Physics Reports vol.309

Physics Reports 309 (1999) 1—116

Quantum mechanics and semiclassics of hyperbolic n-disk scattering systems Andreas Wirzba* Institut fu( r Kernphysik, Technische Universita( t Darmstadt, Schlo}gartenstra}e 9, D-64289 Darmstadt, Germany Received June 1998; editor: G.E. Brown Contents 1. Introduction 1.1. Motivation and historic perspective 1.2. The n-disk repeller: a model for hyperbolic scattering 1.3. Objective 1.4. Outline 2. Semiclassical resonances of the n-disk system 3. The n-disk S-matrix and its determinant 4. The link between the determinant of the S-matrix and the semiclassical zeta function 5. Semiclassical approximation and periodic orbits 5.1. Quantum itineraries 5.2. Ghost contributions 5.3. Semiclassical approximation of a periodic itinerary 5.4. Itineraries in the geometrical limit 5.5. Itineraries with repeats 5.6. Ghost rule 5.7. Itineraries with creeping terms 5.8. More than one creeping section 5.9. Geometrical stabilities 6. Numerical tests of semiclassical curvature expansions against exact data 6.1. Exact versus semiclassical resonances 6.2. Exact versus semiclassical cluster phase shifts 6.3. The quantum-mechanical cumulant expansion versus the semiclassical curvature expansion

4 4 6 8 9 11 14 20 23 24 25 27 29 32 34 37 41 42 47 49 52

7. Conclusions Acknowledgements Appendix A. Traces and determinants of infinite dimensional matrices A.1. Trace class and Hilbert—Schmidt class A.2. Determinants det(1#A) of trace-class operators A A.3. Von Koch matrices A.4. Regularization Appendix B. Exact quantization of the n-disk scattering problem B.1. The stationary scattering problem B.2. Calculation of the S-matrix B.3. The determination of the product D ) C Appendix C. Existence of the S-matrix and its determinant in n-disk systems C.1. Proof that T (kaH) is trace-class C.2. Proof that A(k),M(k)!1 is trace-class C.3. Proof that CH and DH are trace-class C.4. Existence and boundedness of M\(k) Appendix D. Comparison to Lloyd’s T-matrix Appendix E. 1-disk determinant in the semiclassical approximation Appendix F. Semiclassical approximation of two convoluted A-matrices F.1. The Watson contour resummation F.2. The integration paths F.3. Semiclassical approximation of the straight-line integrals

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* E-mail: [email protected].

0370-1573/99/$ — see front matter  1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 8 ) 0 0 0 3 6 - 2

61 67 67 67 69 71 73 74 74 75 79 80 81 82 84 85 85 86 89 90 93 97

QUANTUM MECHANICS AND SEMICLASSICS OF HYPERBOLIC n-DISK SCATTERING SYSTEMS

Andreas WIRZBA Institut fu( r Kernphysik, Technische Universita( t Darmstadt, Schlo}gartenstra}e 9, D-64289 Darmstadt, Germany

AMSTERDAM — LAUSANNE — NEW YORK — OXFORD — SHANNON — TOKYO

A. Wirzba / Physics Reports 309 (1999) 1—116 F.4. Semiclassical approximation of the residua sum F.5. Resulting convolutions

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F.6. Ghost segment Appendix G. Figures of three-disk resonances References

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Abstract The scattering problems of a scalar point particle from an assembly of 1(n(R non-overlapping and disconnected hard disks, fixed in the two-dimensional plane, belong to the simplest realizations of classically hyperbolic scattering systems. Their simplicity allows for a detailed study of the quantum mechanics, semiclassics and classics of the scattering. Here, we investigate the connection between the spectral properties of the quantum-mechanical scattering matrix and its semiclassical equivalent based on the semiclassical zeta-function of Gutzwiller and Voros. We construct the scattering matrix and its determinant for any non-overlapping n-disk system (with n(R) and rewrite the determinant in such a way that it separates into the product over n determinants of one-disk scattering matrices — representing the incoherent part of the scattering from the n-disk system — and the ratio of two mutually complex conjugate determinants of the genuine multiscattering matrix M which is of Korringa—Kohn—Rostoker-type and which represents the coherent multidisk aspect of the n-disk scattering. Our quantum-mechanical calculation is well-defined at every step, as the on-shell T-matrix and the multiscattering kernel M!1 are shown to be trace-class. The multiscattering determinant can be organized in terms of the cumulant expansion which is the defining prescription for the determinant over an infinite, but trace-class matrix. The quantum cumulants are then expanded by traces which, in turn, split into quantum itineraries or cycles. These can be organized by a simple symbolic dynamics. The semiclassical reduction of the coherent multiscattering part takes place on the level of the quantum cycles. We show that the semiclassical analog of the mth quantum cumulant is the mth curvature term of the semiclassical zeta function. In this way quantum mechanics naturally imposes the curvature regularization structured by the topological (not the geometrical) length of the pertinent periodic orbits onto the semiclassical zeta function. However, since the cumulant limit mPR and the semiclassical limit, P0 or (wave number) kPR, do not commute in general, the semiclassical analog of the quantum multiscattering determinant is a curvature expanded semiclassical zeta function which is truncated in the curvature order. We relate the order of this truncation to the topological entropy of the corresponding classical system. We show this explicitly for the three-disk scattering system and discuss the consequences of this truncation for the semiclassical predictions of the scattering resonances. We show that, under the above mentioned truncations in the curvature order, unitarity in n-disk scattering problems is preserved even at the semiclassical level. Finally, with the help of cluster phase shifts, it is shown that the semiclassical zeta function of Gutzwiller and Voros has the correct stability structure and is superior to all the competitor zeta functions studied in the literature.  1999 Elsevier Science B.V. All rights reserved. PACS: 03.65.Sq; 03.20.#i; 05.45.#b

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1. Introduction The main focus of this manuscript is on the transition from quantum mechanics to semiclassics in classically hyperbolic scattering systems, and in particular, on the convergence problems of periodic orbit expansions of n-disk repellers. 1.1. Motivation and historic perspective Why more than 70 years after the birth of textbook quantum mechanics and in the age of supercomputers is there still interest in semiclassical methods? First of all, there remains the intellectual challenge to derive classical mechanics from quantum mechanics, especially for classically non-separable chaotic problems. Pure quantum mechanics is linear and of power-law complexity, whereas classical mechanics is generically of exponential complexity. How does the latter emerge from the former? Secondly, in many fields (atomic physics, molecular physics and quantum chemistry, but also optics and acoustics which are not quantum systems but are also characterized by the transition from wave dynamics to ray dynamics) semiclassical methods have been very powerful in the past and are still useful today for practical calculations, from the detection of elementary particles to the (radar)-detection of airplanes or submarines. Third, the numerical methods for solving multidimensional, non-integrable quantum systems are generically of “black-box” type, e.g. the diagonalization of a large, but truncated hamiltonian matrix in a suitably chosen basis. They are computationally intense and provide little opportunity for learning how the underlying dynamics organizes itself. In contrast, semiclassical methods have a better chance to provide an intuitive understanding which may even be utilized as a vehicle for the interpretation of numerically calculated quantum-mechanical data. In the days of “old” quantum mechanics semiclassical techniques provided of course the only quantization techniques. Because of the failure, at that time, to describe more complicated systems such as the helium atom (see, however, the resolution of Wintgen and collaborators [1]; [2] and also [3] provide for a nice account of the history), they were replaced by modern quantum mechanics based on wave mechanics. Here, through WKB methods, they reappeared as approximation techniques for one-dimensional systems and, in the generalization to the Einstein—Brillouin—Keller (EBK) quantization, for separable problems [3—5] where an n-degree-of-freedom system reduces to n one-degree-of-freedom systems. Thus semiclassical methods had been limited to such systems which are classically nearly integrable. It was Gutzwiller who in the late 1960s and early 1970s (see e.g. [5,6]) (re-)introduced semiclassical methods to deal with multidimensional, non-integrable quantum problems: with the help of Feynman path integral techniques the exact time-dependent propagator (heat kernel) is approximated, in stationary phase, by the semiclassical Van-Vleck propagator. After a Laplace transformation and under a further stationary phase approximation the energy-dependent semiclassical Green’s function emerges. The trace of the latter is calculated and reduces under a third stationary phase transformation to a smooth Weyl term (which parametrizes the global geometrical features) and an oscillating sum over all periodic orbits of the corresponding classical problem. Since the imaginary part of the trace of the exact Green’s function is proportional to the spectral density, the Gutzwiller trace formula links the spectrum of eigen-energies, or at least the modulations in this

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spectrum, to the Weyl term and the sum over all periodic orbits. Around the same time, Balian and Bloch obtained similar results with the help of multiple-expansion techniques for Green’s functions, especially in billiard cavities, see e.g. [7]. For more than one degree of freedom, classical systems can exhibit chaos. Generically these are, however, non-hyperbolic classically mixed systems with elliptic islands embedded in chaotic zones and marginally stable orbits for which neither the Gutzwiller trace formula nor the EBK-techniques apply, see Berry and Tabor [8]. Purely hyperbolic systems with only isolated unstable periodic orbits are the exceptions. But in contrast to integrable systems, they are generically stable against small perturbations [5]. Moreover, they allow the semiclassical periodic orbit quantization which can even be exact as for the case of the Selberg trace formula which relates the spectrum of the Laplace—Beltrami operator to geodesic motion on surfaces of constant negative curvature [9]. The Gutzwiller trace formula for generic hyperbolic systems is, however, only an approximation, since its derivation is based on several semiclassical saddle-point methods as mentioned above. In recent years, mostly driven by the uprise of classical chaos, there has been a resurgence of semiclassical ideas and concepts. Considerable progress has been made by applying semiclassical periodic orbit formulae in the calculation of energy levels for bound-state systems or resonance poles for scattering systems, e.g., the anisotropic Kepler problem [5], the scattering problem on hard disks [10—15], the helium atom [1] etc. (See Ref. [16] for a recent collection about periodic orbit theory.) It is well known that the periodic orbit sum for chaotic systems is divergent in the physical region of interest. This is the case on the real energy axis for bounded problems and in the region of resonances for scattering problems, because of the exponentially proliferating number of periodic orbits, see [5,17]. Hence refinements have been introduced in order to transform the periodic orbit sum in the physical domain of interest to a still conditionally convergent sum by using symbolic dynamics and the cycle expansion [18,19,14], Riemann—Siegel lookalike formulas and pseudo-orbit expansions [20,21], surface of section techniques [22,23], inside-outside duality [24], heat-kernel regularization [3,25] etc. These methods tend to be motivated from other areas in physics and mathematics [26] such as topology of flows in classical chaos, the theory of the Riemann zeta functions, the boundary integral method for partial differential equations, Fredholm theory (see also [27]), quantum field theory etc. In addition to the convergence problem, there exists the further complication for bounded smooth potential and billiard problems that the corresponding periodic orbit sums predict in general non-hermitean spectra. This problem is addressed by the Berry—Keating resummation techniques [20,21] — however, in an ad hoc fashion. In contrast, scattering problems circumvent this difficulty altogether since their corresponding resonances are complex to start with. Moreover, the scattering resonances follow directly from the periodic orbit sum, as the Weyl term is absent for scattering problems. In fact, it is more correct to state that the Weyl term does not interfere with the periodic sum, as a negative Weyl term might still be present, see e.g. [17]. Furthermore, scattering systems allow for a nice interpretation of classical periodic sums in terms of survival probabilities [2,28]. In this respect, it is an interesting open problem why these classical calculations do not seem to generate a Weyl term, whether applied for bounded or scattering systems. For these reasons, the study of periodic sums for scattering systems should be simpler than the corresponding study for bound-state problems, as only the convergence problem is the issue.

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1.2. The n-disk repeller: a model for hyperbolic scattering Hence, one should look for a simple classically hyperbolic scattering system which can be used to address the convergence problem. It should not be too special, as for example the motion on a surface of constant curvature, but reasonably realistic and instructive. Eckhardt [10] suggested such a system, the “classical pinball machine”. It consists of a point particle and a finite number (in his case three) identical non-overlapping disconnected circular disks in the plane which are centered at the corners of a regular polygon (in his case an equilateral triangle). The point particle scatters elastically from the disks and moves freely in between collisions. The classical mechanics, semiclassics and quantum mechanics of this so-called three-disk system was investigated in a series of papers by Gaspard and Rice [11—13], and, independently, by Cvitanovic´ and Eckhardt [14], see also Scherer [17] and Ref. [15]. It belongs to a class of mechanical systems which are everywhere defocusing, hence no stable periodic orbit can exist (see Fig. 1). The classical dynamics with one or two disks is simple, there is either no or one trapped trajectory. The latter is obviously unstable, since a small displacement leads to a defocusing after the reflection from the curved surface of disk [11]. The two-disk system is therefore one of the simplest hyperbolic scattering systems, but it is non-chaotic. However, with three or more disks there are infinitely many trapped trajectories forming a repeller [15]. The periodic orbits corresponding to these trapped trajectories are all isolated and unstable because of the defocusing nature of the reflections. Note that the one-disk and two-disk systems, although classically simple, are nonetheless interesting. The quantum-mechanical one-disk scattering system (since it is separable) has been one of the key models for building up the semiclassical theory of diffraction [29—31]. Similarly, the two-disk system became the toy ground for the periodic-orbit theory of diffraction [32,33]. In fact, the two-disk system has infinitely many diffractive creeping periodic orbits which can be classified by symbolic dynamics similarly to the infinitely many geometrical orbits of the

Fig. 1. The three-disk repeller with the symbolic dynamics of the full domain. The figure is from Ref. [2].

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three-disk system. The symbolic dynamics of a general n-disk system is very simple, see e.g. [2]: periodic orbits can be classified by a series of “house numbers” of the disks which are visited by the point particle which follows the corresponding trajectory. Not all sequences are allowed: after each reflection from one disk, the point particle has to proceed to a different disk, since the evolution between the disk is the free one. Furthermore, for general geometries there may exist sequences which correspond to trajectories which would directly pass through a disk. The sequences corresponding to these so-called “ghost orbits” have to be excluded from the classical consideration. In summary, the geometrical periodic orbits (including ghost orbits) are labelled in the full domain of the n-disk repeller by itineraries ("periodic words) with n different symbols ("letters) with the trivial “pruning” rule that successive letters in the itinerary must be different. The itineraries corresponding to ghost orbits have to be removed or “pruned” with all their subbranches from the symbol tree. Periodic trajectories which have reflections from inside of a disk (i.e. the point particle traverses first through a disk and is then reflected from the other side of the disk) can be excluded from the very beginning. In fact, in our semiclassical reduction of Section 5 we will show for all repeller geometries with n non-overlapping disks that, to each specified itinerary, there belongs uniquely one standard periodic orbit which might contain ghost passages but which cannot be reflected from the inside. There is only one caveat: our method cannot decide whether grazing trajectories (which are tangential to a disk surface) belong to the class of ghost trajectories or to the class of reflected trajectories. For simplicity, we just exclude all geometries which allow for grazing periodic orbits from our proof. Alternatively, one might treat these grazing trajectories separately with the help of the diffractional methods of Refs. [31,35]. The symbolic dynamics described above in the full domain applies of course to the equilateral three-disk system. However, because of the discrete C symmetry of that system, the dynamics can  be mapped into the fundamental domain (any one of the one-sixth slices of the full domain which are centered at the symmetry-point of the system and which exactly cut through one-half of each disk, see Fig. 2). In this fundamental domain the three-letter symbolic dynamics of the full domain reduces to a two-letter symbolic dynamics. The symbol “0”, say, labels all encounters of a periodic orbit with a disk in the fundamental domain where the point particle in the corresponding full domain is reflected to the disk where it was just coming from, whereas the symbol “1”, say, is reserved for encounters where the point particle is reflected to the other disk. Whether the full or the fundamental domain is used, the three-disk system allows for a unique symbolic labelling (if the disk separation is large enough even without non-trivial pruning). If

Fig. 2. Equilateral three-disk system and its fundamental domain.

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a symbolic dynamics exist, the periodic orbits can be classified by their topological length which is defined as the length of the corresponding symbol sequence. In this case the various classical and semiclassical zeta functions are resummable in terms of the cycle expansion [18,19] which can be cast to a sum over a few fundamental cycles (or primary periodic orbits) t and higher curvature D corrections C of increasing topological order m: K 1 "1! t ! C . (1.1) D K f D K The curvature C in Eq. (1.1) contains all allowed periodic orbits of topological length m for K a specified symbolic dynamics and suitable “shadow-corrections” of combinations (pseudo-orbits) of shorter periodic orbits with a combined topological length m. Common to most studies of the semiclassics of the n-disk repellers is that they are “bottom-up” approaches. Whether they use the Gutzwiller trace formula [5], the Ruelle or dynamical zeta function [36], the Gutzwiller—Voros zeta function [37], their starting point is the cycle expansion [18,19,28]. The periodic orbits are motivated from a semiclassical saddle-point approximation. The rest is classical in the sense that all quantities which enter the periodic orbit calculation as e.g. actions, stabilities and geometrical phases are determined at the classical level (see however Refs. [38—40] where leading -corrections to the dynamical zeta function as well as the Gutzwiller— Voros zeta function have been calculated). For instance, the dynamical zeta function has typically the form 1 e  1N#\ LJN . (1.2) f\(E)"“ (1!t ), t "  N N (K N N The product is over all prime cycles (prime periodic orbits) p. The quantity K is the stability factor N of the pth cycle, i.e., the expanding eigenvalue of the p-cycle Jacobian, S is the action and l is (the N N sum of) the Maslov index (and the group theoretical weight for a given representation) of the pth cycle. For n-disk repellers, the action is simply S "k¸ , the product of the geometrical length ¸ of N N N the periodic orbit and the wave number k"(2mE in terms of the energy E and mass m of the point particle. The semiclassical predictions for the scattering resonances are then extracted from the zeros of the cycle-expanded semiclassical zeta-function. In this way one derives predictions of the dynamical zeta function for the leading resonances (which are the resonances closest to real k-axis). In the case of the Gutzwiller—Voros zeta function also subleading resonances result, however, only if the resonances lie above a line defined by the poles of the dynamical zeta function [41—43,15]. The quasiclassical zeta function of Vattay and Cvitanovic´ is entire and gives predictions for subleading resonances for the entire lower half of the complex plane [43]. 1.3. Objective As the n-disk scattering systems are generically hyperbolic, but still simple enough to allow for a closed-form quantum-mechanical setup [13] and detailed quantum-mechanical investigations [44,45], we want to study the structure of the semiclassical periodic sum for a hyperbolic scattering system in a “top-down” approach, i.e. in a direct derivation from the exact quantum mechanics of the n-disk repeller. This is in contrast to the usual semiclassical “bottom-up” studies of the n-disk

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repellers which can be affected by uncontrolled operations during the long and mostly formal derivation from the Gutzwiller trace formula. Especially regularization prescriptions, like the cycle expansion, have to be added from the outside in order to get converging semiclassical predictions. Hence, for any n-disk scattering problem with a finite number of non-overlapping disconnected disks we want to construct a direct link from the defining exact S-matrix to the pertinent semiclassics (in terms of a suitable periodic-orbit expansion) with the following qualifications: 1. The derivation should lead to a unique specification of the periodic orbits for a given n-disk geometry. The method should be able to handle n-disk geometries which allow for ghost orbits, i.e., periodic orbits existing in any of the pertinent “parent” disk-systems (defined by the removal of one or more disks) which are blocked by the return of at least one of the removed disks. 2. Since disk-systems are known where the semiclassics is strongly governed by diffractive orbits (see [33] and especially [63] for the two-dimensional scattering analog of the two-well-potential problem), diffractive periodic orbits should emerge together with their standard partners. 3. The subleading stability structure of the standard periodic orbits should follow from this derivation in order to discriminate between the Gutzwiller—Voros zeta function and other competitors, e.g., the dynamical zeta function of Ruelle [36] or the quasiclassical zeta function of Vattay and Cvitanovic´ [43]; in other words, we want to derive the semiclassical spectral function. 4. The setup of the starting-point, the quantum-mechanical side, should not be plagued by formal or uncontrolled manipulations or assumptions. Especially, if the quantum-mechanical side does not exist without a suitable regularization prescription, the latter should be provided before the semiclassical reduction is performed. This should exclude that the semiclassical sums encounter hidden problems which are already present at the quantum-mechanical level. 5. The link between the exact quantum mechanics and semiclassics should not only allow for the computation of scattering resonances, but should be valid for all values of the wave number, also away from the resonances and from the real axis, modulo the boundary of semiclassical convergence, as this issue can only be addressed during the link-procedure. Branch cuts and singularities on the quantum-mechanical side have to be taken into account of course. 6. The spectral function should not only result in a formal sense, but, if necessary, with a pertinent regularization and summation prescription that should not be imposed from the outside. 7. Most importantly, the derivation should be well-defined and allow for a test of the summation prescription of the period-orbit expansion. If potential problems occur, they should be pinpointed in the derivation. 1.4. Outline The manuscript is organized as follows. We begin in Section 2 with the standard approach relating quantum-mechanical and semiclassical resonances for n-disk repellers. Generalizing the work of Gaspard and Rice [13] to non-overlapping n-disk problems of arbitrary geometry and disk sizes we construct in Section 3 the S-matrix from stationary scattering theory. Details of this calculation are relegated to Appendix B. Utilizing the machinery of trace-class operators which are summarized in Appendix A we construct the determinant of the

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n-disk S-matrix as the product of n one-disk determinants and the ratio of the determinant and its complex conjugate of the genuine multiscattering matrix. It is shown how the latter determinants split under symmetry operations. The proofs for the existence of the determinants are relegated to Appendix C and the comparison to alternative formulations of the multiscattering kernel can be found in Appendix D. In Section 4 we state the link between the exact determinant of the n-disk S-matrix and the Gutzwiller—Voros curvature expansion. We discuss the semiclassical limit of the incoherent part, whereas the actual calculation is reported in Appendix E. It is shown that, under the semiclassical reduction of the quantum traces, the Plemelj—Smithies recursion relation for the quantum cumulants transforms into the recursion relation for the semiclassical curvatures which are known from the cycle expansion. The actual semiclassical reduction is worked out in Section 5. We start with the construction of the quantum cycles or itineraries built from the convolution of a finite number of multi-scattering kernels and show that they have the same symbolic dynamics in the full domain of an arbitrary n-disk system as their semiclassical counterparts, the geometrical periodic orbits. We discuss the case that the quantum-mechanical cumulant sum incorporates geometries which classically allow for non-trivial pruning and hence for periodic orbits which pass undisturbed straight through a disk, see Refs. [54,49]. We show how these ghost orbits cancel against their “parent” periodic orbits. The latter result from itineraries without the disk which is affected by the ghost passage. For the general case of an arbitrary quantum cycle, Section 5.3 mirrors the semiclassical reduction of the convolution of two multiscattering kernels that are studied in detail in Appendix F. This is done there with the help of the Watson contour integration and suitable deformations of the paths in the complex angular-momentum plane. In Section 5.4 the geometrical limit of a quantum cycle is studied which is generalized to the case of r times repeated cycles in the following section. In Section 5.6 the ghost cancellation rule for arbitrary cycles is derived. The semiclassical diffractive creeping contributions are constructed and studied in Sections 5.7 and 5.8. Section 5 ends with the proof that an arbitrary quantum itinerary reduces semiclassically to a periodic orbit of Gutzwiller—Voros stability, such that the link between the exact determinant of the n-disk S-matrix and the Gutzwiller—Voros curvature sum is established. Numerical tests of the semiclassical curvature expansion can be found in Section 6 for the example of the three-disk system in A symmetry-class representation. First, the exact quantum mechanical resonance data are compared to the semiclassical predictions of the Gutzwiller—Voros zeta function, the dynamical zeta function [36] and the quasiclassical zeta function suggested in [43], where all three semiclassical zeta function are expanded in curvatures which are truncated at finite order. Secondly, the exact cluster phase shift (defined by the phase of the determinant of the multiscattering matrix) is compared with the semiclassical predictions of the three zeta functions. Although all three zeta functions seem at first sight empirically equivalent, as they all predict the same leading resonances closest to the real k-axis, this comparison shows clearly which of the three is superior and is hence the candidate for — at least — the FAPP (“for all practical purposes”) zeta function. Section 6 ends with an order-by-order comparison of the exact cumulants with their semiclassical counterpart, the curvatures for the Gutzwiller—Voros zeta function. From these numerical data we extract an empirical truncation rule for the curvature expansion as a function of the wave number. We relate this rule to the uncertainty bound resulting from finite quantummechanical resolution of the exponentially proliferating details of the classically repelling set.

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Section 7 concludes with a summary. Here we emphasis the preservation of unitarity under the semiclassical reduction, the decoupling of the incoherent one-disk from the coherent n-disk determinants, and the particularities, when bounded domains are formed in the case of (nearly) touching disks. Furthermore, the resonance data are correlated with the truncation from the uncertainty bound. We discuss the relevance of those periodic orbits whose topological order exceeds the uncertainty bound. Arguments are presented that the Gutzwiller—Voros zeta function ought to be interpreted in the asymptotic sense as a truncated sum, whether it converges or not. The conclusions end with a discussion on corrections. Note that the contents of Appendices B and C are based on Henseler’s diploma thesis [46], while Section 6 as well as Sections 3, 4 and Appendices A.1—2 have partial overlap with Refs. [47,48], respectively.

2. Semiclassical resonances of the n-disk system The connection between exact quantum mechanics, on the one side, and semiclassics, on the other, for the n-disk repellers in the standard “bottom-up” approach, is rather indirect. It has been mainly based on a comparison of the exact and semiclassical predictions for resonance data. In the exact quantum-mechanical calculations the resonance poles are extracted from the zeros of a characteristic scattering determinant (see Ref. [13] and below), whereas the semiclassical predictions follow from the zeros (poles) of one of the semiclassical zeta functions. These semiclassical quantities have either formally been taken over from bounded problems (where the semiclassical reduction is performed via the spectral density) [15,17] or they have just been extrapolated from the corresponding classical scattering determinant [42,43]. Our aim is to construct a direct link between the quantum-mechanical and semiclassical treatment of hyperbolic scattering in a concrete context, the n-disk repellers. Following the methods of Gaspard and Rice [13] we will construct in Section 3 and Appendix B the pertinent on-shell T-matrix which splits into the product of three matrices, namely C(k)M\(k)D(k). The matrices C(k) and D(k) couple the incoming and outgoing scattering wave (of wave number k), respectively, to one of the disks, whereas the matrix M(k) parametrizes the scattering interior, i.e., the multiscattering evolution in the multidisk geometry. The understanding is that the resonance poles of the n'1 disk problem can only result from the zeros of the characteristic determinant det M(k); see the quantum-mechanical construction of Gaspard and Rice [13] for the three-disk scattering system [10,11,14]. Their work refers back to Berry’s application [49,50] of the Korringa—Kohn—Rostoker (KKR) method [51] to the (infinite) twodimensional Sinai-billiard problem which, in turn, is based on Lloyd’s multiple scattering method [52,53] for a finite cluster of non-overlapping muffin-tin potentials in three dimensions. The resonance poles are calculated numerically by solving det M(k)"0 in a finite, but large basis, such that the result is insensitive to an enlargement of the basis (see, e.g., [44]). On the semiclassical side, the geometrical primitive periodic orbits (labelled by p) have been summed up — including repeats (labelled by r) — in the Gutzwiller—Voros zeta function [5,37]










 1 (zLNt (k))P  zLNt (k) N "“ “ 1! N Z (z; k)"exp ! %4 r 1!KPN KH N N P N H

(2.1)

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the dynamical zeta function of Ruelle [36]





 1 f\(z; k)"exp ! zPLNt (k)P "“ (1!zLNt ) (2.2)  N N r N P N (which is the j"0 part of the Gutzwiller—Voros zeta function) or the quasiclassical zeta function of Vattay and Cvitanovic´ [43]










 1   (zLNt (k))P zLNt (k) H> N Z (z; k)"exp ! "“ “ “ 1! N  r (1!KPN )(1!KP N ) KH>J N N P N H J

(2.3)

which is an entire function. In all cases t (k)"e I*N\ JNp/("K " is the so-called pth cycle, n is its N N N topological length and z is a book-keeping variable for keeping track of the topological order. The input is purely geometrical, i.e., the lengths ¸ , the Maslov indices l , and the stabilities (the leading N N eigenvalues of the stability matrices) K of the pth primitive periodic orbits. Note that both N expressions for the three zeta functions, either the exponential one or the reformulation in terms of infinite product(s), are purely formal. In the physical region of interest, they may not even exist without regularization. (An exception is the non-chaotic two-disk system, as it has only one periodic orbit, t (k).) Therefore, the classical resonance poles are computed from these zeta  functions in the curvature expansion [42,19,15] up to a given topological length m. This procedure corresponds to a Taylor expansion of, e.g., Z (z; k) in z around z"0 up to order zK (with z set to %4 unity in the end), e.g.,



t 2t (t ) z N ! N # N Z (z; k)"z!z %4 1! 1! 1!( N) 2 K K K N N LN LN LN t t NY #2 . N ! (2.4) 1!KN 1!KNY N NY L  L  The hope is that the limit mPR exists — at least in the semiclassical regime Re k<1/a where a is the characteristic length of the scattering potential. We will show below that in the quantummechanical analog — the cumulant expansion — this limit can be taken, but that there are further complications in the semiclassical case. The cycle expansion is one way of regularizing the formal expression of the Gutzwiller—Voros zeta function (2.1). Another way would be the multiplication with a smooth cutoff function, as it is customary in quantum field theories, see e.g. [3]. This is, in principle, allowed. In order to be able to compare quantum mechanics with semiclassics, however, the very same cutoff function has to be introduced already on the quantum level. Candidates for such cutoff functions which work on the quantum side and on the semiclassical side are not so obvious, see e.g. Appendix A.4. They have to be formulated in terms of the T-matrix or the multiscattering kernel and would introduce further complications. Fortunately, the quantum-mechanical side of the present problem exists without further regularization. Thus there is no need for an extra cutoff function. As mentioned above, the connection between quantum mechanics and semiclassics for these scattering problems has been the comparison of the corresponding resonance poles, the zeros of the characteristic determinant on the one side and the zeros of the Gutzwiller—Voros zeta function or



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Fig. 3. The “difference” of two bounded reference systems, where one includes the scattering system.

its competitors — in general in the curvature expansion — on the other side. In the literature (see, e.g., Refs. [12,17,15] based on Ref. [54] or [55]) the link is motivated by the semiclassical limit of the left-hand sides of the Krein—Friedel—Lloyd sum for the (integrated) spectral density [56,57] and [52,53] 1 (2.5) lim lim (NL(k#ie; b)!N(k#ie; b))" Im Tr ln S(k) , 2p C> @ d 1 (2.6) lim lim (oL(k#ie; b)!o(k#ie; b))" Im Tr ln S(k) . dk 2p C> @ See also Ref. [58] for a modern discussion of the Krein—Friedel—Lloyd formula and Refs. [55,59] for the connection of Eq. (2.6) to the Wigner time delay. In this way the scattering problem is replaced by the difference of two bounded reference billiards (e.g. large circular domains) of the same radius b which finally will be taken to infinity, where the first contains the scattering region or potentials, whereas the other does not (see Fig. 3). Here oL(k; b)(NL(k; b)) and o(k; b)(N(k; b)) are the spectral densities (integrated spectral densities) in the presence or in the absence of the scatterers, respectively. In the semiclassical approximation, they will be replaced by a Weyl term and an oscillating sum over periodic orbits [5]. Note that this expression makes only sense for wave numbers above the real k-axis. Especially, if k is chosen to be real, e must be greater than zero. Otherwise, the exact left-hand sides (2.5) and (2.6) would give discontinuous staircase or even delta function sums, respectively, whereas the right-hand sides are continuous to start with, since they can be expressed by continuous phase shifts. Thus, the order of the two limits in Eqs. (2.5) and (2.6) is essential, see e.g. Balian and Bloch [54] who stress that smoothed level densities should be inserted into the Friedel sums. In Ref. [12], chapter IV, Eqs. (4.1), (4.2), (4.3) and (4.4), the order is, however, erroneously inverted. Our point is that this link between semiclassics and quantum mechanics is still of very indirect nature, as the procedure seems to use the Gutzwiller—Voros zeta function for bounded systems and not for scattering systems and as it does not specify whether and which regularization has to be used for the semiclassical Gutzwiller trace formula. Neither the curvature regularization scheme nor other constraints on the periodic orbit sum follow naturally in this way. For instance, as the

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link is made with the help of bounded systems, the question might arise whether even in scattering systems the Gutzwiller—Voros zeta function should be resummed a` la Berry and Keating [21] or not. This question is answered by the presence of the ie term and the second limit. The wave number is shifted from the real axis into the positive imaginary k-plane. This corresponds to a “de-hermitezation” of the underlying exact hamiltonian — the Berry—Keating resummation should therefore not apply, as it is concerned with hermitean problems. The necessity of the #ie in the semiclassical calculation can be understood by purely phenomenological considerations: Without the ie term there is no reason why one should be able to neglect spurious periodic orbits which solely are there because of the introduction of the confining boundary. The subtraction of the second (empty) reference system helps just in the removal of those spurious periodic orbits which never encounter the scattering region. The ones that do would still survive the first limit bPR, if they were not damped out by the #ie term. Below, we will construct explicitly a direct link between the full quantum-mechanical S-matrix and the Gutzwiller—Voros zeta function. It will be shown that all steps in the quantum-mechanical escription are well defined, as the T-matrix and the matrix A,M!1 are trace-class matrices (i.e., the sum of the diagonal matrix elements is absolutely converging in any orthonormal basis). Thus the corresponding determinants of the S-matrix and the characteristic matrix M are guaranteed to exist, although they are infinite matrices.

3. The n-disk S-matrix and its determinant Following the methods of Berry [49] and Gaspard and Rice [13] we here describe the elastic scattering of a point particle from n hard disks in terms of stationary scattering theory. Because of the hard-core potential on the disk surfaces it turns into a boundary value problem. Let t(r) be a solution of the pertinent stationary Schro¨dinger equation at spatial position r which is of Helmholtz type: (e#k)t(r)"0, r outside the n disks , P t(r)"0, r on the surfaces of the disks , where E" k/2m is the energy of the point-particle written in terms of its mass m and the wave vector k of the incident wave. After the wave function t(r) is expanded in a basis of angular momentum eigenfunctions in two dimensions, it reads  t(r)" tI (r)e K pe\ KUI , K K\ where k and U are the length and angle of the wave vector, respectively. The scattering problem in I this basis reduces to (e#k)tI (r)"0, r outside the disks ; P K tI (r)"0, r on the disk surfaces . K

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For large distances from the scatterers (krPR) the spherical components tI can be written as K a superposition of in-coming and out-going spherical waves.  (3.1) [d e\ IP\pKY\p#S e IP\pKY\p]e KYUP , KKY KKY KY\ where r and U denote the distance and angle of the spatial vector r as measured in the global P two-dimensional coordinate system. Eq. (3.1) defines the scattering matrix S which is unitary because of probability conservation. In the angular-momentum basis its matrix elements S describe the scattering of an in-coming wave with angular momentum m into an out-going KKY wave with angular momentum m. If there are no scatterers, then S"1 and the asymptotic expression of the plane wave e k  r in two dimensions is recovered from t(r). All scattering effects are incorporated in the deviation of S from the unit matrix, i.e., in the T-matrix defined as S(k)"1!iT(k). In general, S is non-diagonal and therefore non-separable. An exception is the one-disk problem (see below). For any non-overlapping system of n disks (of, in general, different disk-radii a , j"1,2, n) the H S-matrix can be further split up. Using the methods and notation of Gaspard and Rice [13] this is achieved in the following way (see also Ref. [53] and Appendix B for a derivation of this result): 1 tI (r)& K (2pkr

SL (k)"d !iT L (k)"d !iCH (k)(M\(k))HHYD HY (k) . (3.2) KKY KKY KKY KKY KJ JJY JYKY Here the upper indices j, j"1,2, n(R label the n different disks, whereas the lower indices are the angular momentum quantum numbers. Repeated indices are summed over. The matrices CH and DH depend on the origin and orientation of the global coordinate system of the twodimensional plane and are separable in the disk index j: 2i J (kR ) U K\J H e K 0H , CH " (3.3) KJ pa H(ka ) H H J (3.4) DHY "!pa J (kR )J (ka )e\ KYU0HY , JYKY HY KY\JY HY JY HY where R and U H are the distance and angle, respectively, of the ray from the origin in the H 0 two-dimensional plane to the center of disk j as measured in the global coordinate system (see Fig. 4). H(kr) is the ordinary Hankel function of first kind and J (kr) the corresponding ordinary J J Bessel function. The matrices CH and DH parameterize the coupling of the incoming and outgoing scattering waves, respectively, to the scattering interior at the jth disk. Thus they describe only the single-disk aspects of the scattering of a point particle from the n disks. The matrix MHHY has the structure of a Kohn—Korringa—Rostoker (KKR)-matrix, see Refs. [49,50,53], a J (ka ) J H H (kR )C (l, l) . (3.5) MHHY"d d #(1!d ) H JJY HHY JJY HHY a H(ka ) J\JY HHY HHY HY HY JY Without Ewald resummation [49], as the number of disks is finite. Here R is the separation HHY between the centers of the jth and jth disk and R "R , of course, The auxiliary matrix HHY HYH C (l, l)"e J?HYH\JY?HHY\L contains — aside from a phase factor — the angle a of the ray from the HYH HHY center of disk j to the center of disk j as measured in the local (body-fixed) coordinate system of disk j (Fig. 4).

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Fig. 4. Global and local coordinates for a general three-disk problem.

Note that C (l, l)"(!1)J\JY(C (l,l))*. The “Gaspard and Rice prefactors” of M, i.e., (pa/2i) in HHY HYH [13], are rescaled into C and D. The matrix A,M!1 contains the genuine multidisk “scattering” aspects of the n-disk problem, e.g., in the pure 1-disk scattering case, A vanishes. When (M\)HHY is expanded as a geometrical series about the unit matrix +dHHY,, a multiscattering series in “powers” of the matrix A is created. The product CM\D is the on-shell T-matrix of the n-disk system. It is the two-dimensional analog of the three-dimensional result of Lloyd and Smith for a finite cluster of non-overlapping muffin-tin potentials. At first sight the expressions Lloyd and Smith (see (98) of [53] and also Berry’s form [49] for the infinite Sinai cluster) seem to look simpler than ours and the original ones of Ref. [13], as, e.g., in M the asymmetric term a J (ka )/a H(ka ) is replaced by a symmetric H J H HY JY HY combination, J (ka )/H(ka ). Under a formal manipulation of our matrices we can derive the same J H J H result (see Appendix D). In fact, it can be checked that the (formal) cumulative expansion of Lloyd’s and our M-matrix are identical and that also numerically the determinants give the same result. Note, however, that in Lloyd’s case the trace-class property of M is lost, such that the infinite determinant and the corresponding cumulant expansion converge only conditionally and not absolutely as in our case. The latter fact is based on the trace-class properties of the underlying matrices and is an essential precondition for all further simplifications, as e.g. unitary transformations, diagonalization of the matrices, etc. A matrix is called “trace-class”, if and only if, for any choice of the orthonormal basis, the sum of the diagonal matrix elements converges absolutely; it is called “Hilbert—Schmidt” if the sum of the squared moduli of the diagonal matrix elements converges (see [60,61] and Appendix A for the definitions and properties of trace-class and Hilbert—Schmidt matrices). Here, we will only list the most important properties: 1. Any trace-class matrix can be represented as the product of two Hilbert—Schmidt matrices and any such product is again trace-class. 2. A matrix B is already Hilbert—Schmidt, if the trace of BRB is absolutely convergent in just one orthonormal basis. 3. The linear combination of a finite number of trace-class matrices is again trace-class.

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4. The hermitean-conjugate of a trace-class matrix is again trace-class. 5. The product of two Hilbert—Schmidt matrices or of a trace-class and a bounded matrix is trace-class and commutes under the trace. 6. If a matrix B is trace-class, the trace tr(B) is finite and independent of the basis. 7. If B is trace-class, the determinant det(1#zB) exists and is an entire function of z. 8. If B is trace-class, the determinant det(1#zB) is invariant under any unitary transformation. In Appendix C we show explicitly that the l-labelled matrices SL(k)!1, CH(k) and DH(k) as well as the +l, j,-labelled matrix A(k)"M(k)!1 are of “trace-class”, except at the countable isolated zeros of H(ka ) and of Det M(k) and at k40, the branch cut of the Hankel functions. The K H ordinary Hankel functions have a branch cut at negative real k, such that even the k-plane is two-sheeted. The last property is special for even dimensions and does not hold in the threedimensional n-ball system [46,62]. Therefore for almost all values of the wave number k (with the above-mentioned exceptions) the determinant of the n-disk S-matrix exist and the operations of Eq. (3.6) are mathematically well defined. We concentrate on the determinant, det S, of the scattering matrix, since we are only interested in spectral properties of the n-disk scattering problem, i.e. resonances and phase shifts, and not in wave functions. Furthermore, the determinant is invariant under any change of a complete basis expanding the S-matrix and therefore also independent of the coordinate system:






 i, tr [(CM\D),] det SL"det (1!iCM\D)"exp tr ln(1!iCM\D)"exp ! J J J N J ,  i, "exp ! Tr [(M\DC),] "exp Tr ln(1!iM\DC)"Det (1!iM\DC) * * N * , Det (M!iDC) * . (3.6) "Det (M\(M!iDC))" * Det (M) * We use here exp tr ln notation as a compact abbreviation for the defining cumulant expansion (A.7), since det (1#kA)"exp(!  ((!k),/N)tr(A,)), is only valid for "k"max"j "(1 where j is , G G the ith eigenvalue of A. The determinant is directly defined by its cumulant expansion (see (188) of Ref. [61] and Eq. (A.7) of Appendix A.2) which is therefore the analytical continuation of the e -representation. The capital index ¸ is a multi- or “super”-index ¸"(l, j). In the first line of Eq. (3.6) the determinant and traces are only taken over small l, in the second and third line they are taken over the super-indices ¸"(l, j). In order to signal this difference we will use the following notation: det 2 and tr2 refer to the "m2 space, Det 2 and Tr2 refer to the super-space. The matrices in the super-space are expanded in the complete basis +"¸2,"+"m; j2, which refers for fixed index j to the origin of the jth disk and not any longer to the origin of the two-dimensional plane. In deriving Eq. (3.6) the following facts were used:






1. DH, CH are of trace-class in the +"l2, space (see Appendix C). 2. As long as the number of disks is finite, the product DC — now evaluated in the super-space +"¸2, — is of trace-class as well (see property 3).

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3. M!1 is of trace-class (see Appendix C). Thus the determinant Det M(k) exists. 4. Furthermore, M is bounded (since it is the sum of a bounded and a trace-class matrix). 5. M is invertible everywhere where Det M(k) is defined (which excludes a countable number of * zeros of the Hankel functions H(ka ) and the negative real k-axis as there is a branch cut) and K H non-zero (which excludes a countable number of isolated points in the lower k-plane) — see property (e) of Appendix A.2. Therefore and because of (4) the matrix M\ is bounded. 6. The matrices CM\D, M\DC, are all of trace-class as they are the product of bounded times trace-class matrices and tr [(CM\D),]"Tr [(M\DC),], because such products have the K + cyclic permutation property under the trace (see properties 3 and 5). 7. M!iDC!1 is of trace-class because of the rule that the sum of two trace-class matrices is again trace-glass (see property 3). Thus all traces and determinants appearing in Eq. (3.6) are well-defined, except at the abovementioned isolated k singularities and branch cuts. In the +"m; j2, basis the trace of M!1 vanishes trivially because of the d terms in Eq. (3.5). One should not infer from this that the trace-class HHY property of M!1 is established by this fact, since the finiteness (here vanishing) of Tr(M!1) has to be shown for every complete orthonormal basis. After symmetry reduction (see below) Tr(M!1), calculated for each irreducible representation separately, does not vanish any longer. However, the sum of the traces of all irreducible representations weighted with their pertinent degeneracies still vanishes of course. Semiclassically, this corresponds to the fact that only in the fundamental domain there can exist one-letter “symbolic words”. After these manipulations, the computation of the determinant of the S-matrix is very much simplified in comparison to the original formulation, since the last term of Eq. (3.6) is completely written in terms of closed form expressions and since the matrix M does not have to be inverted any longer. Furthermore, as shown in Appendix B.3, one can easily construct






H(ka ) a J (ka ) J H H (kR )C (l, l) , HY !(1!d ) H (3.7) MHHY!iDH CHY "d d ! JY HHY JJY JKY KYJY HHY JJY H(ka ) a H(ka ) J\JY HHY HHY HY JY HY HY JY where H(kr) is the Hankel function of second kind. The first term on the r.h.s. is just the S-matrix K for the separable scattering problem from a single disk, if the origin of the coordinate system is at the center of the disk (see Appendix B.2): H(ka) S(ka)"! JY d . (3.8) JJY H(ka) JJY JY After Eq. (3.7) is inserted into Eq. (3.6) and Eq. (3.8) is factorized out, the r.h.s. of Eq. (3.6) can be rewritten as





L Det M(k*)R Det [M(k)!iD(k)C(k)] * * " “ (det S(ka )) , (3.9) det SL(k)" J H J Det M(k) Det M(k) * * H * * where +H(z), "H(z ) has been used in the end. All these operations are allowed, since K K M(k)!1, M(k)!iD(k)C(k)!1 and S(k)!1 are trace-class for almost every k with the above mentioned exceptions. The zeros of the Hankel functions H(ka ) are automatically excluded. In K H

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general, the single disks have different sizes and the corresponding 1-disk S-matrices should be distinguished by the index j. At the level of the determinants this labelling is taken care of by the choice of the argument ka . Note that the analogous formula for the three-dimensional scattering H of a point particle from n non-overlapping spheres (of in general different sizes) is structurally completely the same [46,62], except that there is no need to exclude the negative k-axis any longer, since the spherical Hankel functions do not possess a branch cut. In the above calculation it was used that C* (l, l)"C (!l,!l) in general [46] and that for symmetric systems (equilateral HHY HHY three-disk-system with identical disks, 2-disk system with identical disks): C* (l, l)"C (l, l) (see HHY HYH [13]). Eq. (3.9) is compatible with Lloyd’s formal separation of the single scattering properties from the multiple-scattering effects in the Krein—Friedel—Lloyd sum, see e.g., p. 102 of Ref. [53] (modulo the above-mentioned conditional convergence problems of the Lloyd formulation). Eq. (3.9) has the following properties: 1. Under the determinant of the n-disk SL-matrix, the one-disk aspects separate from the multiscattering aspects, since the determinants of the one-disk S matrices factorize from the determinants of the multiscattering matrices. Thus the product over the n one-disk determinants in (3.9) parametrizes the incoherent part of the scattering, as if the n-disk problem just consisted of n separate single-disk problems. 2. The whole expression (3.9) respects unitarity as S is unitarity by itself, because of (H(z))*"H(z*) and as the quotient of the determinants of the multiscattering matrices on K K the r.h.s. of Eq. (3.9) is manifestly unitary. 3. The determinants over the multiscattering matrices run over the super-index ¸ of the superspace. This is the proper form for the symmetry reduction (in the super-space), e.g., for the equilateral three-disk system (with disks of the same size) we have Det M "det M det M (det # M ) , J  J # * U  J 

(3.10)

and for the two-disk system (with disks of the same size) Det M "det M det  M det M det M J  J J * U  J 



,

(3.11)

etc. In general, if the disk configuration is characterized by a finite point-symmetry group G, we have Det M "“ (det AM A(k))BA , * LU  J " A

(3.12)

where the index c runs over all conjugate classes of the symmetry group G and D is the cth A representation of dimension d [46]. For the symmetric two-disk system, these representations are A the totally symmetric A , the totally anti-symmetric A , and the two mixed representations B and    B which are all one-dimensional. For the symmetric equitriangular three-disk system, there exist  two one-dimensional representations (the totally symmetric A and the totally anti-symmetric A )   and one two-dimensional representation labelled by E. A simple check that Det M(k) has been split up correctly is the following: the power of H(ka ) Hankel functions (for fixed m K H with!R(m(#R) in the denominator of “ [det AM A(k)]BA has to agree with the power of the A J "

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same functions in Det M(k) which in turn has to be the same as in “L (det S(ka )). Note that on H H the l.h.s. the determinants are calculated in the super-space +¸,, whereas on the r.h.s. the reduced determinants are calculated, if none of the disks are special in size and position, in the normal (desymmetrized) space +l, (however, now with respect to the origin of the disk in the fundamental domain and with ranges given by the corresponding irreducible representations). If the n-disk system has a point-symmetry where still some disks are special in size or position (e.g., three equal disks in a row [63]), the determinants on the r.h.s. refer to a correspondingly symmetry-reduced super-space. This summarizes the symmetry reduction on the exact quantum-mechanical level. It can be derived from












 (!1),  (!1), Tr [A,] "exp ! Tr [UA,UR] Det M"exp ! * * * N N , ,  (!1),  (!1), "exp ! Tr [(UAUR),] "exp ! Tr [A, ] , (3.13) * *
 N N , , where U is unitary transformation which makes A block-diagonal in a suitable transformed basis of the original complete set +"m; j2,. These operations are allowed because of the trace-class-property of A and the boundedness of the unitary matrix U (see also property (d) of Appendix A.2).








4. The link between the determinant of the S-matrix and the semiclassical zeta function In this section we will specify the semiclassical equivalent of the determinant of the n-disk S-matrix. As det SL in Eq. (3.9) factorizes into a product of the one-disk determinants and the ratio of the determinants of the multiscattering matrix, Det M(k*)R/Det M(k), the semiclassical reduction will factorize as well into incoherent one-disk parts and an coherent multiscattering part. Note, however, that there is an implicit connection between these parts via the removable one-disk poles and zeros. This will be discussed in the conclusion Section 7. In Appendix E, the semiclassical expression for the determinant of the one-disk S-matrix is constructed in analogous fashion to the semiclassical constructions of Ref. [44] which in turn is based on the work of Ref. [29]: (“ [1!e\ LJN lI?]) (4.1) det S(ka)+(e\ L,I?) l J (“ [1!e> LJlI?]) l  with the creeping exponential (for more details, see Appendix E and the definitions of Appendix F.4) ll(ka)"ka#e> L(ka/6)ql#2"ka#ial(ka)#2 ,

(4.2)

lN l(ka)"ka#e\ L(ka/6)ql#2"ka!i(al(k*a))*#2"(ll(k*a))* ,

(4.3)

and N(ka)"(nak)/4n#2 the leading term in the Weyl approximation for the staircase function of the wave-number eigenvalues in the disk interior. From the point of view of the

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Fig. 5. Left- and right-handed diffractive creeping paths of increasing mode number l for a single disk.

scattering particle the interior domains of the disks are excluded relatively to the free evolution without scattering obstacles (see, e.g., [17]). Therefore the negative sign in front of the Weyl term. For the same reason, the subleading boundary term has here a Neumann structure, although the disks have Dirichlet boundary conditions. Let us abbreviate the r.h.s. of Eq. (4.1) for a specified disk j as * * * *   (e\ L,I?H) ZI U  (k aH) ZI U (k aH) , det S(ka )P J H ZI (ka ) ZI (ka ) U   H U  H

(4.4)

where ZI (ka ) and ZI (ka ) are the diffractional zeta functions (here and in the following U   H U  H we will label semiclassical zeta-functions with diffractive corrections by a tilde) for creeping orbits around the jth disk in the left-handed sense and the right-handed sense, respectively (see Fig. 5). The two orientations of the creeping orbits are the reason for the exponents two in Eq. (4.1). Eq. (4.1) describes the semiclassical approximation to the incoherent part ("the curly bracket on the r.h.s.) of the exact expression (3.9). We now turn to the semiclassical approximation of the coherent part of Eq. (3.9), namely the ratio of the determinants of the multiscattering matrix M. Because of the trace-class property of A"M!1, the determinants in the numerator and denominator of this ratio exist individually and their semiclassical approximations can be studied separately. In fact, because of Det M(k*)R"(Det M(k*))*, the semiclassical reduction of Det M(k*)R follows directly from the corresponding result of Det M(k) under complex conjugation. The semiclassical reduction Det M(k) will be done in the cumulant expansion, since the latter is the defining prescription for the computation of an infinite matrix that is of the form 1#A where A is trace-class: z Det[1#zA(k)]"1!(!z)Tr[A(k)]! +Tr[A(k)]![Tr A(k)],#2 2  (4.5) " zLAQ A(A) with Q (A),1 , L  LA where we have introduced here a book-keeping variable z which we will finally set to one. This allows us to express the determinant of the multiscattering matrix solely by the traces of the matrix

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A, Tr[AK(k)] with m"1, 2, 3,2 . The cumulants and traces satisfy the (Plemelj—Smithies) recursion relations (A.16) 1 LA Q A(A)" (!1)K>Q A (A)Tr[AK] for n 51 L \K L A n A K

(4.6)

in terms of the traces. In the next section we will utilize Watson resummation techniques [64,29] which help to replace the angular momentum sums of the traces by continuous integrals which, in turn, allow for semiclassical saddle-point approximations. With these techniques and under complete induction we will show that for any geometry of n disks, as long as the number of disks is finite, the disks do not overlap and grazing or penumbra situations [31,35] are excluded (in order to guarantee unique isolated saddles), the semiclassical reduction reads as follows: t (k)P   (!1)K d n N #creeping p.o.’s Tr[AK(k)]P KPLN N1!(K )P N N P

(4.7)

with inputs as defined below Eq. (2.3). The reduction is of course only valid, if Re k is sufficiently large compared to the inverse of the smallest length scale of the problem. The right hand side of Eq. (4.7) can be inserted into the recursion relation (4.6) which then reduces to a recursion relation for the semiclassical approximations of the quantum cumulants 1 LA t (k)P for n 51 , C A(s.c.)"! C A (s.c.) d Nn N L \K A KPL N1!(K )P L n N A K N P

(4.8)

where we have neglected the creeping orbits for the time being. Under the assumption that the semiclassical limits Re kPR and the cumulant limit n PR commute (which might be problemA atic as we will discuss later), the approximate cumulants C A(s.c.) can be summed to infinity, L A zLAC A(s.c.), in analogy to the exact cumulant sum. The latter exists since A is trace-class. The L L  infinite “approximate cumulant since”, however, is nothing but the curvature expansion of the Gutzwiller—Voros zeta function, i.e.,  Z (z; k)" " zLAC A(s.c.) , L %4   LA

(4.9)

since Eq. (4.8) is exactly the recursion relation of the semiclassical curvature terms [2]. If, in addition, the creeping periodic orbits are summed as well, the standard Gutzwiller—Voros zeta function generalizes to the diffractive one discussed in Refs. [32—34] which we will denote here by a tilde. In summary, we have   ZI Det M(k)P

(k)" %4  

(4.10)

for a general geometry and   ZI (k)" det M A(k)P "A   "

(4.11)

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23

for the case that there is a finite point-symmetry and the determinant of the multiscattering matrix splits into the product of determinants of matrices belonging to the pertinent representations D , A see Eq. (3.12). Thus the semiclassical limit of the r.h.s. of Eq. (3.9) is





L Det M(k*)R * det SL(k)" “ det S(ka ) J H Det M(k) J * H ZI (k*a )* ZI L (k*a )* ZI (k*)*   “ H U  H %4 , (4.12) P (e\ L,I?H) U   ZI (ka ) ZI (ka ) ZI (k) U   H U  H %4 H where, from now on, we will suppress the qualifier 2" . For systems which allow for   complete symmetry reductions (i.e., equivalent disks under a finite point-symmetry with a "a ∀j) H the link reads










ZI (k*a)* ZI (k*a)* L “ (det A M A(k*)R)BA   U  J " det SL(k)"(det S(ka))L A P (e\ L,I?)L U   J J “ (det A M A(k))BA ZI (ka) ZI (ka) A J " U   U  “ (ZI (k*)*)BA ; A "A (4.13) “ (ZI A(k))BA A " in obvious correspondence. Note that the symmetry reduction from the right-hand side of Eq. (4.12) to the right-hand side of Eq. (4.13) is compatible with the semiclassical results of Refs. [65,66]. In the next section we will prove the semiclassical reduction step (4.7) for any n-disk scattering system under the conditions that the number of disks is finite, the disks do not overlap, and geometries with grazing periodic orbits are excluded. We will also derive the general expression for creeping periodic orbits for n-disk repellers from exact quantum mechanics and show that ghost orbits drop out of the expansion of Tr ln(1#A) and therefore out of the cumulant expansion.

5. Semiclassical approximation and periodic orbits In this section we will work out the semiclassical reduction of Tr[AK(k)] for non-overlapping, finite n-disk systems where a J (ka ) (5.1) AHHY"(1!d ) H J H (!1)JYe J?HYH\JY?HHYH (kR ) . J\JY HHY HHY a H(ka ) JJY HY JY H As usual, a , a are the radii of disk j and j, 14j, j4n, R is the distance between the centers of H HY HHY these disks, and a is the angle of the ray from the origin of disk j to the one of disk j as measured HYH in the local coordinate system of disk j. The angular momentum quantum numbers of l and l can be interpreted geometrically in terms of the positive- or negative-valued distances (impact parameters) l/k and l/k from the center of disk j and disk j, respectively, see [49]. Because of the finite set of n disk-labels and because of the cyclic nature of the trace, the object Tr[AK(k)] contains all periodic itineraries of total symbol length m with an alphabet of n symbols,

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i.e. AHHAHH2AHK\HKAHKH with j 3+1, 2,2, n,. Here the disk indices are not summed over and the G angular momentum quantum numbers are suppressed for simplicity. The delta-function part (1!d ) generates the trivial pruning rule (valid for the full n-disk domain) that successive symbols HHY have to be different. We will show that these periodic itineraries correspond in the semiclassical limit, ka G<1, to geometrical periodic orbits with the same symbolic dynamics. For periodic orbits H with creeping sections [44,45,32—34] the symbolic alphabet has to be extended. Furthermore, depending on the geometry, there might be non-trivial pruning rules based on the so-called ghost orbits, see Refs. [7,49]. We will discuss such cases in Section 5.2. 5.1. Quantum itineraries As mentioned, the quantum-mechanical trace can be structured by a simple symbolic dynamics, where the sole (trivial) pruning rule is automatically taken care of by the 1!d factor appearing in HHY AHHY. Thus we only have to consider the semiclassical approximation of a quantum-mechanical JJY itinerary of length m: > > > > > K\HKAHKH (5.2) 2 AHHAHH2AHK\ AHHAHH2AHK\HKAHKH :" JJ JJ J JK JKJ J\ J\ J\ JK\\ JK\ with j 3+1, 2,2, n,. This is still a trace in the angular momentum space, but not any longer with G respect to the superspace. Since the trace, Tr AK, itself is simply the sum of all itineraries of length m, i.e. L L L L L Tr AK" 2 AHHAHH2AHK\HKAHKH , (5.3) H H H HK\ HK its semiclassical approximation follows directly from the semiclassical approximation of its itineraries. Note that we here distinguish between a given itinerary and its cyclic permutation. All of them give the same result, such that their contributions can finally be summed up by an integer-valued factor n " : m/r, where the integer r counts the number of repeated periodic N subitineraries. Because of the pruning rule 1!d , we only have to consider traces and itineraries HHY with n52 as AHH "0 implies that Tr A"0 in the full domain. JJY We will show in this section that, with the help of the Watson method [64,29] (studied for the convolution of two A matrices in Appendix F which should be consulted for details), the semiclassical approximation of the periodic itinerary AHHAHH2AHK\HKAHKH becomes a standard periodic orbit labelled by the symbol sequence j j 2j . Depending on the   K geometry, the individual legs j Pj Pj result either from a standard specular reflection at G\ G G> disk j or from a ghost path passing straight through disk j . If furthermore creeping contributions G G are taken into account, the symbolic dynamics has to be generalized from single-letter symbols +j , G to triple-letter symbols +j , s ;l , with l 51 integer-valued and s "0,$1. By definition, the G G G G G  Actually, these are double-letter symbols as s and l are only counted as a product. G G

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Fig. 6. The ghost itinerary (1, 2, 3, 4).

value s "0 represents the non-creeping case, such that +j , 0;l ,"+j , 0,"+j , reduces to the old G G G G G single-letter symbol. The magnitude of a non-zero l corresponds to creeping sections of mode G number "l ", whereas the sign s "$1 signals whether the creeping path turns around the disk j in G G the positive or negative sense. Additional full creeping turns around a disk j can be summed up as a geometrical series; therefore they do not lead to the introduction of a further symbol. 5.2. Ghost contributions An itinerary with a semiclassical ghost section at, say, disk j will be shown to have the same G weight as the corresponding itinerary without the j th symbol. Thus, semiclassically, they cancel G each other in the Tr ln(1#A) expansion, where they are multiplied by the permutation factor m/r with the integer r counting the repeats. E.g. let (1, 2, 3, 4) be a non-repeated periodic itinerary with a ghost section at disk 2 steming from the 4th-order trace Tr A, where the convention is introduced that an underlined disk index signals a ghost passage (see Fig. 6). Then its semiclassical, geometrical contribution to Tr ln(1#A) cancels exactly against the one of its “parent” itinerary (1, 3, 4) (see Fig. 7) resulting from the 3rd-order trace: !(4A A A A )#(3A A A )"(!1#1)A A A "0 . (5.4)             The prefactors #1/3 and !1/4 are due to the expansion of the logarithm, the factors 3 and 4 inside the brackets result from the cyclic permutation of the periodic itineraries, and the cancellation stems from the rule 2AGG>AG>G>2"2AGG>2 .   

(5.5)

We have checked this rule in Appendix F.6 for the convolution of two A-matrices, but in Section 5.6 we will prove it to hold also inside an arbitrary (periodic) itinerary. Of course the same cancellation holds in case that there are two and more ghost segments. For instance, consider the itinerary (1, 2, 3, 4, 5, 6) with ghost sections at disk 2 and 5 resulting from the 6th-order trace. Its

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Fig. 7. The parent itinerary (1, 3, 4).

geometrical contribution cancels in the trace-log expansion against the geometrical reduction of the itineraries (1, 2, 3, 4, 6), (1, 3, 4, 5, 6) from the 5th-order trace with ghost sections at disk 2 or 5, respectively, and against the geometrical reduction of the itinerary (1, 3, 4, 6) of the 4th-order trace with no ghost contribution: !(6A A A A A A )#(5A A A A A              (5.6) #5A A A A A )      !(4A A A A )"(!1#2!1) A A A A "0 . (5.7)          Again, the prefactors !1/4, #1/5, !1/6 result from the trace-log expansion, the factors 4, 5, 6 inside the brackets are due to the cyclic permutations, and the rule (5.5) was used. If there are two or more ghost segments adjacent to each other, the ghost rule (5.5) has to be generalized to 2AGG> AG>G>2 AG>IG>I>2 AG>L\G>L2     (5.8) "2AGG>2AG>IG>I>2AG>L\G>L2    (5.9) "2AGG>2AG>IG>I>2AG>L\G>L2    "2AGG>L2 . (5.10)  Finally, let us discuss one case with a repeat, e.g. the itinerary (1, 2, 3, 4, 1, 2, 3, 4) with repeated ghost sections at disk 2 in the semiclassical limit. The cancellations proceed in the trace-log expansion as follows: !(4A  A A A A A A A )          #(7A A A A A A A )!(3 A A A A A A )                "(!#1!)[A A A ]"0 . (5.11)      Note that the cyclic permutation factors of the 8th- and 6th-order trace are halved because of the repeat. The occurrence of the ghost segment in the second part of the 7th-order itinerary is taken care of by the weight factor 7.

