"U 2 ,
(137) (138)
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do is the number of "nal laser}molecule states, do "de dq( , (139) < is the operator of dipole laser}molecule interaction for the monochromatic "eld (e, u) (Eq. (32) has no summation over "eld harmonics here). The laser}molecule wave function of the unperturbed initial state "U 2 has the adiabatic form, Eq. (23), given below (we suppose that the molecule is initially in its R-electronic state, X "0): U, (m )"1m "pJM02s (R)R\"N#12 , (140) NT(+ ? ? TN( where 1m "pJMX2 is the molecular electronic-rotation function, Eq. (12), s (R) is the nuclear ? TN( wave function for the "J2 rotation}vibration state in the ground attractive term, "N#12 is the Fock wave function of free laser "eld. The adiabatic molecular wave function, Eq. (140), is normalized as: (m )"U, , (141) (m )2"d d d d d d 1U,Y NNY TTY ((Y ++Y XXY ,,Y NYTY(Y+YXY ? NT(+X ? the adiabatic nuclear wave function s (R) is normalized as TN( 1s (R)"s (R)2"d d d . (142) TYNY(Y TN( NNY TTY ((Y The dissociative laser}molecule wave function for the perturbed "nal state "W 2 has the following non-adiabatic form like Eq. (37): W! (m )"C B!, 1m "pJMX2s(+ (R; N)R\>H (q( )"N2 , (143) # ? B ? NX (+ (X X N(+ , is the phase factor where C is a factor depending on the total wave function normalization, B!, B (X for dissociation, "i(e! N,(X , B!, (X p,X is the dissociative phase shift, s(+ (R; N) is the multi-term nuclear wave function, the spherical ( NX function >H (q( ) allows for the direction of relative momentum of the recoil atoms. (+ The normalization of the non-adiabatic molecular wave function, Eq. (143), is given by Eq. (46) and that of the non-adiabatic nuclear wave function s(+ (R; N) is given as Eq. (47). For the NX normalizations admitted the factor C is given as B C "1 . (144) B The energies of initial and "nal laser}molecule states are expressed for the photodissociation as follows (Fig. 4): E"E #(N#3/2) u , NT( E "; (R)#e#(N#1/2) u#*E , *E is the energy shift of recoil atoms, induced by their interaction with laser "eld after dissociation. The di!erential cross-section of the photoprocess, Eq. (135), is proportional to its probability, Eq. (136): dP dp" . j
(145)
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where j is the density of incident photon #ux which is equal, in our case, to j"c(N#1)/V .
(146)
Here we consider the direct molecular photodissociation in the approximation of two resonant electronic terms. So, there are only two possible sets of the summation indexes p, X in Eq. (143): +p"1, X "0, and +p"2, X ,. Substituting Eqs. (136), (139), (140), (143), (144) and (146) into Eq. (145), integrating it over the relative energy e of the recoil atoms and taking into account the intense-"eld approximation, Eq. (52), we obtain the following di!erential cross-section for the molecular photodissociation in intense laser "eld:
dp(e, q( ; v , J ; M ) 4pu " [B\,]H(M #MU)> (q( ) . (+ (+, (+, (+ c dq( (+,
(147)
where M and MU are the resonant and non-resonant non-perturbative nuclear transition (+, (+, amplitudes: X (e) , (R; N)"D (R)"s (R)2A(+ M "1s(+ ( + T( (+, X
(148)
X (eH) , MU"1s(+ (R; N#2)"D (R)"s (R)2A(+ (+, X T( ( +
(149)
X (eH) D (R) is the modulus of the molecular dipole transition moment, Eq. (43), the factor A(+ ( + allows for the laser "eld polarization and orientation of molecular dipole transition moment, Eq. (41). The di!erential cross-section, Eq. (147), allows for the direction of relative momentum of recoil atoms. The cross-section equation (147) includes a non-linear dependence on laser "eld because the (R; N) and s(+ (R; N#2) non-adiabatic nuclear wave functions of the dissociative channel s(+ X X depend on the "eld occupation number N. Eq. (147) contains also the summation over all molecular rotation quantum numbers J and M and, therefore, it takes into account the impact of the resonant laser "eld upon molecular rotations. We emphasize, that due to selection rules for the Clebsch}Gordan coe$cients in the geometric factor, Eq. (41) [150], this sum in Eq. (147) is done over only three di!erent rotation channels J"J !1, J , J #1, but nevertheless the nuclear amplitudes, Eqs. (148) and (149), include the contributions from all the rotation channels by means of the non-adiabatic dissociative nuclear wave functions which are the solutions of a multi-channel di!erential equation like Eq. (38). If the molecular photodissociation is induced by a laser of UV or shorter wavelength, then the non-resonant nuclear amplitude, Eq. (149), may be omitted in Eq. (147) in comparison with the resonant one, Eq. (148). So, the resonant cross-section equation (147) takes the form
dp(e, q( ; v , J , M ) 4pu " [B\,]HM > (q( ) . (+ (+, (+ c dq( (+,
(150)