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The reader might study more complicated examples and convince him- or herself that the rule (5.10) is sufficient to cancel any primary or repeated periodic orbit with one or more ghost sections completely out of the expansion of Tr ln(1#A) and therefore also out of the cumulant expansion in the semiclassical limit: Any periodic orbit of length m with n ((m) ghost sections is cancelled by the sum of all ‘parent’ periodic orbits of length m!i (with 14i4n and i ghost sections removed) weighted by their cyclic permutation factor and by the prefactor resulting from the trace-log expansion. This is the way in which the non-trivial pruning for the n-disk billiards can be derived from the exact quantum-mechanical expressions in the semiclassical limit. Note that there must exist at least one index i in any given periodic itinerary which corresponds to a non-ghost section, since otherwise the itinerary in the semiclassical limit could only be straight and therefore non-periodic. Furthermore, the series in the ghost cancellation has to stop at the 2nd-order trace, Tr A, as Tr A itself vanishes identically in the full domain which is considered here. 5.3. Semiclassical approximation of a periodic itinerary The procedure for the semiclassical approximation of a general periodic itinerary, Eq. (5.2), of length m follows exactly the calculation of Appendix F for the convolution of two A-matrices. The reader interested in the details of the semiclassical reduction is advised to consult this appendix before proceeding with the remainder of the section. First, for any index i, 14i4m, the sum over the integer angular momenta, l , will be symmetrized as in Eq. (F.3) with the help of the weight G function d(l ) [d(l O0),1, d(l "0)"1/2]. G G G A HH2A HG\HGA HGHG>2A HKH > > > > > > G\HGA HGHG>2A HK H " 2 2 A HH2A HG\ JJ J JG JGJG> JK J J\ J\ JG\\ JG\ JG>\ JK\ > > > > > > " 2 2 d(l )2d(l )d(l )2d(l )  G\ G K J J JG\ JG JG> JK Q2QK\ HG A HGHG> 2A HKKHK   . ;(A HH   2A HG\ Q J Q J  QG\JG\QGJG QGJGQG>JG> Q J Q J 

(5.12)

Furthermore, the angles *a G,a G> G!a G\ G [the analogs of a !a in Eq. (F.3)] will be H H H H H HYYHY HHY replaced by *I a G G"*a G!p 2n where p "0, 2, 1. This will be balanced by multiplying Eq. (5.2) N H G G H with (!1)NGJG where p"p for p "1 and zero otherwise. The three choices for p are, at this stage, G G G G equivalent, but correspond in the semiclassical reduction to the three geometrical alternatives: specular reflection at disk j to the right, to the left or ghost tunnelling. In order not to be bothered G by borderline cases between specular reflections and ghost tunnelling, we exclude disk configurations which allow classically grazing or penumbra periodic orbits [31,35]. Then, the sum over the integer angular momentum l will be replaced by a Watson contour G integration over the complex angular momentum l G



> 1 1 (!1)JG\NGd(l )X G" dl e\ JGLNGX G , G J 2i G sin(l n) J !> G JG

(5.13)

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as in Eq. (F.4). The path C encircles (in the positive sense) all positive integers l . The quantity > G X G abbreviates here J J (ka ) J (ka ) H (kR )H (kR )e QGJG I ?HGNG, JG HG ½ G , X G, JG HG J H(ka ) QG\JG\\QGJG HG\HG QGJG\QG>JG> HGHG> (ka G) J H HG QG\ H JG JG

(5.14)

(ka "J G(ka G)/H (ka G), since l is an where the expression has simplified because of J G G(ka G)/H QGJG HG JG H G J H QJ H integer. The quantity ½ G abbreviates the sum in Eq. (5.14). The next steps are completely the same J as in Appendices F.1 and F.2. The paths below the real l axis will be transformed above the axis. G The expressions split into a sin(l n)-dependent contour integral in the upper complex plane and G into a sin(l n)-independent straight-line integral from iR(1#id ) to !iR(1#id ). Depending on G G G the choice of p , the sum (5.13) becomes exactly one of the three expressions (F.15), (F.16) or (F.17), G where the prefactor ¼HH in Appendix F.2 should be, of course, replaced by all the l -independent G JJ terms of Eq. (5.2) and where j, j, j are substituted by j , j , j . The angular momenta l and l are G\ G G> here identified with s ;l and s ;l , respectively. After the Watson resummation of the G\ G\ G> G> other sums, e.g., of the l sum, etc., l has to be replaced by l and l by l . If the penumbra G\ G\ G> scattering case [31,35] is excluded, the choice of p is, in fact, uniquely determined from the G empirical constraint that the creeping amplitude has to decrease during the creeping process, as tangential rays are constantly emitted. In mathematical terms, it means that the creeping angle has to be positive. As discussed at the beginning of Appendix F.2, the positivity of the two creeping angles for the left and right turn uniquely specifies which of the three alternatives p is realized. In G other words, the geometry is encoded via the positivity of the two creeping paths into a unique choice of the p . Hence, the existence of the saddle point (5.15) is guaranteed. G The final step is the semiclassical approximation of the analog expressions to Eqs. (F.15), (F.16) and (F.17) as in Appendices F.3—F.5. Whereas the results for the creeping contributions can be directly taken over from Eqs. (F.34) to (F.43), there is a subtle change in the semiclassical evaluation of the straight-line sections. In the convolution problem of Appendices F.3 and F.5 we have only picked up second-order fluctuating terms with respect to the saddle solution l from the Q l integration. Here, we will pick up quadratic terms (l !lQ) from the l integration and mixed G G G terms (l !lQ)(l !lQ ) from the neighboring l and l integrations as well. Thus instead of G G G! G! G\ G> having m one-dimensional decoupled Gauss integrations, we have one coupled m-dimensional one. Of course, also the saddle-point equations [the analog toEq. (F.35) or Eq. (F.44)] are now coupled: (1!d G )2 arccos[lQ/ka G]"arccos[(lQ!lQ )/kR G\ G]#arccos[(lQ!lQ )/kR G G>] G G\ H H G G\ HH G H N  !(a G> G!a G\ G!p 2n) , H H G H H

(5.15)

where the saddle lQ of the ith integration depends on the values of the saddles of the (i!1)th G and (i#1)th integration and so on. Indeed, all m saddle-point equations are coupled. This corresponds to the fact that the starting- and end-point of a period orbit is not fixed from the outside, but has to be determined self-consistently, namely on the same footing as all the intermediate points. In order to keep the resulting expressions simple we will discuss in the following subsection just the geometrical contributions, and leave the discussion of the ghost and creeping contributions for later sections.

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5.4. Itineraries in the geometrical limit We will prove that the itinerary (!1)KAHHAHH2AHK\HKAHKH leads, in the semiclassical reduction, to the following geometrical contribution: e I*K\ KL) " , (!1)KAHHAHH2AHK\HKAHKH"  "K "(1!KK  ) K

(5.16)

where the factor (!1)K results from the trace-log expansion Tr ln(1![!A]), as the periodic orbit expansion corresponds to this choice of sign. The quantity ¸ is the length of the periodic orbit K with this itinerary. K '1 is the expanding eigenvalue of the corresponding monodromy matrix K and k "2m is the corresponding Maslov index indicating that the orbit is reflected from K m disks (all with Dirichlet boundary conditions). Thus, for n-disk Dirichlet problems, the Maslov indices come out automatically. [Under Neumann boundary conditions, there arises an additional (ka G),/+(d/dk)H (ka G),K!H (ka G)/H (ka G) in the minus sign per disk label j , since +(d/dk)H H JG H JG H JG H G JG Debye approximation. The minus sign on the right-hand side cancels the original minus sign from the trace-log expansion such that the total Maslov index becomes trivial. Otherwise, the Neumann case is exactly the same.] If the itinerary is the rth repeat of a primary itinerary of topological length p, the length, Maslov index and stability eigenvalue will be shown to satisfy the relations: ¸ "r¸ , k "rk and K "(K )P. K N K N K N Let us define the abbreviations d

, ,(RG\ G![(lQ !lQ)/k]"d G\ G GG\ G\G H H

(5.17)

o ,(aG!(lQ/k) , G G H

(5.18)

¸ ,d !o !o "¸ , G\G G\G G\ G GG\

(5.19)

dl ,l !lQ, G G G

dI l ,dl /d G G G\G

with i evaluated modulo m, especially i"0 is identified with i"m and i"m#1 with i"1. The quantity d is the geometrical length of the straight line between the impact parameter lQ /k at G\G G\ disk j and the impact parameter lQ/k at disk j in terms of the saddle points lQ and lQ. The latter G\ G G G\ G are determined by the saddle-point condition (5.15) which can be re-written for non-ghost scattering (p O1) as a condition on the reflection angle at disk j : G G









lQ!lQ lQ!lQ G\ #arcsin G G> #(a !a !p 2n) . h G,arcsin[lQ/ka G]"arcsin G G H HG>HG HG\HG G H kR G\ G kR G G> H H HH

(5.20)

Thus, o is the radius a G of the disk j times the cosine of the reflection angle and ¸ is the G H G G\G geometrical length of the straight-line segment between the (i!1)th and ith point of reflection. Under the condition that the disks do not overlap, the inequalities ¸ (d (R G\ G hold and G\G G\G H H exclude the possibility that the reflection points are in the mutual shadow region of disks. For each itinerary there is at most one reflection per disk-label j modulo repeats, of course. G

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Then in analogy to Appendix F.5 the geometric limit of the itinerary (5.2) becomes

 





K \ L K 2 e IBG\G\MG ddl “ e L "(!1)K “ (!1)KAHHAHH2AHK\HKAHKH" H   n kd G\G H \\ L G





K e L    \    ;e\ I BJG MG\BG\G e IBG\G\MG\GL BGG>e\ I BJGBJG>BGG> e\ I BJG\BJG BG\G" “ (2nk G



;

\ L



exp(ik K ¸ !imn) \ L G G\G , ddI l 2 ddI l e\ I BI J2BI JKFKBI J2BI JK2"  K "D " K \\ L \\ L (5.21)

where we have used that K 2o " K(o #o ) since o "o . ¸ , K ¸ is the total G  K K G G G G> G G\G geometrical length of the geometrical path around the itinerary, see Appendix G of Ref. [49]. Note that we used the saddle-point condition (5.15) in order to remove not only the linear fluctuations, but all terms of linear order in the lQ’s from the exponents. Only the zeroth-order terms and the G quadratic fluctuations remain. D is the determinant of the m;m matrix F (,F) with K K d 2d F " K!1! K,a ,   d o   2d d F " G\G!1! G\G,a for 24i4m!1 , GG G o d G GG> d 2d F " K\K!1! K\K,a , K KK d o K K

(5.22)

d " G\G"F F ,b for 14i4m!1 , GG> d G>G GG> GG> d F " K\K"F ,b , K K K d K F "0 otherwise GH for m53. [For m"2 the off-diagonal matrix elements read instead d d F "F " # "2 .   d d  

(5.23)

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The corresponding diagonal matrix elements are given as above, but simplify because of d "d .] Thus in general, the determinant reads   a b 0 0 2 0 b   K b a b 0 2 0 0    0 b a b \ 2 $    0 0 b a \ \ $   . (5.24) D "det K $ 2 \ \ \ b 0 K\K\ 0 0 2 \ \ a b K\ K\K b 0 2 2 0 b a K K\K K






Note that determinants of this structure can also be found in Balian and Bloch [7] and Berry [49]. Our task, however, is to simplify this expression, such that the stability structure of an isolated unstable periodic orbit emerges in the end. In order to derive a simpler expression for D , let us K consider the determinant D of the auxiliary m;m matrix F (,F) which has the same K K matrix elements as F with the exception that b "0. The original determinant D can now be K K expressed as d #2(!1)K> , (5.25) D "D ! K\KD K\ K K d K where the last term follows from “K b "1. Here and in the following D is defined as the G GG> JI determinant of the auxiliary (k!l#1);(k!l#1) matrix F with matrix elements F " "F JI JI GH GH ,F ,D ,1. The D determifor l4i, j4k (4m). Furthermore we define D I>I I>I  JI nants fulfill the following recursion relations





 

d d 2d I\I!1! I\I D ! I\I\D , (5.26) JI\ JI\ d d o II> I\I I d d 2d J\J!1! J\J D ! J\JD , (5.27) D " J>I d J>I JI d o JJ> JJ> J such that D can be constructed from all the lower determinants D and D with l4i(j(k. JI JH GH For example, D " JI




2d 2d d #(!1)I 1! KD # D !2 D "! I\ID I\   I o o d   II> 2d 2d 2#(!1)I I\ID , #(!1)G G\GD G\ I\ o o I G 2d 2d d D "! I\ID #(!1)I\J> 1! J\JD # JJ>D JI\ JJ\ JJ JI o o d J J> II> 2d #2#(!1)I\J> I\ID JI\ o I








(5.28)

(5.29)

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as can be shown by complete induction. Note that the product d D is a multinomial in II> JI d /o where, for each index j, the d /o factors appear at most once. HH> G HH> G Replacing the D term in Eq. (5.25) by the r.h.s. of Eq. (5.28) and using the relation (5.27) in K order to simplify the expression d !D ) ! K\K(D K\ K\ d K recursively, we finally find after some algebra that 2d 2d 2d D " K\KD ! K\K\D #2#(!1)G\ K\G\K\GD K K\ K\ K\G o o o K K\ K\G 2d 2d 2d #2#(!1)K\ KD ! K\KD # K\KD !2  K\ K\ o o o    2d 2d #(!1)G\ K\KD #2#(!1)K\ K\KD . GK\ KK\ o o G\ K\

(5.30)

By complete induction it can be shown that D is a multinomial in 2d /o of order m where K G\G H the single factors appear at most once and the highest term has the structure “K 2d /o . Thus, G\G G G all the d ’s are in the numerators, whereas all the o ’s appear the denominators of this G\G G multinomial. We will show in Section 5.9 that




# D "(!1)K K K K K

1



!2 , K

(5.31)

where K is the expanding eigenvalue of the monodromy matrix which belongs to that period K orbit which is given by the geometric path of the periodic itinerary. If the result of Eq. (5.31) is inserted into Eq. (5.21) the semiclassical reduction (5.16) is proven. 5.5. Itineraries with repeats In the following we will discuss modifications, if the periodic itinerary is repeated r times, i.e., let m"p;r still be the total topological length of the itinerary, whereas p is the length of the prime periodic unit which is repeated r times: AHH2AHG\HGAHGHG>2AHKH"[AHH2AHG\HGAHGHG>2AHNH]P .

(5.32)

The length and Maslov index of the itinerary are of course r times the length and Maslov index of the primary itinerary AHH2AHNH, e.g., ¸ "r¸ . The non-trivial point is the structure of the K N stability determinant D . Here we can use that the matrix F has exactly the structure of the K K matrices considered by Balian and Bloch in Ref. [7], Section 6D. Let F be the corresponding N matrix of the primary itinerary with matrix elements as in Eq. (5.22) [where m is replaced by p of

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course]. Following Ref. [7] we furthermore define a new matrix FI (s) with matrix elements N FI (s)" "e\ QF " , N N N N

FI (s)" "e QF " , N N N N

FI (s)" "F " otherwise . N GH N GH The determinant of the total itinerary is then [7] P D " “ DI (e LLP) (5.33) K N L in terms of the rth roots of unity, since after r repeats, the prefactor in front of the (1, m) and (m, 1) matrix elements must be unity in order to agree with the original expression (5.22). Let us furthermore define a ,((DI (0)#(DI (n)), N  N N

b ,((DI (0)!(DI (n)) , N  N N

(5.34)

then, according to Balian and Bloch [7], Section 6D, D "(aP !(!b )P)"aP#bP!2(!a b )P . K N N N N N N In our case we have the further simplification (in analogy to Eq. (5.25)) !a b "!(DI (0)!DI (n))"!(2(!1)N>!+!2(!1)N>,)"(!1)N , N N  N N  as the corresponding two matrices differ only in the sign of their (1, p) and (p, 1) elements. Especially we now have b "(!1)N>/a , such that N N 1 D "(a)P# !2(!1)NP K N (a)P N which corresponds to the usual form






1 !2 D "(!1)NP KP # N KP K N

(5.35)

if K is identified with (!1)Na. Note that from this the structure of Eq. (5.31) follows for the N N special case r"1. Thus we have achieved so far two things: we have proven that the determinant D organizes itself in the same way as a monodromy matrix does and, in fact, that it can be K written in terms of a monodromy matrix M with eigenvalues K , 1/K as follows: N N N D "(!1)NP>det(MP !1) . K N

(5.36)

What is left to show is that M is the very monodromy matrix belonging to the periodic orbit N with the itinerary as in Eq. (5.32). This will be done in Section 5.9. But first, we will complete the study of the geometrical sector by deriving the ghost subtraction rules, and furthermore discuss periodic orbits with creeping contributions.

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5.6. Ghost rule Let us now imagine that the itinerary (5.2) has, at the disk position j , an angular domain that G corresponds to a ghost section, i.e. p "1, i fixed: G AHH2AHG\HG AHGHG>2AHKH .

(5.37)

Because of the cyclic nature of the itinerary we can always choose the label j away from the first G and end position [remember that at least two disk positions of any periodic orbit must be of non-ghost nature]. In this case there are four changes relative to the calculation in Eq. (5.21), see also Appendix F.6: first, the path of the l integration is changed, second, there is a minus sign, G third, the saddle-point condition at disk j is given by Eq. (5.15) with p"1 and not by Eq. (5.20), G fourth, the o terms are absent. As in Appendix F.6, the saddle condition (5.15) at the j th disk G G implies that d #d "d . We can use this in order to express the length of the ghost G\G GG> G\G> segment ¸ between the reflection point at disk j and the next reflection point at disk G\G> G\ j in terms of the quantities defined in Eq. (5.19): G> ¸ "d !o !o "¸ #2o #¸ . G\G> G\G> G\ G> G\G G GG>

(5.38)

Thus, by adding and subtracting the o contributions we get G AHH2AHG\HG AHGHG>2AHKH"



"







\ p

K   ddl “ H \ p HH$G \ 

K ; “  e L  J

\> p > p





ddl (!1)e\ Le IMG> ‚k BJG M‚G G



2 e IBJ\J\MJ   \BJJ>   e\ ‚k BJJBJJ> BJJ>  e\ ‚kBJJ\ BJJ BJ\J e\ ‚kBJJ‚‚MJ\BJ\J n kd J\J



 

K e p K \ p G J\J\MJ “ “ e IB BI J2BI JK2 ddI l e\ ‚k BI J2BI JKFEK H "#e IMG 2nk \ p J H \  K e I KJ*J\J>MG e I G\ J*J\J>*G\G>> JG>*J\J " " . "DEG " "DEG " K K

(5.39)

[Note that the exponent of the ghost itinerary is exactly the same as of one of its parent, the same itinerary without the disk j , whose geometrical path has the length G\¸ #¸ G G\G> J J\J ¸ .] In writing down the last-but-one line we have cancelled the overall minus sign by # K JG> J\J exchanging the upper and lower limit of the dl integration. In addition, the following substitutions G were applied: dl ": d dI l for lOi, i#1, J J\J J




dl ": e p G



d d  G\G GG> dI l , G d G\G>

dl

": d dI l . G> G\G> G>

(5.40)

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35

In this way, the integration path and phase of the ith term agree with the ones of the other terms. DEG is the determinant of the m;m matrix FEG (,FEG) which is affected by the substitutions in K K the following way: F EG J I F EG GG

" F for l,kOi, i#1 , JI " 1,

F EG G\G

" i

F EG GG>

" i







d d  G\G\ GG> "FEG , GG\ d d G\G G\G> d  G\G "iF , "FEG "iF G>G GG> G>G d GG>

F EG " G>G>

d d 2d G\G>! G\G>! G\G> , d d o GG> G>G> G>

" F EG G>G>




(5.41)



d  G\G> , "FEG G>G> d G>G>

where F are the matrix elements as defined Eq. (5.22); i.e., JI




\ \

\ DEG "det(FEG)"det K 2

\

\

$

$

G\G\ FEG GG\ 0

G

FE G\G 1

0

0

FEG GG> FEG G>G> FEG G>G> \

0

F

2

0

FEG G>G 0

$

$

$

FEG G>G> F G>G> \

2 2 2 \ \ \



.

(5.42)

from the (i!1)th and the ith row times FEG form the We now subtract the ith row times FEG GG\ GG> (i#1)th, as both operations leave the determinant DEG unaffected. Using that the ghost segments K add, i.e., d "d #d , the numerators of the terms in the (i, i) and (i#1, i#1) matrix G\G> G\G G G> elements can be simplified. The determinant DEG , expressed via the transformed m;m matrix FI EG, K reads




\

\

\

$

$

2

FI EG G\G 1

FI EG G\G> 0

0

2

\

G\G\ 0

0

2

FI E G>G\ 0

FI E G>G 0

$

$

$

FI E G>G> FEG G>G> \

FE G>G> F G>G> \

2 \

\

DEG "det(FI EG)"det K 2

FI

G

G

G

G

\ \



,

(5.43)

36

A. Wirzba / Physics Reports 309 (1999) 1—116

where d 2d G\G\!1! G\G\ , d o G\G> G\ " 2,

FI EG " G\G\ FI EG G\G

FI EG " G\G>

 

d  G\G\ , d G\G>

(5.44)

d  G\G\ , d G\G> " 2,

" FI EG G>G\ FI EG G>G





" FI EG G>G>

2d d G\G>!1! G\G> . o d G> G>G>

Note that we do not have to specify the elements on the ith row explicitly, as the ones on the ith line satisfy FI EG "d . For the same reason we can remove the ith line and row altogether GJ GJ without affecting the result for the determinant. In doing so, we exactly recover the determinant D and matrix F of the parent itinerary of the considered G\G>K G\G>K “ghost”. [The parent itinerary has the same sequel of disk indices except that the disk j is G missing.]




\

\

\

F G\G\ F G>G\ 0

DEG "det 2 K 2 $ "det F

$

F F F

\

$

G\G>

0

G>G> G>G> \

F F

"D . G\G>K G\G>K

G>G> G>G> \

2 2 \ \ \



(5.45)

The contribution of the ghost segment itself to the total “stability” of the itinerary in the geometric limit, i.e. to the stability factor of the corresponding periodic orbit, is just trivially one. As also the geometrical lengths and signs of both itineraries are the same, we have finally found that "AHH2AHG\HG>2AHKH" , (5.46) AHH2AHG\HGAHGHG>AHKH"   i.e., the ghost cancellation rule (5.5). Of course, the calculation of this section can trivially be extended to itineraries with more than one ghost (with and without repeats) as the operations in Eqs. (5.39), (5.40) and (5.43) are local operations involving just the segments with disk labels j , j and j . Thus they can be performed successively without any interference. FurtherG G\ G> more, as the transformations of the pairs (dl , dl ) in Eq. (5.40) can be done iteratively (and in I I> any order) for k"i, i#1,2, the generalization to the extended ghost cancellation rule (5.10) is

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37

trivial as well: "AHH2AHG\HGAHGHG>2AHKH" AHH2AHG\HGAHGHG>AHG>HG>2AHKH"   "AHH2AHG\HG>AHG>HG>2AHKH" "AHH2AHG\HG>2AHKH"  

(5.47)

etc. 5.7. Itineraries with creeping terms Let us now study an itinerary of topological length m which has, in the semiclassical limit, m!1 specular reflections and a left-handed or right-handed creeping contact with one disk (which we can put without lost of generality at the end position), i.e. AHH2AHK\HK\AHK\HI KAHI K H"

>

.

(5.48)

We mark those disk positions with creeping contributions by a tilde. Using the results and methods of Appendix F.5 and Section 5.3, we find the following result for the itinerary (5.48) AHH2AHK\HK\AHK\HI KAHI K H" >






e LClK  aK  e LJJ lK\NK H "! (ka K) dI "DI  " 1!e LJJ lK l K QK! H K\QKlK K\ ;e QKJJ lK ?HK\LNK>  JQK\\QKlJ lKI0HK\HK \arccos QKlJ lK\l Q I0HKH  K\ ;e I *I QKlK> G *G\G>*I K\QKlK .

(5.49)

(ka K) in the upper complex l plane Here, lJ lK,lJ lK(ka K ) is the l th zero of the Hankel function H H K JK H K [and ClK,ClK(ka K) is the creeping coefficient as given by (F.33), see Appendix F.4], *a K, p and H H K p are defined in Section 5.3, d , o and ¸ are defined in Eqs. (5.17), (5.18) and (5.19) and lQ K G\G G G\G G is the solution of the saddle-point equation (5.15) [where in the cases i"1 and i"m!1, the respective saddles lQ and lQ have to be replaced by s lJ lK]. Furthermore, the following G\ G> K additional definitions have been introduced dI








lQ !s lJ lK   K , RK\ K! K\ #o , ,¸I K\Q H H K\ K\QKlK k l K K








s lJ l !lQ    ,¸I KlK #o dI KlK , RK ! K K Q  Q  HH  k

(5.50)

(5.51)

which correspond to the geometrical lengths to the surface of disk j if lJ l is approximated by ka K H K (see below). Finally, DI  is the determinant of the matrix FI  (,FI ) with the matrix K\ K\

38

A. Wirzba / Physics Reports 309 (1999) 1—116

elements FI GG

"

FI 

"

d 2d G\G!1! G\G d o GG> G 2dI KlK dI KlK Q !1! Q  o d   d 2d K\K\!1! K\K\ d o K\QKlK K\ d  G\G "F G>G d GG> dI KlK  Q  "FI  d 

FI " K\K\





FI GG>

"

FI 

"

FI GH

" 0 otherwise .

, a for 24i4m!2 , G , aJ ,  , aJ , K\

(5.52)

, b for 24i4m!2 , GG> , bI , 

Note that DI  and FI  have exactly the form of the determinant D and the matrix K\ K\ K\ F defined in Section 5.4 if the tilded lengths dI KlK and dI are replaced by the “normal” K\ Q  K\QKlK : geometrical lengths d K and d K\QKK Q K , (5.53) dI ++RK\ K!(s a K!lQ /k),,d K\QKlK H H K H K\ K\QKK dI KlK ++RK !(s a K!lQ /k),,d K . (5.54) Q  K H  Q K HH As discussed in Appendix F.4, this approximation is justified in the leading Airy approximation, where terms of order  of higher are anyhow neglected. To this order we can approximate lJ lK everywhere by ka K, except in the “creeping” exponential, since there the error would be of order H

\. Note that, in order to be consistent, we have to approximate lJ lK+ka K in the saddle-point H conditions for lQ and lQ as well. Thus, in this approximation, the saddles are manifestly real. K\  Hence only in the overall factors in the exponents we keep the O( \) term of




lJ lK"ka K#e L H

ka K  H qlK,ka K#dlJ lK . H 6

(5.55)

For all the other terms the errors from neglecting dlJ lK are, at least, of order O(+dlJ lK,/k)"O( ) or even of O(dlJ lK/k)"O( ). The expansion of the products kdI and kdI KlK in the exponents K\QKlK Q  leads to potentially dangerous linear terms of order dlJ lK. However, they cancel exactly against the terms in the expansion of the arccosines combined with those contributions which result if and lQ . lJ lK+ka K is inserted into the saddle-point relations for lQ H K\  "DI  " corresponds We will show below that in the leading Airy approximation, dI K\QKJK K\ exactly to the effective radius of the creeping periodic orbit R6 defined in Ref. [32]. The latter K K quantity is constructed, as in Eq. (F.41), in terms of the length segments l "¸ between the G\G G\G (i!1)th and ith point of reflection (if 24i4m!1), the length segments l "d !o and K K  l "d !o between the (creeping) impact parameter at disk j and the first or last K\K K\K K\ K point of reflection, respectively, and o ["the radius a G of the disk j times the cosine of the G G H

A. Wirzba / Physics Reports 309 (1999) 1—116

39

reflection angle]: K\ R6 "l “ (1#i l )"R6 (1#i l ), K K K G GG> K K\ K\ K\K G

(5.56)

where the curvature i is given by the recursion relation G 1 2 i" # G i\ #l o G\ G\G G

(5.57)

with 1/i ,0. The proof of the equivalence of dI "DI  " and R6 uses the following  K\QKJK K\ K K relations, which can be derived from Eqs. (5.56) and (5.57) by complete induction:






2R6 2R6 2R6 , R6 "R6 # 1# K # K #2# K H\ l K H\ H\H K H o o o   H\




(5.58)



2R6 2R6 2R6 1 1# K # K #2# K H\ , i " H\ R6 o o o   H\ K H\

(5.59)

where R6 ,0 and R6 "l . For the right hand side of the proof, the recursion relations (5.28) K  K K  are applied to the combinations "DI  "dI +D d ,RI 6 : H\ H\H H\ H\H K H






2RI 6 2RI 6 2RI 6 RI 6 "!RI 6 #(!1)H\ 1! K # K !2#(!1)H\ K H\ d K H H\H K H\ o o o   H\ (5.60) with RI 6 ,0. In the induction assumption one can use that, for 14j(m, the quantities K  RI 6 and R6 are related as K H K H R6 !R6 K H\o . RI 6 "R6 # K H K H H K H ¸ H\H

(5.61)

This follows from the difference between d and l . Note that RI 6 "d "¸ #o and H\H H\H K H K K  R6 "¸ satisfy trivially this induction ansatz. By complete induction it can now be shown that K  K d , can be rewritten as the recursion relation (5.60), applied to RI 6 ,D K\ K\K K K





2R6 2R6 2R6 #l 1# K #2# K G#2# K K\ RI 6 "R6 K K\ K\K K K o o o  G K\



#2l K\K

(!1)K\!1 (!1)K\RI 6 !R6 K #2 K  # o 2 



!R6 (!1)K\G\RI 6 !R6 RI K G#2# K6K\ K K\ , K G # o o G K\

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A. Wirzba / Physics Reports 309 (1999) 1—116

where the last bracket vanishes identically because of !R6 #2R6 RI 6 !R6 RI K H"(1!d ) K6H\ K H\ K H\#d K H H H o o H H\ for 14j(m. This equality can be derived from the induction ansatz Eq. (5.61), if Eq. (5.58) is inserted for the remaining R6 on the left hand side and for the remaining R6 on the right K H K H\ d "R6 is established in the leading Airy hand side. Thus, the equivalence RI 6 ,D K\ K\K K K K K approximation. We therefore get




e LClK a K   H e I*K6KQK +! > (ka K) R6 l K QK! H K K Jl  Q ;e I?HK>BJ K\N K\QKNKL>QK ?HK\  ?HK\QKJ K\I0HK\HK \  ?HK\QKJQI0HKH 

AHH2AHK\HK\AHK\HI KAHI K H"

1 , ; 1!e LI?HK>BJJ lK

(5.62)

where ¸ 6 ,¸ # K\¸ #¸ is the total length of the straight geometrical sections. K K K G\G K\K The impact parameter lQG /k is given by the solution of the saddle-point equation (5.15) where, in the G cases i"1 and i"m!1, the respective saddles lQ /k and lQ /k, have to be replaced by s a K. G\ G> K H In summary, all the quantities entering the semiclassical-creeping limit of the itinerary (5.62) with one creeping section have geometrical interpretations: 1. The integer index l51 enumerates the creeping modes around the boundary of disk j . With K increasing l, the impact parameter (or distance of the creeping path from the surface of the disks) and the “tunneling” suppression factor increases. 2. The index s "$1 distinguishes between creeping paths of positive sense or negative sense K around a surface section of disk j . K 3. The coefficient e LClK/(ka K) is proportional to the product of the two creeping diffraction H constants at the beginning and end of the creeping segment along the boundary of disk j which K parameterize the transition from a straight section to a creeping section and vice versa, see [29,30,32]. 4. The second prefactor is the inverse square root of the effective radius R6 , in units of disk K K radius a K. It is the geometrical amplitude, i.e., the geometrical stability factor. H 5. The lJ l independent terms in the exponents are just ik times the sum of all lengths of the straight #¸ . geometrical segments of the periodic itinerary, i.e. ¸ 6 (s )"¸ # K\¸ K G\G K\K K K K 6. The geometrical length along the creeping section times ik is given by theG sum of all exponential terms that are proportional to a K. H 7. The creeping “tunneling” suppression factor is given by the imaginary part of lJ lK or dlJ lK. 8. The denominator 1!e LJJ lK results from the summation of all further complete creeping turns around the disk j , in terms of a geometrical series [32]. Note that the apparent poles at K 1!e LJJ lK"0 cancel against the corresponding semiclassical poles of one-disk S-matrix, S(ka K). In fact, the zeros of 1!e LJJ lK are given by lJ lK"l(integer) and are nothing but the zeros H of the Hankel function H(ka K) in the Airy approximation. J H

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41

5.8. More than one creeping section The rth repeat of the itinerary (5.48) follows simply as the sum, l , over the rth power of the K QK summands on the r.h.s. of Eq. (5.49). As in the case of geometrical itineraries, this rule is trivial for the occurring prefactors, signs, phases and exponential terms. The non-trivial point is the behavior of the determinant DI  under the rth repeat. However, as the corresponding matrix K\ FI  has zero (1, m!1) and (m!1, 1) matrix elements, such that repeats cannot couple here, K\ the determinant of the rth repeat corresponds exactly to the rth power of the determinant of the primary itinerary. For the same reason, also the determinants and corresponding effective radii of itineraries, with more than one creeping contact (i.e., with at least two disks j and j with creeping G K contacts), decouple from each other. The corresponding semiclassical result for such an itinerary is thus the multiple sum, l l over the products of the corresponding itinerary from disk j to G QG K QK G disk j and the itinerary from disk j to disk j , each individually given by the suitably adjusted K K G summand on the r.h.s. of Eq. (5.49), e.g.: I

AHH2AHG\HI GAHI iHG>2AHK\HI KAHK H">   + l l G QG! K QK! e LClG a G  1 H ;(!1) e I*GKQGQK 1!e LI?HG>BJJ lG (ka G) R H GK Jl  ;e I?HG>BJ G\N G\QGNGL>QG ?HG\  ?HG\QGJQG\I0HG\HG \  ?HG\QGJQG>I0HGHG> 







1 e LClK a K  H e I*K6GQKQG 1!e LI?HK>BJJ lK (ka K) R6 H K G ;e I?HK>BJJ lK\NK\QKNKL>QK ?HK\  ?HK\QKJQK\I0HK\HK \  ?HK\QKJQI0HKH  , ;(!1)

etc. [If there are two (or more) creeping contacts next to each other, e.g., j "j , then, in the G K\ above formula, the corresponding impact parameters lQ /k and lQ /k have to be replaced by G> K\ a K and a K\, respectively.] H H The physical reason for the simple rule of piecing together creeping paths, is the point-like contact at e.g. disk j between the creeping sections on the one hand and the geometrical sections on G the other hand which is mediated by the diffraction constants ClH . [Mathematically, this corresponds to the fact that lJ lH is uniquely determined as the l th zero of the Airy integral and not by H a semiclassical saddle-point equation that would couple with the saddle-point equations at the disks j and j .] Because of this point-like contact [the independent determination of lJ l] the G\ G> semiclassical itineraries multiply for fixed value of the mode numbers l and creeping orientation s . H H Especially, if we limit the mode number to l"1, periodic orbits with common creeping sections can exactly be split up into their primary periodic orbits, see Ref. [32]. Finally, the ghost cancellation works for itineraries with creeping sections in the same way as for itineraries which, semiclassically, are purely geometrical. The reason is two-fold: First, by construction (see Appendix F.5), ghost segments can only occur in the geometrical part of the creeping itinerary. Second, the ghost cancellation rules of Section 5.6 are based on the local properties of the

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A. Wirzba / Physics Reports 309 (1999) 1—116

segments i!1Pi and iPi#1. Let us now assume, for simplicity, that the disk j is cut by the G ghost section. If there is no creeping at the neighboring disks j and j , the reduction of G\ G> the stability matrix FI  and of the phases and lengths of the segments is precisely the same as in the purely geometrical case (see the substitutions (5.40) and the analogous steps of Section 5.6). If there is a creeping contact at disk j or/and disk j , the substitutions (5.40) simplify, as dl or/and G\ G> G\ or/and FEG of Eq. (5.41) are zero and the ith row dl do not exist. Thus, the elements FEG G\G G>G> G> of the determinant (5.42) has only to be subtracted from the (i#1)th or the (i!1)th row or from none, in order that Eq. (5.42) becomes Eq. (5.43)). The reduction in the lengths and phases hold in these cases as before. In summary, the ghost cancellation works for geometrical orbits with creeping sections as well as for purely geometrical orbits, studied in Section 5.6. Semiclassically, neither the ghost itineraries nor their parent itineraries [which have the same symbol sequence except that the ghost labels are removed subsequently] contribute to the semiclassical trace-log expansion and to the cumulant/curvature expansion. Thus, one can omit these “ghost-affected” periodic orbits altogether from the curvature expansion. Deep inside the negative complex k-plane the limitations of the first Airy correction introduce rather big errors, see Ref. [67]. In this case it is advisable to use to the original expression (5.49) for semiclassical “creeping” itineraries with lJ lK and ClK, as given in Eqs. (F.28) and (F.33), instead of Eq. (5.62). To summarize, for the special case of n-disk repellers, the creeping periodic orbits of Ref. [32] have been recovered directly from quantum mechanics, whereas the construction of Ref. [32] has relied on Keller’s semiclassical theory of diffraction [30]. Furthermore, the symbol dynamics has to be generalized from the single-letter labelling +j , to the two-letter labelling +j , s ;l , with G G G G s "0,$1 and l "1, 2, 3,2. G G 5.9. Geometrical stabilities In this subsection we will return to purely geometrical periodic orbits and show that Eqs. (5.35) and (5.36) are correct, i.e. that the determinant D satisfies in fact K






1 # !2 D "(!1)K>det(M !1)"(!1)K K K K K K K

(5.63)

irrespective, whether there are repeats or not. Here M is the 2;2 dimensional real monodromy K matrix of the purely geometrical periodic orbit of total topological length m (m"pr if there are r repeats of a primary orbit of topological length p), that is, the semiclassical limit of the itinerary is unity. For this AHKH2AHK\HK. Because of phase-space conservation, the determinant of M K reason and as the matrix elements of M are real (see below), the two eigenvalues of the matrix K are related as K and 1/K . We do not have to treat repeated orbits explicitly here, as this case K K was already studied in Section 5.5. In Ref. [15] it was shown that, for any two-dimensional scalar billiard problem (whether a bound state problem or a scattering problem), the monodromy matrix M of a periodic orbit with K m collisions with the billiard walls is given by the 2;2 dimensional Jacobian belonging to the infinitesimal evolution of the vector (dp , dx )2 perpendicular to this classical trajectory in phase , ,

A. Wirzba / Physics Reports 309 (1999) 1—116

43

space, i.e., by the product K M "“ T R . K G\G G G Here the matrix T




(5.64)



1

0

" G\G ¸

(5.65)

1

G\G parametrizes the translational (straight ray) evolution of the vector (dp , dx )2 [or rather (dh , dx )2 , , N , with h being the angle of the momentum p, since the modulus of p is conserved anyhow] between N the (i!1)th and ith collision where ¸ is the corresponding length segment. As usual i"0 G\G should be identified with i"m. The matrix




R" G



!1 !2/o G 0 !1

(5.66)

parametrizes the evolution of the vector (dh , dx )2 from immediately before to immediately after N , the ith collision. The quantity o "a cos h is, in general, the product of the local radius of G G G curvature a and the cosine of the reflection angle h at the ith collision with the billiard walls. G G Especially for our n-disk scattering problems, a is of course nothing but a G, the radius of the disk j , G G H whereas h should be identified with the scattering angle h G of Eq. (5.20), the solution of the G H saddle-point equation. Since the determinants det T and det R are trivially unity, the determiG\G G is unity as well, as it should because of Liouville’s theorem. Furthermore, nant of the product M K the matrix elements of M have to be real, since the matrices det T and det R are real, by K G\G G definition. Thus the two eigenvalues of M have the structure K and 1/K . K K K 5.9.1. Monodromy matrix in closed form In the following we will construct a closed-form expression for the matrix elements of the matrix M , 14n, by complete induction. Let us denote these matrix elements as L A B L . (5.67) M ,(!1)L L L C D L L By inserting Eqs. (5.65) and (5.66) into Eq. (5.64), one can show that






2¸ 2 C "¸ , D "1#  B " ,     o o   and that the matrix elements of M and M are related as follows: L> L A "A #B ¸ , L> L L LL> 2¸ 2 B "B 1# LL> #A , L> L Lo o L> L> A "1, 






(5.68)

(5.69) (5.70)

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A. Wirzba / Physics Reports 309 (1999) 1—116

C "C #D ¸ , (5.71) L> L L LL> 2¸ 2 . (5.72) D "D 1# LL> #C L L> L o o L> L> In order to be able to perform the induction step, we do not make use of the cyclic permutation, i.e., in the following we do not replace ¸ with ¸ or ¸ , respectively, and ¸ or  L L> LL> ¸ with ¸ , but keep the original labelling. From Eqs. (5.68) and (5.72) it follows, by L>L> L complete induction, that






L\ 2R GL , (5.73) A "1# L o G G L 2R 2 L R !R 2 L GH " GL> GL , (5.74) B " 1# L o ¸ o o H LL> G G G HG> G C "R , (5.75) L L L 2R G (5.76) D "1# L o G G with R given as in Eq. (5.56) where we identify l with ¸ . In analogy to Eq. (5.58), we HL G\G H>G\H>G can derive the recursion relation











L\\H 2 ¸ , (5.77) R "R # 1# R HH>G o L\L HL HL\ H>G G of Section 5.7 that where R " ,0. Thus R should not be mixed up with the quantity R HL HYL LL L6L rather corresponds here to R with l ,¸ , of course. Note that the first iteration of Eq. (5.77) L   leads to R ,¸ . For later purposes we also define here the effective radius R which is, HH> HH> HL of course, equal to R and which satisfies the recursion relation HL L\\H 2 1# R , (5.78) R "R #¸ L\GL HL H>L HH> o G L\G where again R " ,0, such that R "¸ . HL HYL L\L L\L The second equation of Eq. (5.74) follows trivially from Eq. (5.77). By inserting the Ansa¨tze (5.75) and (5.76) into the induction step (5.71), one can easily show, with the help of the recursion relation (5.77) (for the case j"0), that the result for C is given by Eq. (5.75), with n replaced by n#1. L> Similarly, by inserting the Ansa¨tze (5.73) and the last identity of Eq. (5.74) into the induction step (5.69), one finds that A is compatible with Eq. (5.73). Here we used the identity R ,0. L> LL Applying the recursion relation (5.77) to R , it is easy to show that D is compatible with L> L> Eq. (5.76) as well. Finally, for proving that B is compatible with Eq. (5.74), one inserts the first L> equation of (5.74) and (5.73) into Eq. (5.70), uses Eq. (5.77) for re-expressing R and the fact GL> that ¸ "R . LL> LL> Having a closed form expression for the matrix elements of M we could now construct the K corresponding eigenvalues K! . But, in fact, we only need the linear combination K (!1)K(K #1/K ) which is equal to the sum A #D . K K K K






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In summary, we have now a closed form expression for the right hand sides of Eq. (5.63)






1 (!1)K>det(M !1)"(!1)K K # !2 K K K K K 2 K\ 2 "2(1#(!1)K>)# R # R , G o o GK G G G G where we used the identity R "R in writing down the last relation. GK GK

(5.79)

5.9.2. Stability determinant in closed form In analogy to the definitions of Section 5.7 [see Eq. (5.60)] we define here D d ,RM ,RM . These quantities satisfy, according to Eq. (5.26), the recursion J>I\ I\I JI JI relations






1 ![RM #RM ] d JI\ JI\ d I\I I\ I\I\ and, according to Eq. (5.27), the recursion relations RM #RM " RM JI\o JI JI\




2



2

1 RM ! [RM #RM ] . J>I d J>I J>I J> J>J> By complete induction, these recursion relations can be summed up to give RM #RM "d JI J>I JJ> o






(5.80)

(5.81)

I\J\ 2RM RM "!RM #(!1)I\J\ 1# (!1)G JJ>G d (5.82) JI JI\ I\I o J>G G I\J\ 2RM "RM "!RM #(!1)I\J\d 1# (!1)G I\GI (5.83) JI J>I JJ> o I\G G with RM " "RM " "0. JI JYI JI JYI According to Eq. (5.30) of Section 5.4, the stability determinant D can be rewritten in terms K of the RM ’s as follows: JI 2 K 2 K\ # (!1)G RM . (5.84) D " (!1)K\GRM K G o o GK G G G G Adding and subtracting Eq. (5.79) to Eq. (5.84) we get






K 2 K\ 2 D "2(1#(!1)K>)# R # R K G o o GK G G G G K 2 #2((!1)K!1) ((!1)K\GRM !R ) G G o G G K\ 2 # ((!1)GRM !R ) . (5.85) GK GK o G G The equality (5.63) is established, if we can show that the sum of the last two lines of Eq. (5.85) is identically zero.

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Note that the effective radius RI 6 , in the creeping case, fulfills the recursion relations (5.82) K H and (5.83) as well, see e.g., Eq. (5.60). However, as here d (,d )"o #¸ #o and  K    d "o #¸ #o , whereas in Section 5.7 d "¸ #o and d " K\K K\ K\K K    K\K o #¸ , the relation between the RM ’s and the R ’s have to be modified in comparison to K\ K\K GH GH the relation (5.61) between the RI 6 ’s and the R6 ’s. In fact, instead of Eq. (5.61), we get K H K H R !R j G\o , RM " 1#o R # G (5.86) G j¸ G G ¸ G\G  j R !R G>Ko , RM " 1#o R # GK (5.87) GK Kj¸ GK G ¸ K\K G\G where the differentiations with respect to ¸ and ¸ produce the additional o and o pieces  K\K  K in d and d , respectively, relative to Eq. (5.61). As in Section 5.7, these relations can be  K\K proven by complete induction. Now, by solely inserting Eqs. (5.86) and (5.87) into the second and third line of Eq. (5.85) and collecting terms, we get for this expression

 











K 2 K\ 2 ((!1)GRM !R ) 2((!1)K!1)# ((!1)K\GRM !R ) # GK GK G G o o G G G G j K\ 2R j 2R j K\ 2R G #o GK . K! 2#o "o 1# 1# j¸ j¸ Kj¸ o o o G G K   K\K G G (5.88)













With the help of the recursion relations (5.77) and (5.78) this expression can be rewritten as follows:






j j R !R K K\ #2 R (5.88)"o K j¸ ¸ j¸ K\K   R !R j K K ! 2#o Kj¸ ¸  K\K








"

j R R !R o j j K\!  ! K! K R #2 R # R K K K o ¸ o j¸ j¸ j¸ K K\K K   K\K R ! R R K# K! ! K ¸ o  

(5.89)

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Under o ,o , RM "RM and R "R , the expression simplifies [furthermore, note that  K K K K K R and R are independent of ¸ and ¸ , respectively, if i4m!1]: K\GK G  K\K R !R j R !R j K# K\ R ! K R ! K (5.89)" K K ¸ j¸ ¸ j¸  K\K K\K  j K\ 2R K\ 2R K\GK ! 1# K\GK " R #¸ 1# K  o o j¸ K\G K\G  G G j K\ 2R K\ 2R G ! 1# G # R #¸ 1# K\ K\K j¸ o o K\K G G G G "0 q.e.d. (5.90)





















The identity (5.63) is therefore established. In summary, we have proven that the geometrical semiclassical limit of a quantum itinerary for any non-overlapping n-disk system [see Eq. (5.21)] is exactly the corresponding periodic orbit with the Gutzwiller weight. Hence, the validity of Eq. (5.16) for any non-overlapping finite n-disk system (with the exclusion of the grazing geometries) is shown in the semiclassical limit. Note, however, that this is no general proof of the convergence of the curvature series, since two limits are involved: the semiclassical limit p/ "kPR (or P0) and the cumulant limit mPR. In general, these two limits do not commute. For purely chaotic classical n-disk systems with a positive value for the topological entropy, the exponential proliferating number of orbits and, therefore of classical input, is not compatible to the just algebraically rasing number of operations, needed to solve for the zeros of the quantum determinant of the multi-scattering kernel. In these cases, the curvature sum of the periodic orbits has to deviate from the cumulant sum involving the quantum itineraries. The semiclassical limit and the cumulant limit should better not commute. We will study this numerically in the next section.

6. Numerical tests of semiclassical curvature expansions against exact data In this section which overlaps partly with Ref. [47] we test the predictions of the curvature expanded Gutzwiller—Voros zeta function, the dynamical zeta function [36] and the quasiclassical zeta function of Refs. [42,43] against the exact quantum-mechanical data for the 3-disk-system in the A -representation.  As mentioned in the introduction, the 3-disk repeller [10,11,13—15,42] is one of the simplest, classically completely chaotic, scattering systems and provides a convenient numerical laboratory for computing exact quantum-mechanical spectra as well as for testing the semiclassical ideas. It consists of a free point particle which moves in the two-dimensional plane and which scatters off three identical hard disks of radius a centered at the corners of an equilateral triangle of side length R, see Fig. 8. The discrete C symmetry reduces the dynamics to motion in the fundamental  domain (which is a 1/6th slice of the full domain and which exactly contains one half of one disk), and the spectroscopy to irreducible subspaces A , A and E. All our calculations are performed for   the fully symmetric subspace A [13,44]. 

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Fig. 8. The three-disk system with center-to-center separation R"6a.

The genuine multiscattering data in the A subspace are computed from the determinant  det[1#A (k)] where the multiscattering kernel A (k), expressed in the angular momentum basis   relative to the half-disk in the fundamental regime, reads [13]






n J (ka) cos (5m!m) H (kR) A (k) "d(m)d(m) K K\KY  KKY 6 H(ka) KY n #(!1)KYcos (5m#m) H (kR) K>KY 6








(6.1)

with 04m, m(R and



d(m) " :

(2

for m'0 ,

1

for m"0 .

As A is trace-class for any n-disk geometry, the determinant exists and can numerically be calculated in a truncated Hilbert space. The Hilbert space is here the space of angular momentum eigenfunctions +"m2, on the surface of the half-disk in the fundamental domain which can be truncated by an upper angular momentum m . From the study, in Appendix C, of the asymptotic

 behaviour of A with respect to the angular momentum one can derive the following inequality KKY for the truncation point m :

 e (6.2) m 9 ka+1.5ka .

 2 This agrees, of course, with the numerical findings. The truncated matrix M (k) is then numerically  transformed to an upper triangular matrix and the determinant is calculated from the product of the diagonal elements. This procedure is faster than the computation of the determinant from the product of the eigenvalues of M (k) (see Eq. (A.10)). The numerical results for both ways agree, of  course, up to computer accuracy. The zeros of the determinant, det M (k), in the lower complex  wave-number plane determine the scattering resonances, whereas the phase of the determinant evaluated on the real k-axis gives the cluster phase shift. The cumulants can be constructed either from the Plemelj—Smithies recursion formula (A.14) or from the multinomials of the eigenvalues

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(A.11). The latter procedure is numerically more stable, especially deep inside the negative complex wave-number plane. This concludes the numerical setup for the exact calculation. As shown in Sections 4 and 5 the classical analog of the characteristic determinant (actually of det+1#zA(k), to be precise) is the semiclassical zeta function of Gutzwiller [5] and Voros [37] which, prior to a regularization, is given by Z (z; k) (see (2.1)). However, in the literature there exist %4 other competitors for a semiclassical zeta function, e.g., the dynamical zeta function f\(z; k) of  Ruelle [36] (see Eq. (2.2)) which is the j"0 part of the Gutzwiller—Voros zeta function as well as the quasiclassical zeta-function Z (z; k) of Ref. [43] (see Eq. (2.3)). As usual, for all three choices,  t (k)"e I*N\ JNL/"K " is the pth primary cycle, n its topological length, ¸ is its geometrical N N N N length, l its Maslov index together with the group theoretical weight of the studied C N  representation (in the present case the A -representation), and K its stability (the expanding  N eigenvalues of the stability matrix) — see Refs. [42,43] for further details. The variable z is a book-keeping device for keeping track of the topological order in the cycle- or curvature expansion [18,19] (see Eqs. (2.4) and (4.9)). In the following, the various curvature-expanded zeta functions are truncated at a given curvature (i.e., total topological) order n . The semiclassical A predictions for the scattering resonances are determined from the zeros of these truncated zeta functions, the predictions for the cluster phase shifts discussed in Section 6.2 from the phases on the real k-axis and the curvatures from the terms of order zK in the curvature expansion. Input data for the lengths ¸ , stabilities K and Maslov indices l of the periodic orbits of the 3-disk system in the N N N A -representation have been taken from Rosenqvist [68,69], Scherer [17] and Eckhardt [70].  6.1. Exact versus semiclassical resonances In this chapter we compare the numerically computed exact quantum-mechanical resonances of the 3-disk repeller with the corresponding semiclassical predictions of the three semiclassical zeta functions: the Gutzwiller—Voros zeta function (2.1), the dynamical zeta function (2.2) and the quasiclassical zeta function (2.3). For the 3-disk-repeller with center-to-center separation R"6a, we have computed all exact quantum-mechanical A resonances (numerically determined from the zeros of det M (k))   as well as all the corresponding approximate ones (from the zeros of the at finite curvature order n truncated zeta functions) in the wave-number window: 04Re k4250/a and A 05Im k5!1.6/a. This window contains several hundreds of leading and subleading resonances, from the lowest ones onwards. In Figs. 19—27, for increasing curvature order, the resonances are plotted as the real part of the wave number (resonance “energy”) versus the imaginary part of the wave number (resonance “width”). Some features of the resonance spectra allow for an immediate interpretation [13,15,17]: The mean spacing of the resonances is approximately 2n/¸M , where ¸M is the average of the geometrical lengths of the shortest periodic orbits, namely the lengths ¸ and ¸ of the two periodic orbits of   topologic length one. The data also exhibit various beating patterns resulting from the interference of the periodic orbits of nearly equal length; e.g., the leading beating pattern is of order 2n/*¸, where *¸ is the difference of the lengths ¸ and ¸ .   In Figs. 19—22 a comparison is made from the first to the fourth order in the curvature expansion. Already at fourth order the four leading resonance bands are well approximated by the Gutzwiller—Voros zeta-function (in fact, for Re k:75/a already the second curvature order is

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enough to describe the first two leading resonance bands). This is in agreement with the rule of thumb that any new resonance band is linked with a new curvature or cumulant order. Neither the dynamical zeta-function nor the quasiclassical one perform as well to fourth order. The reason is that the quasiclassical as well as the dynamical zeta-function predict extra resonances which are absent in the exact quantum-mechanical calculation. Thus the third and fourth curvature order of these zeta-functions are distributed over the average of the third and fourth resonance bands and the spurious extra resonances. In the window plotted one can classify the exact data into four leading resonance bands closest to the real wave-number axis and two subleading ones shielded by the leading resonances. Thus, just periodic orbits of topological length up to four are needed in order to reproduce the qualitative trend of the exact data closest to the real axis. The 3-disk-system has 8 periodic orbits up to this topological length. Actually, the 3-disk-system with center-to-center separation R"6a is not very chaotic at these k values. All experimentally accessible spectral data in this regime (which can be extended up to Re k+950/a as only about there the subleading resonance bands mix with the four leading ones) can be parameterized by 16 real numbers, i.e., 8 periodic orbit lengths, 8 stabilities, and 8 Maslov indices. Experimentalists can stop here. The subleading bands are completely shielded (up to Re k+950/a) by the above-mentioned four bands. The subleading bands (below Re k+950/a) are only of theoretical interest, as they can be used to test the semiclassical zeta functions. In Fig. 23 a comparison is made up to fifth curvature order. The Gutzwiller—Voros zeta-function does at least as well as in Fig. 22a for the leading four resonance bands, but now it also describes the peak position of the fifth resonance band for large enough values of Re k. Note the diffractive band of exact resonances from k+(0.0—i0.5)/a to k+(100.0—i1.6)/a which our semiclassical zeta functions fail to describe. As shown in Refs. [32—34] the diffractive band of resonances can be accounted for by inclusion of creeping periodic orbits which have been omitted from our semiclassical calculations. The dynamical and quasiclassical zeta functions show a slight improvement with respect to the four leading resonance bands; however, no agreement with the fifth one. In Fig. 24 a comparison is made up to sixth curvature order. The Gutzwiller—Voros zetafunction fails for the third resonance band below Re k+20/a, for the fourth below Re k+45/a, for the fifth and sixth below Re k+70/a and 80/a, respectively. Below these values, the last two curvature orders try to build up an accumulation line. Above these values, the qualitative agreement with the data is rather good. The dynamical zeta-functions at this order just improves the description of the four leading resonance bands. Furthermore, it builds up a sharp line of accumulation for the subleading resonances, the border of convergence controlled by the location of the nearest poles of the dynamical zeta function, see [41,42]. The quasiclassical zeta function also improves the description of the four resonance bands, although it is still not of the same quality as the Gutzwiller—Voros one even at two curvature orders lower. Note that the quasiclassical zeta function is trying to build up two bands of spurious resonances in agreement with our rule of thumb. In Fig. 25 a comparison is made up to seventh curvature order. The first four resonance bands of the Gutzeiller—Voros zeta function have converged and the accumulation line has moved up. Only above Re k+140/a the fifth and sixth resonance band emerge, now with improved accuracy, however. Also the seventh resonance band is approximated. The dynamical zeta-function now clearly produces its line of convergence (the accumulation line of resonances). Above this line, the resonances (except the ones very close to the accumulation) are approximated as well as in the

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Gutzwiller—Voros case; below, no agreement is found. At this order the quasiclassical zeta-function is doing as well as the Gutzwiller—Voros zeta-function did already at curvature order four. None of the subleading bands are described by the quasiclassical zeta-function. Instead another band of spurious resonances emerges. In Figs. 26 and 27 the comparison is made up to the eighth and twelfth curvature order, respectively. The border of convergence of the Gutzwiller—Voros zeta-function has now moved (in the plotted region) above the fifth and sixth band of the exact resonances. It has moved also closer to the very sharp accumulation line of resonances of the dynamical zeta function. However, these lines are still not identical even at twelfth curvature order. The subleading quasiclassical resonances have stabilized onto the spurious bands. Furthermore, some subleading resonances move further down into the lower complex k-plane. Eventually (see also [68]), starting with curvature order 10 and 12 the fifth and sixth resonance bands are approximated — in addition, to four or six spurious resonance bands, respectively. Thus the quasiclassical zeta-function seems to find the subleading resonance bands, but at the cost of many extra spurious resonances. Note that at these high curvature orders the quasiclassical zeta-function has numerical convergence problems for large negative imaginary k values (especially for low values of Re k). This is in agreement with the expected large cancellations in the curvature expansion at these high curvature orders. Furthermore, periodic orbits of larger topological order than twelve would be needed to falsify the success of the quasiclassical zeta function, since it barely manages to approximate the two bands of subleading resonances at this curvature order. Qualitatively, the results can be summarized as follows. The Gutzwiller—Voros zeta-function does well above its line of convergence, defined by the dynamical zeta-function, already at very low curvature orders where the dynamical zeta-functions still has problems. Below this line we observe that the Gutzwiller—Voros zeta-function works only as an asymptotic expansion. However, when it works, it works very well and very efficiently. This implies that the additional (K -dependent) terms N of the Gutzwiller—Voros zeta function, relative to the simpler dynamical zeta function, are the correct ones. This is of course in agreement with the findings of our semiclassical reduction in Section 5. Eventually, the dynamical zeta-function does as well for the leading resonances as the Gutzwiller—Voros one. As experimentally these are the only resonances accessible, one might — for practical purposes — limit the calculation just to this zeta function, see, however, Section 6.2. The quasiclassical zeta-function seems to find all subleading geometrical resonances. Unfortunately, the highest periodic orbits at our disposal are of topological length 12; the very length where the sixth resonance band seems to emerge. Thus higher orbits would be needed to confirm this behavior. But all this comes at a very high price: The rate of convergence is slowed down tremendously (in comparison with the asymptotically working Gutzwiller—Voros zeta-function), as this zeta-function is producing additional spurious resonance bands which do not have quantum-mechanical counter parts, but only classical ones [47]. Without a quantum calculation, one could therefore not tell the spurious from the real resonances. As a by-product we have a confirmation of our empirical rule of thumb that “each new cumulant or curvature order is connected with a new line of subleading resonances”. This rule therefore  Note that the quasiclassical results of this figure are directly comparable with the results of the so-called QuantumFredholm determinant of Ref. [42] (see Fig. 4b in Ref. [42]) as both calculations involve periodic orbits of topological length up to eight. As we now know, all the subleading resonances of that figure have nothing to do with quantum mechanics.

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relates the curvature truncation limit, mPR, either to the limit Im kP!R, if there is no accumulation of subleading resonances, i.e., if the zeta function is entire [42,43], or to the formation of an accumulation band of resonances. Both facts support our claim that, in general, the curvature limit mPR and the semiclassical limit Re kPR cannot and should not commute deep inside the lower complex k-plane, as the subleading resonances of increasing cumulant order are approximated worse and worse. Only an asymptotic expansion should be possible, in agreement with our findings for the Gutzwiller—Voros zeta function. 6.2. Exact versus semiclassical cluster phase shifts In the last chapter the semiclassical zeta functions were judged by the comparison of their resonances predictions with the exact resonances poles (especially the subleading ones), as was done in the past, see e.g. Refs. [13,15,32,34,42,44,45]. Since the deviations between the zeta functions themselves and from the exact data are most pronounced for the subleading resonances (which are shielded by the leading ones), one could argue that empirically it does not matter which of the three zeta functions are used to describe the measured data, since all three give the same predictions for the leading resonances [42,43]. Below, however, we will show that even experimentally one can tell the three semiclassical zeta functions apart and that, in fact, the Gutzwiller—Voros one is by far the best. 6.2.1. Cluster phase shifts In Section 4 the exact and semiclassical expressions for the determinant of S-matrix for nonoverlapping n-disk systems have been constructed. For the case of the three-disk system they read det M (k*)R det M (k*)R (det M (k*)R) J  J # det S(k)"(det S(ka)) J  J J det M (k) det M (k) (det M (k)) J  J  J # * * * * ** ** **   (e\ L,I)" ZI U  (k ) ZI U (k )  ZI (k ) ZI (k ) ZI #(k )  , (6.3) P ZI (k) ZI (k) ZI (k) ZI (k) ZI (k) U   U    # where the tilde indicates that diffractive corrections have to be included, in general. Especially for the A -representation of the three-disk system we therefore have the relation between the quan tum-mechanical kernels and the Gutzwiller—Voros zeta functions






det M (k*)R   Z (k*)* J  , (6.4) P  det M (k) Z (k) J   where we have now neglected diffractive corrections. As argued in the conclusion Section 7 both sides of Eqs. (6.3) and (6.4) respect unitarity; the quantum-mechanical side exactly and for the semiclassically side under the condition that the curvature expansion converges or that it is truncated. As all the n-disk resonances for non-overlapping n-disk repellers are below the real k-axis, the border of absolute convergence, defined by the closest resonances to the real axis [15,42] is inside the lower complex wave-number plane and unitarity on the real axis is guaranteed. Thus, if the wave number k is real, the left-hand sides and also the right-hand sides of Eqs. (6.3) and (6.4) can be written as exp+i2g(k), with a real phase shift g(k). In fact, we can define a total phase shift for the

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coherent part of the three-disk scattering problem (Fig. 9) (here always understood in the A  representation) for exact quantum mechanics as well as for the three semiclassical candidates by det M(k*)R J : , (6.5) e E I " det M(k) J Z (k*)* %4 e E%4I " : , (6.6) Z (k) %4 f\(k*)*  e EI " : , (6.7) f\(k)  Z (k*)*  e E I " : . (6.8) Z (k)  This phase shift definition should be compared with the cluster phase shift given in Section 4 of Lloyd and Smith [53]. For a separable system, as e.g. the one-disk system (in the angular momentum representation), the cluster phase shift just corresponds to the sum of the partial phase shifts  g(k)" g (k) , (6.9) J J\ as the S-matrix of the one-disk system (evaluated with respect to the center of the disk) reads !H(ka) J S (k)" d "e EJId , JJY JJY H(ka) JJY J such that

(6.10)

> det S(k)" “ e EJI . (6.11) J\ Let us once more stress: the coherent or cluster phase shift is an experimentally accessible quantity: from the measured differential cross sections the elastic scattering amplitudes have to be constructed. This leads to the full phase shift of the three-disk system including the contribution from the single disks. However, the incoherent part can be subtracted by either making reference experiments with just single disks at the same position where they used to be in the three-disk problem or by numerical subtractions as the one-disk phase shifts are known analytically, since the system is separable, see Eqs. (6.10) and (6.11). In this way one can separate the incoherent phase shifts from the coherent ones. Thus g (k) is “measurable” in principle. We next use these cluster phase shifts in order to  discriminate between the various zeta functions. Below, we compare the exact quantum-mechanical cluster phase shift g with  1. the semiclassical cluster phase shift g (k) of the Gutzwiller—Voros zeta function (2.1), %4 2. with the semiclassical cluster phase shift g (k) of the dynamical zeta function (2.2),  3. and with the semiclassical cluster phase shift g (k) of the quasiclassical zeta function (2.3). 

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Fig. 9. (a) The coherent cluster phase shifts of the three-disk scattering system in the A -representation with center-to-center  separation R"6a. The exact quantum-mechanical data are compared to the predictions of the Gutzwiller—Voros zeta function (2.1), the dynamical zeta function (2.2) and the quasiclassical zeta function (2.3) calculated up to 12th order in the curvature expansion. (b) The same for the squared moduli of the exact spectral determinant and the semiclassical zeta functions. The predictions of the Gutzwiller—Voros zeta function and exact quantum mechanics coincide within the resolutions of the plots.

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The zeta functions in the numerator as well as in the denominator of Z(k)*/Z(k) have been expanded to curvature order ("topological length) 12. For the Gutzwiller—Voros zeta function this is an overkill as already curvature order 4 should describe the data below Re k"950/a. In fact, we have not seen any difference in the Gutzwiller—Voros calculation between the curvature order 3 and 12 results for k4120/a and up to figure accuracy. Curvature order 2, however, gives in the regime 100/a4k4120/a noticeable deviations. On the other hand, as mentioned in Section 6.1, the quasiclassical zeta function has problems for lower curvature orders with predicting the (sub-)leading resonances; therefore, these high curvature orders are used in order to give the quasiclassical zeta function as fair a chance as possible. The coherent phase shifts are compared in the window 104/a4k4109/a, which is a typical window narrow enough to resolve the rapid oscillations with k sufficiently large such that diffractive effects can be safely neglected. Furthermore, although we do not have a physical interpretation in terms of the S-matrix, we also compare in the same window the exact quantum-mechanical product det M(k) det M(k*)R with the squared modulus of the Gutzwiller—Voros zeta function Z (k)Z (k*)*, the dynamical zeta function %4 %4 f\(k)f\(k*)*, and the quasiclassical zeta function Z (k)Z (k*)*. Here k is taken to be real and     the case of the three-disk system in the A -representation with center-to-center separation R"6a  is studied. Consider finally the general quasiclassical zeta functions of Ref. [43] and especially the ratio F (, k)F (, k) \ > (6.12) Z(k) " : F (, k)F (, k) > \ with F (b, k; z) and F (b, k; z) being defined as follows: > \  1 (zLNt (k))P N F (b, k; z)"exp ! "KP "\@> , > r (1!KPN)(1!KP  N) N N P (zLN t (k))P  1 KP  N N F (b, k; z)"exp ! "KP "\@> . \ N) N r "KP " (1!KPN)(1!KP N N P Here the subleading factor (1#"KP "@\) of (11) in Ref. [43] has been removed as in (12) of Ref. N [43]. When Eq. (6.12) is used, the corresponding coherent phase shift

 

Z(k*)* e E I" Z(k)





(6.13)

works on the real wave-number axis and in the limit nPR (where n is the curvature order) as well as the original Gutzwiller—Voros zeta function. Hence, it does not matter here whether the Gutzwiller—Voros zeta function is directly expanded in the curvature expansion or whether the individual determinants F (, k), F (, k), F (, k) and F (, k) are each expanded in separate \ \ > > curvature expansions up to the same curvature order and then inserted in the ratio (6.12). Note that the presence or absence of the subleading factor (1#"KP "@\) in the definitions of F (b, k; z) and N > F (b, k; z) does not change the results up to figure accuracy. \ Let us stress that phase shifts are not only of theoretical interest, as are the subleading resonances (which are completely shielded by the leading resonances), but hard data which can be extracted, in principle, from measured differential cross sections.

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In summary, even empirically, one can tell the three semiclassical zeta functions apart and see which is the best. Again the Gutzwiller—Voros one — whether used directly or whether defined as the ratio (6.12) of four quasiclassical determinants as in Ref. [43] — is by far the best. 6.3. The quantum-mechanical cumulant expansion versus the semiclassical curvature expansion In this subsection it will be shown that the Gutzwiller—Voros zeta function approximates its quantum-mechanical counterpart, the characteristic KKR-type determinant [49,51,53], only in an asymptotic sense, such that it always should be understood as a truncated series. As shown in Section 4, the characteristic determinant and the Gutzwiller—Voros zeta-function are related as   Z (k) . det M(k)P %4

(6.14)

Let Q (k) denote the mth cumulant of det M(k) — i.e. the term proportional to zK in the Taylor K expansion of det+1#zA(k), — which satisfies the Plemelj—Smithies recursion relation (4.6) (see also Appendix A). Since the Plemelj—Smithies recursion formula is plagued by cancellations of very large numbers, we have not used the Plemelj—Smithies recursion relations for our numerical calculation of Q (k), but instead we construct this quantity directly from the eigenvalues +j (k), of K H the trace-class matrix A(k), i.e. (6.15) j (k)2j K(k) Q (k)" H H K XH2HK (see again Appendix A for more details). Unfortunately, a semiclassical analog to this exact formula has not been found so far. Thus C (k), the corresponding semiclassical mth-order curvature term, of K Z (k), can only be constructed from the semiclassical equivalent of the Plemelj—Smithies recursion %4 relation (4.8) which exactly corresponds to the standard curvature expansion of Refs. [19,42,15] and is therefore inherently plagued by large cancellations. The cumulant and curvature expansions, truncated at nth order, read: L det M(k)" " Q (k) , (6.16) L K K L Z (k)" " C (k) . (6.17) %4 L K K Let us recapitulate what we already know about these series. From Section 4 together with the Appendix A and Appendix C we deduce that the cumulant sum L lim det M(k)" " lim Q (k)"det M(k) K L L L K is absolutely convergent, i.e.  "Q (k)"(R , K K

(6.18)

(6.19)

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because of the trace-class property of A(k),M(k)!1 for non-overlapping, disconnected n-disk systems. On the other hand, as discussed in Refs. [41—43], the Gutzwiller—Voros curvature sum converges only above an accumulation line (running below and approximately parallel to the real wave-number axis, see Section 6.1) which is given by the first poles of the dynamical zeta function, f\(k), or the leading zeros of the subleading zeta function. However, as shown in Section 6.1, even  below this boundary of convergence the truncated Gutzwiller—Voros curvature sum, Z (k)" %4 L approximates the quantum-mechanical data as an asymptotic series. In addition, a very important property for the discussion of the cumulant and curvature terms is the existence of the scaling formulas (established by us numerically) which relate the mth cumulants or curvatures inside the complex wave-number plane to the corresponding quantities on the real k-axis: (6.20) Q (Re k#i Im k)&Q (Re k)e\K*M ' I , K K (6.21) C (Re k#i Im k)&C (Re k)e\K*M ' I . K K (For this to hold, diffractive effects have to be negligible, i.e. !Im k;Re k.) Here ¸M +R!2a is the average of the geometrical lengths of the shortest periodic orbits, the two orbits of topological length one. The scaling can be motivated by the approximate relation Tr[AK(k)]++Tr A(k),K which, of course, cannot be exact, as otherwise the cumulants would be identically zero. Nevertheless, the overall behaviour follows from this, since Tr[A(Re k#i Im k)]&Tr[A(Re k)]e\*M ' I . From Fig. 10 one can deduce that the deviations between quantum-mechanical cumulants and semiclassical curvatures (as evaluated on the real k-axis) decrease with increasing Re k, but increase with increasing curvature order m. The value of Re k where the quantum-mechanical and semiclassical curves join is approximately given by Re ka&2K>. Approximately the same transition points can be generated from a comparison of the phases of the cumulant and curvatures. By varying the center-to-center distance we have numerically verified that the above limits generalize to the following relations valid on the real wave-number axis (k real and positive): C (k)+Q (k) with 1<"C (k)"+"Q (k)" if ka92K\ ¸M /a , (6.22) K K K K 1<"C (k)"<"Q (k)" if ka:2K\ ¸M /a . (6.23) K K What is the interpretation of Eqs. (6.22) and (6.23)? For fixed k, even in the regime, where Z (k)" %4 L converges, e.g., on the real k-axis, the Gutzwiller—Voros zeta function is only an asymptotic approximation to the true quantum-mechanical cumulant sum, since for m'm , defined by   ka+2K \¸M /a, the exact quantum-mechanical cumulants Q (k) and the semiclassical curvatures K C (k) are grossly different. These deviations can be enhanced by the mth derivative, m'm , with K   respect to the book-keeping variable z, since this operation eliminates all approximately equal terms, such that the corresponding cumulant and curvature series are transformed to completely different expressions. The fact that Z (k)" — even in its convergence regime — is only an asymptotic %4 L expansion to the exact quantum mechanics is normally not visible, as the terms in Eq. (6.22) are exponentially small on or close to the real axis and therefore sum to a tiny quantity. In other words, close to the real axis the absolute error "C (k)!Q (k)" for m'm is still small. The relative error L L  

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Fig. 10. Comparison of the squared moduli (on a logarithmic scale) of the first seven quantum-mechanical cumulant terms, "Q (k)", with the corresponding semiclassical curvature terms, "C (k)", of the Gutzwiller—Voros zeta function (2.1) L L evaluated on the real wave-number axis k. The system is the A three-disk repeller with center-to-center separation  R"6a.

"C (k)/Q (k)" on the other hand is tremendous (see Fig. 10). Deeper inside the negative complex L L wave-number plane, however, under the scaling rules (6.20) and (6.21), the deviations (6.23) are blown up, such that the relative errors "C (k)/Q (k)" eventually become visible as absolute errors L L "C (k)!Q (k)" in the resonance calculation of Section 6.1. If Im k is above the boundary line of L L convergence, these errors still sum up to a finite quantity which might, however, not be negligible any longer, as was the case on or close to the real k-axis. Below the convergence line these errors sum up to infinity. Thus the Gutzwiller—Voros curvature expansion Z (k)" does not suddenly %4 L become an asymptotic approximation to det M(k)" , it always is an asymptotic approximation (as L shown by the relative error "C (k)/Q (k)"), even in its convergence regime above the accumulation line L L and even on the real axis, where the zeta function itself is in its domain of absolute convergence [17]. Thus, the value of Im k where — for a given m — the Z (k)" sum deviates from det M(k)" is %4 L L governed by the real part of k and the scaling rules (6.20) and (6.21). It has nothing to do with the boundary line between the convergence region and the asymptotic region of Z (k), as the %4 asymptotic expansion is given by a finite sum of all terms satisfying (6.22). Therefore, the truncated Gutzwiller—Voros expansion describes the quantum-mechanical resonance data even below the line of convergence of the infinite curvature series, see Section 6.1. On the other hand, the boundary line

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of the convergence regime of the Gutzwiller—Voros expansion is solely governed by those C (k) K which have nothing to do with the quantum analog Q (k), i.e. solely by terms of character (6.23). K The reason is, of course, that the convergence property of an infinite sum is governed by the infinite tail and not by the first few terms. Whether the Gutzwiller—Voros expansion converges or not is therefore not related to whether the quantum-mechanical data are described well or not. The convergence property of a semiclassical zeta function on the one hand and the approximate description of quantum mechanics by these zeta functions are two different things. It could happen that a zeta function is convergent, but not a good description of quantum mechanics (see especially the failure of the entire quasiclassical zeta function to approximate the exact cluster phase shift in Section 6.2). On the other hand it may not converge, in general, but its finite truncations nevertheless approximate — at least to some order — quantum mechanics, as it is the case for the Gutzwiller—Voros zeta function. These findings hold for any re-writing of the Gutzwiller—Voros zeta function, as Z (k)" was %4 L already shown to be asymptotic in a regime where the curvature sum is still absolutely convergent and the limit lim Z (k)" exists. Therefore, any re-writing of Z (k), especially the one of Ref. L %4 L %4 [43] as the ratio of four quasiclassical zeta functions (6.12) will at best work as an asymptotic expansion to the exact quantum-mechanical cumulant expansion. Note that, for finite curvature order n





F (, k)F (, k) F (, k)" F (, k)" \ >  L \  LO >  "Z (k)" . %4 L F (, k)" F (, k)" F (, k)F (, k) > L \ L > L \

(6.24)

If the ratio is evaluated according to the r.h.s. of Eq. (6.24), one obtains exactly the same result as for the original Gutzwiller—Voros expansion using formula (2.1). If, however, the ratio is evaluated according to l.h.s. of Eq. (6.24), the relation to the quantum-mechanical cumulant expansion is lost: the matching of the semiclassical coefficients of zK with the quantum-mechanical ones is spoiled, as the asymptotic terms, resulting from various curvature orders of the Z (k)" calculation, mix. If n is %4 L large enough, also the l.h.s. of Eq. (6.24) will deviate strongly from the quantum mechanics as the original formulation of the Gutzwiller—Voros expansion does — the difference is that this new expression approximates quantum mechanics at slower rate than the original formula, as the asymptotic terms of higher and lower curvature order are mixed. However, at high enough curvature order n also the new l.h.s. of Eq. (6.24) will encounter terms of class (6.23) and will therefore — for large negative imaginary wave numbers — deviate strongly from the quantum— mechanical resonance data. What is the reason for the truncation at ka+2K \¸M /a? This boundary follows from a combination of the uncertainty principle with ray optics and the exponentially increasing number of periodic orbits of the three-disk repeller. For fixed wave number k, quantum mechanics can only resolve the classical repelling set of the periodic orbits up to a critical topological order m . The   quantum wave-packet which explores the repelling set, has to disentangle 2L different sections of size d&a/2L on the “visible” part of the disk surface between two successive collisions with the disk. Since these collisions are spatially separated by the mean length ¸M , the flux spreads by a factor ¸M /a. In other words, the non-vanishing value of the topological entropy for the three-disk system, h&ln 2, is the reason. For comparison, the uncertainty bound on the wave number in the hyperbolic, but non-chaotic two-disk system is independent of the curvature order (in case

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Fig. 11. The moduli of the eigenvalues of the multiscattering kernel A (k) of the A three-disk repeller with R"6a for KKY  the cases k"100/a and k"(100!1.25i)/a on a logarithmic scale. The eigenvalues are displayed as function of their index j in descending order.

diffractive creeping is negligible), as there is only one geometrical periodic orbit and therefore the repelling set is trivial with zero topological entropy. The result that the semiclassical curvature expansion has to be truncated at finite order for a fixed wave number k, is different from the fact that the (in principle infinite) multiscattering kernel A "M !d can be truncated to a finite matrix. The truncation in the curvature order is KKY KKY KKY related to the resolution of the repelling set of periodic orbits of the three-disk system. The truncation in the size of the matrix is related to the semiclassical resolution of the single disks of the three-disk system. The point particle classically only scatters from the disk, if its impact parameter is of the size or smaller than the disk radius a. Note that in the fundamental domain of the A disk  system, one considers only one half-disk. Mathematically, this follows from the asymptotic behaviour of the ratio J (ka)/H(ka) which governs the scaling of the kernel A and which is K K KKY valid for m larger than m , defined in Eq. (6.2), see Appendix C. In order to visualize this, we have

 plotted in Fig. 11 the moduli of the eigenvalues (on a logarithmic scale and in descending order) of the multiscattering kernel A (k) of the A three-disk repeller with R"6a for the cases k"100/a KKY  (on the real wave-number axis) and k"(100!1.25i)/a as function of the eigenvalue index j. The imaginary part of the latter wave number is characteristic for a domain where the subleading resonance bands emerge. The one-disk resolution is clearly visible in the exponentially decreasing tails of both curves above Re k+140/a. In order to exhibit this feature, the matrix itself was

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truncated here at a large value of m"220. Furthermore, from the curves, one can read off that only the first few eigenvalues (six for the upper curve corresponding to the case k"(100!125i)/a and two to four eigenvalues for the lower curves for the case k"100/a) are “essential”, i.e., are of the order unity or bigger. These numbers match very well the minimal topological order needed in the semiclassical calculation to approximate the relevant resonances at the specified k values. Whereas inside the negative complex k-plane one has to go to higher curvature orders in the truncation of the semiclassical zeta function in order to find all the subleading resonances (namely to order six for the specified k value), on or close to the real axis only the leading resonances are “visible”, in agreement with the data for the cluster phase shifts which, for the specified k value, can be well approximate by a semiclassical calculation of order three to four. One can also extract from the figure what happens if the negative imaginary part of k is increased: the curve is basically parallelly shifted upwards. Thus the number of eigenvalues and the minimal curvature order for the semiclassical description of quantum mechanics increases the deeper one “dives” into the lower complex wave-number plane. Although only a few eigenvalues are essential for the computation of the resonances and phase shifts, the size of the matrix is determined by the much bigger number m +(e/2)ka, such that many more eigenvalues are produced by a matrix diagonalization code.

 Unfortunately, one cannot escape this mismatch, as the model space for the matrix M has to be that large in order to guarantee stable numerical results for the leading eigenvalues. In summary, the minimal size of the matrix is determined by the resolution of the single disks, whereas the maximal topological order up to which the semiclassical curvature expansion makes sense, follows from the Heisenberg uncertainty limit on the quantum resolution of the repelling set. The topological exponential rise of the number of periodic orbits, with increasing curvature expansion order n, is the physical reason for the eventual breakdown of the curvature expansion of the semiclassical zeta function (2.1) as compared with the exact quantum-mechanical cumulant expansion which defines the determinant of the multiscattering matrix in an infinite-dimensional Hilbert space.

7. Conclusions Starting from the exact quantum-mechanical S-matrix we have tried to find a direct derivation of the semiclassical spectral function for a rather special class of classically hyperbolic scattering systems, namely the non-overlapping disconnected finite n-disk repellers in two dimensions. We have confined our investigation to these systems as they are on the one hand “realistic” enough to capture the essence of classically hyperbolic scattering problems (or, for certain geometries, even chaotic scattering problems) and on the other hand simple enough to allow for a “top-down” approach from exact quantum mechanics to semiclassics without, and this is the important point, any “formal” step in between. We have reason for this “pedantry”: It is known from the work of Refs. [41—43] that the standard spectral function, the cycle-expanded semiclassical zeta function of Gutzwiller and Voros, is not entire for the three-disk system and therefore fails to describe subleading scattering resonances in the complex wavenumber plane below its boundary of convergence. The question is whether these failures are induced by unjustified formal steps in the semiclassical reduction or whether they are inherently a property of the semiclassics itself. Since we expected that the semiclassical spectral function must follow from the semiclassical reduction of

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the cumulant expansion of the corresponding multiscattering kernel [44,45], we had to avoid any bias or unjustified assumption on the exact kernel as well as on the semiclassical spectral function, e.g., on the existence and regularization of the quantum mechanical expression, on the structure of the period orbits, especially on the structure of their stabilization, on the organization of the spectral function in terms of cycles or curvatures, on the classification of these curvatures by the topological lengths of the orbits, etc. Our first task therefore has been to ensure that the quantum mechanical starting point for the semiclassical reduction is well-defined. The T-matrix of the n-disk scattering systems, derived by the methods of stationary scattering theory [13], was shown to exist on the real k-axis and to be trace-class. Therefore the actual starting point for the spectral classification of the scattering system, the determinant of the S-matrix, exists also and can be manipulated by cyclic permutations, unitary transformations, splitting into sub-determinants, in other words, operations which are non-trivial for matrices of infinite-dimensional Hilbert spaces. With the help of these (then justified) operations, we succeeded in transforming the determinant of the S-matrix into a form that is well suited for the semiclassical reduction step, see Eq. (3.9). It separates into the incoherent superposition of n one-disk scattering determinants and into the ratio of the determinant and the complex conjugated determinant of the genuine multiscattering matrix, M. Furthermore, the determinant of the multiscattering matrix can be decomposed into sub-determinants, if the n-disk system has additional symmetries. All of the above mentioned determinants are shown to exist separately. This is one of the key points for the semiclassical reduction, since the existence of the S-matrix alone would not guarantee that the one-disk aspects can be separated from the multi-disk aspects in a well-defined manner. Note that the standard geometrical periodic orbits (without creeping) can only “know” about the multi-disk aspects, and not about the single disk aspects. As the determinants are taken over infinite dimensional matrices, one has to worry about their very definition. The von-Koch criterion (the existence of the determinant in one orthonormal basis, see Appendix A.3) is not sufficient for this task, since implicitly in the derivation of the S-matrix and explicitly under symmetry-reductions unitary transformations are mandatory. The multiscattering matrix must be reducible to a form “unit matrix plus a trace-class matrix” in order for its determinant to exist. Fortunately, we could prove that the multiscattering kernel A"M!1 is trace-class for any n-disk geometry as long as the disks do not overlap nor touch each other. Furthermore, the determinant over the infinite matrix M is defined as a cumulant expansion which — as shown by us — semiclassically reduces to the curvature expansion. Thus already quantum mechanically the cumulant/curvature “regularization” emerges. Moreover, by working in the full domain of the n-disk system, we could show in Section 5 that the cumulants split via the quantum traces into quantum itineraries which can be classified by the very symbol dynamics of their semiclassical reductions — the semiclassical periodic orbits. Thus the cumulant/curvature ordering in terms of the topological lengths of the quantum itineraries and hence of the topological lengths of the semiclassical periodic orbits is already present on the quantum-mechanical level. One does not need to impose it from the outside (as it would be the case for the semiclassical reduction of the Krein—Friedel—Lloyd sums of two bounded reference systems); but it follows naturally from the

 In fact, even the quantum-mechanical expression is not entire in the whole complex k-plane, since it has a branch cut on the negative real axis and poles which cancel the one-disk singularities.

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properties of quantum mechanics; namely, from the defining cumulant expansion of the determinant of the exact multi-scattering matrix. Thus the classification by the quantum itineraries is a virtue, but it is unfortunately also a vice, as quantum traces and the Plemelj—Smithies recursion formulas are involved. The latter introduce (unnecessarily) large terms which finally cancel in the cumulants themselves. As the cumulant sum of a trace-class operator converges absolutely, a direct semiclassical reduction of a complete cumulant and not of the potentially large quantum itineraries or quantum traces would be highly desirable. Unfortunately, the direct semiclassical reduction of a complete cumulant is not known. It might correspond to an integration of the small differences between the direct motion and the shadowed motion of the quantum wave packet. Instead, the standard calculation for the complete curvatures proceeds through the shadowing of all full periodic orbits of the pertinent topological length by all pseudo-orbits ("products of shorter periodic orbits) of the same total topological length. Because of these large cancellations, the semiclassical reduction on the level of the itineraries is potentially dangerous for the semiclassical equivalent of the quantum-mechanically absolutely converging cumulant sum, the curvature sum. There is no guarantee that it converges as well. As mentioned, in Section 5 we have managed to construct the semiclassical equivalent for each specified quantum itinerary. By working in the full domain and utilizing the pertinent simple symbol dynamics, which is valid under the condition that the number of disks is finite and that the disks do not overlap nor touch, our semiclassical reduction applies for all n-disk geometries, with one exception: we have to veto geometries which allow for grazing periodic orbits. In this way, we could guarantee for any specified quantum itinerary that the sequence of disk labels transforms uniquely to a sequence of non-overlapping semiclassical saddles in the complex angular-momentum plane which corresponds to the standard semiclassical periodic orbit, specified by the same symbolic sequence. The weight of the periodic orbits was shown to be identical to the one derived by Gutzwiller [5]. Furthermore, we have shown that to each itinerary that generates a (non-creeping or creeping) periodic orbit with a “ghost tunneling” section straight through a disk there belong “parent” itineraries, such that the ghost and corresponding parent periodic orbits cancel exactly in the semiclassical curvature sum. This establishes how pruning emerges from quantum mechanics in the semiclassical reduction. We have also shown that, to each quantum itinerary of topological length n, there belong 3L!1 different periodic orbits which contain creeping sections and we have specified a generalization of the symbolic labelling. By the Watson contour method of Ref. [29] we have derived their structure which agrees with the result of the semiclassical construction of Refs. [32,34] which in turn is based on Keller’s semiclassical theory of diffraction [30]. The direct link of the determinant of the exact S-matrix (via the determinant of the multiscattering matrix, via its cumulants and quantum itineraries) to the periodic orbits is therefore established. If the operations are inverted, the right hand side of Eq. (4.12) emerges, modulo the caveat that the semiclassical curvature sum might not converge, in general. What is known about the convergence properties of the curvature expansion of the Gutzwiller—Voros zeta function from the literature? The Gutzwiller trace as well as the zeta-function for n-disk repellers is known to converge (even absolutely) in the complex wave-number plane above a line specified by the resonance with largest imaginary k-value, see e.g. Ref. [17]. As all resonances belong to the lower half of the complex wave-number plane, the zeta function converges at least on the real k-axis. From Refs. [41—43] it is known that the cycle or curvature expansion of the

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Gutzwiller—Voros sum converges even inside the resonance region above an accumulation line defined by the poles of the dynamical zeta function. Thus above this accumulation line (and away from the branch cut and singularities of the exact quantum-mechanical side) our semiclassical limit (4.12) (or Eq. (4.13) for symmetry-reducible problems) is established for the full, untruncated Gutzwiller—Voros zeta function; see, however, below for the discussion of the asymptotic behaviour of the curvature sum. The relations are compatible with Berry’s expression for the integrated spectral density in Sinai’s billiard (a bounded nPR disk system, see Eq. (6.11) of Ref. [49]) and — in general — with the Krein—Friedel—Lloyd sums (2.6). They justify the formal manipulations of Refs. [17,15,71]. Furthermore, for these scattering systems, unitarity is automatically preserved semiclassically (without reference of any re-summation techniques a` la Berry and Keating [21] which are needed in bounded problems). Quantummechanically, unitarity follows from the relation 1 det SL(k)R" det SL(k*)

(7.1)

which is manifestly the case (see the first lines of Eq. (4.12) and (4.13)). Semiclassically, this follows from the second lines of Eqs. (4.12) and (4.13), with the caveat that curvature sums on the right hand sides must exist, i.e., they either converge or are suitably truncated. This is of course a very pleasant property. But, on the other hand, unitarity can therefore not be used in scattering problems to gain any constraints on the structure of ZI , as it could in bounded problems, see [21]. %4 Why are bounded problems special? In the semiclassical treatment of scattering problems the poles of the determinant of the S-matrix result from the zeros of ZI (k) in the lower complex k-plane %4 (where in general — except at the zeros — ZI (k) dominates ZI (k*)* which is the small, but %4 %4 nonzero), whereas the zeros of the determinant of the S-matrix are produced by the ones of ZI (k*)* in the upper complex k-plane (where in turn ZI (k) is the small, but nonzero zeta%4 %4 function). For bounded problems k is real and both zeta-functions become equally important. (A sign of this is the fact that the Hankel functions of either first or second kind which appear in the spectral determinants are replaced by the corresponding Bessel functions.) This obviously calls for a fine-tuning, hence, the re-summation. Note also the symmetry-breaking ie prescription which had to be added to the l.h.s. of the Krein—Friedel—Lloyd sums, see Section 2. As stated above, the incoherent single-disk scattering decouples from the genuine multi-disk scattering. The one-disk poles do not influence the position of the genuine multi-disk poles. However, Det M(k) does not only possess zeros, but also poles. The latter exactly cancel the poles of the product over the one-disk determinants, “L det S(ka ), since both involve the same “numH H ber” and “power” of H(ka ) Hankel functions in the denominator. The same is true for the poles K H of Det M(k*)R and the zeros of “L det S(ka ), as in this case the “number” of H(ka ) Hankel H H K H functions in the denominator of the former and the numerator of the latter is the same — see also Berry’s discussion of the same cancelation in the integrated spectral density of Sinai’s billiard (Eq. (6.10) of Ref. [49]). Semiclassically, this cancellation corresponds to a removal of the addivia the tional creeping contributions of topological length zero, i.e., 1/(1!exp(i2nll)), from ZI %4 one-disk diffractive zeta functions, ZI and ZI . The orbits of topological length zero result  HJ  HP from the geometrical sums over additional creepings around the single disks, U (exp(i2nll))LU L  (see [32]), and multiply the ordinary creeping paths which are classified by their topological length. Their cancellation is very important for situations where the disks nearly touch, as in such cases the

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full circulations of any of the touching disks by creeping orbits should clearly be suppressed, as it now is. Therefore, it is important to keep a consistent count of the diffractive contributions in the semiclassical reduction. What happens to the resonances, when the spacing between the disks becomes vanishing small such that bounded regions are formed in the limit of n'2 touching disks? Because of the ratio Det M(k*)R/Det M(k), to each (quantum-mechanical or semiclassical pole) of det SL(k) in the lower complex k-plane there belongs a zero of det S(k) in the upper complex k-plane with the same Re k value, but opposite Im k. When the bounded regions are formed some of these opposite zeros/poles move onto the real axis (such that their contributions cancel out of Eq. (3.9)). We have convinced ourselves that for the three-disk scattering system with e'0 separation these resonances approach infinitesimally the bound-state eigenvalues of the complementary calculation of the spectrum inside the bounded region, see, e.g., Ref. [17] for the billiard bounded by three touching disks. Semiclassically, this would be a non-trivial calculation as the eigen-energies have to be real which — without resummation a` la Berry and Keating [21] — they are not. In this situation, one really has to think about further resummation techniques. Most of the resonances, however, do not move onto the real axis at all, as n-disks repellers, even with bounded sub-domains, are still scattering systems. The would-be bound states, however, drop out of the exact formula for det SL(k), as they should. Let us come back to the numerical data of Section 6 and the existence of the curvature expansion. In this section we have reported on numerical results for the exact quantum-mechanical A resonances of the three-disk system with R"6a in the complex k-plane in the region:  04Re k4250/a and 05Im k5!1.6/a. The first observation is that the quantum-mechanical resonances in this window can be grouped into (leading and subleading) bands. In addition to the data presented in Section 6.1, where we have related the band structure to the semiclassical curvature expansion, it has been numerically checked that the emergence of a new band is in fact linked to a new cumulant order. The data of this window up to Re k"200/a can be fitted very well with a quantum-mechanical cumulant expansion which is truncated at order seven. This knowledge, together with the fact that any periodic orbit results from the semiclassical reduction of a quantum itinerary with the same symbol sequence (in the full domain), tells us that periodic orbits of topological length eight and higher are completely irrelevant for the description of the presented quantum-mechanical data, for regions below Re k+200/a and above Im k"!1.6/a. Thus any deviation of semiclassical predictions from the exact data cannot be cured by the inclusion of higher topological orbits. At best, they should leave the resonances untouched. This finding seems to be at variance with the result of Section 6.1 where the quasiclassical zeta function of Ref. [43] approximates most of the exact resonances at curvature order twelve. However, this truncated zeta functions finds also six erroneous resonance bands which do not have quantum-mechanical counter parts. This means that its topological expansion does not match the cumulant expansion, as we know of course from the analytical results of Section 5. The semiclassical reduction of a cumulant sum is the Gutzwiller—Voros curvature sum, and not the cycle expansion of the quasiclassical zeta function nor of the dynamical one. In fact, as shown by the comparison of exact to semiclassical coherent phase shifts, see Section 6.2, the latter two zeta functions are very inferior to the Gutzwiller—Voros zeta function which describes the exact phase shifts up to the resolution of the plot. Any competitor zeta function should do at least as well as the Gutzwiller—Voros one in order to be taken seriously. The question whether it converges or not is not a criterion for how successfully it approximates the quantum-mechanical data.

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In Section 6.3 we have finally executed what was already advocated by us in Refs. [44,45]. For the three-disk example we have numerically compared term by term and order by order the quantum-mechanical cumulants with their semiclassical counterparts, the curvatures of the Gutzwiller—Voros zeta function. The numerical data show that, for a fixed value of the wave number, the cumulants and curvatures agree in magnitude and also in phase only up to a finite cumulant order which is determined by the wave number and then deviate strongly. We have interpreted this result from the uncertainty bound on the quantum-mechanical resolution of the details of the classically repelling set which exponentially grow with the topological entropy of the system under consideration. Close to the real axis these deviations are hidden by the smallness in absolute value of the higher-order cumulants and curvatures. Therefore, the semiclassical phase shifts agree very well with the quantum-mechanical ones, even for very high curvature orders. However, with increasing value for !Im k, inside the lower half of the complex plane, the deviations are enhanced by the scaling laws discussed in Section 6.3 such that they eventually become noticeable. This observation matches extremely well the results of the resonance comparison in Section 6.1. The resonances which are located above and away from the boundary of convergence are approximated by the Gutzwiller—Voros curvature expansion as soon as the curvature order is sufficiently high. The resonances at or below the boundary of convergence, however, are approximated only up to the curvature order which respects the uncertainty bound. The curvature expansion works there only as an asymptotic series. Our interpretation is that eventually quantum mechanics and (semi-)classics have to part ways, as the quantum-mechanical spectral data only need power-law complexity, i.e. N operations if the multiscattering matrix can be truncated as an N;N matrix, whereas the resolution of the classically repelling set needs exponential complexity if the topological entropy is non-zero. In other words, whether the curvature expansion converges or not with respect to quantum mechanics it should be truncated at the cumulant order specified by the uncertainty bound. All curvature terms exceeding this order are — from the quantum-mechanical point of view — irrelevant. From this perspective, the semiclassical side of the relation (4.12) (and (4.13)) should be interpreted to be valid just for the truncated Gutzwiller—Voros curvature sums, where the order of the truncation increases with increasing value of Re k (or, since k"p/ , with decreasing ). The semiclassical limit Re kPR and the cumulant limit mPR do not commute, in general, if the topological entropy is non-zero. These facts should be kept separated from the -effects of Refs. [38—40] which investigate the O( ) corrections to the periodic orbits. We discuss here the -corrections to the curvatures which result from the periodic orbits via large cancellations against the pseudo-orbits. Part of the

-corrections of Refs. [38—40] cancel out as well, as can be shown from the comparison of the difference between the mth order exact and semiclassical trace which exceeds by far in magnitude the difference between the corresponding mth order cumulant and curvature. In fact, from the discussion of the subleading Debye corrections in Appendix F.3 one can deduce that each term H(ka)/H(ka) introduces a correction factor of order (1#i /4pa), such that the quantum J J itinerary of topological length m has at least an O(1#mi /4pa) factor relative to the corresponding periodic orbit (assuming that all disks have the same radius a for simplicity). However, the pseudo-itineraries of order m (which are the quantum mechanical analog of the pseudo-orbits) have the same correction factor, such that it cancels in the corresponding cumulant. Thus, the O(m ) terms cancel. But what about the O(m ) terms which might be of the same importance as the O( )

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terms, as the limits mPR and P0 do not commute? Only if the cumulant sum is truncated at a finite order, the O(m ) terms become negligible relative to the O( ) terms. In principle, the uncertainty boundary should be derivable from the semiclassical reductions of Section 5 and Appendix F to the quantum itineraries, if -corrections are taken into account systematically. In practice, however, there is a very long way from the -corrections extracted from the quantum itineraries to the surviving -corrections on the cumulant level because of the very large cancellations of the quantum itineraries with the pseudo-itineraries.

Acknowledgements The author would like to thank the Niels—Bohr-Institute and Nordita for repeated hospitality and the Leon Rosenfeld Scholarship Fund and its board for support of his visit to Copenhagen in the fall of 1996. He is indebted to Predrag Cvitanovic´, Michael Henseler, Per Rosenqvist and Ga´bor Vattay for many helpful contributions. Furthermore numerous illuminating discussions with Andy Jackson, Debabrata Biswas, Bruno Eckhardt, Harald Friedrich, Pierre Gaspard, Bertrand Georgeot, Thomas Guhr, Dieter Gra¨f, Ralph Hofferbert, Ronnie Manieri, Carmelo Pisani, Harel Primack, Achim Richter, Martin Sieber, Uzy Smilansky, Frank Steiner, Gregor Tanner, Hans Weidenmu¨ller, Niall Whelan and the late Dieter Wintgen are gratefully acknowledged. He thanks Bruno Eckhardt and Per Rosenqvist for supplying him with numerical input for periodic orbits of the three-disk system. Part of the present work has overlap with the author’s publications [47,48,62] and with Michael Henseler’s diploma thesis [46] which was prepared and written under the author’s guidance; the author would like to thank all his collaborators and especially Michael Henseler. Encouraging interest of Friedrich Beck, Gerry Brown, Mariana Kirchbach, Achim Richter, Jochen Wambach and all the members of the NHC group of the institute of nuclear physics at Technische Universita¨t Darmstadt is gratefully acknowledged. The author thanks Jochen Wambach for carefully reading the manuscript and Ralph Weichert for his help on the program xfig. Numerical calculations were performed on the DEC-Alpha workstation cluster at the NielsBohr-Institute, on the IBM-Risc workstation cluster at GSI Darmstadt and the IBM-Risc and DEC-Alpha server of the NHC group at the TU-Darmstadt.

Appendix A. Traces and determinants of infinite dimensional matrices This appendix summarizes the definitions and properties for trace-class and Hilbert—Schmidt matrices and operators, the determinants over infinite dimensional matrices and possible regularization schemes for matrices or operators which are not of trace-class. A.1. Trace class and Hilbert—Schmidt class This section is based on Ref. [72] and also Refs. [60,73—75] which should be consulted for further details and proofs. The trace class and Hilbert—Schmidt property will be defined here for linear, in

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general nonhermitean operators A3L(H) : HPH (where H is a separable Hilbert space). The transcription to matrix elements (used in the prior chapters) is simply a "1 , A 2 where + , is GH G H L an orthonormal basis of H and 1 ,2 is the inner product in H (see Ref. [74] where the theory of von Koch matrices of Ref. [76] is discussed). Thus the trace is the generalization of the usual notion of the sum of the diagonal elements of a matrix; but because infinite sums are involved, not all operators will have a trace and, if the trace exists in one basis, it is nontrivial that it exists also in any other basis: 1. An operator A is called trace-class, A3J , if and only if, for every orthonormal basis, + ,:  L "1 , A 2"(R. (A.1) L L L The family of all trace-class operators is denoted by J .  2. An operator A is called Hilbert—Schmidt, A3J , if and only if, for every orthonormal basis, + ,:  L

#A #(R. (A.2) L L The family of all Hilbert—Schmidt operators is denoted by J .  3. Bounded operators B are dual to trace-class operators. They satisfy the following condition: "1t, B 2"4C#t## # with C(R and t, 3H. If they have eigenvalues, these are bounded as well. The family of bounded operators is denoted by B(H) with the norm #B#"sup (#B #/# #) for 3H. Examples for bounded operators are unitary operators ($ and especially the unit matrix. In fact, every bounded operator can be written as a linear combination of four unitary operators [60]. 4. An operator A is called positive, A50, if 1A , 250 ∀ 3H. Notice that ARA50. We define "A""(ARA. The most important properties of the trace and Hilbert—Schmidt classes can be summarized as (see Refs. [60,72]): 1. J and J are *ideals., i.e., they are vector spaces closed under scalar multiplication, sums,   adjoints, and multiplication with bounded operators. 2. A3J if and only if A"BC with B, C3J .   3. J LJ .   4. For any operator A, we have A3J if #A #(R for a single basis. For any operator L L  A50, we have A3J if "1 , A 2"(R for a single basis. L L L  5. If A3J , Tr(A)" 1 , A 2 is independent of the basis used.  L L 6. Tr is linear and obeys Tr(AR)"Tr(A); Tr(AB)"Tr(BA) if either A3J and B bounded,  A bounded and B3J or both A, B3J .   Note that the most important property for proving that an operator is trace-class is the decomposition (2) into two Hilbert—Schmidt ones, as the Hilbert—Schmidt property can be easily verified in one single orthonormal basis (see (4)). Property (5) ensures then that the trace is the same in any

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basis. Properties (1) and (6) show that trace-class operators behave in complete analogy to finite-rank operators. The proof whether a matrix is trace-class (or Hilbert—Schmidt) or not simplifies enormously for diagonal matrices, as then the second part of property (4) is directly applicable: just the moduli of the eigenvalues (or — in case of Hilbert—Schmidt — the absolute squares) have to be summed in order to answer that question. A good strategy for checking the trace-class character of a general matrix A is therefore the decomposition into two matrices B and C where one, say C, should be chosen to be diagonal and either just barely of Hilbert—Schmidt character leaving enough freedom for its partner B or of trace-class character such that one only has to show the boundedness for B. A.2. Determinants det (1#A) of trace-class operators A This section is mainly based on Refs. [61,73] which should be consulted for further details and proofs. See also Refs. [74,75]. Pre-definitions (Alternating algebra and Fock spaces). Given a Hilbert space H, LH is defined as the vector space of multilinear functionals on H with 2 3LH if ,2, 3H dL(H)  L  L is defined as the subspace of LH spanned by the wedge-product 1

2 " e(n)[ 2 ] ,  L (n! P L LL LZ L

(A.3)

where P is the group of all permutations of n letters and e(n)"$1 depending on whether n is an L even or odd permutation. The inner product in dL(H) is given by ( 2 , g 2g )"det+( , g ), ,  L  L G H where det+a ," PLe(n)a 2a . dL(A) LZ GH L LLL dL(AB)"dL(A)dL(B)) on dL(H) with

(A.4) is defined as functor (a functor satisfies

dL(A)( 2 )"A 2A .  L  L

(A.5)

Properties. If A trace-class, i.e., A3J , then for any positive integer k, dI(A) is trace-class, and for  any orthonormal basis + , the cumulant L Tr(dI (A))" (( 2 I), (A 2A I)) G G G G G2GI

(A.6)

is finite and independent of the basis. Trd(A),1. Definition. Let A3J , then det(1#A) is defined as   det(1#A)" Tr(dI (A)) . I

(A.7)

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Properties. Let A be a linear operator on a separable Hilbert space H and + , an orthonormal H  basis. (a)  Tr(dI(A)) converges for each A3J . I  (b) "det(1#A)"4“ (1#k (A)) where k (A) are the singular values of A, i.e., the eigenvalues of H H H "A""(ARA, and "det(1#A)"4exp(Tr"A"). (c) For any A ,2, A 3J , 1z ,2, z 2|det(1# L z A ) is an entire analytic function. G G G  L   L (d) If A, B3J , then  det(1#A)det(1#B)"det(1#A#B#AB) "det((1#A)(1#B))"det((1#B)(1#A)) .

(A.8)

If A3J and U unitary, then  det(UR(1#A)U)"det(1#URAU)"det(1#A) .

(A.9)

(e) If A3J , then (1#A) is invertible if and only if det(1#A)O0.  (f ) If jO0 is an n-times degenerate eigenvalue of A3J , then det(1#zA) has a zero of order n at  z"!1/j. (g) For any A3J ,  ,A (A.10) det(1#A)" “ (1#j (A)) , H H where here and in the following +j (A),,A are the eigenvalues of A counted with algebraic H H multiplicity (N(A) can of course be infinite). (h) If A3J , then  ,dIA Tr(dI (A))" j (dI (A))" j (A)2j I(A)(R . H H H H XH2HIX,A (i) If A3J , then 

(A.11)

 det(1#zA)" zI (A.12) j (A)2j I(A)(R . H H 2 A  I I XH  H X,  (j) If A3J , then for "z" small (i.e., "z"max"j (A)"(1), the series  zITr((!A)I)/k converges and  H I






 zI det(1#zA)"exp ! Tr((!A)I) "exp(Tr ln(1#zA)) . k I (k) ¹he Plemelj—Smithies formula: Define a (A) for A3J by K   a (A) det(1#zA)" zK K . m! K

(A.13)

(A.14)

A. Wirzba / Physics Reports 309 (1999) 1—116

Then a (A) is given by the m;m determinant K



a (A)" K

Tr(A)

m!1

0

2

0

Tr(A)

Tr(A)

m!2

2

0

Tr(A)

Tr(A)

Tr(A)

2

0

$

$

$

$

$ 1

Tr(AK)

Tr(AK\)

Tr(AK\)

2

Tr(A)



71

(A.15)

with the understanding that a (A),1 and a (A),Tr(A). Thus the cumulants Q (A),a (A)/m!   K K (with Q (A),1) satisfy the recursion relation  1 K Q (A)" (!1)I>Q (A) Tr(AI) for m51 . K\I K m I

(A.16)

Note that formula (A.14) is the quantum analog of the curvature expansion of the Gutzwiller—Voros zeta function with Tr(AK) corresponding to the sum of all periodic orbits (primitive and also repeated ones) of total topoloical length m, see Eq. (4.8). In fact, in the cumulant expansion (A.14) (as well as in the curvature expansion there are large cancellations involved: Let us order — without loss of generality — the eigenvalues of the operator A3J as  "j "5"j "525"j "5"j "5"j "52.   G\ G G> This is always possible because of ,A"j "(R. Then, in the standard (Plemelj—Smithies) G G cumulant evaluation of the determinant, Eq. (A.14), there are enormous cancellations of large numbers, e.g., at the kth cumulant order (k'3), all the intrinsically large “numbers” jI , jI\j ,2, jI\j j ,2 and many more have to cancel out exactly until the r.h.s. of Eq. (A.11)       is finally left over. Algebraically, the fact that there are these large cancellations is of course of no importance. However, if the determinant is calculated numerically, the large cancellations might spoil the result or even the convergence. A.3. Von Koch matrices Implicitly, many of the above properties are based on the theory of von Koch matrices [74,76,77]: An infinite matrix 1!A"#d !a #, consisting of complex numbers, is HI HI  called a matrix with an absolutely convergent determinant, if the series "a  a  2a L L" with HI HI H I n"1, 2,2 converges, where the sum extends over all pairs of systems of indices ( j , j ,2, j ) and   L (k , k ,2, k ) which differ from each other only by a permutation, and where j (j (2j . Then   L   L the limit lim det#d !a #L "det(1!A) HI HI  L

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exists and is called the determinant of the matrix 1!A. The matrix 1!A is called von Koch matrix, if both conditions  "a "(R , (A.17) HH H  "a "(R (A.18) HI HI are fulfilled. Then the following holds (see Refs. [74,77]): (a) Every von Koch matrix has an absolutely convergent determinant. If the elements of a von Koch matrix are functions of some parameter k (a "a (k), j, k"1, 2,2) and both series in the HI HI defining conditions, Eqs. (A.17) and (A.18), converge uniformly in the domain of the parameter k, then as nPR the determinant det#d !a (k)#L tends to the determinant det(1#A(k)) uniformly HI HI  with respect to k, over the domain of k. (b) If the matrices 1!A and 1!B are von Koch matrices, then their product 1!C" (1!A)(1!B) is a von Koch matrix, and det(1!C)"det(1!A) det(1!B). Note that every trace-class matrix A3J is also a von Koch matrix (and that any matrix  satisfying condition (A.18) is Hilbert—Schmidt and vice versa). The inverse implication, however, is not true: von Koch matrices are not automatically trace-class. The caveat is that the definition of von Koch matrices is basis-dependent, whereas the trace-class property is basis-independent. As the traces involve infinite sums, the basis-independence is not at all trivial. An example for an infinite matrix which is von Koch, but not trace-class is the following:



i.e.,

2/j

for i!j"!1 and j even ,

A " 2/i GH 0

for i!j"#1 and i even ,




A"

(A.19)

else ,



0

1

0

0

0

0

2

1

0

0

0

0

0

2

0

0

0

1/2

0

0

2

0

0

1/2

0

0

0

2

0

0

0

0

0

1/3

\

0

0

0

0

1/3

0

\

$

$

$

$

\

\

\

.

(A.20)

Obviously, condition (A.17) is fulfilled by definition. Secondly, condition (A.18) is satisfied as  2/n(R. However, the sum over the moduli of the eigenvalues is just twice the harmonic L series  1/n which does not converge. The matrix (A.20) violates the trace-class definition (A.1), L as in its eigenbasis the sum over the moduli of its diagonal elements is infinite. Thus the absolute convergence is traded for a conditional convergence, since the sum over the eigenvalues themselves

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can be arranged to still be zero, if the eigenvalues with the same modulus are summed first. Absolute convergence is of course essential, if sums have to be rearranged or exchanged. Thus, the trace-class property is indispensable for any controlled unitary transformation of an infinite determinant, as then there will be necessarily a change of basis and in general also a re-ordering of the corresponding traces. Nevertheless, the von Koch criteria (A.17) and (A.18) are useful, as any trace-class matrix has at least to meet these simple tests which can be easily performed in any specified basis. A.4. Regularization Many interesting operators are not of trace-class (although they might be in some J with p'1: N an operator A is in J iff Tr"A"N(R in any orthonormal basis). In order to compute determinants N of such operators, an extension of the cumulant expansion is needed which, in fact, corresponds to a regularization procedure [61,73]: E.g., let A3J with p4n. Define N L\ (!z)I AI !1 (A.21) R (zA)"(1#zA)exp L k I as the regulated version of the operator zA. Then the regulated operator R (zA) is trace-class, i.e., L R (zA)3J . Define now det (1#zA)"det(1#R (zA)). Then the regulated determinant L  L L L\ (!zj (A))I ,XA H det (1#zA)" “ (1#zj (A)) exp (R (A.22) L H k H I exists and is finite. The corresponding Plemelj—Smithies formula for det (1#A) results from the L standard Plemelj—Smithies formula (A.14) by simply setting Tr(A), Tr(A),2,Tr(AL\) to zero [73]. See also Ref. [78] where the Fredholm determinant
























j  D(j)" “ 1! j I I is regulated — in the case k,d/m'1 — as a Weierstrass product

(A.23)



j j j j I

 1! exp # #2# . (A.24) D(j)" “ j j 2j [k]j I

I I I I I Here +j , are the eigenvalues of an elliptic (pseudo)-differential operator H of order m on a compact H or bounded manifold of dimension d (with 0(j 4j 42 and j !#R) and [k] denotes the   I integer part of k. Eq. (A.24) is the unique entire function of order k with zeros at +j , and subject to I the normalization conditions d I

d ln D(0)"0 . ln D(0)" ln D(0)"2" dj I

dj

(A.25)

Clearly Eq. (A.24) is the same as Eq. (A.22); one just has to identify z"!j, A"1/H and n!1"[k]. An example is the regularization of the spectral determinant D(E)"det[(E!H)]

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which — as it stands — would only make sense for a finite dimensional basis (or finite dimensional matrices). In Ref. [79] the regulated spectral determinant for the example of the hyperbola billiard in two dimensions (thus d"2, m"2 and hence k"1) is given as D(E)"det[(E!H)X(E, H)] ,

(A.26)

where X(E, H)"!H\e#H\. Thus the spectral determinant in the eigenbasis of H (with eigenvalues E O0) reads L E e##L(R . (A.27) D(E)"“ 1! E L L Note that H\ is for this example of Hilbert—Schmidt character.






Appendix B. Exact quantization of the n-disk scattering problem In this appendix (which is based on Henseler’s diploma thesis [46] where also the corresponding formulas for the three dimensional n-ball scattering problem can be found, see also [62,48]) we will construct the scattering matrix for the scattering of a point particle from n circular hard disks which are fixed in the two-dimensional plane. The basic ideas go back to Lloyd’s multiple-scattering method [53], an application of the KKR-method [51], to three-dimensional band structure calculations as the limiting case of n disjunct non-overlapping muffin-tin potentials (see also Ref. [49] for the translation of these methods to the infinite two-dimensional Sinai-billiard) and to the work of Gaspard and Rice [13], who introduced the techniques reported below to the scattering problem of a point particle from three equal disks in the two-dimensional plane. Here we will present a generalization of these methods to the scattering from n non-overlapping disks of — in general — different sizes. B.1. The stationary scattering problem As stated in Section 3, the quantum-mechanical description of the scattering from n hard disks will be performed in the framework of the stationary scattering theory. Let tk(r) be a solution of the scattering problem (for a fixed incident wave vector k). The decomposition of t into a sum over complex exponential (angular) functions  (B.1) tk(r)" tI (r)e KL\Uk) K K\ (U and U are the angles of k and r, respectively, in the global coordinate system) leads to I P (e#k)tI (r)"0. The corresponding separation of a plane wave in two dimensions into angular P K eigenfunctions reads:  e k  r"e IP  UP" J (kr)e KUPe KL\Uk). K K\

(B.2)

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The ordinary Bessel and Hankel functions (J (z)"(H(z)#H(z))) of integer order satisfy the J  J J expressions (for "z"<1): H(z)&(2/nz e\ X\L J\L incoming , (B.3) J (B.4) H(z)&(2/nz e>GX\L J\L out-going . J The to-be-constructed solution can be written as a superposition of incoming and out-going spherical waves (kr<1)  1 [d e\ IP\L J\L #S e IP\L J\L]e JUP , tI (r)& KJ K (2nkr J\ KJ where the matrix S is the scattering matrix of the two-dimensional scattering problem.

(B.5)

B.2. Calculation of the S-matrix In order to describe a generic configuration of n disks we use the following notation (see Fig. 4): The index j3+1,2, n, labels the jth disk whose radius is a . The distance between the centers of the H disks j and j is called R "R . To specify the n disks we introduce n#1 different coordinate HHY HYH systems. First of all, a global coordinate system (x, y) is chosen with its origin in the neighborhood of the n disks. In case of symmetrical systems, as, e.g., three equal disks at the corners of an equilateral triangle, the origin is best placed in the center of symmetry. In order to fully use the symmetry of such configurations n local coordinate systems (xH, yH) are introduced whose origins are placed at the centers of the n disks, respectively. The axes of these coordinate systems are chosen in such a way that they fully respect the symmetry of the configuration. The spatial vector to the center of the disk j, as measured in the global systerm, is called R , R is its length and H its angle. H H 0 Vectors called s or S are surface vectors. The unit vector RK H ,RH /R is pointing from the center HHY HHY H HHY of disk j to the center of disk j, as measured in the ( j)-system, a is its corresponding angle. In HYH general, vectors with an upper index ( j) are measured in the ( j)-system, vectors without upper index are measured in the global system. The Green’s functions satisfy the differential equation (e#k)G(r, r)"d(r!r). In two P dimensions the free Green’s function reads [13]: i G(r, r)"! H(k"r!r") . 4 

(B.6)

For the following, we will apply the Green’s formula:





dr( (r)et(r)!t(r)e (r))" dS ) ( (S)e t(S)!t(S)e (S)) , P P 1 1 4 w4 where » is the integration volume and j» denotes its boundary. After inserting the expansion coefficients tI (r) from Eq. (B.1) and the (free) Green’s function in the last equation, one finds: K 0 r,» , (B.7) dS ) [tI (S)e G(S, r)!G(S, r)e tI (S)]" K 1 1 K tI (r) r3» . w4 K





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The integration volume is chosen as a big disk whose center is in the origin of the global coordinate system and whose radius is large enough that the asymptotic Eqs. (B.3) and (B.4) hold for the points far away from the origin but inside the integration volume. From the large disk the small n disks (as given in the concrete disk configuration) are excluded; however, the radii of these subtracted disks have been increased by a small increment e'0 in comparison to the original disks. In the end, the case eP0 is considered. In order to construct the S-matrix, one has to work out (B.7) for two different cases [13]. In the first case the point r is on the surface of the (original) scattering disk j, such that it is now outside the integration volume ». In the second case r is in the integration volume; however, so far away from all n disks that the asymptotic equations (B.3) and (B.4) are then valid. The boundary of » splits into n#1 disjunct regions: Into the outer layer of the large disk, j », and into the boundaries j » of the n subtracted disks which contain and cover the scattering  H disks. B.2.1. First case: r"X 3 boundary of disk j H Because of the Dirichlet boundary conditions, the wave function vanishes on the boundary of the scattering disks; however, its gradient does not vanish there > (B.8) tI (X )"0; n ) etI (XH), BH e KYFH . K H H K H KKY KY\ Here the unit vector n is chosen to point perpendicularly to j » into the complementary region H H of ». Note that "XH""a . Furthermore, h labels the direction of XH as measured in the local H H H H coordinate system of the disk j. The coefficients BH are unknown so far. Eq. (B.7) now reads: KKY L 0"IH # IH . (B.9)  HY HY The occurring integrals are



IH " 

dS ) [tI (S)e G(S, X )!G(S, X )e tI (S)] , K 1 H H 1 K w4

(B.10)



(B.11) dsj ) G(sj, XHY)e HYtI (sj) . H Q K wHY4 In the following we will repeatedly apply the addition theorems for Bessel and Hankel functions [80]: IH "! HY

 C (w)e! L@" C (u)J (v)e! J? , L L>J J J\

(B.12)

where w"(u#v!2uv cos a, w cos b"u!v cos a, "ve ?"("u", w sin b"v sin a and C (z)3+J (z), L L ½ (z), H(z), H(z),. L L L Calculation of IH . The calculation is performed in the global coordinate system. The addition  theorem (B.12) is used to rewrite the free Green’s function (B.6). In addition, because of the large value for R , the Hankel function H(kR ) is approximated according to (B.4). The resulting 1 J 1

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expression and the asymptotic expression (B.5) for tI (S) are inserted into IH . The terms proporK  tional to S cancel out, such that KJ  (B.13) (kR )J (ka )e KYFH . IH "e K(0H J K\KY H KY H  KY\ Only the coordinates of the center of the disk j are still expressed in the global system, whereas the coordinates on the disk surface have been transferred to the local coordinate system of disk j. Calculation of IH. Here we work relative to the local coordinate system of disk j. Using the H addition theorem (B.12) for the free Green’s function (B.6) and performing the angular integration under the boundary condition (B.8) we obtain na  (B.14) IH"! H BH H(ka )J (ka )e JFH , KJ J H J H H 2i J\ where all quantities are expressed in the local coordinate system of disk j. Calculation of IH , jOj. Working relative to the local coordinate system of disk j, we have in this HY case: 1  (B.15) H(kXHY)J (ka )e\ J(QHY\?HHY>AHY . G(s , XHY)" J H J HY HY H 4i J\ In writing down the last equation the addition theorem for Hankel functions has been used again. Here HY"a #c , where HY is the angle of XHY, a is the angle of the ray from the center of H HHY HY H H HHY disk j to the center of disk j and c is the difference angle. All three angles are measured relative to HY the local coordinate system of disk j. After insertion into IH and the angular integration we apply HY once more the addition theorems for Hankel functions. Then IH reads: HY  na (B.16) IH "! HY BHY J (ka )J (ka )H (kR )e J?HHY\JY?HYH(!1)JYe JYFH , KJ J HY JY H J\JY HYH HY 2i JJY\ where the entries of IH do not depend on the global coordinate system. The j dependent quantities HY are expressed in the local coordinate system of disk j, the j-dependent ones in that of disk j. The computed integrals are now inserted into the formula (B.9), written as IH "! L IH , HY HY  which leads to (B.17) CK H e JFH" BHY MK HYHe JFH KJY JYJ KJ J HYJY J with CK H abbreviating the terms in Eq. (B.13), whereas MK HYH stands for the terms in Eqs. (B.14) and KJ JYJ (B.16).Eq. (B.17) holds for all points X on the boundary of the disk j. Then, the coefficients CK H and H KJ MK HYH are normalized in such a way, that in the one-disk case the new M-matrix is just the unit JYJ matrix. This corresponds to a division of the l.h.s. and r.h.s. of (B.17) by the diagonal matrix +H(ka )J (ka )na /2i,. Asymptotically (i.e., for "l"<"ka ") the modulus of its matrix elements J H J H H H behaves as "H(ka )J (ka )"&1/(n"l"). Therefore, this division does not affect the “trace-character” J H J H of the matrices CK H, MK HHY and BHY (see Appendix C). Thus one gets the matrix equation CH" BHY ) MHYH HY

(B.18)

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with J (kR ) 2i C H "e J(0H J\K H , JK H(ka ) na K H H

(B.19)

a J (ka ) MHHY "dHHYd #(1!dHHY) H K H H (kR )e K?HYH\KY?HHY(!1)KY , (B.20) KKY KKY a H(ka ) K\KY HHY HY KY HY where R and H are the magnitude and the angle of the ray from the origin of the global H 0 coordinate system to the center of disk j, as measured in the global coordinate system. The angle a is the angle of the ray from disk j to disk j as measured in the local coordinate system of disk HYH j, R "R is the distance of the centers of disk j and j, a , a are their radii, respectively. HHY HYH H HY B.2.2. Second case: point r"r3», r large For this case we obtain from Eq. (B.7): L tI (r)"Ir # Ir . K  H H In analogy to the first case, the following abbreviations have been introduced:



Ir " 

dS ) (tI (S)e G(S, r)!G(S, r)e tI (S)) , K 1 1 K w 4

(B.21)

(B.22)





ds ) G(s , rH)e HtI (s ) . (B.23) H H Q K H wH4 Calculation of Ir . Ir can be calculated in close analogy to IH . A single application of the    addition theorems (B.12) yields Ir "J (kr)e KUP, where U is the angle of r in the global coordinate  K P system. r Calculation of I . We have  1  H(kr)J (kR )J (ka ) e!JY (QH e8 JUP! JU0H , G(s , rH)" (B.24) J J\JY H JY H H 4i JJY\ where the addition theorem for cylindrical functions has been applied twice. The angle H of s is H Q measured relative to the local coordinate system of disk j. After integration over this angle, we get: Ir "! H

na Ir "! H H 2i

 H(kr)J (kR )J (ka )BH e JUP\U0H , (B.25) J J\JY H JY H KJY JJY\ where all quantities, except a , are now defined with respect to the global coordinate system. H Both integrals are now inserted into Eq. (B.21). Taking into account Eqs. (B.3) and (B.4), one gets Eq. (B.5) for kr<1. The S-matrix can now be written as SL"1! iBH ) DH , H

(B.26)

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where we have introduced the superscript (n) in order to indicate that the S-matrix refers to the n-disk scattering problem. The matrix DH in the last equation is given by (B.27) DH "!na J (kR )J (ka )e\ KYU0H . KKY H KY\K H K H Using Eq. (B.18), we finally get the (formal) expression for the S-matrix which will be justified in Appendix C: SL"1! iCH ) (M\)HHY ) DHY . (B.28) HHY The S-matrix S of the scattering of a point particle from a single hard disk is given by H(ka ) Hd , (B.29) S(ka )"! J KJ H H(ka ) KJ J H as can be seen by comparison of the general asymptotic expression (B.5) for the wavefunction with the exact solution for the one-disk problem. B.3. The determination of the product D ) C In order to rewrite the determinant of the S-matrix (see Section 4) we have to determine the product D ) C (see Eqs. (B.19) and (B.27)). We apply once again the addition theorem for Bessel functions (B.12) using R "R !R , where R and R are the magnitudes of these vectors and HYH HY H HY H U HY and U H the corresponding angles, as measured in the global coordinate system. We find the 0 0 following expressions:




a J (ka )  J H J (kR )(!1)JYe J?HYH\JY?HHY , DH I C HI Y "!2i H J JY JJ a H(ka ) J\JY HHY I HY JY HY J\ J (ka )  DH I C HI "!2i J H d . Y J J JJ H (ka ) JJY JY H JI \ Using the expression (B.20) for MHHY we finally get for X,M!iD ) C: KKY a J (ka ) H(ka ) HY dHHYd !(1!dHHY) H J H H (kR )(!1)JY e J?HYH\JY?HHY. XHHY"! JY JJY JJY a H(ka ) J\JY HHY H(ka ) JY HY HY JY HY The r.h.s. of this equation can be reformulated in terms of the scattering matrix of the single-disk problem, S(ka ), and by the complex conjugate of M, namely XHHY(k)"S(ka )(MH HY (k*))*. The HY JJY JJY HY \J\JY matrix X can therefore be expressed as the product, I I XHHY " (MHHI )*YHI HY, \J\JY JJ JY J HI JI I where the second factor is given by YHI HY,dHI HYd I S . Thus we get the formal expression for the JJY JJY \JY\JY determinant of X:






L Det X(k)" “ det S(ka ) Det M(k*)R . * J H * H

(B.30)

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The last step in the formal evaluation of the determinant of SL (as function of the wave number k) is the insertion of Eq. (B.30) into Eq. (3.6) giving the final result (see Eq. (3.9): det SL(k)" J

(B.31)

Appendix C. Existence of the S-matrix and its determinant in n-disk systems This appendix is based on M. Henseler’s diploma thesis [46] and on Ref. [48]. The derivations of the expression for S-matrix (B.28) in Appendix B and of its determinant (see Section 4) are of purely formal character as all the matrices involved are of infinite size. Here, we will show that the operations are all well-defined. For this purpose, the trace-class (J ) and  Hilbert—Schmidt (J ) operators will play a central role. The definitions and most important  properties of these operator-classes can be found in Appendix A. As shown in Appendix B the SL-matrix can be written in the following form (see Eq. (B.26)): SL"1!iT,

T"BHDH .