This simpli"cation is possible because the energy gap between two molecular terms dressed by N and N#2 photons of the UV or shorter laser are very wide and the order of smallness of the
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non-resonant amplitude, Eq. (149), in comparison with the resonant one, Eq. (148), is like e\I S* 0C (see below). (R; N) "ts the following coupled equations: The non-adiabatic nuclear wave function s(+ X D (R) [F>A( +X(eH)s(+ (R; N) [H (R)!E]s( + (R; N#1)# (+ X 2 (+ #F>A( +X(e)s(+ (R; N#2)]"0 , X (+ DH (R) X I (e)s(I + (R; N#1) [F>A(+ [H (R)!E]s(+ X (R; N)# (I +I 2 I I ( + X I (eH)s(I + #FA(+ (R; N!1)]"0 , (151) (I +I X where the geometric factors AN ( + and monochromatic "eld strengths F> are given in N(+ Eqs. (41) and (42). The coupled equations (151) are a simple modi"cation of the general exact system, Eq. (38) for the case of two resonant electronic term model. No other approximations are done in Eq. (151). The multi-channel system, Eq. (151), can be simpli"ed to take into account, that in the laser "elds of UV and shorter wavelengths some of its terms are very weak and may be omitted too. So, the non-adiabatic nuclear wave functions involved in Eq. (151) correspond to the following waves in the multi-term curve crossing picture (the rotation quantum numbers are not shown here): s (R; N#1) is the wave in the term ; (R; N#1), s (R; N) is the wave in the term ; (R; N), s (R; N#2) is the wave in the term ; (R; N#2), s (R; N!1) is the wave in the term ; (R; N!1), where the dressed terms are (152) ;(X (R; N )";(X (R)# u(N #1/2) , N N ;(X (R) is the molecular electronic-rotation term, Eq. (17). N As it follows from conservation of total energy in Eq. (136), the "nal energy of the laser}molecule system E is de"ned by the equation E "E #(N#3/2) u NT( and is a constant during the laser}molecule interaction. Therefore, for the "xed energy E the magnitudes of the wave functions included into Eq. (151) may be estimated as follows: s (R; N#1)&(u )\ C , s (R; N)&(u )\ C , s (R; N#2)&(u )\ Ce\IS* 0C , * s (R; N!1)&(u )\ C , *
(153)
In the long wave lasers (IR or micro-wave ones) the contributions of both amplitudes are comparable with each other and the photodissociation must be calculated using Eq. (147) [59].
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C is a dimensional constant. In the short wave lasers, whose frequency is u &10u (see * Table 1), the last two functions in Eq. (153) are about 10 times and more smaller than the two "rst functions and may be omitted in the coupled equations (151). We must emphasize that this fact is true only for the short-wave lasers and in the IR or optical lasers the full coupled system, Eq. (151), has to be solved. The coupled equations (151) may be simpli"ed more if one were to take into account the properties of molecular rotations. In paper [85] it has been deduced by means of direct numerical integration of coupled equations like Eq. (151), that the multi-channel nuclear wave functions are practically independent of the change of molecular momentum "*J" and one may approximate them as (R; N )+s( + (R; N ) . (154) s(! + NX NX The multi-channel solutions of Eq. (151) depend on magnitude of molecular angular momentum J but do not depend on its small change *J"$1. The radiation coupling of the rotation states is involved in Eq. (151) due to the selection rules for the geometric factor, Eq. (41), and only the geometric factor depends considerably on the molecular momentum change. The approximation, Eq. (154), is true only, of course, for the nuclear wave functions but not for the full molecular wave functions, Eq. (143), which contain the rotation wave function D( X (a, bI , 0) + whose dependence on J is evident and must be allowed for. The approximation, Eq. (154), means X that one has to keep the non-diagonal parts A(+ X (eH), A(+ (e) of multi-channel Hamil(!+ (!+ tonian in Eq. (151). It has been shown in the numerical computations [87], that due to these non-diagonal terms of the nuclear Hamiltonian the higher rotation channels with "*J"""J !J"'1 bring a contribution to the J channel although they are uncoupled with it directly. In other words, as the approximation, Eq. (154), is assumed, this fact does not mean a diagonalization of the coupled equations (151) over rotations, but means the nuclear wave functions may be put before the sums over JM. Taking into account Eqs. (153) and (154) and an approximation of two active electronic terms reduce the full non-adiabatic laser}molecule wave function, Eq. (143), to W! (m )"C (B!,>s(+(R; N#1)1m "1JM02N#12 B ( ? # ? (+ #B!, s(+ (R; N)1m "2JMX 2"N2)R\>H (q( ) . (155) ? (+ (X X It is possible to simplify the coupled equations (151) (as well as the general coupled equations, Eq. (38)) more, if one were to take into account that the rotation momentum J in real diatomics is very often much more than 1, J<1 .