(C.1)

The T-matrix is trace-class on the positive real k-axis (k'0), since it is the product of the matrices DH and BH which will turn out to be trace-class or are bounded there, respectively (see Appendix A.1 for the definitions). Again formally, we derived in Appendix B that CH"BHYMHYH implies the relation BHY"CH(M\)HHY. Thus, the existence of M\(k) has to be shown, as well — except at isolated poles in the lower complex k-plane below the real k-axis and on the branch cut on the negative real k-axis which results from the branch cut of the defining Hankel functions. As we will prove later, M(k)!1 is trace-class, except of course at the above mentioned points in the k-plane. Therefore, using property (e) of Appendix A.2 we only have to show that Det M(k)O0 in order to guarantee the existence of M\(k). At the same time, M\(k) will be proven to be bounded as all its eigenvalues and the product of its eigenvalues are then finite. The existence of these eigenvalues follows from the trace-class property of M(k) which, together with Det M(k)O0, guarantees the finiteness of the eigenvalues and their product. We have normalized M in such a way that for the scattering from a single disk we simply have B"C. Thus the structure of the matrix CH does not dependent on whether the point particle scatters only from a single disk or from n disks. Hence the properties of this matrix can be determined from the single disk scattering alone. The functional form (B.19) shows that C cannot have poles on the real positive k-axis (k'0) in agreement with the structure of the S-matrix discussed in Appendix B. If the origin of the coordinate system is placed at the origin of the disk, the matrix S is diagonal. In the same basis C becomes diagonal. Thus one can easily see that C has no zero eigenvalue on the positive real k-axis and that it will be trace-class there. Thus neither C nor the one-disk (or for that purpose the n-disk) S-matrix can possess poles or zeros on the real positive k-axis. The statement about SL follows simply from the unitarity of the S-matrix which can be checked easily. Since, for real positive k, SR(k)"S\(k), we have "S(k)""1 on the real axis, such that poles (and also zeros) of S are excluded there. Actually, for the exclusion of poles and zeros on

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the real positive k-axis, only the weaker condition that "det SL(k)""1, k'0, is needed. That this is fulfilled for all non-overlapping n disk systems is obvious from the final expression (3.9) for det SL in Section 4. This formula even holds for Det M(k)P0 if k approaches the real positive axis, since then Det M(k*)R approaches zero as well, such that both terms cancel in formula (3.9). Thus the fact that "det SL(k)""1 on the positive real k-axis cannot be used to disprove that Det M(k) could be zero there. However, if Det M(k) were zero there, the “would-be” pole must cancel out of SL(k). Looking at formula (B.28), this pole has to cancel a zero from C or D where both matrices are already fixed on the one-disk level. Now, property (g) and (f) of Appendix A.2 leave for M(k) (provided that M!1 has been proven trace-class) only one chance to cause trouble on the positive real k-axis, namely, if at least one of its eigenvalues (whose existence is guaranteed) becomes zero. On the other hand M still has to satisfy CH"BHYMHYH with C completely determined by the one-disk scattering alone, where Det M"1 everywhere in the k-plane. The fact that CH(k) cannot have zero eigenvalue for k'0 can be used to show that the following inequality holds for the modulus of the diagonal matrix element, "CH (k)"'0, for the state "m2 of any orthonormal basis. Now choose as the basis the eigenbasis of KK MHHY (k) and "m2 as the state there M(k) has a candidate for a zero eigenvalue in the "m, j2 KKY space. Comparing the left and the right-hand side "CH (k)"""BHY MHYH" one finds a contradiction if KK KJ JK the corresponding eigenvalue of M(k) were zero, i.e., the l.h.s. would be greater than zero for k'0, whereas the r.h.s. would be zero. Hence, such a zero eigenvalue cannot exist for k'0, hence Det M(k)O0 for k'0, hence M(k) is invertible on the real positive k-axis, provided M(k)!1 is trace-class. From the existence of the inverse relation BHY"CH(M\)HHY, the trace-class property of CH to be shown and the boundedness of (M\)HHY follows the boundedness of BH and therefore the trace-class property of the n-disk T-matrix, TL(k), except at the above excluded k-values. What is left to prove is (a) M(k)!13J for all k, except at the poles of H(ka ) and for k40,  K H (b) CH(k), DH(k)3J with the exception of the k-values mentioned in (a),  (c) T(ka )3J (with the same exceptions as in (a) and (b)) H  (d) M\(k) does not only exist, but is bounded. Under these conditions all the manipulations of Section 4, are justified and SL, as in Eq. (3.2), and det SL, as in Eq. (3.9), are shown to exist.

C.1. Proof that T(ka ) is trace-class H The S-matrix for the jth disk is given by (B.29). Thus V,!iT (ka )"S(ka )!1 is H H diagonal: !2J (ka ) J H . V "d JJY JJY H(ka ) JY H

(C.2)

Hence, we can write V"U"V " where U is diagonal and unitary, and therefore bounded. What is left to show (see property (1) of A.1) is that "V "3J . This is very simple since we can now use the second  part of property (4) of Appendix A.1: we just have to show in a special orthonormal basis (here the

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eigenbasis) that





> > J (ka ) "V " " 2 J H (R (C.3) JJ H(ka ) J H J\ J\ as "V "50 by definition. The ordinary Bessel and Hankel functions of integer order satisfy J (z)"(!1)LJ (z), H (z)"e LLH(z), H (z)"e\ LLH(z), \L L \L L \L L 2 ez \J 1 ez J , H(z)&!i . lPR, l real: J (z)& J J nl 2l (nl 2l







(C.4) (C.5)

Thus




  1 e"ka " J H ,2 (a )J . (C.6) tr("V "):4 J 2 2l J J These aJ satisfy: aJ (aJ (1 for l'l and l 'e"ka "/2. The series  (aJ )J converges, and hence   H J J J J J also the sum  (aJ )J as it is bounded from above by the previous sum. That means that "V "3J J J  and (because of property (1) of Appendix A.1) S!13J , as well. This, in turn, means that  det S(ka ) exists (see property (i) of Appendix A.2) and also that the product H “L det S(ka )(R in the case where n is finite (see property (d) of the same appendix). The limit H H lim does not exist, in general, as the individual terms det S(ka ) can become large, of course. L H C.2. Proof that A(k),M(k)!1 is trace-class The determinant of the characteristic matrix M(k) is defined, if A(k)3J . In order to show this,  we split A into the product of two operators which — as will be shown — are both Hilbert—Schmidt. Then according to property (2) of Appendix A.1 the product is trace-class. Let therefore A"E ) F, where A follows from Eq. (B.20). In order to simplify the decomposition of A, we choose one of the factors, namely, F, as a diagonal matrix: AHHY"EHHYFHY, FHHY"FHdHHYd , (C.7) JJY JJY JY JJY J JJY (H(kaa ) J H , "a"'2 . FH" (C.8) J H(ka ) J H Already this form leads to the exclusion of the zeros of the Hankel functions H(ka ) and also the J H negative real k-axis (the branch cut of the Hankel functions for k40) from our final proof of A(k)3J . First, we have to show that #F#" (FRF) HH(R. We start with HJ  JJ L  L  "H(kaa )" J H , 2 aJ . (C.9) #F#4 2 J "H(ka )" H H J H J J This form restricts the proof to n-disk configurations with n finite. Using the asymptotic expressions (C.5) for the Bessel and Hankel functions of large orders, it is easy to prove the absolute convergence of aJ in the case "a"'2. Therefore #F#(R and because of property (4) Appendix J J A.1 we get F3J . 

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Using the decomposition (C.7) and the definition of F (C.8), the second factor E, is constructed. We then have to show the absolute convergence of the expression




"J ("ka ")""H ("kR ")" a   L J H J\JY HHY H #E#" "H("kaa ")" a JY HY HHY HY JJY\ H$HY

(C.10)

in order to prove that also E3J . This is fulfilled, if aJ (R, where  JJY JJY "J ("ka ")""H ("kR ")" J>JY HHY . aJ " J H (C.11) JJY "H("kaa ")" JY HY Necessary conditions for the convergence of the double sum over aJ are: aJ (R as well as JJY JY JJY aJ (R. For the case lPR, l fixed, we obtain with the help of the asymptotic formulas (C.5) JJY J expression: the






e"kR " \JY HHY 1 2 1 lPR: aJ & JJY n "H("kaa ")" l JY HY

(C.12)

For any e'0 this yields the estimate:






(1#e)a J l H , l'l with (e . (C.13)  R l HHY  For x,(1#e)a /R (1, the series  xJ converges absolutely. As (2l)JYxJ" J J H HHY (xj/jx)JY xJ(R, the series  b (l) converges absolutely, as well. Therefore we have the J J J absolute convergence of aJ for a (R with fixed l in the limit eP0. In the opposite case, J JJY H HHY lPR, l fixed, the absolute convergence of aJ for  "a"a (R can be proven analogously. JY JJY  HY HHY We must of course show the convergence of aJ for the case l, lPR. Using again the JJY JJY asymptotic behavior of the Bessel and Hankel functions of large order we get the following proportionality for l, lPR: b (l)((2l)JY J







(l (l (l#l)J>JY a J "a" a JY H HY " b . (C.14) aJ J JJY (l#l)l (l#l)l JJY lJlJY R 2 R HHY HHY The double sum  aJ is convergent, if  b converges. In order to show this, we introduce JJY JJY JJY JJY two new summation indices (M, m) as l#l"2M and l!l"m. Hence, we have   + b " c JJY +K JJY + K\+

(C.15)

with









a +>K "a" a +\K (2M)+ H HY . c " +K (M#K)+>K(M!K)+\K R 2 R  HHY HHY 

(C.16)

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For sufficiently large M, the powers occurring in the last expression can be approximately estimated with the help of the Stirling formula, nL&n!eL/(2n. In this way, we get for MPR:














(2M)! a +>K "a" a +\K  H HY . c &2n +K (M#K)!(M!K)! R 2 R   HHY HHY Hence, the total sum reads





(C.17)

 

 a #" ?"a +   >+ H  HY c :2n , (C.18) +K R HHY + K K\+ where the sum over m has been performed with the help of the binomial formula. The remaining series in Eq. (C.18) converges for a # "a"a (R . Therefore, under the stated conditions aJ H  HY HHY JJY JJY converges absolutely, as well. We finally get the desired result: The operator E belongs to the class of Hilbert—Schmidt operators (J ), if the conditions   "a"a #a (R , (1#e)a ((1#e) "a"a (R and (1#e)a (R are met in the limit eP0. HY H HHY HY  HY HHY H HHY  In summary, this means: E(k) ) F(k)"A(k)3J for such finite n disk configurations for which the  disks neither overlap nor touch and for those values of k which lie neither on the zeros of the Hankel functions H(ka ) nor on the negative real k-axis (k40). The zeros of the Hankel functions K H H(k*a ) are then automatically excluded, too. The zeros of the Hankel functions H(kaa ) in H K H the definition of E are cancelled by the corresponding zeros of the same Hankel functions in the definition of F and can therefore be removed. A slight rotation of a readjusts the positions of the zeros in the complex k-plane such that they can always be moved to non-dangerous places. For these (“true”) scattering systems the determinants Det M(k) and Det M(k*)R are defined and can be calculated with the help of one of the cumulant formulas given in Appendix A.2, e.g., by the Plemelj—Smithies formula (A.14) (with Det"e2 , see Eq. (A.13), for small arguments) or by Eq. (A.10) or Eq. (A.12) if M or A can be diagonalized. C.3. Proof that CH and DH are trace-class The expressions for DH and CH can be found in Eqs. (B.27) and (B.29). Both matrices contain — for a fixed value of j — only the information of the single-disk scattering. As in the proof of T3J , we  will go to the eigenbasis of S. In that basis both matrices DH and CH become diagonal: , (C.19) DH "!na J (ka )e\ KU0Hd KKY KKY H K H 1 2i CH "e KU0H d . (C.20) JK H(ka ) na JK K H H Using the same techniques as in the proof of T3J , we can show that CH and DH are trace-class.  In summary, we have DH3J for all k since the Bessel functions which define that matrix possess  neither poles nor branch cuts. The matrix CH3J for almost every k, except at the zeros of the  Hankel functions H(ka ) and the branch cut of these Hankel functions on the negative real k-axis K H (k40). Note that the values of tr DH or tr CH, are finite and the same whether one uses the non-diagonal expressions (B.27)/(B.19) or the diagonal ones (C.19)/(C.20). This is, of course, in agreement with property (5) of Appendix A.1.

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C.4. Existence and boundedness of M\(k) As M(k)!13J except at the zeros of H(ka ) and on the negative real k-axis (k40), M\(k)  K H exists everywhere, except at the points mentioned above and except at k-values where Det M(k)"0. In other words, except at the poles of the SL(k) matrix, see Eq. (3.9). With the exception of the negative real axis and isolated zeros of H(ka ), M(k) is analytic. Hence, the points K H of the complex k-plane with Det M(k)"0 are isolated. Hence, Det M(k)O0 almost everywhere. Thus, almost everywhere, M(k) can be diagonalized and the product of the eigenvalues weighted with their degeneracies is finite, see Appendix A.2 for both properties. Thus M\(k) exists and can be diagonalized as well. Hence, all the eigenvalues of M\(k) (and their product) are finite in the complex k-plane, where Det M(k) is defined and nonzero. Thus M\(k) is bounded (and Det M\(k) eixsts) almost everywhere in the complex k-plane. In summary, the formal steps in the calculation of the n-disk S-matrix (see Appendix B) and its determinant (see Section 4) are all allowed and well-defined, if the disk configurations are such that the disks neither touch nor overlap.

Appendix D. Comparison to Lloyd’s T-matrix As mentioned in Section 3, Lloyd has constructed a formal expression for the T-matrix of a finite cluster of muffin-tin potentials in three dimensions, see (98) of Ref. [53]. Transcribed to the case of a cluster of n disk-scatterers fixed in the two-dimensional plane, Lloyd’s T-matrix reads as TI (k)"CI (k)(MI (k))\DI (k) with






!2iJ (ka ) J H , (D.1) H(ka ) J H (D.2) DI HY "J (kR )e\ KYU0HY , JYKY KY\JY HY J (ka ) MI HHY"dHHYd #(1!dHHY) J H H (kR )C (l, l) , (D.3) JJY JJY H(ka ) J\JY HYH HHY J H where the tilde is discriminating the matrices in the Lloyd representation from the corresponding matrices in the Gaspard—Rice representation, defined in Eqs. (3.3), (3.4) and (3.5). The Lloyd representation allows for a very simple interpretation. The matrix CI H describes the regular propagation (in terms of the homogeneous part of the free propagator) from the origin to the point R and a one-disk scattering from a disk centered at this point, as given by the one-disk T -matrix. H The matrix DI HY describes the (regular) propagation back from the disk j to the origin. The matrix (MI HHY)\ parametrizes the multiscattering chain. If it is expanded around dHHY, it describes the sum of no propagation and no scattering plus the propagation from disk j to disk j (in terms of the full propagator) and a scattering from disk j and so on. The disadvantage of the Lloyd representation is that the trace-class character of AI ,MI !1 is lost, as the terms J (ka ) and (H(ka ))\ K H K H “stabilize” only the asymptotic behavior of the index l, but not of the index l any longer, as the asymmetric Gaspard—Rice form did. The infinite determinant det MI is therefore no longer absolutely convergent, but only conditionally. Any manipulation in the Lloyd representation of the matrix CI H "J (kR )e KU0H KJ K\J H

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MI and the corresponding S-matrix has therefore to be taken with great care. Note, however, that the (formal) cumulant expansions of MI and M are the same as the corresponding traces satisfy Tr (AI L)"Tr (AL). In other words, if the cumulants of MI are summed up according to the Plemelj—Smithies form of M, the result of det MI (k) and det M(k) is the same. In fact, one can derive the Lloyd representations CI H, MI HHY and DI HY from the expressions for CK H, MK HYH and DH of Appendix B.1 (see Eq. (B.17)) by the following formal manipulations: First, CK H and MK HYH are divided by the KJ JYJ diagonal matrix +H(ka )/(!2i),. This produces already CI H. Second, BHY in the (now changed) J H KJY relation (B.17) and in (B.26) is rescaled as !1 , (D.4) BHY "BI HY KJY KJYna J (ka ) HY JY HY such that DI HY and M I HHY emerge. Both manipulations are only of formal nature as they change the “trace-character” of the corresponding matrices.

Appendix E. 1-disk determinant in the semiclassical approximation In Appendix B.2 we have constructed the scattering matrix for the one-disk system (see Eq. (3.8)): H(ka ) Hd . (E.1) (S(ka )) "! K H KKY H(ka ) KKY K H Instead of calculating the semiclassical approximation to its determinant, we instead do so for 1 d ln det S(ka ) , d(k), H 2ni dk

(E.2)

the so-called time delay. Recall that the corresponding T-matrix is trace-class. Thus, according to properties (j) and (c) of Appendix A.2 the following operations are justified:






H(ka ) d H(ka ) 1 d 1 H K H d(k)" tr(ln det S(ka ))" tr K H 2ni dk 2ni H(ka ) dk H(ka ) K H K H a H(ka ) H(ka ) H . H! K " H tr K (E.3) 2ni H(ka ) H(ka ) K H K H Here the prime denotes the derivative with respect to the argument of the Hankel functions. Let us introduce the abbreviation






H(ka ) H(ka ) H! J H s" J (E.4) J H(ka ) H(ka ) J H J H Following Ref. [29], we apply the Watson contour method [64] to (E.3) (see also Section 5 and Appendix F)



a 1 e\ JL a > dl s . d(k)" H s " H K 2ni 2i sin(ln) J 2ni ! K\

(E.5)

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Here the contour C encircles in a counter-clock-wise manner a small semi-infinite strip D which completely covers the real l-axis, but which only has a small finite extend into the positive and negative imaginary l direction. As in Ref. [44], the contour C will be split up in the path above and below the real l-axis such that

 



  





1 >> C e\ JL 1 >\ C e\ JL a dl dl s# s . d(k)" H ! sin(ln) J 2i sin(ln) J 2i 2ni \> C \\ C Then, we perform the substitution lP!l in the second integral so as to get

(E.6)



a 1 >> C e\ JL 1 >> C e> JL d(k)" H ! s! s dl dl 2ni 2i sin(ln) J 2i sin(ln) \J \> C \\ C >> C > a e JL " H 2 s# (E.7) dl dls , J 2ni 1!e JL J \> C \ where we used the fact that s "s . The contour in the last integral could be deformed to pass \J J over the real l-axis since its integrand has no Watson denominator any longer. We will now approximate the last expression semiclassically, i.e., under the assumption ka <1. As the two H contributions in the last line of Eq. (E.7) differ by the presence of the Watson denominator, they will have to be handled semiclassically in different ways: the first will be closed in the upper complex plane and evaluated at the poles of s , the second integral will be performed under the J Debye approximation for Hankel functions. We will now work out the first term. The poles of s in J the upper complex plane are given by the zeros of H(ka ) which will be denoted by ll(ka ) and by H J H the zeros of H(ka ) which we will denote by !lN l(ka ), l"1, 2, 3,2. In the Airy approximation H J H to the Hankel functions, see [80], they are given by Eqs. (4.2) and (4.3):





ll(ka)"ka#e> L(ka/6)ql#2"ka#ial(k)#2 ,

(E.8)

!lN l(ka)"!ka!e\ L(ka/6)ql#2"!ka#i(al(k*a))*#2"!(ll(k*a))* , (E.9) where al(ka ) is defined in [32] and q labels the zeros of the Airy integral (F.27), for details see H J [29,44]. In order to keep the notation simple, we will abbreviate ll,ll(ka ) and lN ,lN l(ka ). Thus H H the first term of Eq. (E.7) becomes finally










e\ JN lL >> C e JL e JlL a  H 2 s "2a # . (E.10) dl Hl 1!e JL J 1!e JlL 1!e\ JN lL 2ni \> C  In the second term of (E.7) we will insert the Debye approximations for the Hankel functions [80]:






 

l n 2 exp $i(x!lGil arccos Gi H(x)& J x 4 n(x!l






2 l H(x)&Gi exp !(l!x#lArcCosh J x n(l!x

for "x"'l , for "x"(l .

(E.11)

(E.12)

 In Appendix F, symmetrized expressions have been Watson transformed. Thus, the corresponding D only has to > cover the real positive l-axis.

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Note that for l'ka the contributions in s cancel. Thus the second integral of Eq. (E.7) becomes H J










a > a >I?H (!2i) d l H dls " H dl (ka!l!l arccos #2 J H 2ni 2ni dk a ka \ \I?H H H



a 1 I?H dl(ka!l#2"! H k#2 , "! H 2 kn \I?H

(E.13)

where 2 takes care of the polynomial corrections in the Debye approximation and the boundary correction terms in the l integration. In summary, our semiclassical approximation to d(k) reads






e JlL  e\ JN lL a d(k)"2a # ! H k#2 . Hl 1!e\ JlL 1!e\ JN lL 2 

(E.14)

Using the definition of the time delay (E.2), we get the following expression for det S(ka ): H ln det S(ka )! lim ln det S(k a ) H  H I




"






I e LJlII ?H akI  e\ LJN lII ?H dkI !i2n #2(i2na ) # I Hl 1!e LJlI?H 1!e\ LJN lII ?H 2  



#2

 I d dkI +!ln(1!e LJlII ?H)#ln(1!e\ LJN lII ?H),#2 , &!2niN(k)#2 dkI l  

(E.15)

where in the last expression it has been used that semiclassically  ll(ka )&  lN l(ka )&a and that I H I H H the Weyl term for a single disk of radius a goes like N(k)"nak/(4n)#2 (the next terms come H H from the boundary terms in the l-integration in Eq. (E.13)). Note that for the lower limit, k P0, we  have two simplifications: First, !H(k a ) K  H d "1;d lim S (k a )" lim KKY KKY  H H(k a ) KKY K  H I I

∀m, m

' lim det S(k a )"1.  H I Secondly, for k P0, the two terms in the curly bracket of Eq. (E.15) cancel. Hence, we finally  obtain the semiclassical result for the determinant of S(ka ) H (1!e\ LJN lI?H) l   e\ L,I“  , det S(ka )P H (1!e LJlI?H) “ l 

(E.16)

which should be compared with expression (4.1) of Section 4. For more details we refer to Appendix F.

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Appendix F. Semiclassical approximation of two convoluted A-matrices In this appendix we introduce the necessary apparatus for the semiclassical reduction of Tr[AK(k)] for the n-disk system where a J (ka ) AHHY"(1!d ) H J H (!1)JYe J?HYH\JY?HHYH (kR ) . J\JY HHY JJY HHY a H(ka ) HY JY H

(F.1)

As usual, a , a are the radii of disk j and j, 14j, j4n, R is the distance between the centers of H HY HHY these disks, and a is the angle of the ray from the origin of disk j to the one of disk j, as measured HYH in the local coordinate system of disk j. The angular-momentum quantum numbers l and l can be interpreted geometrically in terms of the positive- or negative-valued distances (impact parameters) l/k and l/k from the center of disk j and disk j, respectively, see Figs. 12—14. The semiclassical approximation of the convolution of two kernels AHHYAHYH contains all (but one) essential steps JY JJY JYJ necessary for the semiclassical reduction of the quantum cycles and traces themselves. What is missing is the mutual interaction between successive saddles of the quantum itinerary, including the final saddle which “closes” the semiclassical open itinerary to a period orbit. This is studied in Section 5. The idea here is to construct the convolution of the two kernels AHHYAHYH and then to compare it JY JJY JYJ  — in the case jOj — with the single kernel AHH (see Eq. (F.1)) in the semiclassical limit, where the JJ Hankel function H (kR ) is evaluated in the Debye approximation (E.11) to leading order [80]. HH J\J Let us start with a J (ka ) AHHYAHYH"(1!d )(1!d ) H J H (!1)Je J?HYH\J?HYH JJY JYJ HHY HH a H (ka ) H J H JY > J (ka ) ; (!1)JY JY HY H (kR )H (kR )e JY?HHY\?HHY HYH H(ka ) J\JY HHY JY\J JY HY JY > J (ka ) "¼HH (!1)JY JY HY H (kR )H (kR )e JY?HHY\?HHY JJ HYH H(ka ) J\JY HHY JY\J JY HY JY

(F.2)

> J (ka ) "¼HH (!1)JYd(l) JY HY [H (kR )H (kR )e JY?HHY\?HHY JJ HYH H(ka ) J\JY HHY JY\J JY HY JY (F.3) #H (kR )H (kR )e\ JY?HHY\?HHY] , HYH J>JY HHY \JY\J where we have introduced the abbreviations ¼HH for the l-independent pieces and the weight JJ factor d(l)"1 for lO0 and d(0)"1/2. We have symmetrized this expression with respect to l for simplicity using that J (ka )"(!1)JYJ (ka ) and H (ka )"(!1)JYH(ka ), valid for l integer. \JY HY JY HY \JY HY JY HY We will furthermore abbreviate *a ,a  !a where 04*a (2n. However, in order to be HHY HY HY H HY able to get three domains for this angle (which we will later identify with the three different cases: specular reflection from disk j to the right (see Fig. 12), to the left (see Fig. 13) and the ghost “tunneling” case (see Fig. 14) we define *I a ,a  !a !pn where p"0, 2, 1, respectively, and HHY HYN H HY balance this by multiplying accordingly the right-hand side of Eq. (F.3) with the phase factor

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Fig. 12. The geometry belonging to a trajectory, jPjPj, specularly reflected to the right. Shown are the geometrical path (full line) and the shortest allowed right handed (dashed line) and left handed (full line) creeping paths. All paths originate from an “impact parameter” circle of radius "l/k" centered at disk j, then contact the surface of disk j (of radius a) and end on an “impact parameter” circle of radius "l/k" centered at disk j dash. Note that the impact radii do not have to be equal to the disk radii, a and a.

Fig. 13. The same as in Fig. 12 for the case of a specular reflection to the left.

(!1)JYN which is only nontrivial for p"1. We denote this nontrivial phase by (!1)\JYNY where p"p for p"1 and zero otherwise. The three choices for the value of p are still equivalent at this stage. F.1. The Watson contour resummation It will be shown that (F.3) contains in the semiclassical limit ka <1 — depending on the HY pertinent angles and distances — a classical path or a possible ghost path and two creeping paths, all starting under the impact parameter l/k with respect to the origin of disk j and ending at an impact

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Fig. 14. The geometry belonging to a ghost trajectory jPjPj which passes straight through the disk j (of radius a). Shown are the geometrical ghost path (full line and short-dashed line), and the shortest allowed right handed (dashed line) and left handed (full line) creeping paths. All paths originate from an “impact parameter” circle of radius "l/k" at disk j and end on an “impact circle” of radius "l/k" centered at disk j. Note that the impact radii do not have to be equal to the disk radii, a and a.

parameter l/k relative to the origin of disk j. This calculation will be performed with the help of the Watson contour method [64,29], i.e., under the replacement



> 1 1 (!1)JY\NYd(l)X " dl e\ JYLNYX . JY 2i JY sin(ln) !> JY

(F.4)

Here J (ka ) X , JY HY [H (kR )H (kR )e JYDI ?HYN#H (kR )H (kR )e\ JYDI ?HYN] JY H(ka ) J\JY HHY JY\J HYH J>JY HHY \JY\J HYH JY HY J (ka ) (F.5) , JY HY ½ H(ka ) JY JY HY stands for the integrand and ½ for the symmetrized square bracket in Eq. (F.3). The contour C is JY > the boundary of a narrow semi-infinite strip D which completely covers the positive real l-axis. > C has been chosen in such a way that it encircles in a positive sense all poles of the Watson > denominator sin(ln) at l"1, 2, 3,2 exactly once (see Fig. 15). At l"0 the weight factor d(0)"1/2 is taken into account by a principle value description, i.e., by the average of the two contour integrals whose paths cross the real l-axis symmetrically just to the right and left of the point l"0. A precondition on the validity of the Watson replacement is the analyticity of X in JY this strip D . This is the case if D has been chosen narrow enough in the imaginary l direction > > that the poles of X , the zeros of the Hankel function H(ka ) and H (ka ) lie either above or JY JY HY \JY HY below the strip in the complex l-plane [for k real and positive]. The contour can now be split into four parts:







e\ JYLNY > C e\ JYLNY  e\ JYLNY dl X "# X #P X dl dl sin(ln) JY sin(ln) JY sin(ln) JY !> >> C > C \ C e\ JYLNY \ C e\ JYLNY dl dl X # X , #P sin(ln) JY sin(ln) JY  \ C





(F.6)

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Fig. 15. The contour C in the complex l-plane. >

where P2 denotes the principal value integration. The next step in the evaluation is a shift of the contour paths below the real l-axis to paths above this axis by the substitution lP!l in the corresponding integrals:







> C e\ JYLNY > C e\ JYLNY e JYLNY dl X "! X ! X dl dl sin(ln) JY sin(ln) JY sin(ln) \JY > ! > C \> C



#P



1 [e\ JYLNYX !e JYLNYX ] . dl JY \JY sin(ln) > C

(F.7)

We insert X "(J (ka )/H(ka ))½ and use that for general complex-valued angular momenta JY JY HY JY HY JY l, the transformation laws for the Hankel and Bessel functions read H (ka )"e JYLH(ka ) , \JY HY JY HY

(F.8)

H (ka )"e\ JYLH(ka ) , \JY HY JY HY

(F.9)

such that J (ka ) H(ka ) J (ka ) HY , \JY HY " JY HY !ie\ JYLsin(ln) JY H(ka ) H (ka ) H(ka ) JY HY JY HY \JY HY

(F.10)

H (ka ) H(ka ) HY . e JYL \JY HY "e\ JYL JY H (ka ) H(ka ) \JY HY JY HY

(F.11)

Furthermore, by definition, we have ½ "½ . Thus \J >J 1 2i



 

e\ JYLNY e\ JYLNY J (ka ) 1 > C JY HY ½ dl X "! JY sin(ln) sin(ln) H (ka ) JY 2i \> C !> JY HY





H(ka ) J (ka ) 1 >\ C HY !2d JY HY ½ , dle\ JYL\NY JY # P NYH(ka ) JY H(ka ) 4 \> C JY HY JY HY

(F.12)

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where (F.11) and the symmetry of ½ has been used in order to reflect the resulting sin(ln)JY independent integrals at l"0 such that they combine to the symmetric integral: #P











>\ C > C  \ C >\ C dl2, dl2#P dl2#P dl2# dl2 . \> C \> C \> C  \ C

(F.13)

Furthermore, in the case p"1, the identity e JYL"e\ JYL#2i sin(ln) has been employed in order to group the terms resulting from the paths below the real l-axis into the terms belonging to the paths above this axis. Altogether we have so far that

 

1 >> C e\ JYLNY J (ka )  JY HY [H (kR )H (kR )e JY*I ?HYN dl AHHYAHYH"¼HH ! HYH JJ JJY JYJ 2i sin(ln) H(ka ) J\JY HHY JY\J \> C JY HY JY #H (kR )H (kR )e\ JY*I ?HYN] J>JY HHY \JY\J HYH H(ka ) J (ka ) 1 >\ C HY !2d JY HY # P dle\ JYL\NY JY NY H(ka ) H(ka ) 4 \> C JY HY JY HY









;[H (kR )H (kR )e JY*I ?HYN#H (kR )H (kR )e\ JY*I ?HYN] . J\JY HHY JY\J HYH J>JY HHY \JY\J HYH (F.14)

Note that both integrals on the right-hand side exist separately. The one with the Watson “sin”-denominator is finite, because the zeros of the sin(ln) function in the denominator are avoided by the #ie prescription and because the rapid convergence of the ratio Jl(ka )/H(ka ) HY JY HY counterbalances the diverging R and R -dependent Hankel functions, as long as the disks do HHY HYH not touch. This is basically the same argument by which one can show the existence of the sum on the left-hand side. However, we do not have to prove this separately, because we already know from Appendix C that A is trace-class. The existence of the principal value integral follows from the symmetric nature of the path and of the integrand (see below for more details). It will be shown that the term with the Watson “sin”-denominator, !1/[2i sin(ln)]"  e L>JYL, will lead in the semiclassical reduction to paths with left handed and right handed L creeping sections around the middle disk j [where the index n counts further complete turns around this disk]. On the other hand the term without this denominator will give either a semiclassical path specularly reflected from the disk j (to the left or right) or a ghost path passing undisturbed through disk j. F.2. The integration paths Thus the third step is to close the path of the “sin”-dependent integral in the upper complex l-plane (Fig. 16). For given values of a , a  , l/k and l/k, i.e., for a given geometry, this selects which value of HHY H HY p has to be inserted into Eq. (F.14). The reason is that the closing of contour will be performed under the condition that the corresponding semicircular integration arc vanishes, such that the integral is solely given by its residua which are here the zeros ll (l"1, 2, 3,2) of the Hankel function H(ka ) in the upper complex l-plane. At “optical boundaries” this clear separation is JY HY

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Fig. 16. The path for the “sin”-dependent integral. The lines denoting the zeros of H(ka) in the upper and of H(ka) in J J the lower complex l-plane are shown as well.

not possible [31,35]. This is the realm of “penumbra” scattering. In order not to be plagued by these difficulties, we exclude geometries which allow for grazing classical paths from further consideration. In the Airy approximation to leading order, the zeros of these Hankel functions are given by Eq. (4.2), modulo O([ka ]\) corrections. A necessary condition for the vanishing of the semicirHY cular arc, which, in turn, determines the choice among the three values for p, is that the total angle b of the integrand’s “creeping exponential” exp+ilb(l), (including the terms resulting from the Hankel functions) must be positive [and large enough to exclude the penumbra region in the “optical shadow” and “optically lit” region] for l given by Eq. (4.2), i.e., l+ka . A violation of HY this condition would correspond semiclassically to a negative creeping path which has to be excluded for physics reasons: during the creeping the modulus of the wave has to decrease and not increase [29], as tangential rays are continuously ejected, while the path creeps around a convex bending. The positivity of the creeping exponential actually only guarantees the vanishing of the integrand on the arc to the left of the line of zeros ll of the Hankel funcion H(ka ) and to the right JY HY of the line of zeros ll of the Hankel function H(ka ) in the upper complex l-plane. The JY HY vanishing of the remainder of the arc is a consequence of the strongly decreasing J (ka )/H(ka ) JY HY JY HY term which dominates the behavior of the integrand to the right of the ll’s and the left of the ll’s. Whereas the ll line does not cause any problems, the ll line is potentially dangerous as the Hankel function in the denominator is vanishing. The remedy is to put the path right in between two adjacent zeros [29]. As already mentioned, the “sin”-independent integral is symmetric in path and integrand. Because of this the path can be symmetrically deformed as follows (the preserved symmetry takes care of the original principal value description): It is replaced by an arc from !R#ie to #iR(1#id), a straight line from #iR(1#id) to !iR(1#id) and finally a symmetric arc (to

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Fig. 17. The original and the deformed contour of the “sin”-independent integral for the case p"0. The lines of zeros are as in Fig. 16.

the first one) from !iR(1#id) to #R!ie, where in the case p"0, the parameter d is chosen positive and small enough such that "Re l";ka . [This allows later to use the Debye approximaHY tion of the Hankel functions, H(ka ) and H(ka ).] See Fig. 17. The deformation of the path is JY HY JY HY justified as the sum of the new path and the (negatively traversed) original one do not encircle any singularities of the integrand. Since the integrand is symmetric under the exchange lP!l, the integrals over the two symmetric arcs completely cancel, such that only the straight line segment from #iR(1#id) to !iR(1#id) gives a contribution. This expression is finite since it is symmetric under lP!l and since the integrand vanishes rapidly for "l"PR, as long as the slope of the straight line section is negative. In the case p"1 the parameter d has to be chosen negative since the integrand only vanishes rapidly for a straight line section with positive slope (see Fig. 18). The reason for this difference is the presence of the ratio H(ka )/H(ka ) in the first case JY HY JY HY which is replaced by (H(ka )/H(ka ))!2J (ka )/H(ka )"!1 in the case p"1. (See also below JY HY JY HY JY HY JY HY the discussion of the pertinent Fresnel integrals in the semiclassical saddle-point approximation.) As mentioned, the actual result depends on the concrete geometry and on the impact parameters l/k and l/k, i.e., on the value of *a "a  !a , on the value of p and the angles (l, l),arHHY ! HY H HY ccos((l$l)/kR ) and  (l, l),arccos((l$l)/kR ) resulting from the asymptotic Debye apHH ! HYH proximation of the Hankel H (kR ) and H (kR ), respectively. Since p can take three HYH J!JY HHY !JY\J values there exist three mutually exclusive alternatives: The first one corresponds to p"0 and 0(2nG*a ! (l, ka )!  (l, ka )42n [this HY ! HY ! HY geometry allows only a classical path from disk j (under the impact parameter l/k) to disk j (under the impact parameter l/k) which is specularly reflected to the right at disk j, see Fig. 12]:

 

J  (ka )  HY [H (kR )H (kR )e JYL>*?HY>LL dl J AHHYAHYH"¼HH JJ HYH JJY JYJ H (ka ) J\JY HHY JY\J J HY L w!> JY #H (kR )H (kR )e JYL\*?HY>LL] J>JY HHY \JY\J HYH

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Fig. 18. The contour of the “sin”-independent integral in the case p"1 corresponding to a ghost. Note that the lines of zeros from Fig. 16 are absent.



1 \ > B H(ka ) HY [H (kR )H (kR )e JY\L>*?HY # dl JY HYH 4 H(ka ) J\JY HHY JY\J > B JY HY



#H (kR )H (kR )e JY\L\*?HY] . HYH J>JY HHY \JY\J

(F.15)

The second case is p"2 and 0(*a ! (l, ka )!  (l, ka )42n and 0(4n!*a ! HY \ HY \ HY HY

(l, ka )!  (l, ka )42n (this geometry allows only a classical path from disk j to disk j which > HY > HY is specularly reflected to the left at disk j, see Fig. 13):

 

 J (ka ) dl JY HY [H (kR )H (kR )e JY*?HY>LL\L AHHYAHYH"¼HH HYH JJY JYJ JJ H(ka ) J\JY HHY JY\J JY HY L w!> JY #H (kR )H (kR )e JYL\*?HY>LL] HYH J>JY HHY \JY\J



1 \ > B H(ka ) HY [H (kR )H (kR )e JY\L>*?HY # dl JY HYH 4 H(ka ) J\JY HHY JY\J > > B JY HY



#H (kR )H (kR )e JYL\*?HY] . HYH J>JY HHY \JY\J

(F.16)

The third alternative is p"p"1 and 0(*a ! (l, ka )!  (l, ka )42n and 0(2n! HY \ HY \ HY *a ! (l, ka )!  (l, ka )42n (this geometry allows only a “classical” path from disk j HY > HY > HY [under the impact parameter l/k] to disk j [under the impact parameter l/k] which goes directly

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through disk, j, see Fig. 14):

 

 J (ka ) dl JY HY [H (kR )H YY(kR ) e JY*?HY>LL\L AHHYAHYHYY "¼HHYY HYHYY JJY JYJJYY JJYY H (ka ) J\JY HHY JY\J JY HY L w!> JY #H (kR )H YY(kR ) e> JYL\*?HY>LL] J>JY HHY \JY\J HYHYY 1 \ \ B ! dl[H (kR )H YY(kR ) e JY\L>*?HY HYHYY J\JY HHY JY\J 4 \ B





#H (kR )H YY(kR ) e JYL\*?HY] . HYHYY J>JY HHY \JY\J

(F.17)

Note that Eq. (F.17) can also be derived from the “#*a part” of Eq. (F.15) plus the “!*a HY HY part” of Eq. (F.16), by a rearrangement of the corresponding creeping and geometrical terms, i.e., by the addition of an extra term of smaller creeping length than the smallest one before and the subtraction of the very same piece from the geometrical terms. The contour integrals of these three alternatives are evaluated at the zeros ll of the Hankel functions H(ka ), such that JY HY J (ka )  J l(ka ) HY J dl JY HY ½ e L>JYL"2ni j ½ l e L>JlL . (F.18) J J H (ka ) H (ka )" l w!> JY HY HY JYJl  jl JY Up to this point all expressions are still exact. The steps introduced so far just served the purpose of generating the three distinct “classically” allowed angular domains and of transforming the original expression (F.3) into a form ready for the semiclassical approximation. This will be taken next under the condition Re ka <1. Note that this inequality automatically induces Re kR <1 HY HHY and Re kR <1. HYHYY The contour integral (which, in fact, is now a sum over the residua) and the straight line integral are now treated semiclassically in different ways.



F.3. Semiclassical approximation of the straight-line integrals The straight line integrals will be evaluated in the saddle-point approximation at a saddle l where the path crosses the real axis. For evaluating the saddle-point integral, the Debye Q approximation (E.11) will be inserted for the given Hankel functions. For the first and second alternative, an internal consistency check on the validity, is the condition "l /k"(a which in Q HY physics terms means that the impact parameter at disk j has to be smaller than the disk radius a . H For the third alternative the weaker conditions "(l !l)/k"(R and "(l !l)/k"(R are Q HHY Q HYHYY sufficient: The difference in the impact parameters at successive disks should be smaller than the distance between the disks. Its validity is guaranteed by the triangular condition. The saddle-point integral is evaluated by expanding the exponents of the Debye approximates to second order and a successive integration. The reflection angle is determined by the saddle-point condition itself, the geometrical length of the path can be read off from the total exponent at zeroth saddle-point order, i.e. from the sum of square root terms of the Debye exponents divided by k. Under the Gauss’ integration the second-order fluctuations about the saddle determine the stability factor 1/(R and, together with the already present phases, the overall phase. 

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The straight line integral of the first two alternatives corresponds then to the standard geometrical path from disk j [under impact parameter l/k] to disk j [under impact parameter l/k] where there is a specular reflection from the boundary of the disk a either to the right for the first HY alternative (Fig. 12) or to the left for the second alternative (Fig. 13). The slope of the path of this straight line integral, which asymptotically is i(1#id), has to join smoothly the slope of the saddle-point path. This condition determines the sign of the slope. The saddle-point integral, which is of the Fresnel-type, results when the pertinent exponents of the Debye-approximated Hankel functions are expanded to second order around the saddle point l : Q





\ L




2 1 1 ddl exp !i (dl) ! 2 ((ka)!l  ((kR )!(l!l ) \\ L Q HHY Q



1 ! ((kR )!(l !l) HYHYY Q

.

(F.19)

Here dl"l!l is the integration variable. By the substitution dl"e\ Lx this integral becomes Q a standard Gauss’ integral



e\ L

>

\

dx e\V@"e\ L(2n/b

(F.20)

with b"2+(ka)!l ,\!+(kR )!(l!l ),\!+(kR )!(l !l),\ positive as Q HHY Q HYHYY Q a (R !a , R !a . The right-hand side of Eq. (F.20) together with the prefactors and HY HHY H HYHYY HYY phases of the Debye-approximated Hankel functions determine the remaining terms (see below). Perturbative higher-order -corrections (see Refs. [38—40]) result here from higher-order terms in the Debye approximation through expansion terms proportional to (1/kr)L"( /pr)L (with r"a , R or R ) and from the integration of polynomial second- and higher-order ( l/pr)L HY HHY HYHYY terms under the Gauss-type saddle-point integral. The polynomials are generated by a consistent expansion of all prefactors and exponential terms of the Debye series up to a given order. The Debye series reads



2 H(z)& J nz







!




1 1 e! (X\J8 J JX8 L 1Gi l  8z 1! z

9 128z



l

1



5 # 24z l 1! z







 

l l 1 l 231 1155 # # #O , ,2 l  3456z l  l 576z z z 1! 1! 1! z z z 1













l  1! z

,

(F.21)

where the upper signs apply for the Hankel function of first kind and the lower ones for the Hankel function of second kind. Note that this expansion is of asymptotic nature and therefore induces the asymptotic nature of the -expansion itself. Here we will limit our discussion just to the leading term, such that no -corrections arise.

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99

F.4. Semiclassical approximation of the residua sum In the contour integral (or residua sum) the Debye approximation is not justified for the ratio J l(ka ) HY J (F.22) w H(ka )" wJY JY HY JYJl since Re llKka . It is still valid, however, for the R and R -dependent Hankel functions, since HY HHY HYHYY R 'a #a and R 'a #a . Instead, the a -dependent cylindrical functions are evaluated HHY H HY HYHYY HY HYY HY under the Airy approximation. The latter step is justified as we evaluate the ratio (F.22) at the zeros ll of the Hankel function H(ka). In the Sommerfeld representation the contour of a Hankel function H(z) has normally J J two saddles [29]. For Re l;z or Re l




6  2 A(q)#O((ka)\) . H(ka)& e> L J ka n Here

(F.24)





q,e\ L

6  (l!ka)#O((ka)\) , ka

(F.25)

q,e> L

6  (l!ka)#O((ka)\) ka

(F.26)

are the zeros of the Airy integral [29]



A(q),



dq cos(qq!q)"3\nAi(!3\q) , (F.27)  with Ai(z) being the standard Airy function [80]; approximately, ql+6[3n(l!1/4)]/2. Thus Eq. (4.2) arises. Note that this is the first term in an asymptotic expansion where the corrections are of relative order O((ka)\)&O( ) as k"p/ . The first correction to the Airy approximation is therefore more important than the first correction term to the Debye approximation as the latter term scales as O( ). Up to order O((ka)\), the zeros ll read as [67]












ka  6  ql 1 ql ql!e\ L ! 1! ll&ka#e L 180 70ka 30 6 ka #e L

6  1 29ql 281ql ! #2 . 180 ) 6 ka 3150 6

(F.28)

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The l-derivative of the Hankel function H(ka) at l"ll has the form [67] J L 2e\   6  e L 6 ql 6  37 ql  j H(ka)" l& !e\ L A(ql) 1!  J JJ ka ka 630 6 jl n 5 ka 6






!






 

6  37 ql  563 ! #2 , ka 36 ) 5 ) 630 5 ) 630 ) 9 6




(F.29)

where A(ql) is the derivative of the Airy integral A(q) at the position ql [29]. The Airy approxima(ka) reads as tion to the Bessel function J l(ka)"H  Jl J L e 6  L 6  ql !e\ L 29 6  ql  A(q J l(ka)&  l ) 1!e  14 ka J ka n ka 45 45













 

!

31 ql  6  1 ! #2 . 6 45 ka 45 ) 7

(F.30)

Applying the Wronsky-relation A(z)A(z e!L )!A(z) A(z e!L )"!L e8L  one gets for  (with A(q z"q l l )"0) n e\ L . (F.31) A(q l )" 6 A(q l ) Thus, under the Airy approximation, each of the residua in Eq. (F.18) becomes J l(ka ) in J HY 2ni ½ le L>JlL"!e\ LCln\(ka ) ½ le L>JlL J HY w H(ka )" 2 J wJY JY HY JYJl with the coefficient Cl"Cl(ka) n e L ql Cl(ka)"2\3\ 1#  18 5 A(ql)








(F.32)




 

6  1 ql  6  # ka 12 ) 14 5 ka

29 ql  281 1 ! #2 . # (ka) 9 ) 25 ) 14 6 ) 81 ) 14 5

(F.33)

The values of the first zeros ql and the corresponding coefficients Cl, truncated at order O(+ka,\)"O( ), can be found in Ref. [29] and are listed in Table 1. Table 1 The first zeros ql of the Airy Integral A(q) and the corresponding coefficients Cl of the creeping wave under Dirichlet boundary conditions in the leading Airy approximation l

ql

Cl

1 2 3 4

3.372134 5.895843 7.962025 9.788127

0.91072 0.69427 0.59820 0.53974

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We will limit our discussion to the Airy expansion of this leading order, i.e., ll as in Eq. (4.2) and Cl as given by the first term of Eq. (F.33), since all the higher terms vanish at least as fast as  and

, respectively, in the limit P0. dependent Finally, the Debye approximation (E.11) is inserted in ½ l for the R and R HHY HYHYY J Hankel functions. The two square root terms in the exponential of the Debye approximate, e.g. ((kR )!(l!ll), etc., under the approximation llKka , give the length of the two straight HY HHY sections of the path times k. All exponential terms proportional to ll, e.g., ll arccos(2), nnll, correspond to the creeping sections (of mode number l) of the path. The latter include, of course, the creeping tunneling suppression factor linked to the imaginary part of the ll. The product of the two Debye prefactors is just the stability of the path times !i2/n. The latter factor cancels the exposed factor in Eq. (F.32). In summary, the residua of the contour integrals in the Airy approximation correspond to those paths from disk j [under impact parameter l/k] to disk j [under impact parameter l/k] that have straight sections and circular creeping sections of mode number l which join tangentially at the surface of disk j. For the first term of ½ l, the creeping is in the left hand sense and for the second JY term in the right hand sense around disk j. The sum over n counts n further complete creeping turns around this disk. Note that the smallest creeping angle is less than 2n, but larger than zero (see Figs. 12—14). F.5. Resulting convolutions The first alternative (Fig. 11) reads now   AHHYAHYHYY&!¼HHYY e\ Ln\Cl(ka ) HY JJY JYJYY JJYY l  L JY exp i((kR )!(l!ll) exp i((kR )!(ll!l) HHY HYHYY ; [(kR )!(l!ll)] [(kR )!(ll!l)] HHY HYHYY ;e\ J\Jl  J\JlI0HHY \ Jl\JYY  Jl\JYYI0HYHYY > Jl*?HY>L>L



exp i((kR )!(l#ll) exp i((kR )!(ll#l) HYHYY HHY # [(kR )!(l#ll)] [(kR )!(ll#l)] HHY HYHYY



;e\ J>Jl  J>JlI0HHY > Jl>JYY  \Jl\JYYI0HYHYY > Jl\*?HY>L>L #¼HHYY(2/ne\ Le\ J  J\J I0HHY > JYY  J Q\JYYI0HYHYY

JJYY  Q





;exp i((kR )!(l!l )#i((kR )!(l !l) HHY Q HYHYY Q





!2i((ka )!l [(ka )!l ] 2[(kR )!(l!l )] HY Q HY Q HHY Q ;[(kR )!(l !l)] HYHYY Q HY HHY \ . !(l!l )]#[(kR )!(l !l)]) Q HYHYY Q



(F.34)

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Here ll is given as in Eq. (4.2) and Cl as in Eq. (F.33). The value of l follows from the saddle-point Q condition *a #2 arccos[l /ka ]!arccos[(l !l)/kR ]!arccos[(l !l)/kR ]"0 (F.35) HY Q HY Q HHY Q HYHYY which fixes the scattering angle h ,arcsin[l /ka ] as HY Q HY h "*a #arcsin[(l !l)/kR ]#arcsin[(l !l)/kR ] . (F.36) HY HY Q HHY Q HYHYY One might wonder why there do not appear two different geometrical segments corresponding to the two terms of the straight line integral in Eq. (F.15). The answer is that the second term of this integral gives the same contribution as the first one, since the values of the pertinent saddles just differ by a minus sign. [In fact, it is easy to show with the help of the transformation laws (F.8) and (F.9) that the second term of the straight line integrals is identical to the first one.] The effective radius R belonging to Eq. (F.34) results from the prefactors of the Debye-approximated R - and  HHY R -dependent Hankel functions, combined with the r.h.s. of (F.20), and reads HYHYY 2d d !o (d #d ) HY HHY HYHYY (F.37) R " HHY HYHYY  o HY with d ,(R !(+l!l ,/k) , HHY HHY Q d ,(R !(+l !l,/k) , HYHYY HYHYY Q o ,(a !(l /k) . HY HY Q This should be compared with effective radius generated by the standard evolution curvatures in the corresponding classical problem (see Eqs. (5.56) and (5.57))

(F.38) (F.39) (F.40) of the

K R "¸ “ (1#i ¸ ). (F.41)   G GG> G Here ¸ is the length of the leg between the ith and the (i#1)th reflection. The quantity i is the GG> G curvature just after the ith reflection, i.e., 1 2 i" # , (F.42) G i\ #¸ r cos

G\ G\G G G where, in turn, r and are the local radius of curvature and the deflection angle at the ith G G reflection. [Note that i\"0.] By identifying ¸ "d !o , ¸ "d !o , r "a and   HHY HY  HYHYY HY G HY

"h (such that o "a cos h ) one can easily show that R and R give the same result. G HY HY HY HY   The result of the second alternative (Fig. 13) is as in Eqs. (F.34) and (F.35) with *a replaced by HY *a !2n. The third alternative (Fig. 14) reads as HY   AHHYAHYHYY&!¼HHYY e\ Ln\Cl(ka ) HY JJY JYJYY JJYY l  L JY exp i((kR )!(l!ll) exp i((kR )!(ll!l) HHY HYHYY ; [(kR )!(l!ll)] [(kR )!(ll!l)] HHY HYHYY



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;exp+!i(l!ll)arccos[(l!ll)/kR ]!i(ll!l)arccos[(ll!l)/kR ]#ill(*a HHY HYHYY HY exp i((kR )!(l#ll) exp i((kR )!(ll#l) HYHYY HHY #(2n!1)n),# [(kR )!(l#ll)] [(kR )!(ll#l)] HHY HYHYY ;exp +!i(l#ll)arccos[(l#ll)/kR ]#i(ll#l)arccos[(!ll!l)/kR ] HHY HYHYY





2 e\ Lexp+!il arccos[(l!l )/kR ] #ill(!*a #(2n#1)n), #¼HHYY HY JJYY n Q HHY exp+i((kR )!(l!l )#i((kR )!(l !l), HHY Q HHY Q . #ilarccos[(l!l)/kR ], Q HYHYY ([(kR )!(l!l )]#[(kR )!(l !l)] HHY Q HYHYY Q

(F.43)

Here l has to satisfy the saddle-point condition Q *a !arccos[(l !l)/kR ]!arccos[(l !l)/kR ]"0 . (F.44) HY Q HHY Q HYHYY Again, the two terms in the straight line integral of Eq. (F.17) give the same contribution, as the saddle l  of the latter term is !l  of the first one. The minus sign in front of the straight line Q Q integral is cancelled by an additional minus sign [relative to alternative one or two] resulting from the positive slope of the straight-line section (see Fig. 18) and the corresponding changes in the Fresnel integral



e\ L








\ \ B 1 1 1 # ddlexp #i (dl) 2 ((kR )!(l!l ) ((kR )!(l !l) \ B HHY Q HYHYY Q

.

(F.45) The latter, by the substitution dl"e Lx, becomes a negatively transversed Gauss’ integral



e\ L

\

dx e\V@"!e\ L(2n/b ,

(F.46)  where b"+(kR )!(l!l ),\#+(kR )!(l !l),\. In fact, all dependence of the disk HHY Q HYHYY Q j is finally gone from this expression. If the third alternative exists, the pertinent straight line integral corresponds to a “ghost” segment starting at disk j [under the impact parameter l/k] and ending at disk j [under the impact parameter l/k] which is equivalent to the corresponding geometrical segment of the direct term AHHYY ( jOj). Because of the angular conditions, specified JJYY before Eq. (F.17), the ghost path has to cut disk j, i.e. the modulus of the impact parameter l /k has Q to be smaller than the disk radius a (see Fig. 14). HY F.6. Ghost segment Let us now discuss the “ghost” segment, i.e., the non-creeping terms of Eq. (F.43). The ghost cancellation presented here is, of course, related to Berry’s work on the ghost cancellation for periodic orbits in the Sinai billiard, see Ref. [49]. However, here the calculation is based on

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Watson’s method which specifies the integration paths, the signs of the ghost contributions and encodes the geometries (the choice of the three alternatives for p) into the creeping orbits. After restoring ¼HHYY it reads JJYY a J (ka ) ghostJJYY (l )&(1!d )(1!d ) H J H (!1)JYYe J?HYYH\JYY?HHYY HHYY Q HHY HHYY a H(ka ) HYY JYY HYY 2 exp+i((kR )!(l!l )#i((kR )!(l !l), HYHYY Q HHY Q e\ L ; n ([kR )!!l )]#[(kR )!(l !l)] Q HYHYY Q HHY ;exp[#il(a !a !arccos[(l!l )/kR ])] HYH HYYH Q HHY ;exp[!il(a !a !arccos[(l !l)/kR ])] (F.47) HYHYY HHYY Q HYHYY with



a

!a #arcsin[(l !l)/kR ]#arcsin[(l !l)/kR ]"n (F.48) HYYHY HHY Q HHY Q HYHYY which is equivalent to condition (F.44). As this saddle-point condition implies that the impact parameter l /k at disk j lies on the straight line joining the impact parameter l/k at disk j, with the Q impact parameter l/k at disk j, the following relation between the lengths of the segments on this line holds (F.49) (R !+(l!l )/k,#(R !+(l !l)/k,"(R !+(l!l)/k, , HYHYY Q HHYY HHY Q i.e., the length of the straight line from the impact parameter l/k to the impact parameter l/k is the sum of the lengths from l/k to l /k and from l /k to l/k (see Fig. 14). Q Q The “ghost” segment (F.47) should be compared with Eq. (F.1), in the semiclassical approximation (E.11), for the Hankel function H (kR ) J\JYY HHYY 2 exp i((kR )!(l!l) a J (ka ) HHY e\ L AHHYY&(1!d ) H J H (!1)JYYe J?HYYH\JYY?HHYY JJYY HHYY a H(ka ) [(kR )!(l!l)] n HHYY HYY JYY HYY ;exp+!i(l!l)arccos[(l!l)/kR ], . (F.50) HHYY Condition (F.49) implies that the lengths and stabilities of the ghost segment (F.47) and of the direct path (F.50) are the same. The comparison of the phases implies the relations



n/2"a !a #arccos[(l !l)/kR ]#arcsin[(l!l)/kR ] , (F.51) HYH HYYH Q HHY HHYY n/2"a !a #arccos[(l !l)/kR ]#arcsin[(l!l)/kR ] (F.52) HHYY HYHYY Q HYHYY HHYY which are valid under the condition (F.48). Thus, we finally have in the semiclassical approximation ghostJJYY (l ),(AHHY AHYHYY ) KAHHYY HHYY Q JJYY   JJYY under the condition, of course, that the saddle l satisfies Eq. (F.48). Q Appendix G. Figures of three-disk resonances Figs. 19—27 show the three disk resonances.

(F.53)

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Fig. 19. The A resonances of the three-disk system with R"6a. The exact quantum-mechanical data are denoted by  plus signs. The semiclassical ones are calculated up to 1st order in the curvature expansion and are denoted by crosses: (a) Gutzwiller—Voros zeta function (2.1), (b) dynamical zeta function (2.2), (c) quasiclassical zeta function (2.3).

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Fig. 20. The A resonances of the three-disk system with R"6a. The exact quantum-mechanical data are denoted by  plus signs. The semiclassical ones are calculated up to 2nd order in the curvature expansion and are denoted by crosses: (a) Gutzwiller—Voros zeta function (2.1), (b) dynamical zeta function (2.2), (c) quasiclassical zeta function (2.3).

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Fig. 21. The A resonances of the three-disk system with R"6a. The exact quantum-mechanical data are denoted by  plus signs. The semiclassical ones are calculated up to 3rd order in the curvature expansion and are denoted by crosses: (a) Gutzwiller—Voros zeta function (2.1), (b) dynamical zeta function (2.2), (c) quasiclassical zeta function (2.3).

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Fig. 22. The A resonances of the three-disk system with R"6a. The exact quantum-mechanical data are denoted by  plus signs. The semiclassical ones are calculated up to 4th order in the curvature expansion and are denoted by crosses: (a) Gutzwiller—Voros zeta function (2.1), (b) dynamical zeta function (2.2), (c) quasiclassical zeta function (2.3).

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Fig. 23. The A resonances of the three-disk system with R"6a. The exact quantum-mechanical data are denoted by  plus signs. The semiclassical ones are calculated up to 5th order in the curvature expansion and are denoted by crosses: (a) Gutzwiller—Voros zeta function (2.1), (b) dynamical zeta function (2.2), (c) quasiclassical zeta function (2.3).

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Fig. 24. The A resonances of the three-disk system with R"6a. The exact quantum-mechanical data are denoted by  plus signs. The semiclassical ones are calculated up to 6th order in the curvature expansion and are denoted by crosses: (a) Gutzwiller—Voros zeta function (2.1), (b) dynamical zeta function (2.2), (c) quasiclassical zeta function (2.3).

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Fig. 25. The A resonances of the three-disk system with R"6a. The exact quantum-mechanical data are denoted by  plus signs. The semiclassical ones are calculated up to 7th order in the curvature expansion and are denoted by crosses: (a) Gutzwiller—Voros zeta function (2.1), (b) dynamical zeta function (2.2), (c) quasiclassical zeta function (2.3).

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Fig. 26. The A resonances of the three-disk system with R"6a. The exact quantum-mechanical data are denoted by  plus signs. The semiclassical ones are calculated up to 8th order in the curvature expansion and are denoted by crosses: (a) Gutzwiller—Voros zeta function (2.1), (b) dynamical zeta function (2.2), (c) quasiclassical zeta function (2.3).

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Fig. 27. The A resonances of the three-disk system with R"6a. The exact quantum-mechanical data are denoted by  plus signs. The semiclassical resonances are calculated up to 12th order in the curvature expansion and are denoted by crosses: (a) Gutzwiller—Voros zeta function (2.1), (b) dynamical zeta function (2.2), (c) quasiclassical zeta function (2.3).

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Physics Reports 309 (1999) 117—208

Stopping of heavy ions in plasmas at strong coupling Gu¨nter Zwicknagel *, Christian Toepffer
, Paul-Gerhard Reinhard
Laboratoire de Physique des Gaz et des Plasmas, BaL timent 212, Universite& Paris XI, F-91405 Orsay, France
Institut fu( r Theoretische Physik, Universita( t Erlangen, D-91058 Erlangen, Germany Received June 1998; editor: R.N. Sudan

Contents 1. Introduction 2. The projectile—target system 2.1. General considerations and definitions 2.2. Definitions of the stopping power 2.3. Parameters characterizing the projectile—target system 3. Stopping of heavy ions by free electrons in the various regimes 3.1. Linear ion—target coupling 3.2. Semilinear ion—target coupling 3.3. Nonlinear ion—target coupling 3.4. Overview 4. Nonlinear stopping in classical plasmas 4.1. Electronic stopping in classical plasmas 4.2. Nonlinear stopping power of heavy ions 4.3. Screening and electron trapping 5. Quantum effects 5.1. Stopping with effective potentials 5.2. Quantum stopping power

120 121 121 123 126 133 133 144 153 167 169 169 173 182 186 186 188

5.3. Comparison of WPMD and LFC treatments 6. Conclusions Acknowledgements Appendix A. Description of the simulation techniques A.1. MD-simulations A.2. PIC/Test-particle simulations of ion stopping A.3. WPMD-simulations Appendix B. Definitions and list of symbols B.1. Stopping power for infinite projectile mass B.2. Alternative derivations of the parameter of linearity B.3. Definitions used for the Fouriertransformation B.4. Definitions of some important quantities B.5. Notation References

190 192 192 193 193 196 197 198 198 198 199 200 201 203

Abstract Standard approaches to the energy loss of ions in plasmas like the dielectric linear response or the binary collision model are strictly valid only in the regimes where the plasma is close to ideal and the coupling between projectile-ion and the plasma target is sufficiently weak. In this review we explore the stopping power in regimes where these conditions are

* Corresponding author. Present address: Institut fu¨r Theoretische Physik, Universita¨t Erlangen, D-91058 Erlangen, Germany. E-mail: [email protected]. 0370-1573/99/$ — see front matter  1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 8 ) 0 0 0 5 6 - 8

STOPPING OF HEAVY IONS IN PLASMAS AT STRONG COUPLING

Gu¨ nter ZWICKNAGEL , Christian TOEPFFER
, Paul-Gerhard REINHARD
Laboratoire de Physique des Gaz et des Plasmas, BaL timent 212, Universite´ Paris XI, F-91405 Orsay, France
Institut fu( r Theoretische Physik, Universita( t Erlangen, D-91058 Erlangen, Germany

AMSTERDAM — LAUSANNE — NEW YORK — OXFORD — SHANNON — TOKYO

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not met. Actually relevant fields of application are heavy ion driven inertial fusion and the cooling of beams of charged particles by electrons. The conventional linear mean-field treatments are extended by many-body methods and particle simulations to account for strong correlations between the particles and for nonlinear coupling. We report the following important results in connection with the stopping at strong coupling: The energy loss of an ion scales with its charge approximately like Z , the effective screening length depends on Z and is larger than the Debye length. Slow highly charged ions are surrounded by a cloud of electrons trapped by many body collisions. Quantum effects like the wave nature of the electrons and Pauli-blocking reduce the stopping power by mollifying the effective interactions.  1999 Elsevier Science B.V. All rights reserved. PACS: 34.50.Bw; 52.40.Mj; 52.65.!y Keywords: Stopping power; Energy loss; Electron cooling; Strongly coupled plasmas

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1. Introduction The energy loss of fast, heavy particles passing through matter was first considered by Bohr [21] who treated the target electrons as classical particles. As the transfer of energy to the free electrons of the plasma target does not occur in quantized steps the classical approach is valid as long as the plasma is not degenerate. The quantized energy transfer to bound electrons, first considered by Bethe [19] and Bloch [20] leads to a reduction of the stopping power. Also the effective charge state of the ion is higher in a plasma as compared to a cold gas. The energy loss in a plasma is thus much larger than in a cold gas [127]. These theoretical predictions have been confirmed by recent experiments [33,34,42,43,59,94,95,163]. There are two standard approaches to the energy loss of ions in an electron plasma. On one hand one considers the stopping due to the polarization cloud which the moving ion creates in its wake. This is essentially a continuum treatment which ceases to be valid on scales comparable to the inter-particle distance. Alternatively one considers the energy transfer in successive binary collisions between the projectile and the electrons. Here it is essential to include the shielding of the Coulomb potential by the polarization of the plasma on long ranges. Thus both approaches are complementary to each other: The continuum treatment for the polarization requires a cutoff at small distances because the plasma consists of discrete particles and the model involving collisions between particles has an upper cutoff due to shielding. In fact both theories yield nearly identical results for the stopping power as long as the conditions remain standard in the sense that the target plasma is ideal, i.e. hard collisions between the electrons are rare, and that the coupling between the ion and the target is linear. It is the purpose of this report to investigate the energy loss of ions in a plasma beyond these standard conditions, i.e. for strong ion—electron coupling and/or non-ideal targets. Such studies are of considerable actual interest in at least two areas of application: 1. In heavy ion inertial fusion (HIF) one plans to compress a d!t pellet until ignition. The compression is either driven directly by ions or indirectly by X-rays produced through the stopping of ions in a converter. In either case, targets with densities 10—10 cm\ are heated through the energy loss of the ions. The resulting target plasmas eventually reach temperatures of 100 eV, but are nonideal in the initial stages of the heating process. The current status of HIF has been reported in Ref. [143]. 2. In most experiments with charged particle beams it is desired that the particles are concentrated in a small fraction of the single-particle (k)-phase space. One method to reduce the phase space volume of an ion beam is electron cooling as proposed by Budker [28]. The ion beam is mixed with a co-moving electron beam which has a very small longitudinal momentum spread corresponding to a temperature of a few K due to its acceleration from the source. In the rest frame of the beams the cooling process may then be envisaged as a stopping of ions in an electron plasma. Although the density is low, n"10—10 cm\, the electrons are strongly correlated because of the low longitudinal temperature. Moreover the ions in the beam are often highly charged, for example U> in experiments planned to investigate transitions from ordinary hadronic matter to a quark-gluon plasma with the help of the Large Hadron Collider (LHC) at CERN. In such situations the coupling between the ions and the cooler electrons becomes highly nonlinear. Reviews on electron cooling have been given in Refs. [166,142] and

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the proceedings [144,145] present the current status of the physics of cold and highly correlated beams. The report is organized as follows: In Section 2.1 we present the hamiltonian for the ion—target system. The stopping power dE/ds is defined in Section 2.2. There follows in Section 2.3 an identification of various regimes for the ion—plasma system. While the target parameters density n and temperature ¹ define the ideality m and the degeneracy H, the projectile properties charge Z and velocity v determine the strength of the ion—target coupling. In Section 3.1 we discuss methods appropriate for linear coupling with ideal and nonideal targets. In order of increasing perturbation caused by the ion there follows a regime in which the electrons are subjected to weak coupling except for a very small fraction of space where the coupling to the ion is strong. We have coined the name semilinear coupling for this regime which is discussed in Section 3.2. Finally, we present in Section 3.3 methods for the truly nonlinear coupling regime where analytical treatments are no longer possible. Again one follows the lines of either improving the continuum treatment by accounting for higher correlations in the response of the target to the ion or to account for such correlations when calculating effective cross sections for particle collisions in the medium. This involves unavoidably increasing numerical expense. One may then find it advantageous to use numerical methods like molecular dynamics (MD) computer simulations. In Section 3.4 we give a summary of possible approaches to the energy loss in plasmas according to the different regimes of application. In Section 4.1 we present some salient features of stopping in classical plasmas, mainly for purposes of later comparison. Actual results for the nonlinear stopping obtained by classical simulations are presented and compared with other treatments and experiments in Section 4.2. Nonlinear screening and the trapping of electrons by slowly moving highly charged ions is discussed in Section 4.3. Quantum effects, both the wave nature of the electrons as well as Pauli-blocking within the target tend to mollify the effective interaction between charged particle and thus reduce the stopping power. This is investigated in Section 5.1 with the help of effective potentials and confirmed by semiclassical wave packet molecular dynamics (WPMD) computer simulations in Section 5.2. As the WPMD gives some account for quantum correlations it is compared in Section 5.3 with another approach to strong quantum correlations starting from the hypernetted chain (HNC) method with effective interactions. Our results are summed up in Section 6. Finally there are appendices dealing with more technical matters like various simulation techniques, as molecular dynamics (MD) which accounts for particle collisions vs. the particle-incell (PIC) method which is a mean field approach. We further provide lists of definitions of important quantities and a list of symbols.

2. The projectile—target system 2.1. General considerations and definitions For a complete theoretical description of the energy loss of charged particles passing through matter one is confronted with a large spectrum of physical processes. First challenges arise from describing the target, which, depending on density and temperature, may be a solid, fluid, gas or

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plasma. Therefore one has to know the degree of ionization, the degeneracy of the electrons and the various types of ions in the target and the related bound states. This involves already a rather complete solution of many-body and atomic physics. In particular the composition of dense plasmas and their properties as conductivity or opacity are topics of actual theoretical and experimental investigations. Once the target is specified one is left with the additional aspects of the projectile—target interaction. Here the two main topics are: 1. The energy loss of the projectile in elastic collisions with free target electrons and target ions and in inelastic collisions due to ionization of target ions and excitation of bound electrons. 2. The changes in the electronic configuration of the projectile ion by various processes as radiative, dielectronic and three- or many-body recombination of free electrons, collisional ionization and excitation by target ions and electrons as well as charge transfer by bound—bound transitions between target ions and projectile. Most of these processes give also small contributions to the energy loss. Because the time scales are usually quite different, the stopping itself can be separated from the dynamics of the projectile ionization. This results in a much more simplified description of stopping compared to a full scheme where all processes mentioned are considered simultaneously. While the stopping power is now determined for a given electronic configuration of the projectile, the changes of the configuration during the entire process of slowing down are calculated from rate equations which account for all the various atomic processes as discussed e.g. in Refs. [34,116,117,136]. The stopping power for fixed projectile ionization together with the evolution of the charge state finally yields the energy loss over the time or path of interest. If necessary the corresponding amount of energy loss related to the processes which change the electronic configuration of the projectile must be added. There remains the task to calculate the stopping power for a given charge state. This still includes stopping contributions by free electrons, by target nuclei and ionization or excitation of target ions. Nuclear stopping yields relevant contributions only for very small projectile velocities of the order of the thermal velocity of the nuclei in the target. Thus one can neglect if for the largest part of the slowing down process. Then one concentrates on the energy loss to free and bound electrons. The calculation of stopping by bound electrons, see e.g. Refs. [24,26,36,58,79,117,179], requires in general an accurate knowledge of energy levels and related transition probabilities (oscillator strengths) in the target ions. At least a mean ionization potential of the target has to be provided for determining the stopping power at high velocities using the Bethe stopping formula [19]. The existence of bound electrons in the target strongly affects the charge state of the projectile, mainly by the very efficient bound—bound charge transfer. The stopping power, however, is usually dominated by the free electron contribution, provided that the degree of ionization of the target is high enough to supply some amount of free electrons. Stopping by free electrons is very efficient because any amount of energy can be transferred in contrast to the quantized energy transfer for the excitation of bound states. Hence the stopping power in a plasma is higher than in a cold gas. Because decreasing numbers of bound electrons are accompanied by less charge transfer to the target, the charge states of the projectile are usually higher in plasmas compared to a gas. Together with the more efficient free electron stopping this results in an enhanced stopping in plasmas as theoretically predicted [127] and experimentally observed [33,34,42,43,59,94,95,163].

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Here we intend to concentrate on the many-body aspects involved in the stopping at strong coupling rather than the atomic physics of the target ions. Hence we focus only on the free electron contribution to stopping. For discussing the energy loss at strong coupling we thus consider mainly a point-like projectile of given charge in a free electron target plasma. This restriction to free electrons becomes of course more and more realistic for increasing degrees of ionization, that is at high temperatures. Also in dense plasmas bound states disappear through a lowering of the continuum edge. For investigations concerning electron cooling even the real target is a free electron target. Within this simplified model the hamiltonian of the free electron (charge e, mass m) target can be written as e pL  #º , (1) HK " G #   4ne "rL !rL " 2m  G H G H$G G where º is a constant representing the potential energy related to the interaction of the electrons  with a static homogeneous charge neutralizing background as well as to the background—background interaction. Here we restrict ourselves to non-relativistic situations concerning the target conditions — as already assumed in Eq. (1) — as well as the projectile velocities. A review of the stopping of relativistic projectiles can be found in [2]. For treating the energy loss of the charged projectile we assume that the projectile—target interaction is switched on instantaneously at a certain time. This leads to the Hamilton operator for the projectile—target system: PK  (2) HK (t)"HK # !h(t!t ) e (rL !RK ) ,   G  2M G with the position R, momentum P, mass M and potential (r)"Ze/4ne "r" of the projectile and the   step function h(t). On this level of description of the projectile—target system, bound states of the projectile are still included as well as all changes of its electronic configuration as far as free electrons are involved, e.g. in the case of ionization by electron—projectile collisions and recombinations due to three- or many-body interactions. 2.2. Definitions of the stopping power The key observable in experiments exploring the interaction of charged particles with matter is usually the energy loss *E of the projectile ion. It is obtained by comparing the kinetic energy of the ion before and after passing through the target. The more detailed quantity is the stopping power which is defined as the energy change per unit path-length dE/ds and which corresponds to the actual decelerating force on the ion. The knowledge of the stopping power as function of energy allows then to determine the time evolution of ions in matter and further quantities, as e.g. the range of the projectile. The stopping power can directly be derived from the energy loss per path length as dE/ds(v)"*E/*s provided *E and *s remain sufficiently small, as it is often the case for experiments with heavy projectiles and thin targets as well as for simulation studies. For most theoretical approaches the stopping power is more conveniently defined either by the change of its kinetic energy 1d 1P/2M2 , dE/ds" v dt

(3)

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or by the decelerating force as the change in the momentum of the projectile projected on the direction of motion € € d dE/ds" ) F" ) 1P2 . v v dt

(4)

Both definitions are equivalent if the projectile travels along a straight line as it will be the case for sufficiently high projectile energies and/or large masses. Problems show up at very low projectile energy of the order of the mean kinetic energy of the target particles where the motion of the projectile represents a thermalization in the target and takes the character of Brownian motion with stochastically changing momenta. We concentrate now on the proper deceleration processes where the ion travels along a straight or smoothly varying path. This allows for a clearcut definition of a stopping power. In the following we will work out a microscopic expression for the stopping power for the ion—target system Eq. (2). We will do that first for a fully quantum mechanical treatment where the state of the system is described by the density operator oL (t) and reduce it later to classical models. For t(t the ion is still absent and the target as defined by HK , Eq. (1), is assumed in a stationary   state oL "oL (t(0) with [HK , oL ]"0, i.e. typically in an equilibrium state like, e.g.,    oL "exp(!bHK )/Tr exp(!bHK ). For times t't the ion is present and the system evolves     according to i joL (t)/jt"[HK , oL (t)] where HK now denotes the full projectile—target Hamilton operator Eq. (2). With definition (4) the stopping power for t't reads  € 1 € 1 dE € d " ) TroL (t)PK " ) Tr[HK , oL (t)]PK " ) TroL (t)[PK , HK ] v i

v i

ds v dt € € 1 "! ) TroL (t)[PK , e (rL !RK )]" ) TroL (t) e (rL i!RK ) .  G 0  v v i

G G The expectation value recurs only to the ion and one electron coordinate. It is thus useful to introduce the reduced density





o (r, R, t)" dr d(r!r )“ dr 1r ,2, r , R"oL (t)"r ,2, r , R2 .  G G H  ,  , G H$G This allows to express the stopping power finally as

 

dE € "! dr dR o (r, R, t) ) e (r!R) .  P  ds v

(5)

This very general expression allows to determine the stopping power from any kind of theoretical treatment which provides the probability to find at time t an electron at location r and the projectile at R. The expression can be simplified for high projectile mass and energy where we can assume a classical behavior of the projectile with simultaneously known position and velocity. The projectile trajectory is given by the density d(R!€t) where €"€(t) varies only slowly on the time scale of the target—projectile and intra-target interactions. This allows for the approximation o (r, R, t)+o (r, t)d(R!€t) where o (r, t) is the electronic density at location r. Now the stopping   

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power can be expressed in terms of the electric field E at the projectile location R"€t created by the electronic charge density . (r, t)"!eo (r, t)  € € dE " dr . (r, t) ) (r!€t)"Ze ) E(€t, t) , (6) v v P  ds



when employing (r)"Ze/4ne "r".   Expression (6) corresponds to the straightforward definition of the stopping power in a simple classical picture for the ion where the force on the ion is directly related to the electric field. Here it was derived from a fully quantal approach together with the additional assumptions of high projectile mass. The expression (6) becomes a rigorous result in the limit of infinite projectile mass (MPR). For the derivation see Appendix B.1. The ion moves then with constant velocity and acts just as an external moving potential (rL !€t). There is a stationary solution with constant  flow for this type of potential. Thus the stopping power is expected to become time independent after a transient period due to switching on the interaction at t . Hence  dE R € (7) P Ze ) E(€t) , v ds in contrast to definitions (5) and (6), where the stopping power can depend on time due to the feedback of the stopping on the projectile velocity €(t). Several theoretical methods describe the state of a system in terms of a phase space distribution function f (p ,2, p , P, r ,2, r , R, t), e.g. for classical ensembles [108]. The stopping power is then ,  ,  , obtained from a phase space integral which reduces to an integral over the one-particle distribution for the projectile f (P, R, t). Starting from definition (4) we obtain

  



jf dE € " ) dp 2dp dP dr 2dr dR P ,  ,  , jt ds v € jf (P, R, t) " ) dP P dR . v jt

(8)

Here the time evolution of f (P, R, t) is to be determined from the corresponding kinetic equations. The definitions of the stopping introduced above are of course not restricted to the case of an electron target plasma. They apply as well for more complex target systems as gases, solids, fluids and two- or multi-component plasmas. To extend the previous considerations to any kind of target consisting of electrons (q "!e, m "m) and one or more species of nuclei (q "Z e, m ), one has G G G G G just to replace the target hamiltonian HK (1) by the more general  pL  qq G H , (9) HK 夹" G #  2m 4ne "rL !rL " G  G H G G H$G with q "0 for neutral systems. The extended projectile—target system is then described by G G PK  HK 夹(t)"HK 夹# #h(t!t ) q (rL !RK ) . (10)  2M  G  G G

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2.3. Parameters characterizing the projectile—target system The energy loss of an ion impinging on an electron target plasma is determined by the direct interaction of the ion with the target electrons and by the interparticle correlations within the target. Hence, it is necessary to characterize the ion—target system in two respects: first, concerning the strength of the ion—target coupling, and second, concerning the electron—electron correlations in the plasma target. This will be done in the following two subsections. Since we have the impression, that the ion—target coupling is not always paid sufficient attention in the literature, we will address this point in particular. Thus we have a double sorting of strengths distinguishing ideal and nonideal target conditions as well as linear, semilinear, and nonlinear ion—target coupling. Each one of the resulting regimes requires its own theoretical approaches to the stopping power. These will be reviewed subsequently in Section 3. 2.3.1. The electron target The parameter of ideality m for an electron plasma is defined as the ratio of the mean potential energy E of the electron interaction to the mean kinetic energy E of the electrons. The first    one can be estimated by the potential energy for two electrons separated by the distance a"(4nn/3)\, the Wigner—Seitz radius for an electron density n. For the mean kinetic energy, we take a simple interpolation E "E #k ¹ between the Fermi-energy E for a fully degenerate   $ $ plasma with temperature ¹"0 and the thermal energy k ¹ for a hot, nondegenerate plasma. The parameter measuring the ideality becomes thus e 2ar E  , " (11) m" " 4ne a(E #k ¹) 1#H E  $   where r "a/a , a is the Bohr radius and a"(4/9n)"0.5212 . For small m;1 the behavior    of the electron plasma is dominated by the kinetic energy of the electrons and we are in an ideal, collisionless regime where collective plasma phenomena prevail. Increasing m51 corresponds to a strongly coupled, nonideal electron plasma with increasing importance of interparticle correlations. The definition of m in Eq. (11) implied already a further definition characterizing target conditions. There we introduced the degree of degeneracy




a  k ¹ v k ¹ " "2 "2a (ar ) , (12) H" K 13.6 eV  v E $ $ as the ratio of the thermal energy k ¹"mv to the Fermi energy E "mv/2. Alternatively, one  $ $ can view it as the ratio of the interparticle distance a to the thermal wavelength K" /mv where  m is the electron mass, v "(k ¹/m) the thermal velocity and v the Fermi velocity. This  $ parameter H represents a measure for the importance of the Pauli exclusion principle. The electrons have to obey Fermi—Dirac statistics for H;1 and can be treated classically within Boltzmann statistics for H<1. For nondegenerate plasmas, H<1, the parameter of ideality m becomes identical to the classical plasma parameter C, defined as 2ar ;H e  Q m. " C" H 4ne ak ¹ 

(13)

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There is one more criterion to be checked. It is the Coulomb parameter e g"  4ne v   which is concerned with the quantum mechanical wave nature of the interacting electrons. Taking for the relative velocity v the averaged relative velocity 1v 2 between two electrons within the   target we arrive at the averaged Coulomb parameter H< ar e  " P 1g 2"  4ne 1v 2 1v 2(1#(H/2)   



13.6 eV , k ¹

(14)

where we further introduced the dimensionless relative velocity 1v 2"1v 2/1v 2 scaled in units    of the averaged (single) electron velocity 1v 2. The more specific 1v 2"21v 2 can be derived in    the limit of a nondegenerate plasma (H'1) with a Maxwell velocity distribution. Classical motion is a good approximation if g , 1g 2<1 and a quantum mechanical treatment is necessary for   g , 1g 2(1.   The various regimes of target conditions are visualized in the left part of Fig. 1. The different regimes of target conditions are displayed in the n—¹ plane of the electron plasma. The dash-dotted

Fig. 1. The n—¹ regions of different conditions for an electron plasma. ¸eft plot: The nonideal plasmas m'1 are located below the solid curve m"1. The long-dashed line represents C"1. The dash-dotted line H"1 separates the classical plasmas H'1 (above) from the quantum plasmas H(1. Below the dotted curve (1g 2"1) and above the dash-dotted  one (H"1) the electron—electron collisions can be treated classically. Right plot: The dashed curves 1g 2"1, for  different ratios "Z"/1v 2"10,1,0.1 of the ion charge to the relative velocity, demarcate the region 1g 2'1 (to left   bottom) where a classical description of ion—electron collisions is appropriate.

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curve represents the critical degree of degeneracy H"1. The degenerate regime H(1 where the Pauli principle becomes important extends towards higher densities, i.e. below the dash-dotted division line. The solid curve corresponds to the critical ideality m"1. Nonideal plasmas are located in the triangle below the solid curve. The long dashed curve indicates the classical plasma parameter C"1. It deviates from the line m"1 only in the degenerate regime H(1. The electronic Coulomb parameter 1g 2"1 is drawn as dotted line in the nondegenerate region.  A classical treatment of the electrons is possible in the region below that line, and, of course, above the dash-dotted degeneracy line H"1. 2.3.2. The ion—electron coupling For the interaction of an electron and an ion with charge Z, the ratio of the classical collision diameter b ""Z"e/4ne kv and the wavelength " /kv related to the relative motion with      reduced mass k and velocity v defines the Coulomb- or Bloch-parameter  b "Z"e g " " . (15)  4ne v    This parameter is the analogue to g as defined previously. It determines the transition point from  a quantum-mechanical to a classical description of the Coulomb interaction of the particles. While a fast interaction with a large kinetic energy g ;1 can be described by a plane wave in first order  Born approximation, low energetic collisions g <1 allow for a treatment based on classical  trajectories, particularly for high ion charges Z. The line g "1 is the division line between quantal  and classical Coulomb collisions. More precisely, the g should be defined in terms of an averaged  velocity 1v 2 of the relative motion as in the case of 1g 2, Eq. (14). This yields   "Z"ar "Z"e  " . (16) 1g 2"  4ne 1v 2 1v 2(1#(H/2)    Here the relative velocity 1v 2 " : 1"€ !€"2, its dimensionless form 1v 2 and the averaged electron  C  velocity 1v 2 are approximated and interpolated as  1v 2"(1v 2#v) ,   v  1v 2 , (17) 1v 2"  " 1#  1v 2 1v 2   H  1v 2"(v#v )"v 1# ,  $  $ 2











in terms of the Fermi velocity v , the thermal velocity v and the ion velocity v. $  The quantum or classical regimes for the ion—electron interaction are indicated in the right part of Fig. 1. The dashed curves 1g 2"1 are division lines between the classical regime 1g 2'1   extending towards lower temperatures and the quantal regime 1g 2(1 at higher ¹. Three  different dashed curves are shown belonging to different ratios Z/1v 2 as indicated. The scaled  relative velocity is given by 1v 2"1v 2/1v 2. It is to be remarked, however, that the Coulomb    parameters g and g decide only on the validity of a classical treatment for an isolated two-particle  

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scattering event. One has to be aware of the fact that there may appear other quantum interference effects in a many-body system. The most important quantity to classify the ion—target coupling is the measure of the strength of the perturbation on the target caused by the ion. The strength parameter should decide whether one can treat the system in perturbation theory or not. To derive such a measure, we compare the potential energy of an electron in the field of the projectile ion with the electron energy in the absence of the ion but at the corresponding relative velocity. To that end, we need some preconceived knowledge of the potential field around the ion. It turns out that the potential energy can be approximated for the purpose of demarcation between regimes by







r r !Ze 1!exp ! exp ! . (18) » (r)"  j 4ne r   The factor J(1!exp(!r/ ))/r stands for the modification of the ionic Coulomb potential when  explored by an electronic wavepacket centered at distance r and of width " /mv (the reduced   mass is replaced by the electron mass m for heavy projectiles). It parametrizes the quantum diffraction effects at short distances & . The other exponential factor exp(!r/j) parametrizes  the long-range screening. Here we ignore all details of dynamical polarization and assume an exponential behavior for the effective screening with a velocity dependent screening length j. The well-known behavior for ideal plasmas and weak perturbations is j"1v 2/u where   u "(en/me ) is the plasma frequency. More complicated, but still local forms for the screening   may be needed in other regimes. Here, the local potential energy (18) serves to derive a definition of the various ion—target coupling regimes. Its explicit form is of minor importance provided that one properly incorporates both important physical phenomena, the wave nature of the and j electrons and the collective screening effects, and chooses the related parameters  appropriately. While screening results in a decay faster than 1/r on large distances of the order of r&j, the wave nature of the electrons modifies the Coulomb potential for r: and  results for decreasing r in a transition from the 1/r behavior to » (rP0)J1/ , when the   wavepacket is centered on the ion. This behavior is approximated by the simple exponential form » (r)J(1!exp(!r/ ))/r (18).   The relevant strength u of the ion—electron coupling is now given by the ratio of the ion—electron potential to the average electron energy E in the target. We need yet to estimate this electron  energy E . One important ingredient is, of course, the kinetic energy E "m1v 2/2. The potential     energy of the electron in its interaction with the other electrons (and the background) can be expressed in terms of the ideality m as E "m1v 2m/2. Thus we obtain for the average electron   energy E "m1v 2/2#m1v 2m/2 and finally the local strength as    21b 2 1v 2 r r 2"1» (r)2"    " 1!exp ! exp ! , (19) u(r)" r 1v 2#m 1 2 j m1v 2#m1v 2m     where 1b 2""Z"e/4ne m1v 2, 1 2" /m1v 2 and the scaled relative velocity 1v 2"       (1#v/1v 2)51 as introduced in Eq. (17). In an analogue manner, the electron—electron  interaction can be quantified by a local strength u (r)"2"1» (r)2"/m1v 2. Here, the elec   tron—electron potential energy » (r) emerges from the potential energy (18) by replacing the ion  charge Ze with the electron charge !e and taking the averages 122 over the relative









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motion of the electrons only, that is, using 1v 2 instead of the relative ion—electron velocity 1v 2.   The local quantity u (r) is relevant for electron—electron collisions while an average over the  distances r will characterize the global regimes of the electron—electron coupling in the target. Approximating such an average by evaluating u at the typical inter electron distance a yields  e a a 2 2"1» (a)2"  " 1!exp ! exp ! u (r"a)"  4ne a m1v 2 1 2 j m1v 2     e + "m , 4ne a(E #k ¹)  $ noting that a/j(1 and a/ "am1v 2/ 'am v / "1/a+2 (see Eq. (12)) for all densities and   $ temperatures. Thus, the local electron—electron interaction strength u leads directly back to the  parameter of ideality (11) as a measure of the coupling within the target. The dimensionless local strength u(r) of the ion—electron interaction, Eq. (19), on the other hand, allows to distinguish the various regimes of ion—electron coupling. The most simple regime is the case of a weak perturbation where lowest order perturbation theory is applicable. This can be handled with linear response techniques and therefore it is called the regime of linear ion—electron coupling. It requires a weak potential energy at all distances, that is, u(r);1 for all r. Since u(0)5u(r), the corresponding parameter of linearity is









1v 2 1v 2 1#H 21b 2 1v 2     "21g 2 ""Z"m , (20) u(0)"  1v 2#m 1 2 1v 2#m a(1#(H/2) 1v2#m    where linear ion—target coupling is granted for u(0);1. Here we employed for the last step expression (16) for 1g 2 and the definition (11) of the parameter of ideality m to rewrite u(0) (20) in  terms of Z, m and H. Inspecting now the condition for linear coupling u(0);1 and keeping in mind that 1v 251, we conclude that u(0);1 can be achieved only for m;1 or large velocities  1v 2<m resulting in 1v 2/(1v 2#m)+1. Thus small u(0) and small 1g 2 are equivalent:     u(0);1  21g 2;1 . (21)  In summary, the Coulomb-parameter 1g 2"1 separates the regimes of weak and strong perturba tion, that is of linear and nonlinear ion target coupling in coincidence with the regimes of quantum-mechanical and classical ion—electron scattering. To illustrate this additional significance of 1g 2 as a measure of linearity, we reconsider the three dashed curves in the right part of Fig. 1  representing the division lines 1g 2"1. They mark the boundary of the linear regime 1g 2(1   located at higher temperatures and densities for an ion with Z"10 and three different velocities, low v;1v 2, high v+101v 2 and very high v"1001v 2 as indicated. For comparison, for an ion    with Z"1 the last two curves would correspond to v;1v 2 and v+101v 2. The ion—target   coupling is linear in most of the ideal plasma regime (m(1) if Z is sufficiently low or the velocities are sufficiently high. For high charges and/or low velocities the linear regime is reached only for extreme temperatures and densities. As mentioned, the value 1g 2"1 is also the division line  between classical and quantal electron—ion scattering and it is interesting to note that truly linear coupling will occur only in quantal regime. That is understandable from the effective local potential energy (18). The quantum smoothing expressed in the factor [1!exp(!r/ )] is required to  suppress the 1/r singularity of the naked Coulomb potential. Alternative checks for the significance

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of g , 1g 2 as a measure of the ion—electron coupling strength and for the consistency of the linear   coupling Eqs. (18)—(21) are given in Appendix B.2. There fully quantum mechanical approaches are considered which confirm the definitions obtained above which are essentially based on the semiclassical expression for the potential energy (18). The transition to a nonlinear regime with increasing 1g 251 proceeds gradually and there  comes first an intermediate regime where the coupling (19) remains weak almost everywhere except for a small volume next to the origin. Simple approaches can still work very well in this transitional regime and it is thus useful to single it out by its own name. We call it the regime of semilinear ion—target coupling. We reconsider the local condition for a weak perturbation u(r)41 and introduce a critical distance r '0. The ion remains locally a weak perturbation for r'r where   u(r)(u(r ), and represents a strong perturbation only for r(r (and, of course, no perturbation   at all for r'j). The region of a strong perturbation at small r requires special attention in a theoretical description of stopping. But simple solutions are still feasible if the volume of strong perturbation, Jr, remains small. The definition of the semilinear regime is thus  r ;1, u(r )"1, 1g 2'1 . (22)   j The definition of r , i.e. u(r )"1 can be resolved approximately using Eq. (19) for 1g 2'1 (i.e.    r '1 2) and assuming r ;j. This yields r 91b 2 and we can use r +1b 2 for further estimates.        The regime of the semilinear ion—target coupling is then demarcated by 1b 2  (1, j

1g 2'1 

(23)

and larger values of this ratio 1b 2/j hint at an essentially nonlinear coupling. For ideal electron  targets m;1 the effective screening length is j"1v 2/u . For nonideal targets the static screening   will change, but the dynamical screening for high velocities v+1v 2<1v 2 still has the form   j"1v 2/u "v/u because the time for plasma response remains u\ as in an ideal plasma.     Hence, for ideal targets or high velocities the parameter 1b 2/j can be expressed as  "Z"eu 2"Z" ar  1b 2  "   " . (24) 4ne m1v 2 (3n1v 2 1#(H/2) j    Here the semilinear regime (1b 2/j(1, 1g 2'1) is located in the n—¹ plane between the two   boundaries 1g 2"1 and 1b 2/j"1 to the linear regime and the (essentially) nonlinear one,   respectively. This is shown in Fig. 2 for an ion with Z"10, where the dashed curves indicate 1g 2"1 and the dotted curves 1b 2/j"1 for a relative velocity 1v 2"1 (left plot) and 10 (right).    There is a large region of densities and temperatures where the ion—target coupling for highly charged ions is semilinear, in particular for non-degenerate plasmas H<1. In the latter case we have 1v 2"v and for m(1 or v





1 1b 2 (3"Z"C "Z" "Z"  " " " , (1#(v/v )) 4nnj (1#(v/v )) N 3(1#(v/v )) j   "  "

(25)

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Fig. 2. The regions of ion—target coupling for an ion of charge Z"10 in an electron plasma. The dashed curves represent 1g 2"1 and separate the linear regime (1g 2(1) from the semilinear regime (1g 2'1,1b 2/j(1) located between the     linear regime and the boundary (1b 2/j"1, dotted curves) to the proper nonlinear one. The corresponding relative  velocities are 1v 2"1 (left plot) and 10 (right plot) in units of 1v 2"(v#v ). The solid curve is again the boundary   $  m"1 between ideal and nonideal electron targets.

where C is the classical plasma parameter (13) and N "4nnj /3 is the number of electrons in " " a Debye sphere. In condition (23) together with expression (25) we recover the well-known condition to describe the stopping in classical plasmas by the dielectric formalism. There the ion charge state has to be small compared to the number of electrons N in a Debye sphere or, more " precisely, to the number N (1#(v/v )) in the dynamical screening sphere [135], in order to "  ensure that each electron contributes only slightly to the screening or, more generally, to the target response. In summary, there exist for given target conditions and ion charge two critical velocities, v from  Eq. (24) and v 'v from Eqs. (16) and (21): the ion—target coupling is linear for ion velocities   v5v , semilinear for v 4v(v and nonlinear for v(v . Of course, v or both velocities v and       v may be zero in certain cases and only semilinear and linear or only linear coupling occurs,  respectively. In fact, reality is a bit more involved because the effective ion charge depends on the projectile velocity and hence v and v should also vary with velocity. Nevertheless, the critical   velocities serve as a useful sorting criterion delivering this sequence of the three different coupling regimes. The criteria outlined above have been concerned exclusively with the electronic component of the target plasma. The ionic component of the target has been assumed tacitly as a homogeneous static charge neutralizing background. However, the discussed regimes and the corresponding definitions can be applied as well to the target ions by mainly replacing the electron mass and

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charge by the mass and charge of the target ions. Furthermore, the outlined classification of the target coupling regimes in terms of the ratio u(r) (cf. Eq. (19)) of a potential energy » (r) to the  mean energy of an electron can be used for a much larger spectrum of ion—target systems. For instance, the potential energy for electrons in the field of an extended charge distribution related to the bound states of not completely stripped projectile ions can be introduced if the extension of the ion (91 2) does not allow to use the bare Coulomb potential of a pointlike effective charge as  employed throughout the previous discussion. When in addition stopping by bound target electrons is considered, the criterion for ion—target coupling strength should be altered by taking into account the binding energy of the electrons bound to the target ions/atoms. There, the coupling strength at low projectile velocities is measured with reference to the binding/ionization energy of the bound electrons which replaces now the kinetic energy of the free electrons in the expression for the mean electron energy.

3. Stopping of heavy ions by free electrons in the various regimes As outlined above, we distinguish the regimes of an ion—plasma system in two respects: ideality and linearity. The target parameters density n and temperature ¹ define the degree of ideality while the projectile properties charge Z and velocity v have a strong impact on the degree of linearity. Hence stopping at strong coupling can take place in three different scenarios: linear stopping in nonideal plasmas, nonlinear stopping in ideal plasmas, or nonlinear stopping in nonideal plasmas. The description of the energy loss in these different cases deals mainly with the question of how rigorously one has to treat the electron—electron interaction in the presence of the ion. In this section we review the existing methods and recent proposals for the description of the (electronic) stopping in the various regimes. The sections are sorted with respect to the degree of ion—target coupling. 3.1. Linear ion—target coupling It is the main feature of the linear ion—target coupling regime that it is possible to separate the correlations within the target from those between the ion and the target. The projectile can be considered as a weak perturbation such that its effect on the target can be treated within linear response. There the dynamic response for a momentum transfer k and an energy transfer u is given in terms of the dielectric function e(k, u) which contains all necessary information about the internal structure of the target. We will discuss in this subsection the approaches which are based on the dielectric formulation. This is done in the following steps: In Section 3.1.1 we define and discuss the key quantities of linear response and their interrelations. The linear response formulation of the stopping power is then introduced in Section 3.1.2. The RPA formulation for the dielectric function is presented in Section 3.1.3. It is appropriate for ideal targets. Nonideal targets require higher correlations as outlined in Section 3.1.4. These are usually treated by density functional theories which aim to incorporate all higher correlations into an effective interaction while maintaining the simple structure of linear response. There are two mainstreams in these approaches, the Kohn—Sham equations with energy-density functionals derived in local density approximation (LDA) and the technique of the local field corrections (LFC). The Kohn—Sham

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approach is also applicable in the regime of nonlinear coupling and mainly employed there. We thus postpone its description to Section 3.3.5. Here we restrict the discussions to the LFC and spend two subsections to outline two important and widely used classes of approaches to the LFC: the STLS approximations in Section 3.1.5 and the HNC scheme in Section 3.1.6. There exists a somewhat different class of approaches where nonideality effects are described in Markovian approximation through electron—electron collisions, most often further simplified through a relaxation-time approximation to the collision integral. This is discussed in Section 3.1.7. Finally, we comment briefly on perturbative expansions for higher order corrections in Section 3.1.8. A very different approach to the energy loss of an ion rests on a purely kinetic description for the electrons. This is conceivable for an ideal target with sufficiently weak interaction between the electrons. The stopping can then be deduced as result of independent successive binary ion—electron collisions. Collective phenomena as screening and plasmon excitations are included by using an appropriate effective ion—electron interaction which, however, needs to be determined carefully. Since such an approach is a priori not restricted to linear ion—target coupling and turns out to become important mostly in the semilinear regime we postpone the discussion of the binary collision model for stopping to Section 3.2.1. 3.1.1. Different formulations of linear response As basis for all further considerations we recall quickly the main aspects and quantities of linear response theory. Most directly related to the concept of linear response is the density—density response function s which is defined from the relation between an external perturbing potential º and the resulting density response dn as 



dn(r, t)" dr dt s(r, t; r, t)º (r, t) 

(26)

and which is to be computed from first order time dependent perturbation theory (see e.g. [54], Section 13). This involved operator definition becomes a simple algebraic equation for systems with a spatially homogeneous ground state. Then the response function s depends only on the differences r!r, t!t, and Fourier transformation in space and time (see Section B.3) yields dn(k, u)"s(k, u)º (k, u) , (27)  which we will consider as the defining relation for the response s. Note that we are using the same symbols for the space—time dependent quantities as well as for the corresponding momentum— frequency dependent ones. This should not cause confusion as the distinction is usually clear from the context. An alternative formulation of linear response uses the concept of the dielectric function e which is defined from the relation between the external perturbation º and the resulting total potential  º as  1 º (k, u) , (28) º (k, u)"  e(k, u)  which is here already written in its simpler form in Fourier space. This equation is the analogue of Eq. (27) defining the response function s. Both quantities are, in fact, fully equivalent and represent

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merely a different emphasis in their definition. The response density dn of Eq. (27) yields the induced potential º (k, u)"»(k)dn(k, u) , (29)  where »(k) is the Fourier transformed two-body interaction potential, in case of electron systems the repulsive Coulomb potential »(k)"e/e k. The induced potential adds up together with the  external potential to the total potential, i.e. º "º #º . Combining all this information    establishes the expected one-to-one relation






1 1 !1 s(k, u)" »(k) e(k, u)

(30)

or its reverse 1 e(k, u)" . 1#»(k)s(k, u)

(31)

The response functions s, e carry similar physical information as the dynamical structure factor

 

S(k, u)" dr



dt 1dn(r#r, t#t)dn(r, t)2exp(!i(k ) r!ut)) , (32) \ representing the Fourier transformation of the density—density correlation function 1dn(r#r, t#t)dn(r, t)2, where 122 denotes the expectation value in the ground state. In homogeneous systems where the response is independent of the initial location r and time t, the momentum dependence characterizes the spatial transport and the frequency dependence gives access to resonances, relaxation times and other spectral features. The fluctuation—dissipation theorem establishes a direct relation between the dynamical structure factor S(k, u) and the response function s which reads [105,139,140] 2

Im [s(k, u)]"2 f (! u) Im [s(k, u)] S(k, u)"! 1!exp(! u/k ¹)  !2k ¹ Im[s(k, u)] , " u



(33)

where the two first expressions apply for a fully quantum mechanical treatment and the last one for the classical limit. The factor f (x)"[exp(x/k ¹)!1]\ is the Bose function. Equivalently, the dynamical structure factor is related to the dielectric function e through

 

 

1 1 2

 ! 2k ¹ Im f (! u) Im . S(k, u)" " e(k, u) e(k, u) »(k) u»(k)

(34)

Note that both relations (33) and (34) are not directly reversible as the relation between s and e was. The structure function provides only the imaginary part of s or 1/e. The full information is nonetheless contained also in the dynamical structure factor S(k, u) and may be regained by Kramers—Kronig relations.

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Finally, there is one more quantity used to express response properties of a system, the static structure factor S(k) which represents the instantaneous density—density correlation 1dn(r#r)dn(r)2. It is thus connected to the spatial correlations between particles as contained in the pair distribution function g(r) by Fourier transformation (see e.g. [77])



S(k)"1#n dr (g(r)!1)exp(!ik ) r) .

(35)

The same symbol S is used for both, the static and the dynamic, structure factor. The actual meaning becomes apparent from the independent variables. This usage suggests a close correspondence and, in fact, the static structure factor is obtained from the time-dependent (dynamic) density—density correlation as the limit of instantaneous response tP0. This results in the relation



1  du S(k, u) . S(k)" 2n n \

(36)

The unweighted integration over all frequencies is the obvious result from the “instantaneous” limit towards the static structure. It means physically that all dynamical responses are virtually present in the observed static correlations. There is, of course, also a simple relation between the static structure factor and the density—density response function s or the dielectric function e which can be easily established by combining Eq. (36) with the relations (33) or (34). Each one of these four response functions discussed above has a natural relation to a particular observable of the system. The structure factors S have been developed in connection with scattering experiments on homogeneous systems. For example, the dynamical structure function is typically related to the neutron scattering analysis of atomic vibration spectra in solids [8] and applies for any scattering analysis with good energy resolution. It serves as a direct measure for the double differential cross section dp/(dX dE)JS(k, u) where the prefactor depends on the kinematics of the projectile and the scattering angle X is related to the transferred momentum k in the standard fashion. The response function s or the dielectric function e are naturally related to experiments with the optical (or electro-magnetic) response of a system. For example, the cross section for photo-absorption is p Ju Im+s(k"u/c, u),. The dielectric function is, by definition, a measure
 for the electro-magnetic properties of a material. Resonances show up as poles in s or zeroes in e. The broadening of resonances hints at the various damping mechanisms, as e.g. Landau damping or electron—electron scattering. Other transport properties as the electrical resistivity or thermal conductivity can also be related to these basic response functions [89]. A most important observable which is usually looked at before any more detailed analysis is the (Coulomb) interaction-energy E which can be related to the static structure factor S(k) through the exact  relation E /N"(1/16n)dk (e/e k)(S(k)!1). The static structure function dp/dXJS(k) is,   furthermore, related to scattering without energy resolution, which amounts to integrating over all energies ("frequencies). Such a situation is typically met in X-ray scattering analysis of solids, or often deliberately arranged for correlation analysis of liquids by neutron scattering experiments [77]. The most important observable for our purposes is the stopping power dE/ds. The next subsection is devoted to establish and to discuss the relations between the stopping power and the

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response functions in a regime where linear response applies, i.e. for cases of linear ion—target coupling where the ion exerts only a small perturbation on the electron system. 3.1.2. Stopping power by linear response According to its definition (6) or (B.1), the stopping power is determined by the electric field E at the ion position which is caused by the charge density . of the target. For an initially homogeneous system with . ". (t(t )"!en and E "0 the relevant electric field is created by the charge    density . (r, t) " . (r, t)!. induced by the perturbing ion potential (r, t)"h(t!t )Ze/     4ne "r!€t". Rewriting definition (6) in terms of the Fourier transformed . and , the stopping    power on an ion with charge Ze and velocity € is dE "Ze€L ) E(r"€t) ds i dkdu k ) €L . (k, u) (!k)exp(i(k ) €!u)t) . "!   (2n)

(37)

This is still a general result for an initially homogeneous system. Assuming a linear ion—target coupling with sufficiently weak perturbation, we now employ the linear response relations (27) and (30). Identifying . "!edn, º "!e and »(k)"e/e k we obtain the induced charge     density as

.





1 !1 . (k, u)"e k (k, u)    e(k, u)

(38)

Inserting this relation together with the Fourier transformed "2nZed(u!k ) €)/e k into   Eq. (37) and taking advantage of the symmetry eH(k, u)"e(!k,!u) finally yields







Ze k ) €L 1 dE " dk Im . k e(k, k ) €) ds e (2n) 

(39)

This is the basic expression for the stopping power in the regime of linear ion—target coupling 1g 2;1, applicable at any target conditions, i.e. any m, H, so far the corresponding dielectric  function e is available. Note that transient effects caused by switching on the ion—target interaction have been disregarded in deriving relation (39) by considering the limit t P!R. This limit is  practically reached after a transient period of the order of the response time of the electron target u\ [160,161,199]. The induced electric field in the frame co-moving with the ion takes then  a stationary value provided that the ion velocity (vR JdE/ds) varies slowly on the time scale for transient effects. This is usually ensured for a high projectile mass M/m<1. There is an alternative derivation of the stopping power (39) which is based on the concept of a scattering rate R. It illustrates a different aspect of linear response and provides an additional interpretation of stopping [3,6]. The scattering rate R(k, u(k)) depends on the change in momentum

k"p!p and energy u(k)"p/2M!p/2M"((p# k)!p)/2M" k ) €# k/2M for an ion with charge Ze, mass M and velocity €. It can be computed with Fermi’s ‘Golden Rule’ and the

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dynamical structure factor S(k, u) as the density of the elementary excitations of the target. This yields the scattering rate R (per time and k-space volume)














1 1 Ze  2 e k 1  f (! u(k)) Im , (40) R(k, u(k))" "e (k)"S(k, u(k))" e e(k, u(k))

 e k

   where Eq. (34) was used to relate the dynamical structure factor to the dielectric function. The energy loss per time dE/dt is deduced from the scattering rate R as the average over all losses and gains of energy quanta u where u'0 represents a gain for the target and a loss for the projectile. The stopping power dE/ds"1/v dE/dt becomes then



dE 1 dk Ze u(k)f (! u(k)) 1 "!

u(k)R(k, u(k))"! dk Im . (41) ds v (2n) e 4n vk e(k, u(k))  Again we consider here the limit of heavy projectiles (M<m) which allows to neglect the recoil effects, taking u(k)"k ) €. Using the symmetries of e(k, u) as well as the relation f (x)#f (!x)" !1 for the Bose function yields finally the stopping power in the form (39). The present procedure views the energy loss as an average over absorption and emission of energies u by the projectile. This aspect provides also a simple way to describe the straggling, i.e. the deviation of the energy loss per path-length from its mean value. The relative straggling (as compared with dE/ds) is given by the second u-moment of the scattering rate R [6]. The general dependencies on the basic projectile parameters, charge Z and velocity v, are the most important features. One sees immediately from Eq. (39) that the stopping power grows quadratically with Z, which is a trivial consequence from the linear response which depends linearly on the perturbation Z. The dependence on the velocity is more involved. We have to distinguish two regimes. For ion velocities v low compared to the mean electron velocity 1v 2 the  stopping power is linear in v Zen dE "! v [C(m, H)#O(v)], v;1v 2 , (42)  e (2n)u ds   where the dimensionless friction coefficient C(m, H) depends on the target conditions and hence on the interparticle correlations. It can be obtained immediately from Eq. (39) as






 

j 1 4nu   dk k , (43) Im C(m, H)" ju e(k, u) 3n  S assuming e(k, u)"e(k, u) for an isotropic target. In the limit of large velocities, on the other hand, the stopping power goes like






2mv dE Zen 1 "! ln #O(1/v) , v<1v 2 , (44) 

u ds 4nem v   and depends on the target conditions only through the electron density n. The approach to the limit (44) for sufficiently high v follows directly from the f-sum-rule for the dynamical structure factor which is related to particle number conservation ([140], Section 2.2). Thus, only the energy loss at low velocities provides information about the correlations within the target.

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We want to point out once again that the general relation (39) between stopping power and dielectric response applies to any kind of target, ideal and nonideal electron plasmas as well as targets beyond the present considerations like two component plasmas of different degrees of ionization and degeneracy or solids and atomic or molecular gases. In these cases also inelastic contributions from ionization and excitation of bound states are included in the linear response stopping power. It is only required that the ion target coupling is linear or, with some restrictions to be discussed below, semilinear. The crucial task is thus to derive the dielectric function for a given system, including properly all necessary correlations. 3.1.3. The RPA dielectric function for ideal targets For ideal electron targets m;1, the response functions s(k, u) and e(k, u) are derived within the weak coupling limit of interparticle correlations, well-known as the random phase approximation (RPA). Starting point to derive the RPA density—density response function s , denoted by the 0 subscript R, is the free density—density response function s . It is easy to evaluate for homogeneous  systems and yields



dp f (p# k)!f (p)   , s (k, u)"!2  (2n ) u!f(p# k)#f(p)#i0

(45)

where f (p)"1/(exp((f(p)!k)/k ¹)!1) is the Fermi—Dirac distribution with chemical potential  k and the free electron dispersion f(p)"p/2m. The free density—density response function s (45) is  often called the Lindhard function [109]. The RPA density—density response function s is 0 computed from this free response and the two-body Coulomb interaction »(k)"e/e k by the  RPA equation s (k, u)  . (46) s (k, u)" 0 1!»(k)s (k, u)  The RPA dielectric function e then follows from s (k, u) using relation (30) as 0 0 e s (k, u) . (47) e (k, u)"1!»(k)s (k, u)"1! 0  e k   The RPA response functions s (46) and e (47) provide the dynamical response of an ideal electron 0 0 target for any degree of degeneracy, for a cold electron gas in solids (using the jellium approximation) as well as for dilute, hot classical plasmas [7,64]. For the homogeneous and isotropic electron targets which we consider here the response functions s (45), s (46) and e (47) depend only on the  0 0 absolute value of k. 3.1.4. Dielectric functions for nonideal targets, ¸FC and ¸DA The RPA equation (46) incorporates already the most important long range correlations from the Coulomb interactions. In particular, it gives a pertinent description of the response of the collective plasmon mode. There remain, however, several deficiencies from neglecting higher correlations, as, e.g., exchange in the quantum regime, or short range correlations accounting for close two-electron collisions. Density functional methods aim to cure these deficiencies by developing an effective interaction which incorporates correlations beyond mean field while maintaining

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the simple structure of the mean field equations, here the RPA equations (46). Quite general theorems guarantee the existence of such effective interactions [84,102,151], but actually applicable functionals require a few more approximations. Subsequently, there emerges a variety of formulations towards the desired goal of an optimal compromise between theoretical soundness and practicability. There is a widespread literature on this topic and we cannot dwell in all the details of the various approaches. We aim here merely at a short overview of two widely used schemes, the technique of the local field corrections (LFC) and the Kohn—Sham equations with energy-density functionals derived in local density approximation (LDA). We are going to discuss here the LFC while a discussion of the LDA will be presented in connection with applications to the nonlinear coupling regime in Section 3.3.5. The local field correction (LFC) is defined as a correction to an underlying RPA description. Starting point is the RPA equation (46) for the RPA response function s . The exact response 0 function s will differ and contain, of course, more information about higher correlations. But the RPA response has anticipated already a great deal of the most crucial long range correlations. It is thus useful to define the extra correlations with reference to RPA. This is done by s (k, u)  , s(k, u)" 1!»(k)(1!G(k, u))s (k, u) 

(48)

which defines the correlations in terms of a modified residual interaction, »P»(1!G), to be used in the RPA equations. This modification is summarized in the function G(k, u) which is called the local field correction. By relation (30) the modified dielectric function takes the form »(k)s (k, u)  e(k, u)"1! . 1#»(k)G(k, u)s (k, u) 

(49)

The RPA expressions (47) and (46) are recovered for vanishing LFC G(k, u)P0. The form of definition (48) suggests that deeper correlation effects can be treated in terms of an effective interaction for RPA calculations and it hints the direction for the systematic construction of approximations. The two most widely used of the various possibilities will be discussed in the following two subsections. Note that the LFC depends generally on momentum and frequency. This means that the so defined effective interaction »(k)(1!G(k, u)) embodies retardation. The spectral structure of G(k, u) reflects the dynamics of the higher many-body correlations from which one can hope that these are faster than the dominant modes described in RPA (usually the plasmon mode). This suggests already a first step of approximation, the static LFC G(k) where the u dependence is neglected. Rephrasing the relation (48) in terms of the structure functions suggests that a static LFC G(k) should be one-to-one related to the static structure function S(k) and thus to the pair distribution function g(r). The both approximation schemes discussed below try to establish such a relation from different starting points. 3.1.5. The STLS approach to LFC One line of development has been initiated by Singwi—Tosi—Land—Sjo¨lander in [165] (STLS approach). It starts from incorporating the two-particle density in the derivation of the response functions s and s which are based solely on the one-particle density in the usual RPA approach. 