(156)
In this case one may use an asymptotic expression for the Clebsch}Gordan coe$cients [150] in geometric factors, Eq. (41): a, c
(157)
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where h is the angle between the momentum c and the axis of its c-projection, dA is the function ? @ connected with the D Wigner function as d. (h)"exp(!i. a)D. (a, h, c) exp(!ilc) . J J Under the approximations, Eqs. (154) and (156), the summation over rotation quantum numbers in Eq. (38) and Eq. (151) is done analytically. Thus, one obtains Eq. (151) in the form (the UV wave conditions, Eq. (153), are involved too): (158)
H(R)v(R)"Ev(R)
where H(R) is the full matrix Hamiltonian of nuclear motion in the presence of the resonant laser "eld
d #U(R)V(R) H(R)"! 1 2k dR
(159)
U(R) is the diagonal matrix of the dressed laser}molecular terms, Eq. (152), (Fig. 5)
;( (R; N#1) 0 , (160) 0 ;( X (R; N) V(R) is the self-conjugated non-diagonal matrix of the laser-induced non-adiabatic coupling (in our case X "0): 0 < (R; eH) V(R)" , (161) 0
i(+"nJ d. (b)(eH). dX X , \ J . J
(163)
F is the laser "eld strength amplitude, b is the following angle:
M X !arccos , (164) J #1/2 J #1/2 v(R) is the total wave solution of the two-channel nuclear SchroK dinger equation (158), written in the vector form as b"arccos
s( + (R; N#1) , (R; N) s(X+ 1 is the unity 2;2 matrix: v(R)"
1 0 1"""d """ , i, j"1, 2 . GH 0 1
(165)
(166)
The laser-induced non-adiabatic coupling, Eqs. (40) and (41), may be reduced to the simpler form of Eqs. (162)}(164). The non-adiabatic inter-channel coupling, Eq. (162), (as well as Eqs. (40) and (41))
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depends on the inter-nuclear distance R, on the rotation quantum numbers J, M, X, on the electronic states involved in the coupling (via the dipole electronic transition moment, Eq. (43)), on the strength and polarization of laser "eld. Therefore, the laser}molecule non-adiabatic interaction couples together the electronic, nuclear and rotation motions in molecule as well as the quantum states of the laser "eld. The coupling, Eq. (162), changes the electronic, rotation, vibration quantum numbers and "eld occupation number. Therefore, the laser-induced coupling describes as the laser impact upon molecular rotations, electronic and nuclear intra-molecular motions as the response impact of the molecule upon the laser "eld. The approximation, Eqs. (154) and (156), permits to obtain the laser}molecule coupling, Eqs. (40) and (41), in the form of Eq. (162) which has a simple semi-classical explanation. The coupling, Eq. (162), depends on the angle b, Eq. (164), which is like but not exactly equal to the angle between the molecular axis and the OZ axis of the laboratory reference frame. The interaction of the rotating molecule with laser "eld depends, therefore, on the angle being like the angle between the molecular axis and the axis of laser polarization, that is a simple classical analogy of the laser}molecule dipole operator, Eqs. (29) and (30). The laser}molecule coupling, Eq. (162), has the quantum nature which concentrates on the magnitudes 1/2 in denominators of both the arccos terms in Eq. (164). Then, Eqs. (154) and (156) are the quantum approximations and the angle b, Eq. (164), is a quantum magnitude which di!ers from its classical analogy. The approximation, Eq. (156), is often called fast rotations approximation, but this determination is not used in this review because the conditions, Eqs. (154) and (157), are correct for the small rotation numbers J"1, 2, 3, 4, 5,2 too [85,150]. 5.2. Direct photodissociation in multi-channel quasi-classical approximation In this subsection we consider the quasi-classical theory of the direct molecular photodissociation, Eq. (135), developed "rstly in 1987, 1988 in papers [19,58]. The resonant photodissociation cross-section, Eq. (150), is a general equation whose nuclear amplitude, Eq. (148), can be calculated by means of di!erent theoretic techniques. Our method is to integrate the nuclear coupled equations analytically within the multi-channel quasi-classical approach elaborated in Section 4. Therefore, for the cross-section of direct photodissociation, Eq. (150), the non-adiabatic wave (R; N) and the adiabatic unperturbed wave function of the function of the dissociative channel s(+ X bound channel s (R) are presented within the quasi-classical approximation. T( Following Section 4, the regular in zero quasi-classical solutions of the two-channel coupled equations, Eqs. (158) and (165), may be written as (the sub- and superscripts J, M, X, N are omitted). s (R)"j\ (R)+a> exp[i(¸ (R , R)!p/4)] H H H H H #a\ exp[!i(¸ (R , R)!p/4)],, j"1, 2 . H H H
(167)
Here the multi-channel quasi-classical solutions of the two-channel SchroK dinger equation is considered in detail.
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The amplitudes a! are connected with each other by means of the two-channel T-matrix, Eq. (132), as
a! a>"T a\, a!" , T """¹ "" . GH a!
(168)
The T -matrix for the two-term approximation (Fig. 5) may be written in the following form [125}128]: if R(X , then T"1, if R'X , then
Pe R#(1!p)e\ N \( 2i(P(1!P) sin(p ! #t) . T" Pe\ R#(1!p)e N \( 2i(P(1!P) sin(p ! #t)
(169)
In order to "nd the quasi-classical wave amplitudes a! , Eq. (167), one needs to connect the regular in zero solutions, Eq. (167), with the regular at in"nity ones which for the photodissociation are (R'R , Fig. 5): s (R)"A "j (R)"\ exp[!"¸ (R , R)"] , s (R)"j\ (R)+A exp[i(¸ (R , R)#p/4)]#C exp[!i(¸ (R , R)#p/4)], .
(170)
The quasi-classical propagation of the wave functions, Eq. (170), to the classical permitted region X (R(R and the comparison of them with Eq. (167) give the following equation for the non-adiabatic nuclear wave function of the dissociative channel involving the non-adiabatic matrix element, Eq. (148), s (R)"j\ (R)2C i"W"\F (R) ,
(171)
Fig. 5. Dressed molecular terms ;( (R; N#1) and ;( X (R; N), Eq. (152), for direct photodissociation in UV laser "eld. The model of two active electronic terms which brought principal contribution to the non-adiabatic amplitude is admitted. The dashed lines show laser-induced adiabatic curves, Eq. (184).
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where the functions F (R), "W" have been obtained in Section B.1, Eqs. (B.2) and (B.9). The factor C is found from the two-channel normalization condition like Eq. (47) to be k "W" , w ""W"#" (¹) " . (172) "C "" 2p w The quasi-classical adiabatic wave function of the bound channel s (R) has the following form: T( 2k RE NT( cos(¸ (R , R)!p/4) . (173) s (R)" N N TN( p j (R) Rv N Substituting Eqs. (171)}(173) into Eq. (148) one obtains an integral which may be calculated analytically by a stationary phase method (see Appendix C, [151]). Then we obtain the nonadiabatic nuclear integral in the amplitude, Eq. (148), in the form
i 2 RE T ( C (X )g (X ) , (R)"s (R)2+ 1s(+ X (R; N)"D + T( Rv p w where
pk , C (R)"D (R) GH GH
j (R)"q (R)" G GH g (X )"Pr cos(p #t!q )#(1!P)r cos(p !q ) , + ? p "¸ (R , X )!¸ (R , X ) , ¸ (R , R)"¸ (R , R)#p ! , d;(X (R) d;(X (R) G , ! H q (X)" GH dR dR 06 p q " sign(q ) , GH 4 GH
(174)
(175) (176) (177) (178) (179) (180)
the functions r , r are given in Section A.1.1, Eqs. (A.3) and (A.4), p appears in Eq. (120). Eq. (174) simply transforms to the well-known adiabatic quasi-classical matrix element [152,153] in the limit of weak laser}molecule coupling: P"1, 1!P"0,
"p/4, t"0 .
(181)
Therefore, the di!erential photodissociation cross-section, Eq. (150), including approximations, Eqs. (153), (157) and (174), can be expressed as dp(e, q( ; v , J , M ) 8p RE "C (X )g (X )i( + > (q( )" T ( + (+ " c Rv w dq( and the full photodissociation cross-section, being an integral from Eq. (182) over q( , is 8p RE "C (X )g (X )i( + " T ( + . p( + (e, v )" ." w c Rv
(182)
(183)
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The photodissociation cross-sections, Eqs. (182) and (183), allow for an interference of the following non-adiabatic transitions (see Eq. (176) and Figs. 4 and 5): that from the ground electronic term ; (R) to the diabatic term ; X (R), and that from the ground one to the adiabatic laser-induced term ; (R) ; (R)" (;( X (R; N#1)#;( X (R; N) G[(;( X (R; N#1)!;( X (R; N))#4"< (R; eH)"] ) . (184) It must be emphasized that in the non-adiabatic case there are two types of non-adiabatic transitions: the transition to the diabatic molecular potential and another one to the adiabatic laser-induced potential. The non-adiabatic transition probabilities P and 1!P are the damping factors which increase or decrease the contributions of both transitions to the total cross-section in dependence on the laser "eld power. In the intense laser "eld both non-adiabatic probabilities are comparable P&1!P
(185)
and both non-adiabatic transitions bring the comparable contributions to the photodissociation. In the weak laser "eld limit, Eq. (181), the cross-section equations (182) and (183) are the quasi-classical form of the well-known perturbative photodissociation equations [2,4,140}142]. In the strong "eld limit, where the non-adiabatic parameters are as follows: P"0, 1!p"1, "0 ,
(186)
the non-adiabatic cross-sections, Eqs. (182) and (183), transform to the equations for the photodissociation transition to the laser-induced adiabatic dissociative potential ; (R), that corresponds to the Fedorov's strong-"eld approximation. The non-adiabatic photodissociation cross-sections, Eqs. (182) and (183), depend on the molecular rotation momentum J and its laboratory frame projection M due to the dependence of the non-adiabatic parameters on them via the quantum angle b, Eq. (164). Because the photodissociation cross-section depends on the angle b, whose semi-classical analogy is the angle between molecular axis and laser polarization, the photodissociation depends on the molecular orientation in the laboratory frame. But this dependence has a form that is quite di!erent from that in the weak "eld. In the intense resonant "eld the angular dependence involves the geometric factor, Eq. (163) (static angular dependence) and the nuclear wave functions (dynamic angular dependence). Hence, the resultant dependence is more complicated in comparison with the weak "eld cross-section, where all the angular dependence is accumulated into the geometric factor, Eq. (163), and is the static one only. 5.3. High-order perturbation theory limit of the non-perturbative quasi-classical photodissociation cross-section The photodissociation cross-section equation (183) may be applied for the resonant laser "elds of intensity, Eq. (61), I;10 W cm\ . For the details of this estimate see our comment.