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This establishes a relation between the (static) pair distribution function g(r) and the static LFC G(k) given by



1 dq q ) k [S(k!q)!1] , G(k)"! n (2n) q

(50)

where the static structure factor S(k) is uniquely related to g(r) by Eq. (35). The STLS approach was first derived in a classical environment. It has been generalized to quantum systems in [158,159]. To give an idea how relation (50) is derived, we sketch here quickly the simpler case of a purely classical system. Starting point is the parametrization of the two-particle phase-space density f  in terms of the equilibrium correlation function g(r) as f (r , p ; r , p ; t)"f (r , p ; t) f (r , p ; t)g(r !r ) . (51)           Upon integration over the momenta p , p this is an exact relation for the time-independent case. It   is an approximation for a truly dynamical situation based on the assumption that the static correlations are dominated by spatial correlations and that they are not so much affected by the oscillations of the mean field. The further steps are straightforward and tedious: The ansatz (51) is inserted into the lowest of the BBGKY equations determining j f (r , p ; t), see e.g. [14]. The R   one-particle phase space distributions f (r, p; t) in that equation thus emerging are expanded into a stationary part plus a small dynamical perturbation as f"f #df, and linearized in terms of the  df. Rephrasing that equation in terms of the response functions s and s yields a generalized RPA  equation of the type (48) from which one can read off the LFC by comparison resulting in relation (50) after expressing the pair distribution function g by its equivalent static structure function S. Eq. (50) deduces a static LFC from given static structure function. The LFC, on the other hand, determines a structure function from the generalized response equation (48). This closes the circle and suggests a selfconsistent scheme as follows: (1) Start from a given static structure function S(k) and use Eq. (50) to compute the corresponding LFC G(k). (2) The now given G(k) allows to compute the response function s(k, u) using Eq. (48). (3) The response function defines the dynamic structure function S(k, u) according to Eq. (33). (4) Frequency integration in relation (36) delivers a new static structure function S(k) from which one continues from step 1 until sufficient convergence is achieved. This STLS scheme is conceptually obvious and technically still fairly simple. Thus it has been widely used in several applications where a correlated electron plasma plays a role. And it has been developed further to finite temperatures, to a fully quantum mechanical treatment, and to a truly dynamical local field correction [158,159]. There arise, however, the typical problems when constructing approaches to correlations: the emerging response and correlation functions violate basic consistency conditions. The worst problem in the STLS approaches is probably that the spatial correlation function g(r) ceases to be positive semi-definite. i.e. one finds occasionally g(0 at small distances. This does not seem to cause problems for global observables where the STLS approach has much success. But it is a hindrance for studying more detailed observables. The STLS scheme can be simplified by a sum rule approach. The dielectric response is dominated by the free particle response, on one hand, and by the plasmon response, on the other. A simple and reliable approximation can be obtained by constructing the static structure

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function from the free particle structure function and the contribution of the plasmon pole. Such an approach reduces the selfconsistent STLS problem to purely static coupled equations, thus avoiding the cumbersome frequency structure in the full STLS. The model and some applications to the two- and three-dimensional electron gas have been discussed in [63] and references cited therein. It has been applied to compute the stopping power for a two-dimensional electron gas in [181]. It is found that the LFC enhance the stopping power for small ion velocities v(v where $ v is the Fermi velocity of the electrons. For larger velocities, the stopping power converges rapidly $ towards the result from the simple dielectric theory, that is, the high velocity result (44). 3.1.6. LFCs by using information from HNC calculations The STLS approach as reviewed in the previous Section 3.1.5 constitutes a simple and selfconsistent scheme to compute correlations effects which often provides pertinent results. Its correlation function, however, violates basic consistency conditions, e.g. the positivity of g(r). Other approaches spend more effort to achieve compatibility with these conditions. A most widely used scheme amongst these is the hypernetted-chain (HNC) approximation. It has also been much applied to problems in plasma physics, for a review in the context with plasma applications up to computing the stopping power see [90]. The HNC scheme allows to compute the static pair correlation function g(r) for a plasma and subsequently the internal energy or further thermodynamic properties. It is desirable to incorporate this detailed and reliable information into the computation of the dielectric response and its related observables. The connection is established by choosing a static LFC G(k) such that the LFC corrected RPA equation (48) together with the relations (33) and (36) yields the static structure function S(k) as it is prescribed from HNC, i.e.







s (k, u) c S(k)"2 du f (! u)Im  . S&,!(k)" 1!»(k)(1!G(k))s (k, u) n  This relation simplifies dramatically in the classical limit P0 and reduces to [88,90]






1 G(k)"1!kj !1 , " S&,!(k)

(52)

where j is the Debye length. The choice (52) then guarantees the recovery of the static properties " of the HNC approach through the LFC corrected dielectric formulation, summarized in Eq. (49). But now the dielectric formulation can be applied for truly dynamical processes as well. This way one has achieved a dynamical extrapolation of the static HNC correlations. In practical implementations, it is advantageous to employ a properly parametrized form for the static structure function S(k) or the related static LFC G(k) as obtained from HNC, for an example see Section 3.2.1 of [90] and work cited therein. A careful shaping of the ansatz allows to meet most of the important consistency condition for the pair distribution function, i.e. the compressibility sum rule, the positivity condition g'0, and the short wavelength limit limk G(k)"1!g(0). Spin degrees-of freedom are also incorporated and the corresponding spin-susceptibility sum rule is met by construction. There exist also extensive HNC calculations for two-component plasmas consisting of ions and electrons. The inherent instabilities occurring through the attractive ion—electron interaction [112]

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can be circumvented by disentangling the HNC scheme into subsequent steps, solving HNC first for the electrons, then for ions using the electron-screened potential, and then adding carefully the effect of the ion—electron interaction [88,90]. This delivers at the end a 2;2 matrix for the static structure function, and subsequently for the static LFC by a relation similar to Eq. (52). Several further observables can be deduced in the standard fashion using the LFC corrected dielectric theory, see e.g. Section 3.1.1 of [90], once the LFC are given. In particular, one can compute the stopping power using the relation (39). The LFC as defined in Eq. (48) are in general dynamical quantities depending also on the frequency u. The approximation by static LFC discussed above works very well for many purposes. But there are also drawbacks, e.g. in the description of the detailed line broadening of the plasmon peak. A dynamical LFC is thus desirable. Such an extended approach may still be deduced from static HNC calculations by considering limiting cases and interpolation. Guided by dynamical studies of [175], Ichimaru et al. propose an interpolation as [88,90] G(k)!iuq(k)I(k) , G(k, u)" 1!iuq(k)

(53)

where q(k) is the typical relaxation in mode k and the structure component I(k)"lim G(k, u). It S can be obtained from static HNC calculations as a higher weighted average over the static structure function

 



1 dq k ) q k ) (q!k) k ) q I(k)" # [1!S(k!q)] . n (2n) q "k!q" q and it is related to the viscosity [175]. This dynamical LFC approach has been extensively studied by Ichimaru and collaborators with respect to all sorts of observables. A detailed discussion of results using these static and dynamical LFC with HNC input is found in [89—90,123,173,189]. Existing treatments based on HNC and quantum HNC [93], on the STLS (Section 3.1.5) approach and extended STLS schemes [158,159], on informations from Greens-functionMonte—Carlo methods [176] or on diffusion Monte—Carlo results [125,162] are able to provide LFCs at any target conditions H,m. Hence using LFCs to obtain the modified dielectric function (49) and subsequently the stopping power by Eq. (39) allows to study ion stopping in all kind of nonideal, strongly correlated targets, provided, of course, that the ion—target coupling is linear or at least semilinear. Its domain of application is thus the parameter space with m91 and 1g 2:1, i.e.  the regime of low projectile charge states Z or high velocities as can be seen immediately by inspecting definitions (20) and (21). The LFCs generally increase the stopping and they are particularly important at low ion velocity [153,182] while for high ion velocities the simple RPA dielectric description, see e.g. [115], becomes sufficient again. The static LFC includes already most of the corrections and the dynamical LFC adds a bit on top. 3.1.7. Dielectric function in relaxation-time approximation Another possibility is to include correlations in terms of instantaneous (quasi)particle collisions. A particular simple scheme for that is provided by the relaxation-time approximation as suggested by Mermin [121]. The resulting dielectric function for a given electron—electron collision frequency

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l takes the form (u#il)(e (k, u#il)!1) 0 e(k, u)"1# , (54) u#il(e (k, u#il)!1)/(e (k, 0)!1) 0 0 expressed in terms of the RPA dielectric function e (47). The collision frequency l is used as 0 a model parameter. In some investigations on stopping of ions in carbon and silicon the collision frequency l was determined by fitting !Im[1/e(k"0, u)] to experimental optical energy loss functions [1,10,12,133,134]. Treating the collision frequency as a free parameter, the modified dielectric function (54) allows useful qualitative estimates of the influence of electron—electron collisions on the stopping power. The predicted effect is a shorter life time and smaller mean free path of the plasmons resulting in considerable modifications of the wake field behind the passing ion [11]. This is of particular importance for vicinity effects on the energy loss of multi-ion arrangements [12,195,198]. For the stopping of a single ion, the broadening of the plasmon peak with increasing l shifts the threshold for the energy loss by plasmon excitation towards lower projectile velocities. It is now possible that ions with low velocities can also excite plasmons which increases the stopping power at low v compared to the RPA result [10]. The stopping remains almost unchanged for high velocities which are above the threshold for plasmon excitations (at l"0). This is in agreement with the common high velocity limit (44). Recently, relaxation rates l have been deduced from MD-simulations for classical targets [168]. These confirm the kinetic behavior lJu C (rate of Coulomb scattering events with scattering angles '90°) in the weak  coupling regime, but show an anti-kinetic behavior lJu C\  for nonideal systems with C'1.  3.1.8. Perturbative corrections to the linear response The stopping power (39) as obtained from the linear response formalism represents the first order in a perturbation expansion with respect to the external perturbation . This suggests that  improvements in the description of stopping can be achieved by taking into account higher orders in the perturbation series. The second order terms are called the Barkas correction [17]. They are quadratic in and hence their contribution to stopping is JZ. More precisely, the Barkas  correction gives a contribution of the order Z/(1v 2) relative to the leading Z term of the linear P response theory. Thus it is a small correction in the linear, and to some extent in the semilinear, regime. But the perturbation series diverges as soon as one reaches the nonlinear regime. The approach shares the general features of a perturbation series: it is designed to enhance the accuracy of the description within a regime inside the convergence radius of the expansion, i.e. the regime of weak coupling. A general discussion of this line of development is found in [9,51,141]. Investigations on the Barkas contribution in a classical electron target ( P0) and for semilinear ion—target coupling are presented in [37,135,154]. 3.2. Semilinear ion—target coupling In the semilinear coupling regime, the perturbation caused by the ion is just beyond weak. But there arises a spatial distinction such that the perturbation remains truly weak in the largest part of the ion—target interaction region — as ensured by definition (22). Only electrons in a very small volume (Jr;j) near the ion explore a strong perturbation. This suggests an approach which 

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relies on linear response and adds corrections from the small region of nonlinear effects. Such a procedure will be much less expensive than a fully nonlinear description of stopping, and it applies to a rather large area of projectile and target parameters covered by the semilinear regime, see Fig. 2. This justifies our notion of a semilinear regime and a particular treatment of stopping in this regime. These approaches are based essentially on the complementarity of dielectric linear response and binary electron—ion collision. The linear response accounts correctly for all the dynamical polarization effects in the target as long as the coupling is linear, i.e. in the dominant fraction of the interaction zone. Binary collisions in the inner interaction region are governed by the ion—electron interaction at arbitrary strengths g while being insensitive to the dynamical  effects. Some approximation schemes to combine both aspects are presented in the next subsections. First, however, we will discuss the treatment of binary collisions in general. 3.2.1. Stopping by binary collisions For negligible interaction between the electrons in an ideal plasma (m;1), the stopping can be described entirely by independent binary ion—electron collisions. This holds for arbitrary strengths of the ion—electron interaction. This completely kinetic approach starts by describing the system through the momentum distribution fI (P) of the ion and the distribution f (p) of the electrons. The time evolution of the ion distribution is determined from scattering events (P, p)(P, p) of the ion with electrons in terms of a collision integral I as j fI (P)"I . Here, we use the quantum version of  R  a Boltzmann collision integral for ion—electron collisions which includes Pauli blocking factors for the final electron states and allows to treat all degrees of degeneracy of the electron target. The force F on the ion, i.e. the time derivative of its momentum, can be expressed now as the integral (see also the definition (8) of stopping)





dP jfI (P) dP d P "» PI F" 1P2"» (2n ) (2n )  jt dt



2» dP dP dp dp " P w(P, p; P, p) [ fI (P) f (p)(1!f (p))!fI (P) f (p)(1!f (p))] , k (2n )

(55)

where k"mM/(m#M) is the reduced mass, » a spatial volume and w(P, p; P, p)"p(P, p; P, p)d(P#p!P!p)d(fI (P)#f(p)!fI (P)!f(p)) with the energies fI (P)"P/2M of the ion and f(p)"p/2m of the electrons and the cross section p for the collisions (P, p)(P, p). For the projectile ion with a given velocity €, the corresponding distribution is fI (P)"[(2n )/»]d(P!M€). This allows to integrate over two momenta. Assuming a cross section with the symmetry p(P, p; P, p)"p(P, p; P, p) and substituting (p!p)Pq, the stopping power dE/ds"€L ) F becomes



dE 2 dp dq " €L ) q p(M€, p; M€#q, p!q)d(fI (M€)#f(p)!fI (M€#q)!f(p!q)) ds k (2n ) ;f (p)(1!f (p!q)) .

(56)

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Alternatively to this expression in terms of the transferred momentum q, one can obtain a more common representation by a transformation to center of mass momentum K"M€#p"K and relative momenta p "p!(k/M)K"(k/m)p!k€, and p "p !q. The cross section p is usually P P P independent of the center of mass motion, that is, p"p(p ; p ). This yields P P





 






2 m  dp dp p p  dE P P d P ! P € ) [ p !p ]p(p ; p ) f " P P P P (2n ) 2k 2k ds k k !f



m m 1# p #m€ f p # p #m€ P P M M P

.





m 1# p #m€ M P

(57)

The cases of heavy ions and spherical scattering potentials allow further simplifications. For a heavy ion, the reduced mass becomes k+m or m/MP0 and the quadratic term f ((1#(m/M))p #m€) f (p #(m/M)p #m€) connected with Pauli-blocking cancels out for differenP P P tial cross sections p(p ; p ) which are invariant under an exchange of p and p . There remains P P P P





p dp dE P f (p #m€) P dX €L ) [pL !pL  ]p(p ; p pL  ) . "2 P P P P P P m (2n ) ds

(58)

It is worth to remark, that the stopping power (58) obtained under these conditions within the present kinetic-probabilistic description agrees completely with a rigorous, fully quantum mechanical treatment developed in [25,156]. There one starts with the definition of the energy transfer (B.1) and evaluates it in the limit tPR for non-interacting electrons, taking as initial state an ensemble of plane waves for the single electron states with occupation numbers according to the distribution f. For spherical symmetric scattering potentials, where p(p ; p )"p(p ,0) the stopping power P P P appears finally in the very compact form



dE p dp P f (p #m€) P €L ) p p (p ) . "2 P P  P ds m (2n )

(59)

Here the transport cross section p (p )"2n d cos(0)[1!cos(0)]p(p ,0) has been introduced P  P \ after integrating over the scattering angle 0, with cos(0)"(p ) p )/p. Usually, one is interested in P P P ion stopping in an electron plasma in thermal equilibrium and the electron distribution f is the Fermi—Dirac distribution f for free electrons. As a remarkable feature, the final results for the  stopping power Eqs. (58) and (59) depend linearly on the electron distribution f"f for any degree  of degeneracy H. That is, the influence of the quadratic Pauli-blocking term in Eq. (57), f ((1#(m/M))p #m€) f (p #(m/M)p #m€), becomes negligible in the limit of small mass ratios  P  P P m/MP0. In the general case of arbitrary m/M this Pauli-blocking term becomes unimportant only in the high temperature limit H<1, when f merges into a Maxwell distribution and this quadratic  term can be neglected. This case of arbitrary mass ratios and high temperatures H<1 is commonly used as starting point for the energy loss in terms of binary collisions [53,164,60,131]. There, one has just to retain the mass factors of Eq. (57) to obtain expressions analogous to Eqs. (58) and (59) with f (p #m€)/m replaced by (m/k)f ((1#m/M)p #m€)/k. P P For spherical potentials simple and compact expressions in terms of the transport cross sections can be easily derived in the low and high projectile velocity range, respectively. At low velocities, an

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expansion with respect to v of the shifted electron distribution in the stopping power (59) yields



dE  1 jf "v (60) dp p  (p) p (p), v;1v 2 .  C ds 3nm  jf(p)  For high velocities an even simpler expression is obtained by substituting p P(p!m€) and P approximating p ""p "+mv. This leads to P P dE "!mnvp (mv), v<1v 2 , (61)  C ds where the density n emerges through 2dp f (p)"n(2n ).  To express the stopping power in terms of the electron distribution and an effective electron—ion cross section goes in fact beyond a mere binary collision approximation and is employed in various different regimes, see e.g. Section 3.3.6 for applications to the case of nonlinear electron—ion coupling and highly degenerate target systems. The key issue is then to find an appropriate effective ion—electron potential and to compute the related effective cross section. A main ingredient to that are collective effects of the electron cloud like dynamical screening and plasma waves. These effects are, of course, best under control in the linear response regime. We thus regard, as a first exercise and test, the cross section for the case of linear ion—target coupling. The required effective ion—electron potential is then for ideal plasmas (m;1) simply provided by the linear response  of the medium in terms of the RPA dielectric function with (k, u)" (k)/e (k, u)"   0 Ze/[e ke (k, u)], see Eq. (28). Because we deal with linear coupling 1g 2;1, the cross section  0  p can be evaluated in first order Born approximation, yielding Ze k k "e (k, u(k))"" , (62) p(" k")" 4n  ek"e (k, u(k))" 4n    0 for a momentum transfer k"q and the related energy transfer u(k)"fI (M€)!fI (M€# k)" f(p! k)!f(p). The cross section in first order Born approximation depends only on the transferred momentum q and it is thus favourable to insert it into Eq. (56). Rewriting the product of Fermi—Dirac distributions as f (p)(1!f (p! k))"![f (p)!f (p! k)]f (f(p)!f(p! k)) by     using the Bose distribution f (x)"[exp(x/k ¹)!1]\ yields

 

dp dE Ze "! 2 dk d( u(k)#f(p)!f(p! k)) ds 4ne (2n )  Ze Im[e (k, u(k))] [f (p)!f (p! k] 0  "! dk €L ) k f (! u(k)) . ;f (! u(k)) €L ) k  4ne k"e (k, u(k))" k"e (k, u(k))"  0 0 (63)



In the last step, the imaginary part of the RPA dielectric function (47)



dp e d( u#f(p)!f(p! k))[f (p)!f (p! k)] Im[e (k, u)]"2n   0 e k (2n ) 

(64)

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was introduced. It is to be remarked, that for the derivation of the stopping power in the present form of Eq. (63) the assumption of a target in thermal equilibrium with f (p)"f (p) is essential.  The mass ratio between projectile and target particles may, however, be arbitrary. In the limit of heavy projectiles, m/MP0, projectile recoil can be neglected and yields u(k)" fI (M€)!fI (M€# k)+! k ) €. This allows to exploit the symmetry of the dielectric function e (k, u)"eH(k,!u) and the relation f (x)#f (!x)"!1, to obtain in this limit the stopping 0 0 power in the form of Eq. (39) for e"e , now called Born-RPA stopping power [115]. Thus the 0 linear response stopping power (39) in an ideal target can alternatively be interpreted as result of binary ion—electron encounters in the linear coupling regime and with the effective interaction e (k, u)"Ze/e ke (k, u) containing the electron screening from linear response. This agree  0 ment is of course not unexpected as binary collisions and linear response are treated on the same level of approximation, that is in first order of the perturbation. The binary collision approach Eqs. (55)—(59) applies a priori to any strength of the ion—electron interaction, i.e. for any 1g 2, provided that the full many-body dynamics can be mapped into an  effective interaction. This should be feasible at least for situations where the electron—electron coupling is small compared to the ion—electron coupling at small distances as it will be the case for nonlinear coupling, 1g 2<1, in an ideal plasma, m(1, with weak electron—electron correlations.  However, one has yet to determine this effective interaction which includes the dynamical and nonlinear response of the medium, for instance, by solving the ladder T-matrix equation with a fully dynamically screened interaction. Generally applicable tools for this task are not available at present, and we have to consider approximations. Static effective interactions are sufficient at low ion velocities where the strongest nonlinear effects are expected anyway. The well developed techniques of density functional theory can be employed in that limit, as will be outlined in Section 3.3.5. A widely used, simpler alternative is to parameterize the screened Coulomb potentials in terms of Yukawa functions. This concept allows also to include in a phenomenological way dynamical effects by using velocity dependent screening lengths. Surely, such heuristic treatments can be justified only a posteriori in truly nonlinear and dynamical situations, but they provide a very useful tool to supplement the linear response description by small nonlinear contributions in the more relaxed case of semilinear coupling. The resulting corrected linear response treatments are well suited to get fair approximations of the energy loss at semilinear coupling. This will be discussed further in the next subsections. In any case, of course, all approximations for the semilinear regime and extensions into the nonlinear one which are based on some appropriate assumptions and models for an effective potential need to be checked by fully dynamical manybody treatments as e.g. the simulation techniques presented in Section 3.3. 3.2.2. Corrections of linear response stopping using cutoffs As shown above, the linear response expression (39) can be viewed as result of the binary collision approach in the weak coupling regime. We have seen, furthermore, that systems in the semilinear coupling regime violate the linear coupling only in a small vicinity of the ion where r(r . Small distances correspond to high momenta. This suggest to approximate the stopping  power by using again the linear response, i.e. Born, result but to cut the k-integration at an upper limit +1/r and to add a correction term which accounts for the excluded inner region of nonlinear  coupling. This line of development has been introduced by Bloch for ion stopping by bound electrons in a gas target [20]. For the inner strong coupling region, the projectile—electron

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interaction is much stronger than the atomic bonding. Thus he regarded the corresponding projectile—electron collisions with small impact parameter as collisions with a free electron and calculated them from a binary collision description using a pure Coulomb potential. The finally resulting well-known Bloch correction has then to be added to the Bethe formula for gas targets or in the case of an electron plasma to its analogue, the linear response stopping power (39). As Bloch assumed a pure Coulomb interaction for r(r the strong coupling region and hence in our case  r /j must be very small. For a recent, quantitative discussion of the regimes of applicability of the  Bloch correction see [120]. Extensions to higher ratios r /j are provided by more elaborated  correction terms which are discussed in the next subsection. Here we concentrate on the simpler, widely used approximation scheme of using cutoffs. Guided by the observation that the Bloch correction or similar correction terms reduce the linear response stopping by some amount the k-integration in the stopping power expression (39) is restricted by an upper limit k , i.e. K







1 Ze k ) €L dE " dk Im , e (k, k ) €) k ds e (2n) IIK 0 

(65)

where the cutoff parameter k 91/r has to be defined suitably in order to exclude the small region K  r(r of strong perturbation and to add the remaining effects of the close collisions in the average.  This can be achieved by comparison with the binary collision approach using typical screened potentials, as will be discussed now. The first step for this comparison is to work out a simplified closed expression for the stopping power (65). By interpolation between the known exact results in the limits of high and low velocities [115] one can deduce dE +Zh(v)ln(k j) . K ds

(66)

This approximation is valid for k j<1 and any H, v. The function h(v) is linear in v for v;1v 2 K  and goes Jv\ for v<1v 2 while j can be identified as an effective velocity dependent screening  length. Note that the validity condition k j<1 together with the typical value k &1/r nicely K K A recovers the definition (22) of the semilinear regime. The approximation (66) applies also to the linear regime where k can be related to the averaged deBroglie wavelength as K k "2/1 2"2m1v 2/ .  P K The effective screening length j(v) read off from the dielectric approach can now be used to approximate the ion—electron interaction by (r)" (r)exp(r/j)"(Ze/4ne r)exp(r/j). In the limit    of infinite projectile masses the projectile velocity is not a dynamical variable of the system and the velocity dependent potential interferes only as an external potential which can be employed in a binary collision treatment. The transport cross section for this Yukawa-like potential and the related stopping power have been much studied over the last decades, see for example Refs. [86,113,114,70,60,71,53]. It turns out that one can derive an approximation for the binary collision stopping power Eq. (59) which reads


 dE ds






2j 1 +Z hI (v) ln ! ln(1#c1g 2) ,  1 2 2
 

(67)

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where ln c"0.5772 is Euler’s constant and hI (v) is very close to the h(v) from Eq. (66) having exactly the same behavior for small and large velocities. The approximation (67) is valid for large arguments of the logarithm, i.e. 2j/1 2(1#c1g 2)<1, and for any coupling strength 1g 2.    We now compare the binary collision approach with screened potential (67) with expression (66) obtained by a cutoff dielectric description. Both expressions are indeed very similar and we find the desired cutoff as 1 2 . (68) k " K 1 2 (1#c1g 2   In the quantum mechanical plane wave regime 1g 2;1, k becomes identical to the intrinsic  K cutoff 2/1 2, as provided from the behavior of the dielectric function e (47), and Eqs. (65) and (66)  0 merge into the exact linear response or Born-RPA result. In the classical limit 1g 2<1 of  ion—electron collisions with nonlinear coupling in the inner interaction region, the cutoff (68) tends to k "2/c1 21g 2+1/1b 2, see Eq. (15). From the definition of r via the potential (19) and for K     the conditions 1b 2<1 2 (i.e. 1g 2<1) and r ;j, which define the semilinear regime (see     Eq. (22)), we obtain r 91b 2. This yields finally k +1/b 91/r in agreement with the initial   K   assumption when introducing the cutoff k . K Classical derivations of the stopping power [135] treat the electrons as classical particles and employ accordingly the Fried-Conte plasma dispersion function [55] which is the limit P0 of the RPA free density—density response (45) and provides the classical dielectric function. The classical dielectric function fails at large k-values and a cutoff is needed to render the k-integration in Eq. (39) finite. We identify k +min(1/1b 2, 2/1 2) as the commonly used cutoff in these classical  K  dielectric approaches. It has to be emphasized, however, that the cutoff (68) for the linear response stopping power in Eq. (65) is introduced to account for physical effects and not just as a mathematical tool to avoid divergences. The general linear response (Born-RPA) expression (39) yields a convergent integral by virtue of quantum effects which provide naturally an intrinsic cutoff at 2/1 2 at any degeneracy  H including the case of classical targets H<1. The linear response stopping, however, is a valid approximation only for linear coupling 1g 2;1 and becomes invalid in the semilinear and more  so in the nonlinear regime. The physical cutoff discussed here intends to correct for a small region of strong perturbation at r(r and to extend the validity of the linear response expression into the  semilinear regime. We remind that the linear regime never appears in an entirely classical system since P0 implies both HPR and 1g 2PR. The need of a cutoff is decided by the question  whether we are in the linear or the semilinear regime, that is, by the parameter 1g 2, and  independent of the degree of degeneracy H. 3.2.3. Corrections of linear response by binary collisions So far we have seen that the comparison of the linear response description with cutoff k and K the binary collision treatment with screening length j provides a simple recipe to extend the linear response formalism into the semilinear regime with a minimum of calculational effort. The whole scheme, however, is based on several approximations as expressions (66) and (67) which are well justified only close to the linear regime. In order to proceed further into the semilinear coupling regime one should prefer more complete approaches to incorporate the binary collision

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contributions from the strong coupling region. A first attempt in this direction was the Bloch correction mentioned above. Recently more elaborated schemes have been proposed in [61,132,124]. They are based on an idea of Gould and DeWitt [65] in context of deriving convergent collision integrals as needed for the calculation of transport properties in the framework of kinetic theory. Motivated by the corresponding diagrammatic expansion, see also e.g. [146], the cross section in the collision integral is estimated by composing three terms: The linear response or first order Born approximation (BI) for the dynamically screened Coulomb interaction and the binary collision result for a statically screened interaction are added while the linear response cross section for the statically screened interaction is subtracted to compensate for its appearance in the other terms. More precisely, the last term is the first order Born approximation of the binary collision term as well as the static limit of the dynamically screened linear response contribution. In other words, the composed cross section represents a T-matrix expression with a dynamically screened first order Born approximation but statically screened ladder terms of higher order. Translated to stopping powers we arrive at a combined expression (dE/ds)  consisting of the following three contributions



 dE ds





dE  dE dE  " ! # . ds ds ds 
 ' '

(69)

Here (dE/ds) represents the Born-RPA, i.e. Eq. (63) taken in the limit of heavy projectiles ' (m/MP0, u(k)"k ) €), which is equivalent to the linear response stopping power (39) with RPA-dielectric function e . The static limit of this expression is obtained by replacing the 0 dynamically screened interaction used in the cross section (62) and subsequently for deriving the Born-RPA stopping power (63) by the statically screened effective ion—electron interaction Ze e (k)  " , » (k)"  e (k, u"0) e ke (k, 0)  0 0

(70)

which results in the third term






dE  Ze Im[e (k, k ) €)] 0 " dk €L ) k . ds e (2n) k"e (k, 0)" '  0

(71)

The binary collision contribution (dE/ds) is equal to expression (59) where the transport cross
 section p (p ) has to be determined from the exact differential cross section p(p ,0) for the statically  P P screened potential » (70) calculated, for instance, from a phase shift analysis. A specification of this  scheme to entirely classical systems is discussed in Section 4 together with a comparison to simulation results. More general expressions for arbitrary mass ratios are given in [61,124] where the different contributions are evaluated numerically for some specific target conditions and projectile parameters. The combined expression (69) has several nice features. First, no external parameter appears in contrast to the cutoff approach. Furthermore, a smooth merging into the correct high velocity limit of linear coupling is guaranteed. For increasingly high velocities, where 1g 2P0, the complete  binary collision result turns into its first order Born approximation. Thus the last two terms (dE/ds) !(dE/ds) cancel gradually and leave solely the dynamical linear response contribution
 '

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(dE/ds). At low velocities the dynamically screened potential turns into the statically screened ' one and both the first order Born approximations (dE/ds) and (dE/ds) become identical. The ' ' remaining binary collision term takes into account all the strong coupling effects. The resulting stopping power can thus be expected to be rather close to the exact one provided that linear response accurately reproduces the true statically screened interaction. In general, the whole scheme is essentially based on the assumption that the collective effects, static ones as well as dynamical ones, are well described by linear response. Since the collective phenomena take place on the scale of the actual screening length, i.e. at small momenta k, the strong interaction region must thus be small on this scale and hence restricted to high momenta. This immediately leads back to the definition of the semilinear regime, Eqs. (22) and (23), as the proper domain for such type of approximations. It may be extended to (weakly) nonideal plasmas by replacing the RPA dielectric function by the appropriate one which includes correlations within the target, see Section 3.1.4. 3.2.4. Binary collisions with effective screening lengths In the previous approaches effective interactions served to provide binary collision contributions for correcting the linear response energy loss. Relying on binary collisions only widely used treatments are based on some parametrized screened Coulomb potentials typically in terms of Yukawa functions or related approximations. There, dynamical effects are frequently included in the average by employing velocity dependent screening length as, for instance, in deriving approximation (67). This should be easily feasible for low ion velocities where adiabaticity is guaranteed as well as for sufficiently high velocities where an effective screening length j(v) can be simply adjusted in the binary collision treatment to meet the exact high velocity result (44). The choice of an appropriate screening length becomes, however, crucial for imitating the dynamical effects at medium velocities and no assurance exists that the real dynamical situation can be treated in a satisfactory manner by a velocity dependent, but still statically screened ion—electron interaction. Here the combined expression (69) can provide a good starting point for constructing an effective screening length. First the statically screened potential is transfered, e.g. into a Yukawa potential by replacing e (k, 0) with 1#1/(kj). Now one can interpret Eq. (69) in a different manner by 0 considering the difference (dE/ds)!(dE/ds) as a correction to the binary collision term ' ' (dE/ds) . Demanding that the stopping power is given entirely by (dE/ds) the desired screening

 length j(v) can be read off from a numerical solution of (dE/ds)"(dE/ds) . For this, admittedly ' ' quite phenomenological, approach again no external parameter is needed and the same restrictions to the semilinear regime apply as discussed in the previous subsection. It is resumed later on for classical systems and compared with numerical simulations in Section 4. Similar ideas have been proposed and investigated for atomic stopping by bound electrons in [118,119]. As it turns out in these investigations, however, this type of description yields for increasing coupling strength improved approximations compared with the theoretically better founded scheme, Eqs. (69)—(71). 3.2.5. Charge dependence at semilinear coupling It emerges as a new feature in the semilinear regime that the Z-charge dependence of the stopping power is modified for low and medium velocities by correction terms which carry the charge dependence of the binary collision cross section, see e.g. [132]. Considering for instance again a Yukawa like screened Coulomb potential with screening length j. There the transport cross section can be written as p (p )"bh(b /j, g "b /1 2) [120] where b and g depend linearly    P     

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on Z. For g ;1 the function h turns into a function of 1 2/j only. This results in the quadratic   Z dependence of the cross section in first Born approximation. Increasing g , however, is connected  with a quite complex behavior of h as function of Z. For values of b /j and g as characteristic for   the semilinear regime h depends logarithmically on Z. This behavior is then recovered in the stopping approximations for semilinear coupling. It can be immediately read of from expressions (66) and (67) in connection with the definitions of k (Z) (68) and 1g 2(Z) (16). To be more specific, K  the low velocity limit of the corrected linear response stopping power Ze n dE "! v [CI (m, H, "Z")#O(v)], v;1v 2 , (72)  e (2n)u ds   is now described by a dimensionless friction coefficient CI (m, H, "Z") with a logarithmic dependence on Z




CI (m, H, "Z")Jln



8ne m1v 2j (v"0)    . c"Z"e

(73)

for the semilinear coupling conditions 1b 2/j;1 and 1g 2<1.   For velocities larger than 1v 2 but still small enough to fulfil 1g 2<1 we get the classical high   velocity result of Bohr [21]






8ne mv Ze dE  &! n ln , v<1v 2, 1g 2<1 , (74)   c"Z"eu 4nemv ds   whereas still higher velocities lead back to linear coupling and the high velocity limit (44). We want to point out that the conditions for semilinear coupling imply large arguments of the logarithms in the above expressions. 3.3. Nonlinear ion—target coupling The regimes of linear and semilinear coupling allowed to separate the problems of the ion—target coupling from the interactions within the target. The common concept was the linear response formulation which allows to describe the energy loss for a large variety of target conditions by the same basic relation (39) just adapting the dielectric function to the particular target condition. This changes if we proceed to nonlinear ion—target coupling. Now each parameter regime for projectile and target requires its own methods. In this section, we are going to discuss some of these methods to compute stopping in the nonlinear regime. We start in Section 3.3.1 with the Vlasov—Poisson equation which is designed to study nonlinear coupling effects in classical, ideal targets. The appropriate tool for classical, but nonideal targets is given by molecular dynamics (MD) simulations which are presented in Section 3.3.3. The wave nature of the electrons becomes crucial in two-component plasmas. This can be taken into account in a generalization of MD simulations, the wave-packet molecular dynamics (WPMD) where the pointlike electrons of the classical MD simulations are replaced by Gaussian wave-packets. This will be discussed in Section 3.3.4. The WPMD adds quantum effects of wave nature and allows to treat weakly degenerate targets (H:1). For fully degenerate targets (H;1), density functional methods are a widely used tool in all fields

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of electron physics. The basic aspects of density functional approaches, the local density approximation (LDA) and its time dependent extension (TDLDA) are outlined in Section 3.3.5, an application to stopping is reported in Section 3.3.6. There are, furthermore, some promising concepts for future work. An extension of the Vlasov equation to treat ideal and degenerate (H41) electron plasmas is sketched in Section 3.3.2. In Section 3.3.7, we present an expression for the stopping power at low velocities in terms of a force—force correlation function, which offers an access to nonlinear stopping by means of the simpler equilibrium descriptions of the ion—target system. 3.3.1. Vlasov—Poisson for classical ideal targets We first consider the Vlasov equation in the regime of classical (H<1, 1g 2'1) and ideal  (C;1) electron plasmas. Ideal targets allow a mean-field treatment of the electrons because correlation effects are negligible. The electron cloud is described in terms of a phase-space distribution f (r, u, t) and it is propagated by the Vlasov equation jf e j jf jf #u ) # ) "0 jr m jr ju jt

(75)

using the static electric potential as given by the Poisson equation





. . Ze e f (r, u, t) du!n . (76) D (r, t)"! ! "! d(r!€t)# e e e e     Here n is the number density of the target electrons in absence of the ion. It serves here as a homogeneous neutralizing background. This electrostatic approximation is valid for nonrelativistic ion and electron velocities. The induced electric field E "!

at the ion position or   alternatively the induced electron density provided by a solution of Eqs. (75) and (76) yields the stopping power by virtue of Eq. (6). The Vlasov—Poisson equation requires ideal targets but is applicable for any strength of the ion—electron potential. The regime of weak coupling is also accessible and it is interesting to look first at the Vlasov equation in that well understood coupling regime. The relative strength of the ionic perturbation can be estimated as




e j jf ) m jr ju

jf v r Q , +(3ZC  (77) jr u "r!€t" Q where we have employed a typical velocity scale u , a typical length scale r "u t "u /u , and as Q Q QQ Q  typical time scale one plasmon cycle t "u\"(me /en) together with the relation Q   C"u/4n(3nv for the classical plasma parameter (13). The factor (3"Z"Cv /u character   Q izes obviously the strength of the perturbation by the ion. A natural and often used choice for the velocity scale is the thermal velocity, u "v . The strength of the perturbation becomes then Q  (3"Z"C""Z"/3N where N is the number of particles in the Debye sphere, see Eq. (25). Weak " " perturbations are defined by "Z"/3N ;1. They allow an expansion in orders of the perturbation " Z/N . The linear terms of the Vlasov—Poisson equation yield the classical dielectric description " ([87], Section 3.4) and its stopping power [135]. The linearized Vlasov—Poisson equation can be u)

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solved formally using Fourier transformation (see Appendix B.3). There results the dielectric function e (k, u) for classical electron plasmas [135,55]  en 1 jf (u) e (k, u)"1# du k )  , (78)  me k ju u!k ) u#i0  where f (u)"(m/2nk ¹)exp(!mu/2k ¹) is a Maxwell distribution. This classical dielectric  function e is the limit P0 of the RPA dielectric function e (47) and reproduces correctly the  0 long wavelength (small k) behavior of e at H<1 where quantum effects can be safely neglected 0 [7]. One thus obtains from the linearized Vlasov—Poisson equations the classical version of all linear response quantities and observables connected with e (see Section 3.1.1) and the linear  response stopping power (39). The second order of the expansion in Z/N yields the Z-Barkas " contribution mentioned in Section 3.1.8. A closer inspection reveals that a more relevant energy scale is associated with the mean relative velocity between ion and electron, u "1v 2"v 1v 2. This choice suggests that the essential Q    expansion parameter is Z/N (1v 2) instead of Z/N [51,154]. There is, however, a basic problem "  " with the classical approach to the stopping power. It yields diverging integrals which require the introduction of an upper cutoff k in the final quadratures of, e.g., Eq. (39). This feature can be K nicely seen from Eq. (77). The Coulomb potential of the ion diverges at rP€t however small the prefactor may be. This demonstrates that the linearized Vlasov—Poisson equation is only applicable for relative distances r/j'(3"Z"C/(1v 2), where j"1v 2/u is the dynamical screening    length in an ideal plasma. The condition is, of course, only relevant for radii r(j which, in turn, requires (3"Z"C/(1v 2);1. We thus recover the definition of the semilinear regime (25) where  the parameter (3"Z"C/(1v 2)"1b 2/j marks the boundary between the semilinear and the   nonlinear coupling regime. In summary, the linearization or, more general, the expansion of the Vlasov—Poisson equations is applicable in a parameter region which coincides with the semilinear coupling regime and provides approximations for the stopping power depending on a cutoff as discussed earlier in Section 3.2.2. We see also that the condition for linear coupling is here always violated in a sufficiently small vicinity of the ion. Numerical solutions of Eqs. (75) and (76) are unavoidable in the regime of nonlinear coupling. The Poisson equation Eq. (76) is usually solved on a spatial grid. There are various methods available for that task. A particularly efficient method employs fast Fourier transformation to obtain the potential for a given charge density . [106]. There are two different schemes for   the propagation of the Vlasov equation (75), either by finite element techniques on a grid in phase space or by particle-in-cell (PIC) techniques [80]. The direct solution of the Vlasov equation in six-dimensional phase space is extremely demanding concerning memory requirements, still beyond nowadays technical limits. There exist, however, solutions of the Vlasov—Poisson equations for a reduced four dimensional phase space (r, z, v , v ) corresponding to a cylindrical simulation P X box, achieved with the help of massively parallel supercomputers [23]. The more efficient alternative is the PIC treatment where the phase-space distribution is represented by a swarm of pseudo-particles. These are propagated according to Newton’s equations of motion in the potential

. The (moderate) effort to solve the Poisson equation on the spatial, i.e. three-dimensional, grid remains the same. But the propagation of the Vlasov equation is much accelerated because the numerical expense depends only linearly on the number of pseudo-particles. And this number is to



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be adapted to the spatial grid (rather than the full phase-space grid) to achieve a sufficiently smooth representation of the electron density on the grid. Typically, a few pseudo-particles per grid cell are necessary. Two basically different scenarios are conceivable. In the one case, one pseudo-particle stands as a “macro particle” for many physical electrons. This is the genuine PIC scheme. In the other case, one physical electron is represented by many pseudo-particles. This is the regime of test-particle simulations, typically applied in dense fermion systems where the Vlasov equation is thought to represent the classical limit of a quantal phase-space distribution [18]. The PIC simulations are usually very efficient and allow to perform the computation on work stations at the expense of some fluctuations, or noise, however in the results. The noise can be suppressed, if necessary, by enhancing the number of pseudo-particles. The PIC technique was applied recently to determine the stopping power in cylindrical 2-D spatial geometry [81,38] and in full 3-D space [196,199,200]. The problem with the Coulomb singularity at small distances is also present in these numerical solutions. One usually smoothes the ionic 1/r potential for rP0, either explicitly [22] or implicitly by the finite resolution of the spatial grid. The PIC simulations of [196,199,200] use the full ionic Coulomb potential by switching to piecewise analytical solutions for the 2-body Coulomb problem of a pseudo-particle close to the ion. This scheme was taken over from the MD simulations for ion stopping which will be discussed in Section 3.3.3 and in Appendix A.2. There is, however, a price to be paid for this achievement: High resolution at small distances limits the size of the simulation box if the number of test particles is kept fixed (which is usually the case due to technical limitations). The box size, in turn, sets an upper limit on the wave length in the description. Thus, a high spatial resolution can run into a conflict with long-range phenomena. The appropriate compromise depends on the actual problem. The stopping power at high velocities requires a reasonable treatment of the dynamical screening at long wavelengths, whereas high resolution of the order of the static screening length is necessary for studies in the low velocity regime. Results are presented and discussed in Section 4. We mention in passing that it is not guaranteed that a solution for the Vlasov—Poisson equations (75) and (76) exists, because of the singular external potential J1/r. However, for any smooth  potential related to an external charge distribution with a continuous first derivative solutions with a stable time evolution in the neighborhood of stationary solutions have been proven to exist [150]. 3.3.2. Fermionic Vlasov equation The Vlasov equation (75) will fail for dense Fermion systems where the degeneracy is high, i.e. H;1. This regime calls, in principle, for fully quantal methods as time-dependent Hartree—Fock or TDLDA (see Section 3.3.5, for an example of an application to dense hydrogen see [85]). The full treatment of all quantum aspects, however, grows quickly very expensive and inhibits all calculations with large samples. But one can take advantage from noting that there are two different quantum effects: wave nature of the particles and Pauli principle. And there are cases where the wave nature of the fermions can still be well approximated by classical propagation but where the phase-space density grows so large that the Pauli principle becomes crucial [148]. For electrons, this is the regime of high degeneracy, H(1, but still large Coulomb parameter 1g 2'1.  The most widely used solution for that problem is to continue to use the Vlasov equation but to start from an initial distribution which respects the Pauli principle, often chosen as the Fermi

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distribution instead of the classical Maxwell distribution. This strategy provides reliable results for a certain (small) time interval. And so, the Vlasov equation has been successfully applied for studies of nuclear dynamics in energetic heavy-ion reactions [18]. Nevertheless, the violation of the Pauli principle develops with time and sooner or later the Vlasov results become unreliable. The problem can be traced back to the finite resolution in any numerical treatment which induces artificial dissipation. This dissipation then drives relaxation towards the classical Boltzmann equilibrium distribution which, of course, violates the Pauli principle at low enough temperature [148]. The actual time scale of the relaxation depends on the number of test particles. But it is often interfering with the time scale of the studied processes, see [148] for the case of nuclear dynamics and for the case of electrons in metal clusters [44]. A possible solution of this problem is to complement the Vlasov equation by a dedicated collision term which scans the violations of the Pauli principle, i.e. occurrences of f'1, and collides fermions out of these dangerous areas in phase space. It was shown that such a scheme is manageable and successful, for the case of nuclei in [149] and for electron systems in [45], respectively. The modified Vlasov equation respects the Pauli principle to a good approximation and over arbitrarily long times. This opens the way to test-particle simulations of large systems in degenerate, but otherwise ideal, plasmas. 3.3.3. Molecular dynamics simulations We now consider entirely classical projectile—target systems which are nondegenerate, H<1, have a large Coulomb parameter for the electrons, 1g 2'1, and nonlinear ion—target coupling  corresponding to classical ion—electron collisions, (1g 2'1). We want to have no restriction on  the coupling strength within the target, i.e. on the ideality m(H<1)"C. Molecular dynamics (MD) simulations provide a suitable tool to study the full many-body dynamics of these systems, see e.g. [4]. They have been successfully applied to various strongly coupled systems during the last decades, in particular, to classical Coulomb systems as the one component plasma (OCP) or (binary) ionic mixtures [73,75]. They are also a powerful tool to compute stopping of a heavy ion in an electron OCP, i.e. to study the ion—electron system as described by the Hamilton operator (2) [191—197,199—201]. The procedure is conceptually very simple: the ion and the electrons just follow the classical equations of motion with mutual Coulomb interactions





dr G"u , G dt

du 1 j e Ze G"! ! , dt m jr 4ne "r !r " 4ne "r !R"  G H  G G H$G

dR "€, dt

d€ 1 j Ze " , dt M jR 4ne "r !R"  G G

(79)

where +r , u , i"1,2,N, are the electronic and (R, €) the ionic coordinates. These equations are G G solved with standard numerical algorithms. The events where an electron comes close to the ion need to be singled out and propagated separately in order to cope with the Coulomb singularity at rP0. Observables are measured as time averages over a certain time interval or as ensemble averages over different initial configurations. To compute the stopping power, an equilibrium state of the electron system is produced first by a MD propagation without the ion. Then the ion is

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added and its energy loss *E as well as travelling distance *s are determined over a sufficient long time q. The ratio *E/*s(q) is the time averaged stopping power. Heavy ions have negligible change in velocity during the measuring time interval and thus *E/*s represents immediately the stopping power for the given (initial) ion velocity v. Further technical details of the MD-simulations are outlined in Appendix A.1 and the results are discussed in Section 4. The MD simulations contain without restriction all correlation effects, dynamical screening, close collisions, and even multi-particle correlations. They do not rely on a spatial grid for the Coulomb force and thus there is no restriction in the spatial resolution. On the other hand the expense grows quadratically with the particle number N because the Coulomb force is to be evaluated separately for each pair of particles. The available computing power thus limits the computation to at most a few thousand particles. Typical MD simulations use 500 electrons and can still be performed on small work stations. In order to approximate an infinite plasma by this finite particle number, the electrons are packed into an elementary cubic simulation box and this box is continued periodically in all three spatial directions. The limited simulation box hence introduces a largest length scale on which collective phenomena can be explored. This limitation is unimportant for strongly coupled targets (C'1) and low projectile velocities because the screening length is then small compared to the size of the simulation box. It becomes a real problem, however, at high projectile velocities where a spatially extended wake field is induced by the ion. But then one leaves anyway the regime of nonlinear coupling and can continue with the simpler linear response methods. This example shows that MD simulations are particularly suited for the case of strong coupling and become inefficient, if not impossible, just where linear response is valid. We mention in passing that there are attempts to overcome the quadratic growth of expense in the MD simulations by a mixed strategy. The particles are collected in spatial cells and the Coulomb force from remote cells is computed globally. The necessary book-keeping is managed using tree structures, immediately suited for coding in what one calls “tree codes” [16,152,167,137]. The method succeeds in cutting the quadratic growth of expense. But there comes first an enormous overhead of book keeping which makes these tree codes competitive only for very large ensembles around 8000 particles and larger. The MD-simulations are based on a pure classical treatment. This implies the restriction to parameters where the ion—target system behaves classically. As a consequence, no ground state of the classical ( P0) ion—target system exists and the dynamics at long time scales ends up in a Coulomb collapse where some electrons get deeper and deeper bound in the ion potential by transferring energy and momentum in interactions with the surrounding electrons. This problem is not explored in the dynamical non-equilibrium simulations of stopping where only a short time interval after switching on the ion—electron interaction is of interest. The Coulomb collapse, however, inhibits straightforward classical MD simulations to a two component target plasma for which no classical equilibrium state exists. A way out of this dilemma is to introduce effective interactions which account for quantum effects by an appropriate smoothing of the Coulomb potential at short distances [39—41,97,74,76,56]. These effective interactions are deduced from an exact solution of the two-body quantum problem at thermal equilibrium. This restricts the method to situations close to equilibrium. Truly non-equilibrium simulations require still other approaches, as e.g. the wave-packet molecular dynamics which is presented in the next subsection.

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3.3.4. Wave-packet molecular dynamics (¼PMD) in the quantum, nonlinear, nonideal regime The classical MD simulations, as outlined in the previous subsection, are conceptually simple and provide nonetheless a fairly complete description including all sorts of correlations. But often enough one cannot completely neglect quantum effects. Dense systems are likely to run into the degenerate regime H:1. Such a situation can be mastered when using the fermionic Vlasov equation as outlined in Subsection 3.3.2. But one often needs also to take care of the wave nature of the electrons, g :1, e.g., to reach a stable equilibrium for two-component plasmas. There are  several techniques for a fully quantum mechanical description of coupled electron—ion systems, e.g. the path-integral Monte Carlo method [32,111] or ionic MD on electronic Born—Oppenheimer surfaces [30,85]. These powerful techniques allow detailed calculations with compromises resulting mainly from exchange, e.g. treatment of nodes in the density matrix. But they are extremely expensive and limited to situations close to equilibrium. In order to deal with quantum systems (g :1) which also may be far from equilibrium, one has  to use a time-dependent quantum mechanical propagation for the electrons. But this is an extremely demanding task. The aim is an approximate quantum treatment which comes from the classical molecular dynamics side, thus maintaining its simplicity as well as its correlation content, and adds just the most basic quantum features, as e.g. finite extension and uncertainty principle. This is achieved by using localized wave packets with a simple analytical form. Such a wave packet dynamics based on Gaussians has been explored in [78,98,99]. It is called wave-packet molecular dynamics (WPMD). The use of Gaussian wave packets yields simple analytical expressions which allow an efficient coding, and it provides an exact solution in two limiting cases: a free particle and a particle in an harmonic oscillator potential. It is still a very good approximation if the width of the wave packet is smaller than the typical length scale of variation of the potential, i.e., if the potential can be approximated locally by a harmonic oscillator [78]. We discuss here the WPMD for a Coulomb system of ions, treated classically, and electrons. The state at a given time t is thus specified by +R (t), P (t), W(x ,2, x ; t), , ' '  ,

(80)

where R (t) and P (t) describe the positions and momenta of the ions and W(x ,2, x ; t) is the ' '  , many-particle wave function for the electrons. The dynamics of this sytem is governed by the Hamiltonian HK "H (R , P )#HK  (xL , pL )#HK U (R , xL ) . ' ' G G ' G

(81)

The Hamilton function 1 Ze P H (R , P )" ' # ' ' "R !R " 2M 4ne  '( ' ( '

(82)

determines the dynamics of the ions with mass M and charge Ze, and the Hamilton operator pL  e 1 HK  (xL , pL )" G # G G 2m 4ne "xL !xL " H  GH G G

(83)

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the dynamics of the electrons with mass m. Finally, the electron—ion interaction is given by 1 Ze HK U (R , xL )"! . (84) ' G 4ne "R !xL "  'G ' G The equations-of-motion are derived from the principle of stationary action [104] which yields RQ "jP'1W"HK "W2, PQ "!jR'1W"HK "W2 , (85) ' ' R R d d dt W i W "d dt1W"HK "W2 . (86) dt R R These equations are still equivalent to the time dependent Schro¨dinger equation if the variational wavefunctions are allowed to exhaust the full Hilbert space. But then the system of equations is not tractable for a true many-body system. A very simple approximation for the electron propagation is needed to allow for long-time studies for large probes. To this end, the electrons are parametrized as Gaussian wave packets, i.e.

   





W(x ,2, x ; t)+A “ u (x ; t) , G G  , G 3  3 ip ip exp ! # A (x!r)! (x!r) , uq(x)" 2nc 4c 2 c











(87) (88)

with the variational parameters q"(p , c, p, r), where p and r are the classically conjugate paraA meters for the position and p and c for the width. The antisymmetrization operator is denoted by A A. The complete antisymmetrization limits the approach to small systems [52,130]. A manageable compromise is obtained if one expands antisymmetrization in orders of exchanges and keeps only terms up to 2-body exchange [98—100]. Thus WPMD permits at this stage studies in the regime H:1, but further extensions are necessary for the completely degenerate case. With the parametrized ansatz (88) the dynamics of the electron wavefunction is reduced to a “classical” dynamics of the parameters q. Application of the variational principle (86) to a parametrized wave function yields jH . (89) N qR " JI I jq J I which differs from a straightforward Hamiltonian equation by the occurrence of the norm overlap [147] N "i 1jn Jjo I!jn Ijo J2 . O O JI O O Hamiltonian dynamics is recovered only if the norm overlap takes a symplectic form




0 N " JI !1ª



1ª 0

,

(90)

(91)

where 1ª is the unit matrix in the 4N;4N subspace. It is one of the advantages of the particular form (88) for the single electron wavepacket that the norm overlap N without any exchange is 

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independent of time and always symplectic which makes the equations-of-motion for q particularly simple. Including 2-particle exchanges, one obtains a correction dN to the norm matrix. The correction to the inverse norm matrix is calculated according to N\+N\!N\dNN\ .   

(92)

One thus finds that the expectation value of HK over the wavepackets H(R , P , r , p , c , p G)"1HK 2 ' ' G G G A

(93)

acts as a classical Hamilton function. It comes as an additional advantage of the ansatz (88) that this expectation value can be performed analytically for a Coulomb system leading to an expression which contains only elementary functions. This is crucial for a fast propagation algorithm. Taking all pieces together, one can describe the dynamics of the plasma by the generalized classical Hamilton function H"H (r , p , c , p G)#T.  (r , p , c )  G G G G G G A

(94)

#H (R , P )#H U (R , r , c ) , ' ' ' G G

(95)

where













1 "r !r "(3 1 pG 9  p e A# # G # H erf G H (r , p , c , p G)" G G G A "r !r " m 2 8c 2 4ne (2(c#c) G H G  GH G G H



(96)

 3(c!c)#4ccD 3c c G H G H GH G H , T.  (r , p , c )" exp(!D ) GH  G G G m (c#c) (c#c)!8(c c )exp(!D ) H G H GH G H G GH

(97)

#V (r !r ) , #  G H





P 1 Ze #V (R !R ) , H (R , P )" ' # #  ' ( ' ' 2M 4ne "R !R " ' (  '( '








(98)



Ze 1 "R !r "(3 G H U (R , r , c )" erf ' #V (R !r ) , ' G G #  ' G 4ne "R !r " c (2  'G ' G G V






1 1 ! . (r)"  #  "r#l¸" "l¸" l

(99)

(100)

Evaluating Eq. (86) in terms of two body exchanges in the two body center of mass frame yields the expression (97) where the phase space distance 2(c c ) 3 G H D " (p !p )# (r !r ) GH 3(c#c)  G H H 2(c#c) G G H G H

(101)

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shows that the expansion parameter for the hierarchy of n-body exchanges is e\D. By measuring this parameter, the validity of the approximation is checked during propagation. Technically, the two-body exchange effects are incorporated in Eq. (97) as an additional effective electron—electron potential. Note the occurrence of the Ewald potentials (100) in every Coulomb term. This accounts for the Coulomb interactions with the particles in the periodically continued simulation boxes which are used to simulate infinite systems with a finite sample. Details on the solution of the pseudo-hamiltonian equations (89)—(101) are given in Appendix A.3. The criterion for the validity of a classical approach is that the thermal wavelength K" /(mk ¹ should stay safely below the average inter electron distance, i.e. the Wigner—Seitz radius a"(3n/4n)\. The limit of purely classical dynamics would be recovered if the forces vary sufficiently slowly over the width of the wave packet, c » P ;1 , ([(E!») ( /mc)]

(102)

where c is the width of the wavepacket and » a typical value for the curvature of the force. Note P the order of the derivative. Up to a purely harmonic potential »Jr the WPMD propagation is exact. On the other hand Eq. (102) cannot be fulfilled everywhere for a Coulomb potential. The WPMD includes at least the same amount of correlations as classical molecular dynamics. This is seen, e.g., by making the width of the wavepacket proportional to the thermal wavelength and keeping it fixed during the simulation. Then, the method yields the same results as traditional molecular dynamics simulations employing effective potentials. The purpose of the WPMD reaches farther out to true non-equilibrium situations. This is achieved by allowing the width of the wave packets its own dynamics which can adjust itself to the temperature and density of the system. The WPMD contains enormous simplifications in order to allow for the handling of large ensembles. This implies, of course, severe approximations. We have already seen that the expansion of exchange limits the regime of acceptable degeneracy to H:1, in practice to H'1/10. There is, furthermore, the requirement that the widths have to stay below the size of the simulation box. This is guaranteed for systems which have predominantly localized electrons, as e.g. a dense hydrogen plasma consisting of H -molecules, atoms and possibly larger clusters. In other cases an unphysical  growth of the delocalization can be prevented by adding to Eq. (96) a suitably tailored potential DH (c ) whose parameters are fixed by fitting a known observable of the OCP, e.g. the potential G energy, see for example Chapter 3.2 of Ref. [91]. 3.3.5. Local energy-density functionals: ¹he Kohn—Sham equations, ¸DA and ¹D¸DA As outlined in Section 3.1.4, where we discussed the LFCs, density functional approaches aim to incorporate higher correlations into mean field calculations by virtue of an effective interaction, where quite general theorems guarantee the existence of such effective interactions [84,102,151]. While the LFC deals with homogeneous many-electron systems and inserts nonideality effects in the linear response, inhomogeneous systems, as atoms, molecules, surfaces, or in our particular case the inhomogeneity from the projectile ion º , add further complications and different (even  simpler) approaches are needed as starting point. A very efficient and widely used scheme are the Kohn—Sham equations with exchange and correlations in Local-Density Approximation (LDA), for extensive reviews see [46,96]. We discuss it here for the fully degenerate case (H"0) which is

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the typical regime of applications for the LDA. It is based on a variational formulation starting from an expression for the total energy in the form



E "E (+u ,)#E (n(r))#E (n(r))# dr n(r)º (r)    ? !  

(103)

and referring to the local electron density n(r)" "u (r)" . (104) ? ? Note the dependencies: the Coulomb energy depends naturally on the local density, the exchangecorrelation energy is approximated by a local density form, and the full dependence on the single electron wavefunction u is carried through for the kinetic energy which serves to maintain the ? quantum shell structure. The Kohn—Sham equations are obtained by variation with respect to the uH which yields ? pL  #º (r)#º (r)#º (r) u "e u , (105) !   ? ? ? 2m






dE (n(r)) º (r)"  ,  dn(r)

(106)

where d/dn(r) is the functional derivative. This is a nonlinear equation because the potentials depend themselves on the wavefunctions through the density (104). In that respect, the Kohn—Sham equations are similar to the Hartree—Fock equations, but simpler because only local potentials occur. The existence of a local-density representation is guaranteed by the Hohenberg—Kohn [84] and Kohn—Sham theorem [102]. The actual form of the exchange-correlation functional E , however, can only be deduced in an approximate way. Most popular is the LDA  whose construction is conceptually very simple and can be summarized as



E (n(r))" dr n(r)e (n(r)),  

E (n) e (n)"  .  N

(107)

In practice, one computes the binding energy E /N of a homogeneous electron cloud at various  fixed densities n, and transfers the result point by point independently to the actual local density n(r). Key quantity is thus the exchange-correlation energy e (n) to be obtained from a careful  calculation of correlated electron matter. Early attempts employed just the RPA correlations [15]. Further electron—electron correlations in a Galitskij approach have been added in the widely used Ref. [69]. The very elaborate Green’s function Monte Carlo calculations of [31] have been parametrized in the exchange-correlation functional of [180]. In any case, the Kohn—Sham equations with exchange-correlation functionals in LDA constitute a versatile and robust approach to mean-field calculations of inhomogeneous electron systems for which there exists an enormous body of applications in all fields of electron physics. The success of this Kohn—Sham scheme in static applications suggests a naive and minimal extension to dynamical situations by using the time-dependent analogue of Eq. (105). This yields

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the time-dependent LDA (TDLDA) with the basic equation






pL  #º (r, t)#º (r, t)#º (r, t) u "i j u , !   ? R ? 2m

(108)

where the exchange-correlation potential º is again defined as in Eq. (106). TDLDA has been  successfully applied, e.g., to compute the plasmon response of metal clusters [27,50] and applications are still showing up, nowadays even in the highly excited regime [29]. The TDLDA, consisting out of the dynamical Kohn—Sham equation (108) together with the instantaneous exchange-correlation potential (106), is not only local in coordinate space but also local in time because the º refers to the density n(r, t) at the same instant in time as the  wavefunctions u . That may be very well justified for slow processes. But there may arise equally ? well situations where the time scale of the mean-field motion interferes with times for the correlation processes hidden in º . An appropriate definition of º should then include retarda  tion effects, i.e. should recur also to the density n(r, t) at past times t4t and possibly other places rOr. Such generalized concepts for a dynamical exchange-correlation potential have been much discussed over the past decades. The essence is that one can extend the definition of the exchangecorrelation potential to become nonlocal in space and time, i.e. º "º (+n(r, t), t4t,) .  

(109)

The construction of such generalized effective potentials is very intricate because there are now further, dynamical, boundary conditions to be obeyed. There is, for example, the causality requirement t(t. This inhibits a variational formulation in terms of an energy-density functional E and subsequent functional derivatives. Modelling has thus to take place at the level of º and   a consistent definition of energy or related quantities needs to be constructed separately. Nonetheless, considerable progress has been made, particularly in recent years. For a detailed discussion of this wide and involved topic, we refer the reader to the review articles [66—68]. For our present purposes, it suffices to know that such generalized potentials (109) do exist. A truly dynamical Kohn—Sham scheme has the same capabilities as the technique of the LFC discussed above. There are, in fact, close relations between the both schemes. These become obvious when linearizing the dynamical Kohn—Sham equations and expressing the results in terms of the response function s. We skip the lengthy and straightforward derivation and just present the essentials. The density is split in the standard manner as n(r, t)"n#dn(r, t) into stationary density and small dynamical perturbation. Variation with respect to dn thus introduces the exchangecorrelation part of the residual interaction dº (r, t) . » ("r!r", t!t; n)"   dn(r, t)

(110)

Switching to density-operator representation of the quantum state and subsequent linearization yields the response equations as s (q, u)  s(q, u)" , 1![»(q)#» (q, u)]s (q, u)  

(111)

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where »(q)"e/e q is the Coulomb potential as usual. The pure RPA consists in letting » "0   in the above expression. The sum »#» is thus the effective interaction for RPA calculations as  derived from the dynamical Kohn—Sham scheme. It takes the same place as the »(1!G) in the definition (48) of the local field correction. We thus can identify » (q, u)"!»(q)G(q, u) . (112)  This expression shows that there is a simple one-to-one correspondence between the timedependent Kohn—Sham scheme and the LFC, at least for homogeneous systems. The experience in designing LFC helps in the construction of º , and vice versa, the experience with treating  inhomogeneous systems can be carried over for extensions of LFC. This is what actually takes place in the present development of density functional theory [138]. The TDLDA is the most simple of these density-functional theories. It is not only local in coordinate space but also local in time because the potentials as defined in Eq. (108) refer to the same instant in time as the wavefunctions u . Relating E to its kernel ne (n) by Eq. (107) then ?   yields the doubly local residual interaction j(ne (n))  d(r!r)d(t!t) »*"("r!r", t!t)"  jn

(113)

with a most simple Fourier transform j(ne (n))  "constant . »*"(q, u)"  jn As mentioned above, this simplistic approach suffices for many purposes, in particular for slow processes and weak spatial dependence, but it often succeeds if these slowness and softness conditions are not visibly met. As an example, remember the success of the static LFC which leads to locality in time, » Jd(t!t). In fact, a nonlocality in coordinate space can still be imple mented into an energy-density functional in a fairly simple manner. One merely has to replace the local density dependence by a folded density dependence E (n(r)) P E (nJ (r)),  



nJ (r)" dr F(r!r)n(r)

(114)

and carry through the variational derivations as usual. The folding function F is related to the momentum structure of the LFC, or » , simply as  F(q)J» (q) .  Such an approach has been pursued, e.g. for an electron liquid [92] and in connection with He systems where finite range effects are crucial already at a static level [183,184]. A similar folding approach for the time profile of the interaction cannot be so simple because a second functional derivative would deliver a residual interaction which is symmetric in time t!t, in contradiction to the causality requirement t4t. 3.3.6. Stopping power using LDA and TDLDA The most demanding case comes with highly degenerate, H;1, and nonideal, m91, targets. They require a fully quantal treatment and inclusion of correlations. But a detailed quantum

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many-body calculation is even nowadays prohibitively complicated when spatial inhomogeneities are involved. Density functional methods as LDA and TDLDA offer a way out of this dilemma for moderately inhomogeneous systems in the nonlinear coupling regime. They allow to take up a great deal of correlations which have been computed extensively in homogeneous electron systems [32] and to carry them over to the inhomogeneous systems by means of an energy-density functional at the level of the local density approximation (LDA). Particularly simple is the static case with the ionic charge at rest for which successful LDA calculations for the nonlinear screening of a proton were performed [5]. The computation of the stopping power in these schemes employs the general results (59), or (58), respectively, obtained in the binary collision picture and expressed in terms of the transport cross section p and a shifted electron (Fermi) distribution [25,156]. For  H"0 and v;v the friction coefficient has been calculated using Eq. (60) for positively charged $ ions and antiprotons, see e.g. [49,126], and an application for the stopping power of an antiproton in the low velocity regime v(v is found in the recent paper [190]. The transport cross section is $ evaluated with selfconsistent potential built around the ion due to the screening by the electronic scattering states. The electronic exchange-correlation effects in these selfconsistent calculations are computed in LDA. The calculations used the relation (59) which is restricted to spherical scattering potentials. It works well in the regime v:v where deformations remain small. This encourages $ future work without restriction to spherical potentials, i.e. using expression (58) for general potentials instead of Eq. (59). 3.3.7. Nonlinear response at low ion velocities An alternative access to the stopping power in the fully nonlinear regime and at low projectile velocities is provided by relating it to the force—force correlation function [47,48] which, in turn, can be computed by dynamical schemes as TDLDA, Vlasov—Poisson or simulation methods (see Sections 3.3.3 and 3.3.4). Starting point is the definition (B.1) of the stopping power taken in the limit of infinite projectile mass. The projectile then represents an external moving potential at position €t and the Hamilton operator is given by HK (t)"HK !h(t) e (rL !€t). Here HK acG  G   counts for the kinetic energy of the electrons and their mutual Coulomb interactions and e (rL !€t) for their coupling to the external potential. One introduces the operator FK for the G  G component of the force along the direction of motion FK (€, t)"e (€/v) ) G (rL !€t) and considers G P  G the time average. This yields the stopping power as





1 O dE 1 O (q)"! dt 1FK (€, t)2"! dt Tr oL (t)FK (€, t) . (115) q ds q   This is an exact result for constant ion velocities €, i.e. for MPR, where dE/ds reaches a time independent value for times large compared to the initial transient period (typically &u\). In the  limits vP0 and qPR, the stopping power Eq. (115) can be expressed in terms of a force—force correlation function as [47,48]





v  dE T v  dt 1FK (0)FK (t)2" dt Tr oL FK (0) exp(iHK t/ )FK (0) exp(!iHK t/ ) , (116) (q) J  2 2 ds \ \ where the force FK (t"0) is taken at the origin r"0. The density operator oL describes the  equilibrium state of the fully correlated projectile—target system with the projectile at rest as defined

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by the time independent Hamilton operator HK "HK ! e (rL ) and e.g. oL "exp(!bHK )/ G  G   Tr exp(!bHK ). Note that the stopping power depends linearly on the velocity v as it should be in the regime of small velocities. The factor next to v is the friction coefficient of the system. The result (116) thus relates the friction coefficient to the force—force correlation function at equilibrium. This allows to recur to any theoretical method which is appropriate to describe the equilibrium properties of the ion—target system without restrictions on the projectile—target coupling, degeneracy, ideality, or other state conditions. This offers an alternative access to the stopping at low velocities where the strongest nonlinearity effects are expected. The relation (116) holds under very general conditions. Nonetheless, it is interesting to look at the regime of linear response and to see how the known results in that regime are recovered. The connection is established by neglecting the ion—target coupling in the equilibrium dynamics of the target, i.e. HK PHK and oL PoL "exp(!bHK )/Tr exp(!bHK ). Now Tr oL 2 only accounts for       the unperturbed electron system and with a Fourier representation of FK the correlation function becomes





 





  dt 1FK (0)FK (t)2P dt Tr oL FK (0)FK (t)J dk "F(k)" dt Tr oL nL (k, t)nL (!k, 0)   \ \ \ J dk "F(k)"S(k, u"0) ,

(117)

using the definition of the dynamical structure factor Eq. (32). Inserting expression (117) into Eq. (116), connecting S(k, u) to e(k, u) via relation (34) and replacing F(k) by the derivative of the ion potential F(k)"i€L ) ke (k) directly leads to the linear response result for low ion velocities  Eqs. (42) and (43). 3.4. Overview Table 1 summarizes the presentation of methods in keywords sorted in tabular form. The presentation is, hopefully, self-explanatory. Nonetheless, a few more remarks are in order. There are basically two opposite approaches to the problem: On the one hand, the continuum approach formulated in terms of the dielectric constant e(k, u), and on the other hand, the particle aspects pronounced in the binary collision approaches. These both classes provide complementary points of view. The collisional approaches motivate appropriate cutoffs in the dielectric formulation or provide more sophisticated correction terms for the linear response approach whereas the dielectric linear response approach gives the appropriate screening for computing the effective cross sections. The more elaborate the treatment, the more does it contain from both aspects. And the most complete theories, as e.g. MD simulations or WPMD, finally contain each side with equal weight. Before closing this section, we ought to mention that there exist a few more approaches which we have not reviewed here. For linear coupling and ideal or moderately nonideal plasmas a Fokker— Planck equation can be used as starting point [110]. A more general approach is provided by (quantum) kinetic theory [61,103,172], which allows successive improvement by incorporating various levels of approximations for the used collision integrals. The usually employed Lenard—Balescu or Boltzmann collision integrals are designed to describe stopping at linear and

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Table 1 Summary of the methods sorted according to the appropriate regime of application

semilinear coupling in ideal targets. Non-Markovian extensions include higher correlations which provide access to nonideal targets. Finally hydrodynamic descriptions have been used to determine the energy loss [35]. Their validity is a question of time-scales rather then coupling strengths: a hydrodynamic approach is suitable whenever the relaxation times are fast enough such that the ion—target system is in local equilibrium. Furthermore, our presentation and the classification is biased on one-component plasmas, although more general cases are discussed occasionally. The theories for plasmas with two or more

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components are complicated by new conditions and regimes showing up. A thorough discussion of these aspects goes beyond the scope of the present review.