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In this subsection we discuss the perturbative limit of the photodissociation cross-section obtained within our multi-channel approach. Here one may use the Landau}Zener model due to its greatest convenience to describe the non-adiabatic transitions in the photodissociation curve crossing problem. Then, the nonadiabatic parameters are written as P"e\B ,
(187)
t"0 ,
(188)
d d p d !1 !arg C 1#i ,
" # ln p p 4 p
(189)
d""Fi( + C (X )/2" , where the functions i( + and C (X ) are de"ned in Eqs. (163) and (175). Let us expand Eq. (183) over the following parameter:
(190)
1!P ;1 , P
(191)
which is small in the weak laser "eld, and only the "rst and second terms of the expansion remain. The cross-section obtained is the quasi-classical form of the following high-order perturbation theory expression
4pu 2pI p( + (e, v)" 1s "D "s 2# 1s "D G D G D "s 2 , ." C( T( C( T( c c
(192)
where s (R) is the wave function of in"nite nuclear motion in the unperturbed dissociative C( dressed term ;( X (R; N) 2k cos(¸ (R , R)!p/4) , (193) s (R)" N N CN( p j (R) N normalized as
1s (R)"s (R)2"d(e!e )d d , (194) CYNY(Y CN( NNY ((Y D (R) designates the following product: D (R)"D (R)i( + H , G(R, R ; E), p"1, 2, are the one-channel Green's functions of the unperturbed nuclear HamilN tonian, Eqs. (64) and (72), with the dressed terms, Eq. (152). The quasi-classical equations for the unperturbed nuclear Green's functions of attractive and dissociative terms are presented in Section B.3, Eqs. (B.40) and (B.41). Therefore, the weak "eld expansion of the non-perturbative photodissociation cross-section, Eq. (192), is a simple sum of the odd terms of the perturbation theory. So, the "rst term in Eq. (192) is the "rst perturbative order photodissociation amplitude and the second one is the third-order photodissociation amplitude. The second perturbative term in Eq. (192) describes the three-photon resonant transition of the type `absorption}emission}absorptiona. In dependence on the total
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energy, this term can bring a positive or a negative contribution to the photodissociation cross-section. Hence, it is possible to see a decrease of the dissociation cross-section if the "eld increases. This is a perturbative explanation for the non-linear dependence of the photodissociation probability on laser power. If the laser intensity increases more and more, the perturbative equation (192) ceases working and we have to use the non-perturbative one, Eq. (183). The use of the weak "eld expansion for the G(R, R ; E) component of the non-perturbative two-channel Green's function for the term system `attractive#dissociativea, Eq. (B.33), permits to rewrite the perturbative cross-section Eq. (192) as
4pu F 1s "D "s 2# 1s "D GD "s 2 . p( + (e, v)" ." C( T( T( c 2 C(
(195)
Therefore, the non-perturbative Green's function G(R, R ; E) sums the appropriate orders of the perturbation theory (Fig. 6). 5.4. Over-excited molecule and laser-induced quasi-bound nuclear states As it follows from Eqs. (176) and (183), the perturbation theory approach is inapplicable for the radiation transitions in intense resonant laser "elds due to a non-linear creation of quasi-bound states of dissociative molecule under the resonant "elds. In other words, the molecule AB, which absorbed the resonant quantum u, before dissociating lives some time as a bound complex in an " over-excited quasi-bound state AB whose energy is higher than the molecular dissociation
Fig. 6. Diagram representation of multi-channel and unperturbed Green's function of nuclear motion. The diagonal component GNN(R, R ; E) of the two-channel Green's function is pictured as a broad solid line (a), the unperturbed Green's function G(R, R ; E) is drawn as the narrow one (b), the non-diagonal component G(R, R ; E) of the two-channel N Green's function is presented as diagram (c), the second matrix element in Eq. (195) 1s "D GD "s 2 has view of C( T( diagram (d).
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potential " AB#(N#1) uNAB#N uNA#B#N u . As a result, one can observe a time delay (or suppression) of the photodissociation process. Due to the laser-induced coupling, the attractive laser dressed term ;( (R; N#1) evolves (in the strong "eld limit) to the dissociative adiabatic term ; (R). In the same time, the dissociative laser dressed term ;( X (R; N) evolves to the attractive adiabatic term ; (R) whose laser-induced potential pit (Fig. 5) smooths out more and more as the "eld increases. As a result of radiation coupling, the pure attractive and dissociative dressed potentials ;( (R; N#1) and ;( X (R; N) are modi"ed and the exact states of the molecule are not more as the pure vibration or pure dissociative ones. The molecular states get an energy width induced by the resonant laser}molecule interaction. Therefore, quasi-bound states of nuclear motion arise in the pure dissociative nuclear continuum. These states are created by the laser-induced interaction of the resonant attractive and dissociative molecular terms and, as the "eld increases, some of them vanish to continuum and others are transformed to vibration states of the attractive adiabatic potential ; (R), Eq. (184) (Fig. 5). The molecular laser-induced quasi-bound states are those which can be formed in resonant laser "eld only. As the molecule AB absorbs a quantum of resonant laser radiation, it transfers to the laser-induced quasi-bound state and lives there during either very short or a reasonable time, that depends on the strength of the laser}molecule resonant interaction and on the type of the state. So, if the "eld is weak, I(10 W cm\, there are no laser-induced quasi-bound states in the molecules, the diatomic, which absorbed the photon, transfers to the dissociative continuum without reaching a quasi-bound state. There is no delay of the molecular photodissociation. But if the "eld is intense, I<10 W cm\, the laser-induced coupling of the resonant molecular terms takes place and the molecular photodissociation is not more as a pure bound-continuum transition. The molecule, which absorbed the resonant laser photon, transfers to the quasi-bound state and only this state decays to the dissociative continuum. There is a delay of molecular photodissociation. It is important to emphasize that the quasi-bound states are generated before as the molecule transfers to either a bound state of the laser-induced attractive adiabatic term ; (R) or a con tinuum state of the laser-induced dissociative adiabatic term ; (R). The molecular states in these new laser-induced terms are only the limit of the quasi-bound ones, which are reached in strong "elds I<10 W cm\ only. 5.5. Non-linear photodissociation of noble gas dimers. Numerical results The population of the quasi-bound states under thermodynamic equilibrium is given by the "eld-dependent Boltzmann distribution [58] f (F, b)"Z\e\#L $ @ I 2 , L For this estimate see our comment in Footnote 1.
(196)
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where E (F, b) is the total energy of the laser}molecule system, which is the real part of the complex L solution of the following spectral equation: (197) Pe R cos S (E)#(1!P)e (\N cos S (E)"0 , where S (E)"¸ #t, S "¸ !p # are the quasi-classical actions along the diabatic and adiabatic attractive terms. In the weak "eld limit, Eq. (181), the energies E (F, b) coincide with the unperturbed ones L E (F, b)PE # u , L T ( k in Eq. (196) designates the Boltzmann constant, ¹ is the temperature of molecular gas, Z is the statistical sum
Z" e\#L $ @ I 2 d cos b . L In order to calculate the photodissociation measured in an experiment, the cross-section, Eq. (183), (or Eq. (182)) must be averaged with the distribution Eq. (196) as follows:
(198) p (u, F)" f (F, b)p( + (e, v ) d cos b . ." L ." L The absorption coe$cient K (u, F) and the partial laser power absorbed during the photodissoci ation Q (u, F) are given as ( u
, Ne>, Xe>, on their D R> QAR> electronic transition in the resonant "eld of linear polarized XeF-laser with wavelength j"351 nm [19,58]. In this case we have *X""X !X ""0, n "(0, 0, 1), e"(0, 0, 1) . Then the laser}molecule coupling, Eq. (162), is F < (R)" D (R) cos b , 2
(199)
the Landau}Zener non-adiabatic parameter, Eq. (190), and photodissociation cross-section, Eq. (183), are proportional to the cosine of the quantum angle, Eq. (164): d&cos b , &cos b . (200) p( + (e, v )" ." . . The attractive molecular terms have been "tted to the Morse-type potentials and the dissociative terms have been approximated by the exponent-type ones (see Appendix D).
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309
Fig. 7. Photodissociation cross-section, Eq. (198), for the Ar> molecule, Eq. (201), versus "eld strength F of linear polarized laser with (solid line) and without (dashed line) molecular rotations. j"351 nm, ¹"300 K.
The cross-sections calculated for all those molecules have the same qualitative behaviour and only the photodissociation calculation for the Ar> molecular gas Ar>(AR> )#(N#1) uNAr>(DR> )#N u NAr>(P)#Ar(S)#N u (201) is presented in this review. The laser "eld strength dependence of the Ar> photodissociation, Eq. (201), is shown in Fig. 7. In the very weak laser "eld I410 W cm\ (F(5;10\ a.u.) the photodissociation cross-section is a constant (in accordance with the "rst order of perturbation theory). There are no laser-induced e!ects in such "elds. As the intensity increases the cross-section decreases (in accordance with the higher orders of perturbation theory, see Eq. (192)) and reaches its minimum at I&(10}10) W cm\ (F((5}12);10\ a.u.). If the "eld increases further, then the photodissociation increases rapidly in the intensity interval I&(10}10) W cm\ (F((0.5}5);10\ a.u.). Therefore, there is a laser-induced delay or suppression of the Ar> photodissociation under the intensities I"(10}10) W cm\. The photodissociation cross-section has been calculated with and without molecular rotations in order to consider the role of molecular rotations. Both results are shown in Fig. 7. The photodissociation without rotations has been calculated with the equations written above, where the rotation momentum was equal to zero J"0 and the calculations have been carried out for a "xed angle b between the non-rotating molecule and laser polarization vector (the model of molecule frozen into a crystal matrix).
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One can see in Fig. 7 that the molecular rotations are important and must be taken into account in the "elds of intensity 10 W cm\(I(10 W cm\. So, under I&10 W cm\ (F"5;10\ a.u.) the rotations strengthen the photodissociation cross-section by more than 100 times. In the very weak and strong "elds I(10 W cm\, I'10 W cm\ the contribution of rotations to the photodissociation is not so important and does not change signi"cantly its cross section. The molecular rotations impact considerably upon the photodissociation in a resonant "eld of moderate intensity. So, if the minimum of the zero-rotation cross-section is less than the weak "eld cross-section by more than 550 times, then the inclusion of the rotations depresses this rate by 35 times. Moreover, the minimum moves to the more intense "eld: for the frozen molecules the minimum is at F"5;10\ a.u., for the rotating molecules one is moved to F"12;10\ a.u. Fig. 8 shows the photodissociation cross-section as a function of the quantum angle, Eq. (164), in the weak, moderate and strong "elds. As one can see, the cross-section in moderate "eld is the smallest one and its angular dependence is smoother than one in both limits. The angular dependences of the weak and strong "eld cross-sections have an exact cos b-like form, as follows from the limit of the general equation (183), Eq. (200). For a moderate intensity the angular dependence of the photodissociation is determined not only by the geometric factor, Eq. (163), but also by the nuclear transition amplitude, Eq. (174), which is also an angular-dependent function and impacts upon the total cross-section more signi"cantly than the geometric factor. The behaviour of the photodissociation cross-section received may be explained classically as follows. The molecular rotations cause the centrifugal forces acting upon the molecular fragments within the molecular reference frame. The forces increase if the rotations become faster. The centrifugal forces accelerate molecular decay, counteract laser-induced stabilization of the photodissociation and increase the molecular photodissociation. This e!ect takes place but only in the
Fig. 8. The Ar> photodissociation cross-section, Eq. (201), versus quantum angle b, Eq. (164), without angular averaging. j"351 nm, ¹"300 K. p !F"10\ au, p !F"2.5;10\ au, p !F"5;10\ au.