4. Nonlinear stopping in classical plasmas This section is dedicated to the presentation and discussion of the nonlinear stopping power in classical systems studied by numerical methods as MD-simulations (Section 3.3.3) or by solving the Vlasov—Poisson equations (Section 3.3.1). 4.1. Electronic stopping in classical plasmas Before presenting and discussing the results, we want to specialize several important relations to the classical case as discussed here. This serves to define key quantities for the subsequent discussions. The basic scenario throughout this section is a completely classical system with classical ion—electron collisions 1g 2<1, see Eq. (16), nondegenerate targets H<1, see Eq. (12),  and classical electron—electron collisions, 1g 2<1, see Eq. (14). The singularity of the Coulomb  potential will not be removed by quantum effects and the projectile represents always a strong perturbation on the electrons for sufficiently small distances. According to the classification scheme for the ion—target coupling, Section 2.3, the regime of linear coupling is essentially related to quantum effects and does not exist for a classical ion—target system. There we can deal only with semilinear and nonlinear coupling. Hence, the weak coupling limit for a classical system is the semilinear regime and any discussion of nonlinear features in ion stopping has to be compared with the behavior of stopping at semilinear coupling as reference point. The decisive parameter to characterize the (non)linearity is thus the demarcation between semilinear and nonlinear coupling in classical targets, as defined in Eq. (25), 1 (3"Z"C "Z" 1b 2  " " . (118) (1#(v/v )) 3N (1#(v/v )) j   " Classical weak coupling, i.e. the semilinear coupling regime, is given for 1b 2/j;1, and strong  coupling, of course, in the opposite case. No length scale exists for classical Coulomb systems and thus the dynamics of the ion—target system is completely characterized by C and Z. This can be seen easily after rescaling the equations of motion, as demonstrated in the Appendices A.1 and A.2. The stopping power for the classical weak coupling regime has to be deduced from the approximations discussed in context with the semilinear regime like linear response with cutoffs (Section 3.2.2), the combined scheme of Section 3.2.3, or binary collisions with effective screening length (Section 3.2.2). We start with discussing the stopping power in the binary collision picture as outlined in Section 3.2.1. The stopping power for spherical symmetric potentials, Eq. (59), can be rewritten in dimensionless form using for classical systems as natural units for time, velocity, length and energy, the inverse of the plasma frequency, the thermal velocity, the Debye length and the thermal energy, respectively (see Eq. (A.1)). Furthermore, we employ the relations C\"4n(3nj , j "e k ¹/en and e/e "4n(3Ck ¹j . After replacing the Fermi distri" "   " bution f (p) by the Maxwell distribution f (v )"(m/2nk ¹)exp(!mv/2k ¹) the dimensionless    

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binary collision stopping power resulting from Eq. (59) reads








(3 (€ #€) v €L ) € p (v ) dv dE j ""  exp !      . 2v v v 12nj ds k ¹ C (2n)v     " It reduces in the limit of high velocities (see Eq. (61)) to the compact expression

(119)




(3 v  p (v) dE j ""  , v



(3 v dE j " "! R(Z, C), v;v .  C v ds k ¹  In case of the binary collision stopping power (119) this friction coefficient is






(121)



4n  dv v v p (v )   exp !    , R(Z, C)" (122) 3 (2n)v 2v 12nj   "  and represents together with definition (121) the scaled version of the general result (60), now applied to classical systems. The binary collision description is, of course, incomplete as long as the transport cross section p is not specified. It has to be computed from an effective projectile—electron interaction.  A suitable and very common choice is the Yukawa-like shielded Coulomb interaction







r (3ZC r Ze exp ! "!k ¹ exp ! . (123) » (r)"!  j j r/j 4ne r "  Here, the screening length j enters as an external parameter and may depend on the projectile velocity v which itself is an external parameter and not a dynamical variable of the projectile—target system in the considered limit of infinitely massive projectiles. In the classical description, the related transport cross section





 dp(v ,0)  "2n db b[1!cos(0(b))] d cos(0)[1!cos(0)] (124) dX \  is defined in terms of the classical mechanics relation between the center-of-mass scattering angle 0 and the impact parameter b ([62], chapter 3) p (v )"2n  



 



» (r) b \ dr » (r ) b 1!  ! (125) , where 1!  K ! "0 E r r r E K PK and E"kv/2+mv/2 is the relative kinetic energy. The transport cross section given by the   integrals (124) and (125) must in general be found numerically. For g <1 and /j;1, as granted   in the classical limit, the cross section obtained from Eqs. (124) and (125) completely agrees with the

0(b)"n!2b

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fully quantum mechanical p calculated from a numerical solution of the stationary Schro¨dinger  equation with the scattering potential (123) and a subsequent phase shift analysis [120]. An approximate analytical expression exists for high energies (E"mv/2<"Ze/4ne j" [107]. Fur  thermore a fit formula is available which is valid up to "Ze/4ne jE"+3 for negatively charged  projectiles Z(0 [131]. An inspection of the expressions (123)—(125) in terms of scaled quantities (A.1) shows that the transport cross section depends on v /v , ZC, and j/j . In particular, it is   " function of the sign of the ion charge at variance with the linear response approach which depends on "Z" only. The difference in the binary collision stopping power for different signs of Z is, however, only important for nonlinear coupling "Z"C91 while it is unimportant in the truly semilinear regime ("Z"C;1) where the binary collision treatment is used to provide corrections to the linear response description as outlined in Sections 3.2.2 — 3.2.4. The first of these approximations, the correction by cutoffs, will be discussed now. Using the classical scaling parameters (A.1), the linear response stopping result with cutoff, Eq. (65), takes the dimensionless form




v dE j " "Z(3Cg k , , Kv ds k ¹  1 v j k ) €L g k , " " dk Im , (126) Kv e (k, k ) €) k 2n IIK   where the RPA dielectric function e (47) and the cutoff k (68) are replaced by their classical limits 0 K e (78) and  2 2 1 v k " " 1# , (127) K c1b 2 j c(3"Z"C v  " 















respectively. The dimensionless function g(k , v/v ) depends only on the quantities "Z"C and K  v/v . It neither depends on Z and C separately nor on the sign of the ion charge.  The high velocity limit of the stopping power (126) reads


 




v  2 v  dE j " &!Z(3C  ln , v
(128)

and represents Eq. (74) rewritten in terms of the dimensionless system parameters C, Z, and v/v .  At low velocities, the friction coefficient R (121) deduced from the classical linear response stopping power with cutoff k , expression (126) is a function only of the combination "Z"C and takes the K form [3,135,194]







 

< k j 1 2 (ZC) K  2(ZC) I" ln(k j #1)! K " ln ! . R("Z"C)" K " k j #1 2 3(2n c(3"Z"C 3(2n K "

(129) The last step is justified because we have k <1 in the semilinear coupling regime ("Z"C;1) at K low velocities. The classical friction coefficient R (129) replaces the general friction coefficient

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CI (m, H, "Z") which was defined in Eq. (72) and which carries the full dependencies for arbitrary target conditions. To establish next the classical limit of the combined expression (dE/ds) , Eq. (69), some  additional considerations are necessary. We start with the statically screened interaction written in terms of the classical scaling parameters » (k)"!4nk ¹j  "

( ZCk#1/j

(130)

which corresponds for j"j to the general expression (70) when replacing the RPA dielectric " function e (k, 0) (47) with its classical limit e (k, 0) (78). Simultaneously, » (k) (130) represents the 0   Fourier transform of the screened scattering potential (123) for arbitrary j used in the binary collision stopping power. The binary collision contribution (dE/ds) which enters into the
 combined stopping power (69) is thus directly given by Eq. (119) using » Eq. (130) or Eq. (123).  However, the two first order Born stopping powers, (dE/ds) and (dE/ds), for the statically ' ' screened potential (130) and the dynamically screened counterpart (3ZC » (k)"!4nk ¹j ,  " ke (k, u"k ) €)  respectively, do not exist in the classical limit as discussed at the end of Section 3.2.2. A way out is given by calculating the difference (dE/ds)!(dE/ds) and not each contribution separately. ' ' This cures the problems of the classical description related to its invalid predictions for high transferred momenta k, since the statically and dynamically screened potentials become identical for high k and cancel out. In dimensionless form and using the classical scaling the difference of the linear response contributions which enter (dE/ds) reads 



 
   dE  dE  ! ds ds ' '




v j " "!Z(3Cg j, , Q v k ¹ 



v j k ) €L Im [e (k, k ) €)] Im [e (k, k ) €)]   " " dk ! . g j, Q v 2n k "1#1/(kj)" "e (k, k ) €)"  

(131)

Hence, the sum of the linear response term (131) and the binary collision stopping power (119), when using the statically screened interaction (123), (130) with j equal the Debye length j , " represents the classical version of the combined expression (69) presented in Section 3.2.3. To finish this discussion of the several approximations for semilinear stopping in the classical case we turn now to the binary collision stopping with velocity dependent screening length as sketched in Section 3.2.4. There, the dynamical polarization effects are taken into account in an approximative manner by parametrizing them in terms of a velocity dependent screening length, that is, calculating the stopping power from expressions (119) and (123) with j"j(v) as function of the projectile velocity v. It remains, however, to find an appropriate function j(v) to complete this phenomenological approach and various choices of j(v) can be considered. Here, we propose to demand equal linear response stopping powers for dynamically screened and effective, velocity

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dependent interactions. Then, the screening length j(v) is defined implicitly using this difference, Eq. (131), by



 

v v , "0 , v v   which has to be resolved numerically for j(v/v ).  g

Q

j

(132)

4.2. Nonlinear stopping power of heavy ions We now discuss the results from a large number of MD and test-particle simulations reported in [191—197,199—202] and compare it with recent experimental [170,185—188] as well as other theoretical results [23,38]. The common important observation is that the stopping power in the regime of nonlinear ion—target coupling depends on the ion charge as ZV, with x&1.5. This is clearly different from the Zln(const./"Z")-behavior in the semilinear coupling regime, see Eq. (129). This reduced growth with Z in the nonlinear regime is connected with a charge dependence of the screening length. 4.2.1. Velocity dependence Figs. 3 and 4 show the stopping power dE/ds as function of the ion velocity v. The stopping power dE/ds is normalized by division through Z and scaled in units of 3Ck ¹/j as " suggested by the linear response expressions Eqs. (126)—(129). The ion velocity v is scaled in units of the thermal velocity v "(k ¹/m). For graphic reasons we split up the forthcoming consider ations into two parts. We start with a comparison of the different theoretical expressions for the stopping power in the semilinear regime as described in the previous section. This serves as reference for the following presentation and discussion of the simulation results. The solid curves in Fig. 3 represent the combined expression (dE/ds) given by the sum of the  linear response term (131) and the binary collision stopping power (119) for a statically screened interaction (123) with j"j . The dotted curves show the linear response stopping power (126) " with cutoff (127) while the dash-dotted curves are the results from the binary collision expression (119) with the velocity dependent screening length j(v) provided from Eq. (132). In all cases a positively charged ion is considered. For the lowest displayed coupling strength ZC"0.11 where we are within the semilinear regime for all velocities the combined expression and the binary collision stopping power with velocity dependent screening are almost identical except of a small deviation around the maximum of the stopping power. This deviation vanishes quickly for still smaller coupling. The linear response result with cutoff exhibits higher stopping powers at medium velocities, is rather close to the other approaches but a bit lower at low velocities and agrees with them for high velocities. This is simply a consequence of the approximative derivation of the cutoff k , see Section 3.2.2, which is best substantiated for large k , that is, for high projectile velocities K K and/or for decreasing coupling parameter. In fact, the difference further diminishes for still smaller "Z"C. For increasing coupling strength, i.e. when proceeding towards and into the nonlinear regime, increasing deviations develop between the three descriptions. At ZC"0.36 we are at the fringe to the nonlinear regime at low ion velocities, but well within semilinear regime for higher velocities while for the highest shown coupling strength ZC"11.2 nonlinear coupling is

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Fig. 3. Normalized stopping power dE/ds/Z in units of 3Ck ¹/j as function of the ion velocity v in units of the " thermal velocity v "(k ¹/m) for a positively charged ion at various ZC. The solid curves represent the combined  expression, composed from Eq. (131) and Eq. (119) with interaction (123) and j"j . The dotted curves are the linear " response stopping power (126) with cutoff (127) while the dash-dotted curves exhibit the binary collision result (119) with the velocity dependent screening length j(v) from Eq. (132).

expected for v(2.5v if we assume 3"Z"C/(1#v/v )"1 as the division line between   semilinear and nonlinear coupling. At medium velocities the binary collision approximation with velocity dependent screening (dash-dotted) stays always below the combined expression (solid)

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Fig. 4. Normalized stopping power dE/ds/Z in units of 3Ck ¹/j as function of the ion velocity v in units of the " thermal velocity v "(k ¹/m) for ZC"0.11, 0.36, 2 and 11.2 corresponding to a fixed ion charge state Z"10 and  plasma parameters C"0.05, 0.11, 0.34 and 1.08, respectively. The diamonds are results of MD-simulations, the circles of Vlasov (PIC/test-particle) simulations. The errorbars for the MD results are suppressed for reasons of clearness, their typical size is indicated right top. For the test-particle simulations the errors are of the same size as the symbols. The dash-dotted curves shows the binary collision result with velocity dependent screening length, see Fig. 3, here corrected for the actual size of the simulation box as explained in the text.

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with an increasing deviation for growing ZC, whereas both approaches agree in all cases at low and sufficiently high velocities. At sufficiently high velocities they also meet the linear response description with cutoff (dotted curves), which, however, considerably deviates at low velocities and for increasing coupling where it exhibits a drastically smaller stopping power. Its apparent cubic behavior at low velocities for ZC"2 and 11.2 is deceiving insofar as there exists a non-vanishing linear term, however very small. This can be easily read of from the friction coefficient Eq. (129), which decreases RJk for small k (127). We remind however, that the assumption of large k was K K K essential for establishing the correction of the linear response stopping power by cutoffs, see Section 3.2.2. Agreement with the other approaches could be achieved of course by treating k in K Eq. (126) as a free, velocity dependent fit function. The results of the stopping power dE/ds as function of the ion velocity v obtained from the simulation studies are shown in Fig. 4 for coupling strengths ZC"0.11, 0.36, 2 and 11.2 corresponding here to a fixed ion charge state Z"10 and plasma parameters C"0.05, 0.11, 0.34 and 1.08 of the electron target. The diamonds are results of MD-simulations. The typical size of the errors is indicated by the errorbar right top. These errorbars are related to an ensemble averaging over different microscopic initial configurations. They represent as a conservative estimate the standard deviations and not the (smaller) mean error of the mean, see Appendix A.1. The circles represent the results from test-particle simulations of the Vlasov equation. Here the large number of test particles, which are used to represent a smooth charge density, considerably reduces the fluctuations and the errorbars are of the order of the size of the symbols. As reference for comparison with the theoretical approaches of Fig. 3 we chose the predictions of the binary collision treatment (119) with the velocity dependent screening length from Eq. (132) which is shown as dash-dotted curves. It is to be remarked however, that the MD and test-particle simulations are inapplicable at high velocities because the screening length increases and becomes larger than the finite length ¸ of the simulation box. To allow nevertheless some comparison at high v, one can imitate the finite simulation box in all the theoretical predictions by excluding stopping contributions which are originated only large wave length excitations. Compared to Fig. 3 where an infinitely extended target was assumed the dash-dotted curves in Fig. 4, corresponding to the binary collision model with velocity dependent screening length, are thus modified by subtracting the linear response stopping power related to k values smaller than 2n/¸, that is, by subtracting the correction term (dE/ds) j /k ¹"Z(3Cg(k "2n/¸, v/v ), Eq. (126). Due

   " to the relation between the screening length and the plasma parameter C, see Section A.1, Eq. (A.3), this correction becomes significant only at high ion velocities v
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Vlasov simulations (circles) for judging on their range of validity. For the lowest coupling strength ZC"0.11, Fig. 4, the binary collision model with velocity dependent screening (dash-dotted curve) agrees well with the Vlasov results (circles) within the errors. These are of the same order as the difference between the binary collision model and the combined expression, see Fig. 3. Thus both approaches are in very good agreement with the simulation results while the linear response result with cutoff, on the other hand, shows too large stopping values around the stopping maximum. For increasing ZC"0.36 the binary collision treatment reproduces well the Vlasov results at high and low velocities but underestimates the stopping power for medium velocities where dE/ds reaches its maximum. There is, however, almost perfect agreement with the combined expression which provides a bit higher values at medium velocities as can be seen from the difference of the solid and the dash-dotted curve in Fig. 3. The linear response result with cutoff, on the other hand, becomes already rather unreliable except for large v. At still larger coupling ZC"2 the binary collision stopping excellently fits the simulation results while the combined expression would predict around 30% to large values at the stopping power maximum. For the largest coupling strength ZC"11.2 where the low velocities lie well within the nonlinear regime the binary collision treatment still reproduces satisfyingly the typical dimension of the stopping power at medium velocities but clearly underestimates it at low velocities. Here, the simulation results (MD and Vlasov) exhibit a much more pronounced linear increase in v which extends up to higher velocities, in the MD simulation (diamonds) practically up to the stopping maximum. At this high coupling the combined expression grossly overestimates the stopping power by a factor +2. From this we can conclude that the combined expression, Eq. (69), which can be considered as the theoretically best founded treatment works well up to a coupling parameter ZC:1 in agreement with the definition of the semilinear coupling regime for which it was designed. For large coupling, more precisely in the range ZC+1—5, the more phenomenological binary collision treatment with velocity dependent screening length yields a much better prediction for the stopping power. For even larger coupling all discussed approximations fail. The effects of the nonideality of the target are revealed by comparing the results from MD with those from the Vlasov simulations, because collisions between the electrons are included in the full MD-simulation but are excluded in the mean-field description. There is good agreement for the smallest plasma parameter C"0.05, see Fig. 4 for ZC"0.11. For the next higher C"0.11 some deviations begin to appear at low velocities and are clearly visible around the stopping power maximum. With increasing nonideality of the target C"0.34 and C"1.08 this trend proceeds and a more and more pronounced difference develops at low velocities up to the velocity where the stopping is maximal. This clearly indicates an influence of the nonideality of the target plasma, which tends to reduce the nonlinear stopping power at low and medium velocities. The high velocity stopping remains unaffected. This influence of the nonideality of the plasma has been investigated for a large number of simulations at various Z and C. They allow to conclude that nonideal effects start to appear at C90.1. All these simulations show the appropriate linear increase of the stopping power with v at low v almost up to the maximum of dE/ds at v9v . The  position of the maximum shifts towards higher velocities with increasing coupling "Z"C as it is the case also in the simpler models, Fig. 3. But a simple functional dependence of this shift on ZC cannot be detected. In view of the linear increase of dE/ds at low v, the discussion of nonlinear stopping can be restricted to an investigation of the friction coefficient R(Z, C), as defined in Eq. (121). We remark, that a similar linear increase up to v+v was also found for nonlinear $

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stopping in the completely different target regime of fully degenerate electrons [190], using density functional methods (see Section 3.3.6). Finally it is to be noted that the low velocity stopping power dE/dsJR(Z, C)v obtained for a one-component electron target provides already the stopping power in a (fully ionized) two component plasma for v (v:v —3v where v  is the thermal     velocity of the target ions. Due to the high ion mass v ;v .   4.2.2. Charge dependence We now concentrate on stopping at low velocities, parametrized in terms of the friction coefficient R(Z, C), see Eq. (121), and investigate the dependence on the ion charge Z, particularly in the regime of nonlinear ion—target coupling. The left part of Fig. 5 shows the Z dependence of R computed with MD simulations for positively charged ions and various target correlations C, namely C"0.11 (circles), C"0.34 (triangles), C"1.08 (diamonds) and C"3.41 (crosses). The typical errorbars for the MD analysis can be looked up in the more detailed Fig. 6. At first glance, the MD results seem to line up with one common dependence on ZC. But a closer look reveals that results for different C have a different offset. Thus one has to conclude that R(Z, C) depends on C and Z separately as expected from a full many-body dynamics. For comparison we have drawn a pure ZC-behavior as it corresponds to a naive extrapolation of dielectric theory to arbitrary C, see the dashed line in Fig. 5. It is obvious that the friction coefficient from MD increases slower

Fig. 5. Dimensionless friction coefficient R(Z, C) (121) as function of ZC. The dotted curve represents the linear response description (129) and the dash-dotted curve the binary collision treatment (122) for the screened interaction (123) with j"j . To guide the eye and for comparison, a pure ZC dependence (dashed line) is added. ¸eft plot: " MD-simulation results for various ion charge states Z and target plasmas with C"0.11 (circles), C"0.34 (triangles), C"1.08 (diamonds) and C"3.41 (crosses). Right plot: The circles are the results of Vlasov simulations which only depend on ZC. The diamonds are the MD results given in the left part now not further distinguished with respect to C.

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Fig. 6. Dimensionless friction coefficient R(Z, C) (121) as function of "Z"C. MD-simulation results of Fig. 5 for different Z'0 and C"0.11 (circles), 0.34 (triangles), 1.08 (diamonds), 3.41 (crosses), now in more detail and with errorbars (standard deviations, see Appendix A.1) together with MD results for different Z(0 and C"1.08 (stars). The various curves show the friction coefficient R as obtained in the linear response description, Eq. (129) (dotted) and the binary collision treatment, Eqs. (122) and (123) with j"j , for positively (dash-dotted) and negatively charged ions (dashed). "

than Z. The Z-dependence for targets with C50.34 fits well on a simple power law RJZV with x:1.5 for each fixed C. This is compared with the theoretical predictions for R of the linear response description Eq. (129) (dotted curve) and the binary collision approach, Eq. (122), for the screened interaction (123) at j"j (dash-dotted curve), which both depend only on ZC. Here, " the binary collision result represents the friction coefficient corresponding to both the previously discussed combined expression (dE/ds) and the binary collision expression (119) with the  velocity dependent screening length from Eq. (132), which become identical in the low velocity limit. The linear response result as well as the binary collision prediction deviates from the simple Z law as expected for the stopping power in the semilinear regime. The linear response results follows the trend of the Z dependence for quite a while, but then it fails grossly for nonlinear coupling with "Z"C91. The curve bends over and finally falls off like (ZC)\ for large "Z"C. The charge dependence of the binary collision description agrees generally better with the MD simulation results, but for large ZC'10 it also underestimates the friction coefficient R. The right part of Fig. 5 shows a comparison between the MD (diamonds for all C) and Vlasov (circles) simulation results again for various Z and C. The friction coefficient R(Z, C) of the Vlasov simulations depend only on the product ZC as expected for a mean-field description. The friction coefficient and thus the stopping power exhibit a very similar charge dependence in the mean-field treatment as compared to the full MD which includes electron—electron correlations. Thus the nonlinear effects which lead to the typical trend JZ  are already present at the mean-field level. There remain, however, differences between the mean-field description and the MD simulations due to correlations, as seen from the separate Z and C-dependence of the MD results. Fig. 6 enlarges a part of Fig. 5 (left) with MD results for R(Z, C) now shown with errorbars. In addition, MD results for negatively charged ions at C"1.08 are shown (stars). Results from the binary collision description, Eq. (122) with interaction (123) and j"j , are shown for Z'0 "

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(dash-dotted) and Z(0 (dashed). We see larger stopping powers for positive ions than for negative ions in both, the MD and the binary collision results. This difference will vanish, of course, for weak coupling "Z"C;1. The binary collision results for positive ions (dash-dotted curve) show a bump at ZC+2—10 which is absent for the negative ones. This is due to electron scattering events with scattering angles larger than 180° (resonant states). To judge on the agreement between the simulation results and the binary collision description we better compare with the Vlasov simulations. There the particles move, after a transient period, in a constant mean field as it is assumed in the binary collision approach. Both descriptions depend only on ZC whereas the MD simulations additionally distinguish different degrees of internal correlation. This better agreement of the binary collision results (dash-dotted) with the Vlasov (circles) than MD results (diamonds) can be seen in the right plot of Fig. 5 or more clearly when comparing Fig. 6 with the left part of Fig. 7. The remaining differences of the Vlasov simulations and the binary collisions at large ZC are due to the fact that the selfconsistent potential in the Vlasov calculations differs in these cases from the effective potential (Eq. (123) with j"j ) " assumed for the binary collisions. There nonlinear screening becomes more involved with increasing ZC. This is taken into account for obtaining the improved solid curve in Fig. 7 (left plot) by

Fig. 7. Dimensionless friction coefficient R(Z, C) (121) as function of ZC. The dotted curves represents the linear response description Eq. (129). ¸eft plot: Vlasov (PIC/test-particle) simulation results (circles) as shown in the right part of Fig. 5 and the results (stars) of numerical solutions of the Vlasov—Poisson equations on a grid in phase-space from Ref. [23]. The dash-dotted and the solid curves represent the binary collision treatment (122) with interaction (123) for j"j " and j"j(ZC)"j (1#0.09(ZC) ) (see Section 4.3), respectively. For the dashed curve, the next order term of the " target response, as calculated in Ref. [135], was added to the linear response contribution. Right plot: The diamonds are the MD-simulation results for various Z and C of Fig. 5. The dashed line is a pure ZC dependence. The large symbols are the experimental results from cooling force measurements by Wolf et al. [188] (circles) and Winkler et al. [185] (stars) adapted to the case of an isotropic plasma assuming an effective temperature of ¹"3.1¹ and ¹"7.1¹ , respectively , , (see text).

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employing instead of j a screening length j(ZC) extracted from the effective potential observed " in the simulations. This will be discussed in more detail later on in Section 4.3. 4.2.3. Comparison with other approaches and experiments The left part of Fig. 7 compares R(Z, C) obtained from test particle Vlasov simulations (circles), the binary collision model for two different choices for j (dash-dotted and solid), the linear response approach with cutoff (dotted), linear response plus perturbative corrections (dashed), and a numerical solution of the Vlasov—Poisson equations on a grid (stars). The solution on a grid employed cylindrical symmetry which yields a reduced treatment in a four dimensional phase space, see Ref. [23]. Even then the method is very expensive and only few points are available. These points exhibit a slightly reduced friction and are practically identical with the results of a test-particle solution of the Vlasov equation also in cylindrical symmetry reported in Ref. [38]. The smaller stopping power found in these investigations may be in part due to the reduced dimensions which excludes several excitation modes but the major cause is probably the different treatment of the ion—electron interaction at short distances. In typical grid based numerical schemes the finite grid size automatically introduces a short distance cutoff of the Coulomb potential, whereas a proper description also of the close ion—electron collision requires additional efforts and a special treatment as included in the test-particle simulations (circles) reported here, see Appendix A.2. The result of a perturbative treatment to the next order beyond linear response is taken from Ref. [135]. It is based on a classical treatment and requires a cutoff k at large momentum transfer. K With the standard choice of this cutoff given by Eq. (127) the friction coefficient including the next order term (dashed curve) thus bends over and falls off for "Z"C91 like the linear response result (dotted). The example demonstrates the character of any perturbative description. The higher order corrections remain negligibly small for very weak perturbations "Z"C;1 and improve the results for increasing strength of the perturbation. But it is only a small range of intermediate coupling where this next term suffices for a satisfying description. Very soon an increasing number of perturbative terms is needed which make the whole perturbative expansion obsolete for strong perturbations "Z"C91. Experiments on the stopping of highly charged heavy ions in dense and highly ionized target plasmas are in progress [82,83,42,43,59,163,33,34,94,95,171]. But the plasma densities are not yet sufficiently high to reach the regime of nonlinear projectile—target coupling. There exist, however, experimental results from measuring the cooling force for electron cooling in heavy ion storage rings. In the electron cooler the ions of the stored ion beam are merged with a cold electron beam once in each cycle. Thermal energy is transferred from the ions to the cold electron plasma. This produces colder ion beams of lower phase-space volume [142]. The cooling of the ions can be related to the energy loss of an ion when moving through the electron cloud and the measured cooling force represents the stopping power averaged over the velocity distribution of the ions. The cooling forces at low ion—electron relative velocities observed experimentally show a ZV dependence with x between 1.5 and 1.7 [170,185,186,188], more recent experiments at the experimental storage ring (ESR) at GSI, Darmstadt, yield even smaller values x+1.3 [187]. This agrees qualitatively with the presented MD and test-particle results for isotropic electron plasmas. In the electron cooler, however, the situation is more complex because one has two different electron temperatures in the transversal and the longitudinal direction and because there is a longitudinal magnetic field which guides the electron beam. The typical parameters in the electron cooler at the

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test storage ring (TSR) in Heidelberg are n+3;10 cm\, k ¹ +0.1 eV, k ¹ +1—5;10\ eV , , and Z"1,2,16 [188], where ¹ denotes the temperature transversal to the beam direction and , ¹ longitudinal with the beam. These parameters stay safely in the regime of ideal target and linear , ion—target coupling for the transversal temperature whereas the longitudinal temperature yields C :0.7 and ZC:9. The ESR in Darmstadt has the parameters n+10 cm\, k ¹ +0.15 eV, , , , k ¹ +5;10\ eV and Z"6,2, 92 [170,185]. That is again an ideal target and the ion—target , coupling is linear for the transverse temperature but C :0.4 and ZC:23 for the longitudinal , , degree of freedom. The longitudinal conditions in both devices corresponds to a target close to the nonideal regime and to nonlinear coupling. In addition there comes the longitudinal magnetic field. This hinders the transversal motion of the electrons and thus freezes the hot transverse electronic degrees of freedom. There should remain an effective temperature for the ion—electron interaction which stays close to the longitudinal temperature. We compare the measured friction coefficients with MD simulations of an isotropic plasma by interpreting the experiments in terms of an effective temperature. The result is shown in the right part of Fig. 7. An effective temperature of ¹"7.1¹ , (stars) had been used to adjust the data from [185] and of ¹"3.1¹ (circles) for comparison with , [188]. The experimental trends are well reproduced which provides an experimental confirmation for the typical Z-dependence as seen in the MD simulations. Furthermore, the effective temperatures remain close to the given ¹ which supports the picture of nearly frozen transversal degrees , of freedom. 4.3. Screening and electron trapping We discussed in Section 4.2.2 the Z-scaling of the stopping power and found a trend JZV, x+1.5. This trend is related to the development of screening of a highly charged, slow ion. We have investigated that in detail in specific simulations. The Coulomb interaction between an ion at rest (v"0) and the electrons was switched on at time t"0 and the ion subsequently attracts electrons. The electron density about the ion grows and reaches an almost stationary state on a time scale of plasma periods q . The screening function ¶(r) is then extracted as the total electric N potential about the ion divided by the bare ion potential. The linear response theory yields ¶(r)"exp(!r/j ) (Debye—Hu¨ckel). The screening function deduced from MD and Vlasov simula" tions for nonlinear coupling is also rather close to an exponential function ¶"exp(!r/j) but with a much different screening length j"j(Z, C)'j . This suggests that the corresponding effective " ion—electron interaction is well approximated by the potential (123) when j is replaced by j(Z, C). The results for the observed screening length are shown in Fig. 8. The rather large errorbars reflect here not only the different initial states but mainly the imperfectness of matching the real potential to the simple exponential form exp(!r/j)/r. One sees that the screening length increases with the coupling strength, particularly for ZC'10. This trend of the static screening is in good agreement with the trend of the friction coefficient. Remind the Figs. 5 and 7 (left part) where the friction coefficients at strong coupling ZC<1 lie well above the results from the binary collision model where an interaction potential (123) with the linear response screening ¶(r)"exp(!r/j ) " was used. To be more quantitatively, a rough estimate for j(ZC) can be deduced from the data in Fig. 8 as j(ZC)"j (1#0.09(ZC) ). Taking this screening length as an input for the " ion—electron interaction (123) in the binary collision approximation Eq. (122) results in the solid curve in the left plot of Fig. 7 which agrees fairly well with the Vlasov simulation results. The

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Fig. 8. Static screening length j in units of the Debye length j as function of ZC. MD-simulation results for different " ion charge states Z and electron targets C"0.34 (triangles) and C"1.08 (diamonds) and Vlasov simulation results (circles) obtained from the observed approximatively exponential decay of the total electrical potential about the ion. The typical errors as indicated left top are mainly due to the imperfect agreement of the actual potential with the simple functional form Jexp(!r/j)/r. The dotted line represents the constant screening length in the linear response description.

enhanced friction is clearly related to the enhanced screening length. Of course, the simple binary collision picture can only provide the qualitative trend of nonlinear stopping in an ideal target. The hard electron—electron collisions which are present in the MD simulations and which become increasingly important for increasing C give an additional contribution due to the target nonideality as documented by the separate dependence of j(Z, C), Fig. 8, and R(Z, C), Fig. 6, on Z and C. A further interesting observable related to screening is the electron density near the ion. We characterize it by the number of electrons N(r(r ) in a sphere of radius r "0.3a (a"(4nn/3)\)   about the ion because this quantity can be sampled with reliable statistics in our MD simulations. Fig. 9 shows N(r(r ) for Z"10 and C"1.08 as function of the ion velocity v. The results  represent the ensemble averages over the same simulation runs from which the stopping powers in Fig. 4 have been computed. MD-simulation results (diamonds) are compared with results from the Vlasov equation (circles) and the ideal gas value N"n4nr/3"(0.3) (dotted line). MD simula tions produce a huge enhancement of N(r(r ) at low velocities compared to the ideal gas, about  a factor of 80. The enhancement decreases for increasing velocity and vanishes completely at large ion velocities which agrees with the restoration of linear response at large velocities. But note how large the velocities have to be to recover that limit (v/v '5). The Vlasov simulation results (circles)  behave similar, however, with a considerably smaller enhancement at low velocities. Most of the electrons which contribute to this enhancement at low ion velocities are trapped in the ion

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Fig. 9. Number of electrons N within a sphere of radius r "0.3a (a"(4nn/3)\) about an ion of charge state Z"10  as function of the ion velocity v in units of the thermal velocity v "(k ¹/m) in a plasma with C"1.08. The diamonds  are results of MD-simulations, the circles show Vlasov simulation results. The dotted line indicates the ideal gas value N"n4nr/3"(0.3). 

potential as can be concluded from the negative energies of these electrons. Here, this energy is determined as sum of the kinetic energy of relative motion and the potential energy in the field of the ion and the other electrons or in the mean-field. The trapping of electrons is a typical many-body effect, caused by an exchange of energy and momentum with the ion and the surrounding electrons in multi-particle collisions involving the ion and two or more electrons. Such processes are sensitive to the nonideality of the plasma, i.e. to the collisions and correlations between the electrons. While the electron—electron interaction and hence the nonideality effects are correctly treated in the MD-simulations, the trapping and the enhanced densities are suppressed in the Vlasov treatment. The behavior of the density near the ion is of course connected to screening and to the stopping power although in a rather intricate manner. While the electron densities close to the ion are quite different in the MD and Vlasov treatments, that is, in nonideal and ideal targets, respectively, the corresponding screening lengths behave very similar, see Fig. 8. In particular for negative ions where the charge density near the ion is reduced, the screening lengths is nevertheless enhanced over the Debye length at nonlinear coupling as we have observed in the simulations. This also contributes to the behavior of the friction coefficient in Fig. 6 for Z(0 (stars) which lies for increasing coupling clearly above the binary collision prediction (dashed curve) for j"j . " The electron trapping is also of interest with respect to electron—ion recombination. The trapping of electrons by multi-particle collisions corresponds to recombinations into high Rydberg states of the ion. The enhanced electron densities and the population of loosely bound electron

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states are one crucial mechanism to explain the enhancement of radiative recombination rates by factors of 4—5 as observed for a large number of ions in recent experiments with highly charged ions in heavy ion storage rings [13,57,157,177]. These experiments used the electron beam in the cooler as an electron target to study spontaneous or laser-induced radiative electron—ion recombination. It should be emphasized that these many-body effects can occur only for strong coupling. They are different from the usual three-body-recombination in an ideal target where a sufficiently strong interaction is only possible in the very unlikely event that an ion and two electrons come very close together. Therefore, the theoretical description of these processes always require a full MDsimulation for a proper description of the essential multi-particle collisions. One needs, of course, also to take into account the anisotropic electron temperatures and the presence of the magnetic field. Such extensive MD-simulations covering all these aspects were performed recently for an ion with charge Z"51 at rest (v"0) for typical target parameters in an electron cooler, here specifically those of the TSR at Heidelberg [169]. The resulting enhancement factor n(r)/n, here for the density which yields rather high errorbars, is shown in Fig. 10 for various strengths of the magnetic field. It represents the average over an interaction time of three plasma periods according to the typical time of flight through the cooling section of a storage ring. As a general trend the expected growing density enhancement with decreasing distance from the ion is clearly visible. But the magnetic field introduces further quite intricate influences and dependencies. At close distances an increasing magnetic field progressively reduces the density enhancement considerably while it seems to support the enhancement slightly farther away from the ion. We have to be aware, however, that we are here in an transient stage of the buildup of the electron cloud about the ion.

Fig. 10. Electron density about an ion with respect to the homogeneous density n as function of the radius r in units of a"(4nn/3)\ and averaged over q"3q +35 ns after switching on the ion—electron interaction. The ion of charge  state Z"51 is at rest in an anisotropic electron plasma as typical in an electron cooler with n"10 cm\ and k ¹ "15 meV, k ¹ "0.5 meV and for magnetic field strengths B"0 (solid bars) 0.1 T (dotted) and 1.0 T (dashed). , ,

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A closer look on the time evolution of the local density n(r, t) then reveals the influence of the magnetic field on this buildup, see [169]. It becomes slower with increasing magnetic field which indicates a reduction of the mobility of the electrons connected to the freezing of the transversal degrees of freedom. Further MD studies on the local electron density around an ion are reported in [197,200,201].

5. Quantum effects 5.1. Stopping with effective potentials MD simulations with classical point particles are orders of magnitude less expensive than fully fledged quantum mechanical methods. It is therefore tempting to map quantum features into a classical simulation by means of an effective potential. These are often derived from the quantum mechanical density matrix of the electrons in equilibrium and they thus incorporate all the needed smoothing from the quantum effects [76]. An effective potential between charges q and q , whose ? @ shape and parametrization have been adjusted to quantum chemical calculations has been proposed by Kelbg [97] and Deutsch et al. [39—41]. It is parametrized in simple local form as qq (133) » (r)"! ? @ [1!exp(!r/j )] , ?@  4ne r  where all quantum effects are mediated through the short-range cutoff parameter j . The WPMD ?@ simulations, on the other hand, aim at a direct simulation of the quantum system. But they also allow to determine an effective potential “a posteriori” by measuring the potential energy for every electron—proton pair as a function of the distance. This means essentially that one averages the electron—proton interaction (99)




 

R !r 3 e G erf ' (134) » " 'G 4ne "R !r " c 2 G  ' G for all pairs of ions and electrons and over observation time. The results are shown in Fig. 11. The “measured” effective potential from WPMD agree with the conventional effective potential in the gross features. Both models introduce a substantial reduction as compared to the Coulomb potential. At second glance, however, the WPMD result turns out to be softer than the potential (133). Of course the evaluation of » from a simulation contains statistical fluctuations about an 'G average potential which are due to the influence of the surrounding medium in the evolution of the width of the wave packet. The fluctuations are indicated by errorbars in Fig. 11. Furthermore, calculations at higher densities indicate a density dependence of the effective potential, in contrast to the ansatz (133) and to the results in Ref. [76]. One could also expect that the effective potential depends on the velocity. This has been checked by making two alternative samplings, one taking into account only particles with more than twice of the thermal velocity and another one with a velocity window on less than half-thermal velocity. Both samplings give potentials which are very similar to the unbiased average drawn in Fig. 11. The WPMD data do therefore not require a velocity dependent effective interaction.

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Fig. 11. (Effective) potentials as function of the distance. The solid line is the Coulomb potential, the dashed line is the effective potential (133) [97], and the points are averages obtained from Eq. (134) in WPMD-simulations. The errorbars reflect the fluctuations in the simulations.

The lack of velocity dependence also carries through to the effective potential for a moving ion as it is required for computing the stopping power. One expects that the wavelength j in Eq. (133) ?@ depends on the average relative velocity between ion and electron plasma. The quantum smoothing should decrease for v
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Fig. 12. Dimensionless friction coefficient R(Z, C) (121) as function of ZC. Classical MD-simulations for various ion charge states Z with the Coulomb potential for C"0.11 (circles) and C"1.08 (diamonds) and with effective potentials of the functional form (133) for C"0.11 (crosses) and C"1.08 (stars) and a temperature of k ¹"10 eV. The ranges of the potential (133) are j " /(2nmk ¹) for the electron—projectile interaction and j "2j for the interaction    between the electrons. The dotted curve is the linear response description (129).

5.2. Quantum stopping power Qualitatively one expects that quantum effects reduce the stopping power. On one hand, quantum indeterminacy mollifies the Coulomb interaction near the origin, thereby reducing the probability of hard collisions with large momentum transfer. Furthermore, collisions are suppressed by Pauli-blocking in the degenerate regime. In order to exhibit the transition into the quantum regime we performed WPMD simulations of the stopping power for target parameters (n,¹) shown as working points in Fig. 13 [101]. The plasma parameter was kept fixed near the dashed line at C"1.08. With increasing temperature one moves then along this line from a nonideal, classical regime into the quantum regime. The ion has a charge number Z"5 and mass M"80 000m. Fig. 14 shows the stopping power !dE/ds as a function of the ion velocity v for the various working points shown in Fig. 13. On the scale employed the classical stopping power depends only on C, but not separately on n,¹, as the quantum stopping power obviously does. The comparison of the classical simulation with the WPMD simulations at the working points of Fig. 13 shows indeed the trend envisaged above: The magnitude of the stopping power decreases when moving into the quantum regime. On the other hand the shape of dE/ds as a function of the

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Fig. 13. Working points in the n—¹-plane. See also Fig. 1.

Fig. 14. Normalized stopping power for Z"5, M"80 000m as a function of the ion velocity. The crosses are results from a classical simulation for C"1.08. The results of the WPMD-simulations for C"1.08 at the different temperatures corresponding to the working points given in Fig. 13 have been joined by curves to guide the eye.

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Fig. 15. Normalized stopping power dE/ds/Z in units of 3Ck ¹/j as function of the ion velocity v in units of the " thermal velocity v "(k ¹/m) for Z"5 and at the working points of Fig. 13 (C"1.08) with k ¹"1.25 eV (solid  curves), k ¹"10 eV (dotted) and k ¹"40 eV (dashed). The left plot shows the linear response stopping power (39) with the RPA dielectric function (47), the right plot the semilinear stopping power (65) using the cutoff (68).

ion velocity remains unchanged, in particular we always observe a linear increase for small ion velocities v4v . For large velocities the simulation results should not be taken too quantitatively,  as it becomes increasingly difficult to represent the infinite system by finite simulation boxes, see Appendix A.1. We use this occasion to explore the range of validity of the RPA expressions (39), (47) which were obtained in the linear, ideal regime and extended into the regime of semilinear coupling by the cut-off (65)—(68). In the left plot of Fig. 15 we show the stopping power obtained from Eqs. (39) and (47). Due to the blocking factors in the dielectric function (47) the stopping power decreases as the working point moves into the quantum regime. While the shape of these curves is quite similar to the WPMD results in Fig. 14, the RPA results of Fig. 15 (left) turn out much too large. Indeed the RPA does not account for the wave nature of the electrons in their interaction with the projectile, which is therefore over-estimated if the coupling becomes strong. This can be corrected in the semilinear regime by introducing the cutoff (68), which reduces the stopping power as shown in the right part of Fig. 15. However, also this treatment fails for small velocities v/v 41, where  one enters the truly nonlinear regime 1g 251 (16).  5.3. Comparison of WPMD and LFC treatments We discussed in Section 3.1.4 the LFC extension of the RPA. Higher correlations beyond the RPA are contained in the Local Field Correction G(k, u) which effectively introduces a modified

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residual interaction (48), (49). The LFC has been calculated by Ichimaru and coworkers for hydrogen in the regime of a liquid metal, e.g. r "1 and C"2. The protons were treated classically  in the framework of the HNC. The electron—electron and the electron—proton correlations were studied in the framework of the modified convolution approximation (MCA) [174,91] for the LFC. In Fig. 16 the correlation functions of this HNC MCA scheme are compared with those obtained

Fig. 16. Proton—proton g (r), electron—proton g (r) and electron—electron g (r) pair correlation functions (radial    distribution functions) for a hydrogen plasma with C"2, r "1, k ¹"13.6 eV, n"1.61 A> \, k "3.627 A> \. Points  $ with errorbars: WPMD results for the spin averaged g (diamonds) and for parallel (circles) and antiparallel (triangles) IJ electron spins, respectively. Solid curves: HNC MCA results from Ref. [91].

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in the WPMD simulation. Quite generally the WPMD correlation function show for r'a some  long-range structure, which is not visible in the HNC MCA results. For r:a the correlation holes  in the WPMD are somewhat more pronounced. The correlation function for electrons with like spin approaches zero for rP0, i.e. the Pauli-principle is observed. 6. Conclusions In this report on the stopping of ions in plasmas at strong coupling we have drawn attention to some recent important results and also emphasized some known, but hitherto rather neglected points. 1. We presented a careful and detailed classification scheme for various stopping regimes according to the target parameters density and temperature, or equivalently, ideality and degeneracy and according to the ion properties charge and velocity. 2. We identified three coupling regimes: Linear, semilinear and nonlinear. The semilinear regime is distinguished by the fact that the nonlinearity occurs only for a negligible fraction of space inside the typical shielding length. Rather universal methods like the RPA and the binary collision approximation are applicable in the linear and semilinear regimes. Extensions to nonlinear coupling require detailed investigations at large numerical expense. We found molecular dynamics and test-particle computer simulations very useful in this regime. 3. In agreement with experiments we find that the friction coefficient for slow ions in a plasma scales approximately like Z . This is of large importance for the electron cooling of highly charged ions in storage rings. 4. This change in the scaling is partly due to nonlinear shielding. The effective shielding length is larger than the Debye length and depends separately on the charge of the ion and the plasma parameter. 5. A highly charged slowly moving ion traps electrons, so that the electron density around the ion is increased. This may result in an enhanced recombination rate, as has been observed at several storage rings. For their effective operation it is essential that the cooling rate remains larger than the recombination rate. 6. Quantum effects like the wave nature of the electrons and Pauli-blocking tend to mollify the effective interaction and to reduce the stopping power. Finally we want to point out some areas where further progress on the stopping power is highly desirable: Bound states and the variation of the charge state of the projectile should be calculated in a self-consistent manner. The WPMD should be generalized by admitting larger variational spaces than Gaussian wave packets in order to describe better the delocalization of the electrons, e.g. in metallic target states. For more realistic applications to electron cooling the influence of a temperature anisotropy in the plasma and of external magnetic guiding fields should be accounted for explicitly. Such work is in progress. Acknowledgements This work could not have been completed without the enormous help we experienced through clarifying and illuminating discussions with many colleagues and coworkers, among them:

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C. Deutsch, J.W. Dufty, D.H.H. Hoffmann, S. Ichimaru, D. Klakow, M. Knaup, G. Maynard, P. Mulser, D. Schwalm, M. Seurer, Q. Spreiter, M. Steck, C. Ullrich, T. Winkler and A. Wolf. It has been supported by grants from the Bundesministerium fu¨r Bildung und Forschung (BMBF), Gesellschaft fu¨r Schwerionenforschung (GSI) and the European Community (HCM Network Ringnet). G.Z. thanks the ‘Laboratoire de Physique des Gas et des Plasmas’ (LPGP), Orsay, for its hospitality during a two year stay and acknowledges a grant from the ‘Deutsche Forschungsgemeinschaft’ (DFG).

Appendix A. Description of the simulation techniques A.1. MD-simulations As outlined in Section 3.3.3 the MD-simulation technique provides the full many-body dynamics of a classical system by numerical integration of the Newtonian equations of motion. This allows in particular to study non-equilibrium properties of highly correlated systems without restrictions on the strength of the interaction. In this section, we sketch the simulation scheme as it was used to investigate the stopping power [191—197,199—201]. It is convenient to cast the equations of motion (79) into dimensionless form by introducing characteristic units for time, velocity, length and energy, given by the inverse of the plasma frequency, the thermal velocity, the Debye length and the thermal energy, i.e. t "u\, v "v "(k ¹/m), r "j "v /u ,      "   This leads to the set of coupled equations



E "k ¹ . 

(A.1)



duJ j 1 (3ZC drJ G"! G"uJ , (3C ! , G dtI jrJ "rJ !rJ " "rJ !RI " dtI G G H G H$G d€J m j 1 dRI "€J , "(3ZC , dtI M jRI "rJ !RI " dtI G G rJ "r/r , uJ "u/v , RI "R/r , €J "€/v , Q Q Q Q

(A.2)

tI "t/t , (3C"eu /(4ne k ¹v ) . Q    In this form, one can read off immediately the coupling parameters. The plasma parameter C characterizes the importance of the electron—electron interactions, and "Z"C the strength of the ion—electron coupling. But one has to remind the strong Coulomb singularity at the ion position which produces large forces however small "Z"C may be. The proper measure for linearity is u(0) or 1g 2, see Eqs. (19)—(21) and the discussion in Section 2.3. The classical equations of motion (A.2)  come never into a completely linear regime as there is always a small vicinity around the ion where fields are large and the condition for linearity, u(r);1, is violated. In that case, weak perturbations correspond already to semilinear coupling and we recover in the ion—electron coupling strength "Z"C the parameter which distinguishes for low ion velocities the semilinear regime ("Z"C;1) from the nonlinear regime ("Z"C51), see Eqs. (24) and (25).

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A particular MD-simulation based on an integration of Eq. (A.2) depends on C, Z, the mass ratio m/M and the initial positions and velocities of the particles. Global observables, as e.g. the stopping power, are obtained with ensemble averaging. These depend only on C, Z and the initial ion velocity €. The exact value of the ion mass M remains unimportant as long as M<m. The target conditions, temperature ¹ and density n, enter only through the plasma parameter in the combination n/¹JC such that simulation results for a given C (and Z, v) yield the stopping power for a variety of target parameters n and ¹Jn. Finally, we want to point out that the stopping power in MD simulations depends on both, target condition C as well as ion—electron coupling ZC, whereas the Vlasov—Poisson equations (75)—(77) depend only on ZC (and €). The parameter C which accounts for the correlations in the target becomes meaningless in a mean-field description. It is obvious that one can only afford to propagate a finite number of electrons in a MD simulation. The expense grows quadratically with the particle number because the long-range Coulomb force needs to be evaluated between each pair. On a workstation, one can typically afford a system of 500 particles. The actual plasmas, however, contain a macroscopic number of particles. Such a practically infinite system is approximated by treating a finite number of electrons in a cubic simulation box of length ¸ and continuing this box periodically in each direction. A particle is thus replaced by a cubic lattice of particles and the Coulomb interaction by the Ewald sum of Coulomb interactions [72,128]. The neutralizing background of positive charges is simplified in the jellium approach where the positive background is distributed homogeneously. The actual definition and calculation of the Ewald force are given in Ref. [72], Appendices A and B. One has to be cautious to avoid artifacts from the artificial periodic structure. Results become unreliable as soon as longrange correlations are propagated outside a simulation box into the next one. This is avoided if the screening length j remains below the box length ¸. The volume of the box, ¸, is related to the particle number N through ¸"N(4na/3), where a denotes the Wigner—Seitz radius of the electrons. In terms of the classical plasma parameter C and the screening length j"j (1#(v/v )) (which can be used as an estimate also for strong coupling, see Section 4.3) "  we obtain


 







4n  3C  C  c ¸ +22.2 , N 1( " 3 1#(v/v ) 1#(v/v ) j  

(A.3)

where we used N " 500 in the last step. For an application of MD-simulations to heavy ion stopping, the favorable regime lies at low ion velocities with nonlinear coupling and at nonideal targets C50.1 where one easily meets the conditions ¸/j'1. This is confirmed by simulations using N " 250 and N"5000 electrons, where the observed stopping power at low velocities remains unchanged compared to the standard number N"500. For high ion velocities, on the other hand, one has to be aware of artifacts from the periodic simulation lattice, as e.g. interference effects in the wake fields excited by the ion and its images. Such unwanted side-effects limit the use of the MD technique. They can be turned into a truly physical effect when considering the ion—ion correlations in the stopping of a lattice of ions, as has been done in [192,195,198]. MD simulations with the full Coulomb force run into a technical difficulty. Occasionally, particles come very close to each other, in particular for the attractive ion—electron interaction. There arise large forces and high velocities which require to reduce the time step. This slows down

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the numerical propagation and one can easily get stuck in an intractable situation. To overcome this deadlock, we extended the conventional MD-simulation scheme by a separate treatment of close collisions, now taking advantage of the fact that the timescales in the close collision region are much smaller than those for the system as a whole. Hence, we check in each time step the occurrence of clusters of close particles which are defined by setting an appropriate chosen critical distance d , where a particle belongs to a cluster if its distance to at least one other particle in the  cluster is less than d . While all particles outside a cluster are propagated with the standard time  step Dt, the most frequent binary clusters are propagated with the analytically known trajectories of the Rutherford problem. The few remaining larger clusters are propagated numerically but with a much reduced time step as compared to Dt. The interaction between the cluster and the rest of the particles is computed at the slower time-scale of the whole system by mediating to the cluster particles once a momentum shift according to the external forces (i.e. the sum of the forces from all other particles) and taking into account, of course, the reverse effect on the other particles. This suffices to guarantee momentum and energy conservation of the whole simulation. The outlined technique of a separate treatment of clusters of close particles considerably reduces computing time and ensures high stability of the simulations. The numerical integration of the equations of motion for all particles outside a cluster with the time step Dt as well as for those within higher clusters and with a much smaller time step is performed by a standard Velocity—Verlet algorithm [4,178]. Actual simulations to compute the stopping power proceed in two phases. First, the pure electron system is relaxed to an equilibrium stage without the ion. Then the ion with velocity € is added suddenly at an arbitrary location, and in the second phase the full system of ion and electrons is propagated. The initial equilibration starts from a random sampling of positions and velocities and relaxes towards the equilibrium distribution of desired temperature by dynamical propagation with velocity rescaling. The second phase aims to simulate the full propagation with the ion. In contrast to a truly infinite system, the energy loss of the ion is distributed over a “very finite” number of electrons and thus induces some heating of the electrons. We need a scheme for repeated cooling, in order to obtain the stopping power for a well defined target temperature ¹. To that end, we select a part of the electrons which is far away from the ion, i.e. outside a given sphere with a radius r9j about the ion, and restore the initial temperature ¹ by drawing new velocities " from the corresponding Maxwell distribution. This is done in regular time intervals of 1—3 plasma periods q "2n/u . A complete simulation run takes typically 10—15q , which requires    &10 000—15 000 time steps Dt and around 1—3 h on a work station. The accuracy and stability of the runs are monitored using the total energy. The propagation thus stabilized is accompanied by the analysis of the stopping power. The energy loss DE and the travelled path Ds of the ion are “measured” in each time step resulting in an instantaneous stopping power (DE/Ds)(t). The stopping power is finally obtained as the time average 1DE/Ds2 over the instantaneous DE/Ds. Since the projectile mass is large compared to the electron mass (usually M"50 000—10 0000 m), the projectile velocity changes only slightly ((1%) and the ion travels almost perfectly along a straight line and we have 1DE/Ds2"1DE2/1Ds2, which represents the stopping power dE/ds for the given velocity v. These time averaged values include the transient period after adding the ion where the stopping power has not yet reached a stationary value. But this period is small, usually (q [160,161], and affects the results only weakly. From several runs with the same C, Z, €, but  different samplings of the initial state, an ensemble average over the time averaged 1DE/Ds2 is performed and yields finally the stopping power. The error can be deduced from the fluctuations in

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the ensemble of events. The errorbars shown in Figs. 4, 6, 9, 12 and 14 represent the standard deviations and not the (smaller) mean error of the mean. A.2. PIC/¹est-particle simulations of ion stopping In addition to the previously described MD-simulations we performed particle-in-cell (PIC) simulations with test particles, a method commonly used for numerical solutions of nuclear dynamics [18,148]. The PIC simulations use particle propagation similar as the MD simulations. But the aim is much different. PIC is designed to solve the Vlasov—Poisson equation (see Section 3.3.1) for the smooth phase-space distribution f (r, u, t), thus carrying only the mean-field aspects and discarding any correlations. It turns out, however, that the most efficient way to solve the Vlasov equation recurs to the concept of particle propagation in that the distribution is represented as a superposition N (A.4) f (r, u, t)" w(r!r (t))w (u!u (t)) , G S G N 2G where w and w are smooth and fairly well concentrated, normalized wave-packets in phase space. S The representatives i are pseudo-particles, N is the number of pseudo-particles and N the number 2 of physical particles. We use here the PIC simulations to compare the mean-field propagation with the fully correlated MD simulations. To that end, we suppress the interparticle correlations by splitting each physical target electron into a large number of test particles, typically N /N"350. 2 The test particles have smaller mass m "mN/N and charge q "eN/N , but the physical charge 2 2 2 2 to mass ratio q /m "e/m. The test particles follow the Newtonian equations of motion in the ionic 2 2 field and in the mean-field produced from their own charges. Using the same natural units (A.1) as in Eq. (A.2)





duJ j (3ZC drJ G"! G"uJ ,

I (rJ )! , G G "rJ !RI " dtI jrJ dtI G G 1 d€J mN j dRI "€J , "(3ZC , dtI MN jRI "rJ !RI " dtI 2 G G N .J (rJ , t)"! wJ (rJ !rJ ) , G N 2G DI I (rJ , t)"!.J (rJ , t) ,

(A.5)

. w e

e

.J " , wJ " , I " " , DI "rD .  en n E k ¹  One sees immediately from these scaled PIC equations that propagation now depends only on the combined coupling parameter ZC. The explicit dependence on C, which was present in the MD simulations and which characterized target correlations, has disappeared as it should be for a proper mean-field dynamics. A few remarks on the technical aspects of the PIC scheme are in order. Note that compared to the MD-simulations for N electrons (79) the equations (A.5) now run over N ) N test particles and 2

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the direct electron interaction is replaced by the mean-field potential (r). In addition to the numerical integration of the particle propagation, the charge density . is sampled on a spatial grid from the sum of the test-particle weight functions w and the selfconsistent field is determined by solving the Poisson equation D (r, t)"!. (r, t)/e on the grid. The forces on the particles are  retrieved from the grid using the same weight functions again, for details see [18,80,148]. The expense of accumulation of the density . and of retrieval of the forces grows only linearly with the number of test particles. Thus one can afford a large number of those. Our PIC simulations, e.g., have been done with N "175 000 test particles. The density and mean field have usually been 2 handled on a 32;32;32 grid in coordinate space. Both numbers have been varied to check the stability of the results. Particle propagation and separate handling of close electron—ion collisions remains as in the MD simulations, see the previous subsection. The Poisson equation is solved in Fourier space using fast Fourier transformation to switch forth and back. The periodic boundary conditions, and the related Ewald summation, are automatically taken care of by using the Fourier transformation with complex exponentials. We close this section with a remark on the versatility of the PIC method. We used it here for comparison with MD simulations in order to work out effects of the nonideality of the target. To that end, electron—electron correlations had been suppressed by using a large number N /N of test 2 particles per electron. One can equally well use PIC simulations in the regimes where correlations are unimportant in the target. Thus one can go for the opposite strategy to pack several electrons into one numerical macro-particle, i.e. N /N;1. This allows to use large simulation boxes, larger 2 than the (large) screening length of those loosely correlated systems. Thus the PIC simulations can penetrate into the regime C;1 which is inaccessible to MD simulations for reasons of the numerical effort. A.3. WPMD-simulations With the help of the variational principle the quantum problem was mapped into a classical one with additional degrees of freedom, see Section 3.3.4. This is solved according to the general methods outlined in Appendix A.1. The Ewald sum (100) was calculated and stored on a lattice of 75 points. The actual values required during the simulation were obtained by linear interpolation ([129], ch. 3) according to V (r)"a(r) ) r#r ) @ B ) r#c(r)xyz # 

(A.6)

with r"(x, y, z). As the Ewald sum is smoothly varying over the width of the wave packet, the same expression is used for the electrons as for the ions. The resulting set of differential equations is solved with a second order Runge—Kutta algorithm. As the computation of the forces between the pseudo-classical particles in Eqs. (89)—(101) requires most of the computational effort, it is essential to set up the code in a hierarchical manner [101]. The ith particle is propagated with a time step






Dt "q min 2\LGH , G GH

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where q is an arbitrary scale factor and n 3- is chosen in such a manner that GH

 





F , n "min n3- q2\L(b min GH F$ (i, j) I GHI where F (i, j) is the kth (k"(x, y, z, c)) component of the force between the ith and the jth particle, I and F is a constant average force. The desired accuracy is controlled by the parameter b(+10\).