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311
moderate laser "elds I410 W cm\, where the contributions from both non-adiabatic parts of nuclear matrix element, Eqs. (174) and (176), to the cross-section, Eq. (183), are comparable with each other. In the weak and strong "elds I(10 W cm\, I'10 W cm\ this e!ect is suppressed by another one. In the weak and strong "elds there is no interference of the two non-adiabatic transitions. In the weak "eld the molecule, which absorbed the laser photon, transfers to the diabatic electronic continuum ; X (R). In the strong "eld the molecule transfers to the laser induced adiabatic continuum ; (R). The probability to populate the rotation molecular states is distributed in accordance with Eq. (196) and, if the molecule rotates faster, then the probability to populate such a rotation state is lower and the molecular rotations may to some extent (but not signi"cantly) depress the photodissociation. 5.6. Stable and unstable quasi-bound states. Softening and hardening of bonds in over-excited molecule In the weak "elds the quasi-bound molecular states are exactly equal to the continuum molecular states in the dissociative term ;( X (R). The molecule dissociates to its fragments without delay. In the moderate "elds, I410 W cm\, the contribution of the pure attractive term ;( (R) increases (see Eq. (192) and the discussion after it) and the quasi-bound state di!ers from the dissociative one. So, the probability to emit coherently the laser photon by the excited molecule increases and the over-excited molecule is formed. Due to the coherent interaction of nuclear continuum with the bound state, the dissociation probability decreases, which can be observed experimentally as the delay or suppression of the photodissociation. In order to analyse the behaviour of the energy levels in resonant "eld we have calculated the energy spectra of the molecules Ar>, Ne>, Xe> as the numerical solutions of the spectral equation (197) (Tables 2}4, some of these results had been "rstly presented in [19,154,192]). The real and imaginary parts of the complex energies E"E !iC/2 under a "xed laser P frequency u and molecular orientation in the "eld (the angle b) depend on the laser strength non-linearly. As the laser strength increases, only few levels shift: these are Ar>!v "8, 10, 12, 13, 15, 16, 18, 19, 21 , B (i) Ne>!v "6, 13, 14, 18, 19 , B Xe>!v "15, 18, 20 , B but other states shift "rstly and disappear under higher "eld: these are Ar>!v "7, 9, 11, 14, 17, 20 , B (ii) Ne>!v "7, 8, 9, 10, 11, 12, 15, 16, 17, 20 , B Xe>!v "9, 10, 11, 12, 13, 14, 16, 17, 19, 21, 22, 23 . B Such behaviour of the energy levels can be explained by the crucial modi"cation of the form of the potential under resonant radiation coupling (for the same dissociative potential the more narrow adiabatic term ; (R) contains less number of levels than the wider diabatic one ;( (R; N#1), see Fig. 5). It is interesting to note that as the "eld strength increases the energies of few levels increase too (for the Ar> molecule these are the levels v "7, 9, 12, 15, 18, 21; for Ne> these are the levels B
1.05!0 1.46!0 2.59!1 3.01!1 3.05!1 4.17!1 3.18!1 8.99!1 9.95!1 6.14!1 2.23!1 8.84!1 2.92!1
3 !2 3 0 !4 0 !1 1 !2 0 !1 !6 1
C/2
2.48!0 8.84!1 1.08!0
2 !2 4
2.34!0 4.39!0
1.73!0 1.71!0
2 !3 2 !6
!3
1.17!0
!1
7
4 !5
4 !7
6 !8
!6
D
6.24#0
C/2
4;10\
!7
D
2;10\
1.91!0
2.05!0 2.26!0
3.56!0 4.87!1
2.43!0 2.35!0
7.09#0
6.55#0
C/2
9
7 !8
5 !11
8 !10
!4
!8
D
5;10\
2.59!0
4.41!0 4.05!0
3.31!0 3.62!1
1.95!0 7.86!2
5.21#0
1.30!0
C/2
17
12 21
8 !15
17 !17
!16
!15
D
9;10\
3.15!1
8.38!1 2.57!1
2.93!0 4.09!2
6.25!1 9.85!3
2.43!0
5.25!2
C/2
The spectrum is obtained from numerical solution of spectral equation (197). j"353 nm, the laser radiation is linear polarized, D"E !E , b"0, E ! u is the energy of vibration level v in ground electronic term AR> . The vibration number in ground diabatic term is shown P B B in parentheses. 8.62!3 designates 8.62;10\.
1.00!2 1.56!2 2.99!3 2.98!3 3.02!3 4.53!3 3.65!3 2.40!3 8.62!3 9.78!3 1.69!3 6.70!3 2.96!3 8.95!3 2.83!3
D
D
C/2
1;10\
1;10\
F (a.u.)
30 472(7) 0 30 743(8) 0 31 010(9) 0 31 274(10) 0 31 534(11) 0 31 792(12) 0 32 045(13) 0 32 296(14) !1 32 542(15) 0 32 786(16) 0 33 026(17) 0 33 263(18) 0 33 496(19) 0 33 726(20) 0 33 952(21) 0
E B (cm\) (v ) B
Table 2 Energy spectrum of molecule Ar> in dependence on laser "eld strength F
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31 956(6) 32 468(7) 32 967(8) 33 454(9) 33 929(10) 34 392(11) 34 843(12) 35 281(13) 35 707(14) 36 121(15) 36 523(16) 36 913(17) 37 290(18) 37 655(19) 38 008(20)
E B (cm\) (v ) B
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
3.79!3 9.07!3 5.51!3 2.89!3 1.71!4 2.50!3 4.93!4 9.92!3 1.53!3 1.05!3 9.71!3 3.25!3 1.54!2 2.07!3 2.70!3
4.10!1 1.00!0 1.50!1 2.51!2 3.31!1 5.04!2 1.36!0 1.21!1 1.23!1 3.42!1 1.77!0 2.34!0 9.65!1
1 !1 !4 2 1 !1 !2 1 1 !1 !2
C/2
1 !3
D
D
C/2
1;10\
1;10\
F (a.u.)
4 2 !2 !3
3 !3 !12 11 2 !2 !5
3 !6
D
2;10\
1.62!2 2.21!0 9.99#0 6.69!1
1.73!1 2.06!0 2.14!1 1.86!1 9.25!1 1.30!0 2.44!1
1.19!0 5.47!1
C/2
Table 3 Energy spectrum of molecule Ne> in dependence on laser "eld strength F
1.09!0 6.00#0 1.90!2
1.33!1 7.28!1
6 !5
5 !3 !8
3.94!1 7.23!2
1.46!0 2.54!1
C/2
10 !8
9 !23
D
4;10\
12
7 !4
9 !8
15 !14
D
5;10\
4.77!1 3.99!0
9.71!2 6.21!1
7.91!1 9.92!1
1.52!0
C/2
17
17 !17
21 !28
D
9;10\
7.71#0 5.38!1
1.73!0 2.61!1
1.87!0
C/2
A.I. Pegarkov / Physics Reports 336 (2000) 255}411 313
29 441(9) 29 520(10) 29 628(11) 29 735(12) 29 843(13) 29 949(14) 30 054(15) 30 158(16) 30 262(17) 30 365(18) 30 467(19) 30 568(20) 30 668(21) 30 768(22) 30 867(23)
E B (cm\) (v ) B
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
3.87!3 2.51!4 6.32!3 5.82!3 3.81!4 4.21!3 1.95!4 3.05!3 3.13!4 5.52!4 1.08!3 6.02!4 6.01!6 4.51!4 1.57!3
3.98!1 1.85!2 6.19!1 5.76!1 3.69!2 1.89!2 3.22!1 5.30!1 5.61!2 9.60!2 6.54!2 5.94!4 4.92!2 1.84!1
0 !1 1 !1 !2 0 0 !2 0
C/2
!3 0 !1 2 !1
D
D
C/2
1;10\
1;10\
F (a.u.)
3.12!0 7.26!1 9.65!1 1.46!1 1.00!1 7.50!1
2.78!1
0
2.61!0 9.34!1 8.55!1 2.34!0
C/2
1 !2 2 !1 !3 1
2 !2 2 !2
D
2;10\
Table 4 Energy spectrum of molecule Xe> in dependence on laser "eld strength F
1
4.32!1
1.01!2
2
1
2
!2
2.78!0
!2
6.55!2
3.67!1
1.99!0
2.18!0
1.91!0 !3 1
6.09!1
C/2
2
D
2.47!0 1.03!1
1.23!0 2.66!1 3.28!1 8.99!1
C/2
5;10\
1 !3
2 !3 4 !2
D
4;10\
2
5
!4
D
9;10\
2.94!1
1.65!0
2.77!1
C/2
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v "6, 9, 12, 13, 17, 18; for Xe> these are the levels v "10, 12, 15, 17, 20, 23), but the energies of B B other levels decrease (for the Ar> molecule these are the levels v "8, 10, 11, 13, 14, 16, 19, 20; for B Ne> these are the levels v "7, 10, 11, 14, 15, 19, 20; for Xe> these are the levels v " B B 9, 11, 13, 16, 18, 19, 22). There is no regularity in the dependences. One can see from Tables 2}4 that if the laser "eld increases more and more, so that I<10 W cm\, then the quasi-bound states transform to the states of both adiabatic terms, Eq. (184). Therefore, some quasi-bound states are stabilized and their laser-induced widths reach maximum decrease (these are those of the group (i) above). These states may be named as stable quasi-bound states. The other quasi-bound states do not reach the stabilization but transform to the pure continuum ones (these are those of the group (ii)). Such states are unstable quasi-bound states. In the strong "eld limit the "rst type of states transform to the vibration states of the attractive adiabatic term ; (R) and the second type of states transform to the continuum states of the dissociative term ; (R). Although the stable quasi-bound state evolves to the vibration state of the attractive adiabatic term, it is not a pure bound one and has a radiative width because the laser-induced adiabatic potentials exist during only the half-period of laser wave. If the free molecule is in the vibration state which corresponds to the stable quasi-bound state, then as the "eld switches on, the molecule transfers to this stable quasi-bound state, that can be observed as the laser-induced suppression (decrease) of molecular photodissociation. Such a suppression of photodissociation may be explained as vibration trapping to the stable quas