Appendix B. Definitions and list of symbols B.1. Stopping power for infinite projectile mass The limit of infinite projectile mass (MPR) allows an alternative derivation of expression (6) for the stopping power. The ion moves with constant velocity and thus the Hamiltonian (2) reduces to the simpler one HK (t)"HK !h(t!t ) e (rL !€t), where now acts as an external moving    G  G potential at position €t. Since we have no change in momentum or energy of the projectile, the stopping power must be derived from the energy transfer to the target. Guided by expression (3) the stopping power can be defined for t't through  dE 1d 1d 1 j "! 1E2"! TroL (t)HK (t)"! TroL (t) HK (t) ds v dt v dt v jt j € 1 " TroL (t) e (rL !€t)"!Tr oL (t) e ) G (rL !€t)  G jt v P  G v G G € "! dr 2dr 1r ,2, r "oL (t)"r ,2, r 2 e ) G (r !€t) .  ,  ,  , v P  G G Rewriting the last term by introducing the charge density





(B.1)



. (r, t)"!e dr d(r!r )“ dr 1r ,2,r "oL (t)"r ,2,r 2 G G H  ,  , G H$G and the electrical field E we immediately recover the result (6). We observe in addition that the steady excitation leads to a final steady state with constant stopping which depends only on the velocity of the inert ion, see Eq. (7). B.2. Alternative derivations of the parameter of linearity The expression for the local strength of the ion—target coupling Eqs. (19) and (20), and the subsequent definition of linear coupling by definition (21) are based on the semiclassical expression (18) for the potential energy where quantum effects are included in an approximative manner only. Here we sketch two fully quantal approaches for deriving a criterion for linear coupling to substantiate the previous considerations in Section 2.3.2.

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One alternative way to characterize the coupling strength starts from a look at ion—electron Coulomb scattering. There the stationary Schro¨dinger equation reads (for Z'0)






2g DrI #1#  t (rJ )"0  rJ

with rJ "r/ and the pure Coulomb wave t (rJ , g ) [122,155]. Here, a measure of the coupling    strength is established by comparing the probability of finding the electron at the ion position with the free particle case g "0. From the known behavior of the Coulomb wave one easily derives the  relative change ; "t (rJ "0, g )" 2ng      ng , !1" !1 EP  "t (rJ "0, g "0)" 1!exp(!2ng )   

which turns out to be small for g ;1.  A further check for the strength of the perturbation is provided by inspecting the change of the electron density when an ion is added to a free electron target of density n. As a particular critical test we regard the induced density dn(r)"n(r)!n at the ion location for the most severe case of an ion at rest (v"0). In framework of the linear response treatment in Section 3.1, the induced density can be computed explicitly from the linear response relation (38) and by using the RPA dielectric function e (47). A numerical survey yields 0

  



Z 1 "dn(r"0)" " dk !1 "1g 2h(m, H)+1g 2 ,   n(2n) e (k, u"0) n 0 with 1g 2 as defined in Eq. (16), and a function h(m, H) which is of the order of unity for any degree  of degeneracy H and any m(1. The small relative change of the density resulting for 1g 2;1  indicates a weak perturbation by the ion and linear coupling. B.3. Definitions used for the Fourier-transformation For the Fourier transformation of a function f in space and time we used throughout this review the definitions

   

f (r, t)Pf (k, u)" dr f (k, u)Pf (r, t)"



dt f (r, t)exp(!i(k ) r!ut)) , \

dk  du f (k, u)exp(i(k ) r!ut)) , (2n) 2n \

with the transform of a constant f (r, t)"1Pf (k, u)"(2n)d(k)2nd(u) .

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B.4. Definitions of some important quantities ¼igner—Seitz radius a"




4nn \ 3

Bohr radius e h a "  "5.29;10\ m  nem r parameter  n \ a C r" +  a 1.6;10 cm\  Parameter of degeneracy








4  v k ¹ k ¹ "0.521 "2 " (ar ), a" H" 9n v 13.6eV  E $ $ Parameter of ideality 2ar  m" 1#H Plasma parameter




4nn  e 2ar e  " " C" 4ne k ¹a H 4ne k ¹ 3   Plasma frequency




en  u"  me  Plasma period 2n q"  u  Debye length




e k ¹  v a " " j "  " ne u  (3C ¹hermal velocity




v " 

k ¹  m

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Fermi velocity

(3nn) v " $ m Fermi energy 1

(3nn) E " mv" $ 2 $ 2m Averaged velocities (approximations)






1v 2 v  1v 2"(1v 2#v), 1v 2"  " 1# ,    1v 2 1v 2   H  1v 2"(v#v )"v 1#  $  $ 2






Classical collision diameter "Z"e ITK6T7 "Z"e "Z"ar  b " P 1b 2" " a  4ne kv  4ne m1v 2 1v 2(1#(H/2))       The collision diameter b is equal to the impact parameter for a classical collision in the (bare)  Coulomb potential with 90° deflection angle at relative energy kv/2.  Ion—electron Coulomb parameter "Z"e "Z"ar "Z"e T6T7  P 1g 2" " g"   4ne v 4ne 1v 2 1v 2(1#(H/2)      Electron—electron Coulomb parameter e T6T7 ar e  g" " P 1g 2"  4ne v  4ne 1v 2 1v 2(1#(H/2)      B.5. Notation a a  b ,1b 2   C, CI E E e dE/ds F

Wigner—Seitz radius a"(4nn/3)\ Bohr radius 0.529;10\ m classical collision diameter dimensionless friction coefficients energy electric field elementary charge, distinct from exponential function stopping power force on the ion

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f f  f (x) G g HK , H I Im[2] k, k k k K m M N N " N IJ n n(r) dn p, p, P, P r, R R(Z, C) r  S S&,! ¹ t º » V #  €, v""€" €L "€/v v ,v ,v   $ v , u, u  Z

phase space distribution function phase-space distribution in ground state, usually Fermi or Maxwell Bose function local field correction spatial pair distribution function Hamilton operator, function high frequency limit of LFC (used in dynamic interpolation) imaginary part wave number Boltzmann constant cutoff parameter electron mass projectile mass particle number number of electrons in a Debye sphere norm-matrix bulk ground state electron density distribution of particle density small deviation from equilibrium density momenta coordinates in space dimensionless friction coefficient for classical systems a/a  structure factor (function), dynamic as well as static static structure factor from HNC electron temperature time one-body potential, mean field two-body interaction, usually Coulomb Ewald sum ion velocity unit vector of ion velocity relative, thermal, Fermi velocity electron velocities effective ion charge state or nuclear charge state

a C c, ln c c, c G d(x), d(x) e e 0 e 

(4/9n)"0.5212 (classical) plasma parameter ln c"0.5772, Euler’s constant width parameter in the WPMD scheme (double usage of c!) Dirac delta function dielectric function RPA dielectric function dielectric constant in vacuum

G. Zwicknagel et al. / Physics Reports 309 (1999) 117—208

e ? f, fI g , 1g 2   g , 1g 2   H h(x) i K ,1 2   j, j " k l m . oL p, dp/dX p  q ¶





 u u , u , uq ? G s s 0 s  "W2, W u u 

single-electron energy kinetic energy, free particle dispersion Coulomb parameter for electron—electron collisions Coulomb parameter for projectile-electron collisions degree of degeneracy step function cutoff parameter thermal wavelength de Broglie wavelength /kv , /k1v 2   screening length, Debye length reduced mass electron—electron collision frequency parameter of ideality charge density density operator differential cross section transport cross section relaxation time screening function electric potential bare ion potential effective electron ion potential parameter of linearity of ion—target coupling single-electron wavefunction denisty—density response function RPA density—density response function free particle density—density response function wave function frequency plasma frequency

[2,2] 122

commutator quantum and/or ensemble averaging

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Density functional calculation of semiconductor surface phonons J. Fritsch*, U. Schro¨der Institut fu( r Theoretische Physik, Universita( t Regensburg, D-93040 Regensburg, Germany Received February 1998; editor: A.A. Maradudin Contents 1. Introduction 2. Microscopic calculation of surface phonons 2.1. Slab lattice dynamics 2.2. Molecular-dynamics simulations 3. Total-energy calculations 3.1. Density-functional theory 3.2. Expansion in plane waves 3.3. Special points 3.4. Pseudopotentials 3.5. Other minimization techniques 3.6. Slab-supercell approach 3.7. Relaxation and reconstruction 4. Density-functional phonon calculations 4.1. Ab initio molecular-dynamics simulations 4.2. Frozen-phonon approaches 4.3. Density-functional perturbation theory 4.4. Comparison of the methods 5. Phonon dynamics of III—V(1 1 0) semiconductor surfaces 5.1. Atomic structure of the III—V(1 1 0) surfaces 5.2. Density-functional phonon calculations for III—V(1 1 0) surfaces 5.3. Phonons in GaAs(1 1 0) 5.4. Phonons in InP(1 1 0) 5.5. Comparison of the III—V(1 1 0) phonon spectra 6. Phonon dynamics of diamond-structure (0 0 1) surfaces 6.1. Application of DFPT to Si and Ge (0 0 1)

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6.2. Surface reconstruction in Si(0 0 1) and Ge(0 0 1) 6.3. Surface phonons in Si(0 0 1) 6.4. Phonon dispersion in Ge(0 0 1) 6.5. Surface vibrations in C(0 0 1) p(2;1) 7. Structure and dynamics of the Si(1 1 1) surface 7.1. Atomic surface geometry of Si(1 1 1) (2;1) 7.2. Phonons in Si(1 1 1)(2;1) 7.3. Phonons in the Si(1 1 1)(7;7) surface 8. Adsorbate covered semiconductor surfaces 8.1. Hydrogen coverage on III—V(1 1 0) surfaces 8.2. Other adsorbates on III—V(1 1 0) surfaces 8.3. Hydrogen coverage on Si(0 0 1) and C(0 0 1) 8.4. Arsenic on Si(0 0 1) 8.5. Hydrogen coverage on Si(1 1 1), Ge(1 1 1) and C(1 1 1) 8.6. As on Si(1 1 1) 8.7. Group-III elements on Si(1 1 1) 9. Conclusions and outlook Acknowledgements References

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* Corresponding author. E-mail: [email protected]. 0370-1573/99/$ — see front matter  1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 8 ) 0 0 0 3 4 - 9

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DENSITY FUNCTIONAL CALCULATION OF SEMICONDUCTOR SURFACE PHONONS

J. FRITSCH, U. SCHRO® DER Institut fu( r Theoretische Physik, Universita( t Regensburg, D-93040 Regensburg, Germany

AMSTERDAM — LAUSANNE — NEW YORK — OXFORD — SHANNON — TOKYO

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Abstract Vibrations on a surface directly reflect the atomic arrangement and the force constants between the atoms in the outermost layers of a solid. Semiconductor surfaces are particularly interesting, because of the covalent bonding nature of the bulk material. A thorough analysis of the vibrational modes therefore contributes essentially to a better understanding of the driving forces and processes that determine the geometry and the electronic properties in the surface region. In this report, we summarize recent developments in density-functional theory which have substantially intensified theoretical investigation of semiconductor surface dynamics. We briefly describe the conceptional details of ab initio linear-response theory, molecular-dynamics simulations, and frozen-phonon-like approaches which are the most successful computational schemes applied in the past few years. A collection of the most important results obtained on the basis of such computations will demonstrate how density-functional calculations of this kind contribute to a clear and detailed insight into the structural and dynamical properties of semiconductor surfaces.  1999 Elsevier Science B.V. All rights reserved. PACS: 68.35.Bs; 68.35.Ja; 71.15.Mb Keywords: Density-functional theory; Semiconductor surfaces; Surface structure; Surface dynamics

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1. Introduction Surface phonon modes are, generally speaking, vibrational excitations in which the displacements of atoms are large near the surface and decrease essentially exponentially into the bulk. The investigation of localized phonon modes therefore provides information of the atomic geometry, bonding structure, and the interatomic force constants in the topmost layers. With the refinement of experimental techniques such as inelastic electron scattering and inelastic He-atom scattering it has become feasible to measure phonon dispersion curves for a large variety of systems with high precision. Particularly, semiconductor surfaces have been studied with increasing intensity during the last decade, mainly supported by the remarkable advances achieved in the development of computational schemes based on density-functional theory (DFT) and the availability of supercomputers. It is now possible to determine complete phonon-dispersion curves within ab initio calculations that are free from any adjustable parameters. While ionic crystals and their surfaces are sufficiently described by lattice-dynamical models that include the Coulomb interaction, short-range repulsion terms, and the deformability of the electron charge density, vibrations in the bulk and surfaces of semiconductors require a more sophisticated theoretical treatment. The elastic moduli show large deviations from Cauchy’s relation indicating the importance of three-body interactions and especially long range interactions that involve deformations of bond angles. These particularities are resulting from the covalent bonding structure of the material. The situation is even more complicated on a semiconductor surface. The break-up of directed bonds often results in the formation of new bonding units and charge redistribution from energetical higher to more favorable dangling bonds. Model calculations have been performed taking into account bond stretching and angle bending force constants. However, the charge distribution on the surface can be described only by some of the lattice dynamical models and only to a limited extent. Density-functional calculations based on the local-density approximation or a more generalized gradient expansion for the exchange correlation are now widely used to determine the structural and electronic properties of the bulk and the surfaces of semiconductors. Important developments have been made in the calculation of forces as the first derivatives of the total energy and force constants which are the second derivatives of the total energy with respect to small displacements. Hence, DFT has been successfully applied to the determination of bulk phonons and with recent extensions also to surface dynamics. Particularly the last years have brought a large number of new results which led to a detailed insight into vibrational excitations in semiconductor surfaces. In many cases DFT is able to provide precise information about the existence and the dispersion of surface phonon modes prior to experiments. As the calculations are free from any parameters, the obtained results are reliable guidelines for experiments and in general agree very well with existing experimental data. It is now timely to summarize the development of such ab initio calculations and to compile the most important results. In this report, we illustrate the main ideas of the commonly used approaches, starting with a general introduction into DFT and its applications to determine the structure and dynamics of semiconductor surfaces. Then we focus on density-functional linearresponse theory, ab initio molecular-dynamics simulations, and frozen-phonon-like approaches, including a comparison of the conceptional details which characterize the different approaches. The main results obtained from such calculations for nonpolar surfaces of many III—V compound

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semiconductors and low-index surfaces of elemental semiconductors are presented and compared to experimental data. New and important insight into the interplay between the atomic, electronic, and bonding structure of a semiconductor surface and its manifestations in the phonon spectrum is achieved. This illustrates that the investigation of surface vibrational excitations essentially contributes to a better understanding of the microscopic nature of structure forming processes in a semiconductor surface. The advances in the calculation of semiconductor surface phonons represent only a small part of the achievements made to investigate surface vibrational excitations. Let us briefly summarize the basic theoretical concepts developed in this field. A comprehensive introduction and a more detailed illustration of the progress in the computation of surface phonons is given in a series of review articles, including those of Wallis [383], Kress and de Wette [205], Maradudin [236], and other compilations [217,291]. The problem of surface vibrational excitations on crystal solids was first addressed by Lord Rayleigh in 1887 [300], who investigated the propagation of acoustic waves on a stress-free planar surface of an elastic isotropic medium. For the limit of large wavelengths, the solution of the equation of motion yields a vibrational state which has atomic displacements in the sagittal plane defined by the wavevector of the Rayleigh mode and the surface normal. A generalization to anisotropic media, which is necessary already for most of the cubic materials, was given by Stoneley [349]. In many cases, surface acoustic waves occur in the form of generalized surface waves [130] or leaky waves (pseudosurface waves) [226] rather than as pure Rayleigh modes. Another macroscopic surface vibration is the Fuchs—Kliewer phonon mode [128]. Its dispersion can be explained on the basis of Maxwell’s equations as pointed out in Ref. [200]. The Rayleigh wave and the Fuchs—Kliewer phonon are vibrational solutions obtained within a description that neglects the discrete atomic character of a crystal and its surfaces. This approximation is justified only for phonon modes of this kind and only in the limit of long wavelengths. It is obvious that the crystalline structure and the exact atomic positions of the lattice constituents are important as soon as the wavelength of the phonon modes becomes comparable to the interatomic distances. To achieve a microscopic treatment of surface vibrations Clark and co-workers [81] suggested to apply the concept of lattice dynamics to thin crystal films (slabs). First slab phonon-dispersion curves have been published in Refs. [14,74,363]. In this approach, which has been applied since then in many calculations, the normal modes of a slab can be classified as surface modes, bulk modes, and mixed modes. A systematic description including the introduction of a general terminology for the lattice dynamics of thin crystal films was given by Allen and co-workers [15]. The polarization of surface phonon modes was analyzed in dependence of the symmetry. For propagation along a high-symmetry direction, the eigenvectors can be characterized as sagittal modes, in which the atomic displacements lie in the sagittal plane, and shear-horizontal modes which have atomic displacements only normal to the sagittal plane. Detailed calculations for monoatomic fcc crystals with (0 0 1), (1 1 0), and (1 1 1) surfaces were presented by the same authors in Ref. [16]. In addition to generalized Rayleigh waves, a number of other surface-phonon branches was found in the phonon dispersion of the crystal films. It was demonstrated that localized modes can appear in gaps of the bulk continuum and also within bulk subbands for certain symmetry directions. Some of the surface modes identified in Ref. [16] are primarily localized in subsurface layers rather than in the first layer. Modifications of surface force constants were found to significantly change the qualitative features in the surface phonon spectra. Based on

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the slab method, increasingly sophisticated lattice-dynamical models were applied to determine the phonon dynamics of many different crystal surfaces [40,75,76,204,302,333]. Experimentally, surface-phonon frequencies can be determined with great detail by means of inelastic He-atom scattering (HAS) [116] and high-resolution electron-energy loss spectroscopy. Successful measurements of phonon-dispersion curves by means of HAS were first reported by Brusdeylins, Doak, and Toennies [51] in 1981 for LiF(0 0 1). Since then, inelastic He-atom scattering has been applied to the surfaces of many ionic crystals [41,42,52—54,362], semiconductors [55,92,156,157], and metals [49,176,359]. First signatures of surface phonons probed by inelastic scattering of electrons were reported by Ibach in 1971 for Si(1 1 1)(2;1) [178]. From then on electron-energy loss spectroscopy has evolved to an experimental technique which has been often used to measure phonon-dispersion curves of clean surfaces [250,252,261,262,277,278,393] as well as the vibrations of adsorbed atoms and molecules [29,263,264,288,289]. Two additional experimental techniques, which are used to investigate the phonon spectrum of a crystal surface, are the measurement of optical surface vibrations by Raman spectroscopy [163,175] and the determination of the dispersion of surface acoustical modes by Brillouin light scattering [99]. Recent developments in theory largely intensified the computation of semiconductor surface phonon modes in the framework of density-functional theory. In this review, we describe the methods used in these calculations and summarize the results obtained for a large number of systems. The covalent nature of semiconductors requires a particularly careful treatment of interatomic force constants. Because of the formation of directed bonds, three-body interactions that comprise angle deformations over a large range have to be included. Moreover, the surfaces of semiconductors are characterized by broken unsaturated bonds. This causes substantial changes in the atomic equilibrium positions which allow to decrease the energy of the electronic system and the surface. As a result characteristic electronic states and microscopic vibrations occur, which are closely related to the atomic geometry and the bonding structure of a relaxed or reconstructed surface. Early calculations carried out for GaAs(1 1 0), InP(1 1 0) [87,141], and Si(1 1 1) [139—141] relied on parametrized potentials taking into account bond-stretching and bond-bending force constants. Molecular-dynamics simulations were performed for the (0 0 1) surfaces of Si and Ge [387,402], applying the widely used two- and three-body potential suggested by Stillinger and Weber [348]. In addition to bond-stretching and bond-bending force constants, the bond-charge model (BCM) [388] accounts for the covalent nature of semiconductors by introducing massless pseudoparticles that describe the charge between neighboring atoms and the electrons in the occupied dangling bonds. The BCM was applied to determine the phonon dispersion of GaAs(1 1 0) [316,317,368,370], InP(1 1 0) [369,370], and other III—V(1 1 0) surfaces [370], Ge(1 1 1)(2;1) [317], as well as the free and As-covered Si(1 1 1) surface [244,318]. Semi-empirical total energy schemes based on the tight-binding approach [69,237] incorporate electronic properties more thoroughly, using model parameters which are adjusted not only to measured phonon frequencies but also to electronic band-structure data. Beside some computations for the (1 1 0) surfaces of III—V compounds [138,194,322,385], extensive tight-binding calculations were performed for Si(0 0 1) [8,11,12,147,238,241,293] and the free and hydrogen-covered Si(1 1 1) surface [9,146,246,311]. Recent investigations have been focused on hydrogen-covered C(1 1 1) [312,313]. Particularly for free and adsorbate covered surfaces of elemental semiconductors, the tightbinding scheme was applied with great success. However, similarly to all other model calculations, the tight-binding approach cannot treat the electronic degrees of freedom in a rigorous way. In

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many cases, the degeneracy of dangling bonds is removed by the reconstruction of a semiconductor surface [70,208,273]. The resulting charge transfer has to be described self-consistently. To account for charge redistribution in the broken bonds, Alerhand and Mele applied a modified tight-binding scheme which includes a Hubbard-like term [8]. Their results for the relaxation and the lowest gap mode determined for the asymmetric dimer reconstruction of the Si(0 0 1) surface vary as a function of the magnitude of the on-site electron—electron repulsion. In the bond-charge model the position and the amount of the charge occupying the dangling bonds has to be fixed at the beginning of the calculation. Usually, results from self-consistent structure calculations are taken to obtain the respective model parameters [317,368]. In order to obtain correct force constants, it is necessary to determine in a self-consistent way the atomic equilibrium positions and the charge redistribution, due to relaxation or reconstruction by going beyond parametrized model potentials. This is achieved by calculations based on the density-functional theory (DFT) [166,203], which are known to yield reliable results for the atomic and electronic properties of the bulk as well as the surfaces of crystals [273,373]. The first successful steps in the application of DFT to the determination of surface phonons within slab calculations were achieved for simple metals. Ho and Bohnen have used a frozenphonon-like approach to compute the surface phonons of Al(1 1 0) [165]. Interlayer coupling constants were derived from the forces determined for atomic off-equilibrium geometries. Since then similar calculations have been applied to many different systems. Recently, the frozen-phonon method was used to determine surface vibrations in Cu(1 1 0) [305], Rh(1 1 1) [45], and for the hydrogen covered Mo(1 1 0) and W(1 1 0) surfaces [202]. An alternative approach which is based on the calculation of the density-response function for interacting electrons was introduced by Eguiluz and co-workers [106]. This method which combines linear-response theory with DFT was applied to compute the phonon-dispersion curves of Al(1 1 0). The results for the unrelaxed Al(1 1 0) surface compare very well with the findings of Ho and Bohnen. However, for the relaxed surface the lowest frequency surface mode has a higher energy than the one of the previous calculation. The approach presented in Ref. [106] was also used to determine the phonon dispersion of other low-index surfaces of Al [107,119,120]. While the density-response function was determined using a jelliummodel in Ref. [106], the electron-ion interaction is taken into full account in the density-functional perturbation scheme presented by Quong and co-workers [298]. Computations were carried out for Na(0 0 1), however, no experimental data are available for this system. An important stimulus to determine phonon-dispersion curves of semiconductor surfaces in the framework of ab initio calculations was provided by Baroni, Giannozzi, and Testa [32], who developed a new linear-response technique which combines first-order perturbation theory with DFT in a very efficient way. The method allows one to calculate dynamical matrices at arbitrary wavevectors with a computational effort comparable to that of a self-consistent calculation for the unperturbed system. As demonstrated by Giannozzi and co-workers [134], accurate phonondispersion curves can be determined for essentially all important elemental and compound semiconductors [134]. Recently, theoretical investigations based on the efficient linear-response formalism of Baroni and co-workers have been successfully extended to the calculation of surface phonon-dispersion curves. The results obtained for GaAs(1 1 0) [121], InP(1 1 0) [122], GaP(1 1 0), InAs(1 1 0) [100—102], InSb(1 1 0) [58], and free and adsorbate covered Si(1 1 1) [19,167,169] are in excellent agreement with the data available from HAS [92,157,318] and HREELS [256,261,262,286,326]. New and important insights have been obtained from DFPT for the

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Ge(0 0 1) [346] and Si(0 0 1) [124] surfaces. Another important technique which has been established combines DFT with molecular-dynamics simulations. The (1 1 1) and (0 0 1) surfaces of silicon [18,336] as well as the free and hydrogen covered GaAs(1 1 0) surface [111] have been investigated by means of the Car-Parrinello approach [60]. Recently, conjugate-gradient dynamics simulations [285] have been applied to investigate the dynamics of the Si(1 1 1)(7;7) surface [343,344]. Finally, some density-functional calculations have been carried out on the basis of the frozen-phonon approach. Systematic studies exists for the free and Sb-covered (1 1 0) surfaces of III—V compounds [329,330]. Moreover, vibrational modes have been determined by frozenphonon calculations for H : GaAs(1 1 0) [43], H : C(100) [10], and H : Si(1 1 1) [192,225].

2. Microscopic calculation of surface phonons With the exception of some macroscopic excitations like the Rayleigh mode [300] and the Fuchs—Kliewer phonon [128], most of the surface vibrational states cannot be explained by continuum theory. The vast majority of localized modes is only accessible by means of microscopic calculations [15,16]. We therefore briefly introduce the formalism. The discrete atomic character of a crystal is conveniently described by primitive translations a , a , and a defining the Bravais    lattice and the vectors R , a"1,2, n giving the basis positions of all n atoms in the unit cell ? relative to its origin. Periodic repetition of the primitive unit cell generates the entire crystal. The time dependent position vector of an atom in the solid is represented by (1) Rl (t)"Rl #ul (t) , ? ? ? where ul (t) is the displacement of the ath atom in the lth unit cell (l"(l , l , l )) from the ?    equilibrium position Rl "Rl#R "l a #l a #l a #R . (2) ? ?       ? The integers l , l , and l range in the interval from !R to #R for the bulk.    With an appropriate choice of a and a parallel to the surface, a semi-infinite crystal is described   by !R(l (#R, !R(l (#R, and !R(l 40 or 04l (#R. As proposed     in the early calculations [14—16,74,363], surfaces can also be described on the basis of a thin crystal film which comprises a sufficiently large (but finite) number of atomic layers with infinite extension in the plane of the slab. As discussed in more detail in the next section, electronic-structure calculations are often carried out for configurations of identical thin crystal films which are periodically repeated in the direction normal to their surfaces. Neighboring crystal films have to be separated by vacuum with sufficient extension to exclude interaction. Under such conditions, the dynamics of a single-slab and of a periodic-slab configuration are essentially the same. In this section we restrict the discussion to a single slab (l "0). The lattice vectors  (3) RM lM "l a #l a     form a two-dimensional lattice in the (x, y)-plane with vanishing z-component as indicated by the abbreviations RM "(R , R , 0) and lM "(l , l , 0) .    

(4)

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Fig. 1. Surface Brillouin zones for the (1 1 0), (1 1 1), and (0 0 1) surfaces of zincblende-compound semiconductors. The lower part shows the Brillouin zones of (1;1) and (2;1) surface geometries.

The atomic equilibrium positions in the crystal film are defined by RlM "RM l#R , ? ?

(5)

where the basis vectors R specify the positions of all atoms in the slab unit cell. The index ? a indicates the type of atom and the layer of the crystal film in which it resides. The primitive translations a and a correspond to the lattice constant determined for the bulk material and have   to be chosen in accord with the reconstruction of the surface. We define the primitive translations b and b of the reciprocal space according to the relations   a ;(a ;a ) a ;(a ;a )   .   and b "2n  (6) b "2n    "a ;a " "a ;a "     Since a and a lie in the (x, y)-plane, the z-component vanishes for b and b and hence for all     reciprocal lattice vectors GM "(G , G , 0) which are generated by   GM "n b #n b (7)     with n and n being integers in the range from !R to #R. Fig. 1 illustrates the two  dimensional Brillouin zones of the most commonly investigated zincblende-compound semiconductor surfaces. The (0 0 1) and (1 1 1) surfaces of elemental semiconductors often exhibit (2;1) reconstructions as indicated in the lower part of Fig. 1. For all cases, standard notations for high-symmetry points are given.

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2.1. Slab lattice dynamics Atomic vibrations in the crystal slab reflect the increase of the potential energy induced by lattice distortions. Since the electron mass is much smaller compared to the atomic masses, the electrons are assumed to follow instantaneously the motions of the atoms. Hence, the adiabatic approximation is used to calculate the potential energy » (+RlN ,) as a function of the atomic positions +Rl , as ? M?  given by Eq. (1). The adiabatic potential » (+RlN ,) can be represented by a sum of an electronic ?  part and a pure ionic contribution according to (8) » (+RlN ,)"» (+RlN ,)#E (+RlN ,) . ?  ?  ?  Model calculations describe the electronic and ionic part in terms of parametrized potentials. Recent developments in total-energy schemes go beyond the original approaches and determine the structure and dynamics of semiconductor surfaces from first principles. Sections 3 and 4 focus on the application of density-functional theory to the calculation of E (+RlN ,) and its first and ?  second-order derivatives. The adiabatic potential (Eq. (8)) determines the vibrational properties of the crystal. Its Taylor expansion has a vanishing first-order term for small atomic displacements +ulN , from the equilib? rium positions +RlN , and is given by ? 1 » (+RlN #ulN ,)"» (+RlN ,)# U (lM a, lM a)ul ul #2 . (9) ? ?  ?  GH 2 lN lN ?G Y?YH ?G Y?YH The indices i and j indicate the three Cartesian coordinates. The coefficients U (lM a, lM a) are the GH harmonic force constants defined as the second-order derivatives of the adiabatic potential with respect to the atomic displacements taken in the equilibrium positions (+ul ,"0): ? j»  U (lM a, lM a)" . (10) GH julN julN +ulM ?, ?G Y?YH  In the harmonic approximation, the equations of motion for the atomic displacements can be written as



d (11) u "! U (lM a, lM a)ulM , Y?YH GH ?dt lN ?G lN Y?YH where M is the mass of the ath atom in the unit cell. The atomic force constants determine the ? vibrational properties of the crystal film. For atoms in the center of the crystal film, the coupling constants are the same as in the bulk, as long as the number of atomic layers in the slab is large enough. The force constants of surface near atoms, however, reflect that the translational symmetry normal to the surface is broken. Moreover, rebonding, atomic relaxation, charge redistribution and reconstruction can drastically change the coupling constants U (lM a, lM a) for atoms close to the GH surface. Translational periodicity is still assumed parallel to the surface for all planes of the crystal film. We therefore search for solutions for the equations of motion (11) in the form M

1 ulM " v (qN ) exp[i(qN )RM lN !ut)] , ?G (M ?G ?

(12)

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where qN "(q , q , 0) is a wavevector lying in the surface Brillouin zone. Substitution of Eq. (12) into   Eq. (11) yields the eigenvalue problem u(qN ) v (qN )" D (aa, qN ) v (qN ) ?G GH ?YH ?YH with the dynamical matrix defined by

(13)

1 U (la, la) exp[!iqN )(RM lM !RM lM Y)] . (14) D (aa, qN )" GH GH lN Y (M?M?Y For n atoms in the slab unit cell the dynamical matrix has the dimension (3n;3n). Diagonalization of the hermitian matrix for each wavevector qN in the SBZ yields 3n real and non-negative eigenvalues u(qN ) corresponding to 3n eigenvectors vQ (qN ) which obey the equations Q ?G u(qN ) vQ (qN )" D (aa, qN ) vQ (qN ), s"1,2, 3n . Q ?G GH ?YH ?YH

(15)

2.1.1. Slab dispersion curves The 3n normal modes of the slab vQ (qN ) determined for each qN can be classified with respect to ?G their polarization and the square of the amplitudes in the different layers of the crystal film. While bulk modes do not change significantly the vibrational amplitudes when comparing the outer with the inner planes of the slab, the atomic displacements associated with true surface vibrations decrease essentially exponentially approaching the central layers of the crystal film. Resonances do not decay approaching the inner region, however, exhibit larger amplitudes in the surface layers. For propagation along high-symmetry directions, the polarization of the modes is either strictly sagittal with non-vanishing components of the vectors € "(v , v , v ) (a"1,2, n) only in the ? ? ? ? plane defined by the surface normal and the wavevector qN or shear horizontal with all atoms moving exclusively perpendicular to the sagittal plane. If qN is not oriented along a high-symmetry direction, the two kinds of polarization are mixed in general. The eigenfrequencies u (qN ) are usually plotted as a function of the wavevector qN . Typical Q slab-dispersion relations comprise all branches (s"1,2,3n). The majority of the 3n branches is representing bulk phonon modes. Only a small number is related to localized and resonant surface vibrations. While the number of surface phonon modes is characteristic and does not change, the number of bulk related branches increases the more atomic layers are used in the slab calculation. All bulk related branches lie in a frequency range determined by the vibrational spectrum of the material. In slab calculations the bulk phonon dispersion perpendicular to the plane of the crystal film is effectively folded back to a corresponding wavevector qN in the SBZ. This results from the fact that the number of layers per crystal film usually is much larger compared to the minimum number required to describe the bulk using the same lattice vectors a , a as for the surface (slab adapted   bulk). This is schematically illustrated in Fig. 2 for the bulk phonon dispersion of InP folded onto wavevectors lying in the (1 1 0) plane. Bulk-phonon branches appear in the entire range of the surface-projected band structure. The density of bulk branches increases the more atomic layers the single slab comprises. Gaps often appear in the bulk mode spectrum, especially for wavevectors qN different from zero. No bulk

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Fig. 2. Schematic illustration for the projection of the bulk band structure of InP onto the (1 1 0) plane in the CX direction of the SBZ. The surface normal is parallel to the z-direction. Fig. 3. Phonon-dispersion along the CX direction of a InP(1 1 0) slab comprising 25 atomic layers. Surface localized states are indicated by heavy lines, while all other vibrational states of the crystal film are represented by thin lines.

vibration is allowed such regions. Similar to the case of InP, many diatomic crystals exhibit a gap between the acoustical and optical bulk modes for all wavevectors of the Brillouin zone. The existence of such gaps is interesting, since surface vibrations can occur in regions forbidden for bulk modes. This can be seen from the phonon dispersion of InP(1 1 0) which is shown in Fig. 3 for the CX direction. In such a case, surface phonons are well-localized modes. True surface vibrational states can also occur in bulk mode frequency regions as long as the surface mode eigenvector has another symmetry than the eigenvectors of the nearby phonons of the bulk. As surface near atoms have less neighboring bonding partners, the effective force constants are generally smaller in the outer layers. Often, surface vibrations appear at lower frequencies compared to the bulk modes in which they originate. One example is the Rayleigh wave which peels off from the bottom of the transverse acoustical bulk bands in many cases. Optical surface-phonon modes can occur in gaps of the surface-projected bands of the bulk. In addition, for many systems so-called longitudinal surface resonances are known to appear just below the lower edge of the continuum formed by the longitudinal bulk modes. Especially among semiconductors some important examples exist, in which surface phonon frequencies are larger compared to those of corresponding bulk modes. Because of the elastical anisotropy, the branch of the lowest surface acoustical phonon mode can lie above the branch of

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the respective bulk phonon band for certain directions of propagation [131,374]. Rebonding and charge transfer are interesting particularities that characterize semiconductor surfaces [341]. Through the formation of new chemical bonds (such as dimers on Si(0 0 1) (2;1) [70]) or a change in the bonding configuration through relaxation (like in the (1 1 0) surfaces of III-V compounds [126]), effective force constants of surface near atoms can be increased compared to bulk atoms. Hence, phonon modes can appear even above the optical bulk continuum as shown for GaAs(1 1 0) by theory [121] and experiment [261]. 2.1.2. Slab filling Often surface localized modes exhibit large vibrational amplitudes in many layers. To identify this kind of vibrational states or also resonances by means of the slab method, crystal films have to be used that comprise a sufficiently large number of atomic layers. The determination of the dynamical matrices D (aa, qN ) (Eq. (14)) for a crystal film is very time consuming especially in GH the framework of ab initio calculations. Computational labor can be minimized by calculating the dynamical matrix of arbitrarily large slabs using the matrix elements determined for a smaller system. The force constants U (lM a, lM a) (Eq. (10)) and hence the elements of the dynamical matrices GH can be significantly different for surface near atoms compared to bulk values. For atoms in the central region, however, the force constants of even considerably thin crystal films are essentially the same as the corresponding coupling constants of the bulk. Since the force constants decay rapidly enough (in a range of about four atomic layers), the results obtained for a thin crystal film can be used to construct the dynamical matrix of larger slabs. This is schematically illustrated in Fig. 4. The force constants in the two surface regions of the large crystal film are the same as those

Fig. 4. Modelling of the dynamical matrix of a large crystal film (spanning 25 layers) using the force constants of a smaller system (comprising 9 layers).

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of the smaller system. The force constants of the bulk can be determined in a separate calculation or can be taken from the central layers of a thin crystal film including appropriately the symmetry of the bulk. 2.1.3. Green’s function approach Another approach to identify surface phonon modes with large penetration depth is provided by the Green’s function method [235] which was applied to alkali halide surfaces [40] and semiconductor surfaces [139—141] in the context of model calculations. The surface is treated as a twodimensional perturbation of an ideal crystal. Hence, the Green’s function formalism allows one to determine the surface vibrations of a semi-infinite system. With denoting the force constant matrix of the unperturbed crystal by U and introducing the atomic masses by the tensor M, the Green’s
  function of the ideal crystal is given by G"(U !Mu)\ , (16)
  which can be expanded with respect to the eigenvectors and eigenfrequencies of the unperturbed lattice. The changed coupling parameters U introduced by the truncation of the crystal between adjacent layers define a defect matrix J: J"U !U .
  For the semi-infinite medium the equations of motion are given by

(17)

Mu( #U u"Ju .
  The solutions of Eq. (18) can be represented by

(18)

u"u #GJu , (19) F with u being a (homogeneous) solution of the ideal crystal. In the solution for a true localized F surface phonon mode, the homogeneous part u is zero. Therefore, the frequencies of localized F vibrations are determined by the condition Re det[1!G(u, qN )J(qN )]"0 .

(20)

A convenient method to obtain bulk resonant solutions of Eq. (18) is to compute the density of states of the semi-infinite system, which is given by the trace of the imaginary part of the full Greens’s function G: 1 D(u)"! Im Tr[G(u)] . n

(21)

Resonances are related to peaks in the density of states. The Green’s function of the semi-infinite crystal is determined by means of Dyson’s equation G"G#GJG .

(22)

With the exception of deeply penetrating and resonant phonon modes, which can be identified more easily and more precisely by means of the Green’s function approach, the direct analysis of all slab normal modes as illustrated in Section 2.1.1 provides a more straight forward and efficient method to determine surface localized vibrations [56].

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2.2. Molecular-dynamics simulations The methods discussed so far describe the dynamics of the system using the interatomic force constants which are defined as the second-order derivatives of the adiabatic potential with respect to atomic displacements from the equilibrium positions. Diagonalization of the dynamical matrices or the solutions obtained within the Green’s function formalism yield vibrational eigenstates in the harmonic approximation. In contrast, molecular-dynamics simulations extract the dynamical properties from finite-temperature particle trajectories which are computed for a sufficiently large period of time by solving the equations of motion for all atoms in the slab. Temperature effects and anharmonicity are directly accessible within this method. Calculations of this kind have been performed for the dynamics of several semiconductor surfaces on the basis of model potentials applied to Si(0 0 1) and Ge(0 0 1) [387,402] and the tight-binding formalism used for the (1 1 0) surfaces of GaAs and InP [322]. Recently, efficient density-functional schemes have been developed [60,285,314,343,344] which solve the equations of motion for the atoms treating the electronic degrees of freedom rigorously in parameterfree total energy calculations. A standard way to extract the vibrational frequencies of an atomic configuration from the particle trajectories [315] is to compute the Fourier transform



1 g(u)" g(t) ¼(t) cos(ut) dt q

(23)

of the velocity autocorrelation function L 1uR (t) ) uR (0)2 ? ? , g(t)" 1uR (0) ) uR (0)2 ? ? ?

(24)

where q is the total simulation time, ¼(t) is an appropriate window function [154], uR (t) is the ? velocity of the ath particle at the time t, and 1uR (t) ) uR (0) 2 is the corresponding ensemble average. ? ? Periodicity is assumed within the range of one unit cell to which the summation is restricted. Hence, an indication of the Bravais lattice index l is not needed. For the calculation of frequencies at the zone boundary or other qN points of the SBZ, the size of the unit cell has to be extended in correspondence to the wavelength of the wavevector. The resolution of the Fourier transform approach is limited by the length of the time period q of the simulation. It is given by *u'2n/q. Another limitation is that weakly excited modes can be hidden in the main peaks or first aliases of a close, strongly excited mode in the frequency spectrum. A more sensitive spectral analysis can be obtained with the help of the multiple signalclassification method (MUSIC) [267,325] which is able to detect also slightly excited modes without the need to reach thermal equilibration. This is achieved by a separation of the received data into signal and noise subspaces as follows: The trajectory comprises the displacements +u (t ), ? J of all atoms in one unit cell at the times t (l"0, 1, 2,2, M). The displacements can be expanded J by means of 3n eigenmodes +€Q , and eigenfrequencies u according to ? Q L +u (t )," +(1/(M )€Q ,(a cos(u t )#b sin(u t ))#+C (t ), , QJ Q QJ ? J ? J ? ? Q Q

(25)

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where a and b are the amplitudes of each eigenmode, M are the particle masses and +C (t ), Q Q ? J ? represents additive noise. Vectors comprising all n atoms have the dimension 3n and are indicated by braces. With the abbreviations º "+u (t ),, C "+C (t ),, A "a cos(u t )#b sin(u t ), and J ? J J ? J JQ Q QJ Q QJ » "+(1/(M )€Q ,, Eq. (25) can be written in the form Q ? ? (26) U"AV#C , in accord with Eq. (1) of Ref. [235]. Since the noise and the signal are not correlated, the autocorrelation matrix 1(U, U )2 of the vector U is separated into a signal autocorrelation matrix and a noise autocorrelation matrix: 1(U, U )2"A1(V, V )2AR#1(C, C )2 .

(27)

The representation of 1(U, U )2 in its eigenvectors and eigenvalues can be written as L +> 1(U, U )2" j T TR# k N N R , (28) G G G H H H G HL> where the j and T are the 6n largest eigenvalues and respective eigenvectors of 1(U, U )2, while the G G k and N correspond to the noise [267,344]. Since the subspaces formed by the eigenvectors T and H H G N are orthogonal, ARN has to be essentially zero. Based on the structure of the columns of the H H matrix A, the vector e (u)"(1, exp(iu(q/M)),2, exp(iuM(q/M))) is used to define the MUSIC+ frequency estimator, which is given by




 

+> \ N N R e (u) . (29) H H + HL> The frequency estimator has pronounced peaks for u"u , while it is small for all other frequenQ cies. The extension to a self-consistent variant of the MUSIC algorithm [201] provides the possibility to extract not only phonon frequencies but also phonon eigenvectors from the trajectories [111,336]. Moreover, the signal-to-noise separation is further improved [344]. In the self-consistent scheme displacements +uL (t ), given as functions of +€Q ,, +u ,, +a ,, and +b , are defined ? J ? Q Q Q according to Eq. (25). The eigenvectors and amplitudes are determined by minimizing the cost function P (u)" e (u) + +






+ (30) N(+€Q ,, +u ,, +a ,+b ,)" " u (t )!uL (t )"! K €G ) €H !d , ? ? GH ? Q Q Q ? J ? J GH ? J ? GH which implies the constraint that the eigenvectors +€Q , are orthogonal and normalized (the K are ? GH the Lagrange multipliers of the constraint). To minimize N the eigenvectors and amplitudes are determined using a steepest descent method, while the frequencies +u , are provided by an estimate Q through the MUSIC algorithm. By projecting the original trajectory onto the fitted eigenvectors, partial trajectories of the form +uQ (t ),"+€Q , €Q ) u (t ) ? J ? ?Y ?Y J ?Y

(31)

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are extracted. Subsequently the MUSIC scheme is applied to the projected trajectories yielding a new set of frequencies which are used to achieve a better esimate of the eigenvectors. The results are improved until the spectral estimator (Eq. (29)) gives one well-defined peak for each partial trajectory. The self-consistent MUSIC scheme (SCM) has been successfully applied to extract eigenfrequencies and eigenvectors from trajectories determined for complicated systems containing a large number of atoms [111,336,344]. Especially in the case of the GaAs(1 1 0) surface, the eigenvectors and phonon frequencies extracted with the help of the SCM method [1 1 1] are in excellent agreement with the results obtained from the diagonalization of dynamical matrices. By decomposing the original particle trajectories into the representations of the corresponding symmetry of the crystal film, the numerical labor can be reduced along with improving the resolution of the self-consistent scheme [344].

3. Total-energy calculations The methods described in Sections 2.1 and 2.2 require to calculate forces and dynamical matrices which are first and second-order derivatives of the total energy. In order to achieve a rigorous treatment of the covalent bonding structure in semiconductors and the charge redistribution on the surface, due to relaxation or reconstruction, the quantum-mechanical many-particle problem including all electrons and nuclei has to be solved by minimizing the total energy of the system with respect to the electronic and atomic coordinates. In its general form, however, the many-particle problem is too complicated to be solved. Hence, a series of simplifications and approximations is necessary to sufficiently reduce the complexity of the problem which then can be solved by efficient numerical methods: 1. The adiabatic approximation or Born—Oppenheimer approximation uses the fact that the mass of the electrons is much smaller compared to the nuclear masses. Therefore, the electrons follow essentially instantaneously the motion of the nuclei. Because of this, the coordinates of the electrons and nuclei are decoupled. Consequently, the many-particle problem is reduced to determine the electronic ground-state for a given (and fixed) atomic configuration. 2. In the pseudopotential approach, the electronic system is divided into the valence electrons, which are influenced by the chemical environment, thus contributing to the chemical bonds, and the tightly bound core electrons which are inert. With the constraint of norm-conservation and the constraint not to change the atomic eigenenergies as well as the wave functions outside the core region compared to the real eigenstates of the isolated atom [26], it is possible to construct soft pseudopotentials, which describe correctly all salient features of the electrons in the solid. 3. The remaining many-particle problem of interacting valence electrons, which feel the field of the ionic cores in terms of the pseudopotentials, can be solved on the basis of mean-field theory. One of the most important approaches which is successfully used to investigate the solid state under many different aspects is density-functional theory (DFT) [166,203]. The central idea is to express the electronic energy as a functional E[n(r)] of the single-particle density n(r) which is minimal only if n(r) equals the ground-state density n (r) of the electrons [166]. 

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4. Kohn and Sham [203] have found a representation for E[n(r)] by mapping the interacting many-electron system onto a system of noninteracting electrons moving in an effective potential. By treating the exchange-correlation energy on the basis of the local-density approximation (LDA) [203], the exchange-correlation potential and hence the effective single-particle potential are given as functions of the single-particle density n(r) within the mean-field approach. The variation of the charge density in inhomogeneous systems can be taken into account on the basis of generalized gradient corrections (GGC), where the charge density and its gradient are used for the expansion of the exchange-correlation energy [82]. 5. Density-functional calculations for semiconductor surfaces are often carried out on the basis of a slab geometry. Particularly the use of periodically repeated thin crystal films allows one to extend the methods developed for the bulk directly to surfaces. Beside traditional diagonalization techniques, new iterative procedures are available for minimizing the total energy of the electronic system [60,285]. The development of new efficient algorithms allows one to perform calculations for complex systems which contain a large number of atoms. In the following, we describe in more detail the above summarized concepts standing behind electronic structure calculations based on DFT. First, we illustrate the main ideas of DFT and its application to periodic systems in conjunction with the use of norm-conserving pseudopotentials. Then we focus on recently developed energy minimization techniques. Finally, a brief illustration of the periodic-slab model is given including the application of total energy schemes to determine optimized atomic structures comprising surface relaxation and reconstruction. 3.1. Density-functional theory The concept of density-functional theory (DFT) was developed by Hohenberg and Kohn [166] and Kohn and Sham [203]. It provides in many cases an efficient formalism to treat exchange and correlation in a system of interacting electrons which are under the influence of a static external potential, such as created in a crystal solid by the field of the nuclei as long as the adiabatic approximation is valid. The electronic system is divided into the valence electrons, which are influenced by the chemical environment, and the tightly bound core electrons, which are inert showing no changes with respect to the isolated neutral atom ( frozen-core approximation). Hence, the problem is reduced to determine the ground-state of N interacting valence electrons which feel an external potential stemming from the ionic cores comprising the nuclei and the core electrons. 3.1.1. Hohenberg—Kohn theorem In the framework of DFT [91,135,166,186,203,223,224] the electronic energy is represented by a functional E[n(r)] of the single-particle density n(r). Levy suggested to define the total energy functional in the following way [222]: The Hamiltonian of N interacting electrons (l, k"1,2, N) is given by 1 H "¹#º#» " t(l)# u(l, i)# v (l) ,  ,  2 J$G J J

(32)

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with ¹, º, and » being the operators that describe the kinetic energy, the electron-electron  interaction, and the interaction of the electrons with the external potential. The functional E[n(r)] can be represented by



E[n(r)]" dr v (r)n(r)#F[n(r)] . 

(33)

The functional F[n(r)] is defined as F[n(r)]"min 1t " ¹#º " t2 , (34) RL where the minimum is taken over all many-particle wave functions that give the density n(r). The Hohenberg—Kohn theorem states that the total energy functional is minimal only if n(r) is the ground-state density n (r) of the electrons, and that the minimum value of E[n(r)] is equal to the  ground-state energy [166,186,223]: E[n(r)]5E

for n(r)On (r) and E[n (r)]"E . (35)     The variational behavior of the functional E[n(r)] as indicated by Eq. (35) provides a general method to calculate ground-state properties as a function of the single-particle density and justifies to work with the density instead of wave functions. 3.1.2. Kohn—Sham equations The applicability of DFT is dependent on the construction of an adequate functional for the total energy or more precisely for the functional F[n(r)] (Eq. (34)). Kohn and Sham have suggested a simple approximation for the total energy functional by mapping the interacting many-electron system onto a system of noninteracting electrons moving in an effective potential v (r) [203]. The  fundamental assumption is that it is indeed possible to find an effective potential v for nonin teracting electrons which yields the same ground-state density as the potential v in the case of  interacting electrons. The ground-state wave function of the noninteracting electrons can be represented by a Slater determinant of the N lowest eigenstates t ,2,t of the Hamiltonian H "!( /2m) #v (r),   ,  where v (r) is the effective single-particle potential and m is the free electron mass. The single particle density has the representation , n(r)" "t (r)" . J J In the Kohn—Sham formalism, the total energy is given by

(36)

E[n(r)]"¹ [n(r)]#º [n(r)]#» [n(r)]#E [n(r)] , (37)  &   where ¹ [n(r)] is the kinetic energy of noninteracting electrons and º [n(r)] the Hartree contribu & tion to the electron-electron interaction. Comparison with Eq. (34) shows that the exchangecorrelation energy E [n(r)] has to be defined as follows:  E [n(r)]"F[n(r)]!¹ [n(r)]!º [n(r)] . (38)   &

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The variation of the total energy in the representation of Eq. (37) yields the Kohn—Sham equations which are a set of coupled single-particle equations:






 ! #v (r) t (r)"e t (r), l"1,2, N  J J J 2m

(39)

with



v (r)"v (r)#  

dr



e n(r) #v ([n(r)]; r) ,  "r!r"

(40)

MAA (41) n(r)" t (r)t*(r) . J J J Beside the external potential stemming from the ionic cores, the effective potential v (r) also  incorporates the electron-electron interaction in terms of the Hartree potential and the exchangecorrelation potential v ([n(r)]; r) which is defined as the variation of the exchange-correlation  energy with respect to the density: dE [n] . v ([n(r)]; r)"   dn(r)

(42)

As the eigenfunctions t (r) and the effective potential are related to each other by the single-particle J density given as a sum over the occupied electronic states (Eq. (41)), the Kohn—Sham equations have to be solved self-consistently. 3.1.3. Local-density approximation In their general representation (Eqs. (37) and (38)) the total energy functional and the exchangecorrelation energy are correct. For practical purposes, however, E [n(r)] and v ([n(r)]; r) have to   be given in an explicit form. A great simplification was suggested by Kohn and Sham [203]. For slowly varying charge densities, the exchange-correlation energy can be approximated by





E [n(r)]" dr e [n(r); r]n(r)+ dr e (n(r))n(r) ,   

(43)

where e (n(r)) is equal to the exchange-correlation energy per electron in a homogeneous electron  gas with the (constant) density n"n(r). This is the so-called local density approximation (LDA) which is purely local since e (n(r)) is a function of r (or the charge density at r, respectively). The  determination of the exchange-correlation potential is straight forward: d[e (n(r))n(r)] . v*"([n(r)]; r)"   dn(r)

(44)

Several parametrizations have been suggested for the exchange-correlation energy [33,112,290,403]. All approaches interpolate exact results for a high-density electron gas with those obtained for intermediate and low particle densities. Many total energy calculations employ the representation of e (n(r)) (and v*", respectively) introduced by Perdew and Zunger [290] which   was derived from the results of Ceperley and Alder who performed Monte-Carlo simulations for an

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interacting homogeneous electron gas [63]. All results that are presented in this review were obtained with the parametrization of Perdew and Zunger. Although effects related to nearby inhomogeneities in the charge density are in principle ignored, LDA can be successfully applied to a wide range of systems including molecules and crystal surfaces. From a series of calculated physical properties, limitations of the LDA become obvious. The computed band gap in semiconductors is generally too small [177], the high-frequency dielectric constant usually is overestimated [134]. An improvement of electronic band energies can be achieved by means of the GW approximation which corrects the LDA results in terms of a first-order perturbation scheme [177]. The variation of the charge density in inhomogeneous systems is taken into account by generalized gradient corrections (GGC), however, no systematic improvement was achieved comparing the results obtained from GGC schemes with those from LDA [82]. 3.2. Expansion in plane waves The considerations of Section 3.1 are valid for essentially arbitrary external potentials v (r). In  periodic systems, like the crystalline bulk or a periodic slab configuration (see Section 3.6), the external potential acting on an electron is given by (45) v (r)" v (r!Rl ) , ? ?  l ? where Rl represents the atomic positions (Eq. (2)). Hence, v (r) and consequently the effective ?  potential v (r) (see Eq. (40)) are periodic with respect to the primitive translations of the system.  Therefore, the Kohn—Sham eigenstates t (r) are Bloch states t k(r) which are characterized by J L a respective band index n and wavevector k: 1 kr t k(r)" e  u k(r) . L L (»

(46)

Periodic boundary conditions have to be assumed in the calculation. Hence, the wavevectors k are restricted to a discrete mesh in the Brillouin zone, » denotes the volume which corresponds to the periodic boundary conditions. The lattice periodical part u k(r) has the Fourier L expansion u k(r)" t (k#G) e G  r . L L G

(47)

With this, the Kohn—Sham equations (Eq. (39)) can be written as





 "k#G"dGG #v (G!G) t (k#G)"e k t (k#G) . (48) Y  L L L 2m G Y If the Fourier transforms of the effective potential and the eigenstates converge rapidly enough, the solution of Eq. (48) requires to determine the eigenvectors and eigenvalues of a relatively small matrix.

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The size of the matrix conveniently is determined by the choice of an appropriate cutoff energy E . Only plane waves (1/(») e k>G r with a maximal kinetic energy E are used:  

 "k#G"4E . (49)  2m If valence and core electrons have to be treated, a relative high cutoff energy is required, because of the larger gradient of the core electron charge density. To minimize E and hence the number of  plane waves, norm-conserving pseudopotentials are used. The strong potential of the ionic cores is replaced by a much weaker potential which leads to a fast convergency of the Fourier expansion for the valence electron wave functions and the effective potential. The expansion in plane waves is advantageous under several aspects. In particular, most of the interaction terms in the electronic structure calculation can be represented by compact and simple expressions. Since plane waves are basis functions that are independent on atomic positions (in contrast to localized orbitals), derivatives with respect to atomic displacements do not affect the basis set. Therefore, forces acting on the ions can be easily calculated on the basis of the Hellmann—Feynman theorem [159,114], using only the Kohn—Sham eigenstates. In contrast, basis functions that depend on the atomic positions require to compute the so-called Pulay forces [285,297,321]. As a consequence of the formal simplicity of the first-order terms, plane waves lead also to a compact representation of higher-order derivatives like interatomic force constants. For ionic compounds and for systems with a strong localization of electrons, the expansion in plane waves may require very high cutoff energies. In this case, the use of more appropriate basis functions can reduce the numerical labor to a substantial extent. In the mixed basis approach [45,165] the electronic wave functions are expanded in a mixed representation which uses for the description of the wave functions localized orbitals and in addition plane waves to describe delocalized contributions. Another approach is to use Gaussian orbitals with s, p, d, s* symmetry like suggested in Refs. [207,208,309]. The total energy can also be computed by means of the Harris-Foulkes functional [118,155]. In this method, the accuracy can be significantly increased by using slightly confined pseudo atomic orbitals (PAO) [10,90,314], which simulate the contraction of the atomic charge density as observed in solid state systems [115]. 3.3. Special points The computation of the single-particle density, energies, potentials, and hence the Hamiltonian of Eq. (39) requires to perform integrations over the first Brillouin zone which are carried out by summations over a sufficiently dense grid of wavevectors. In principle, the eigenvectors "t k2 have L to be determined on this grid by solving Eq. (48) for all k-points on the grid of wavevectors in the Brillouin zone. The density of the grid corresponds to the number of unit cells for which periodic boundary conditions are assumed. The solution of Eq. (48) can be very time consuming, especially if a large cutoff energy is used or if the size of the unit cell is large. Special points schemes [30,68,127,251] use suitable grids of wavevectors and the symmetry of the system to minimize the number of diagonalizations needed to solve the Kohn—Sham equations. Symmetry related k-points in the grid are reduced to corresponding points which form a smaller set of wavevectors in the irreducible part of the Brillouin zone.

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Since the individual points in the irreducible wedge can be equivalent to a different number of wavevectors in the complete grid, each of the so-called special points k ,2, k has to be  P characaterized by its weight m (l"1,2, r). With this, the integration of a wavevector dependent J function F(k) over the first Brillouin zone is simplified to the summation over only a few special points:



P dkF(k)+ m F(k ) , (50) J J 8 8 J where X is the k-space volume of the Brillouin zone. Hence, the time consuming calculation of the 8 Kohn—Sham eigenstates can be restricted to the determination of a few electronic wave functions corresponding to a small number of special points. Typically, ten special points are used in the calculation of bulk properties [134], while four to six special points are used in slab-supercell calculations [85,124,121,273]. 1 X

3.4. Pseudopotentials The number of plane waves which are necessary to expand the electronic wave functions and the effective potential in the momentum space representation of the Kohn—Sham equations depends on the character of the ionic potentials and to which extent tightly bound electrons have to be included in the self-consistency procedure. In many cases, the core electrons are essentially inert, not being influenced by the chemical environment. This justifies the incorporation of the core electrons into the potential of the ionic cores. The valence electron wave functions are smooth outside the core region of the ions. Close to the nuclei, however, rapid oscillations appear related to the strong potential of the ionic core and the orthogonality of the valence wave functions to the core states. The main idea of the pseudopotential approach is to replace the strong ionic potential near the nuclei by a much weaker potential. The softening of the potential in the core region is chosen to yield pseudo-wave-functions that are free from oscillations. Outside the core region (r5r ) the pseudo-wave-functions have to be the same as those of the original strong potential.  Different kinds of pseudopotentials have been suggested: The method of orthogonalized plane waves (OPW) [229] modifies strongly attractive ionic potentials by introducing suitable repulsive contributions, using the orthogonality of valence and core states. Based on the OPW method several types of pseudopotentials have been developed. Beside pure model potentials [298], empirical and semi-empirical pseudopotentials have been applied [72,73,324]. With the introduction of norm-conserving pseudopotentials, a new generation of potentials became available that are transferable to many different chemical environments [26,27,143,150,196,199,366,376,377]. For the construction, the electronic ground-state configuration of an isolated atom is determined by solving the Kohn—Sham equations. The all electron calculation yields wave functions t# "R#(r)½ (0, u) and atomic energy levels e#, where n is the LJ LJK LJ JK principal quantum number, while l and m characterize the angular momentum. In contrast to the inner electrons, which are assumed to be inert, the valence electrons are influenced by the type of neighboring atoms and the particular bonding configuration. Pseudopotentials are constructed for the electronic states in the outer shell. Hence, it is not necessary to specify the principal quantum number. The potential acting on the outer electrons has to satisfy the following conditions:

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1. The atomic energy levels have to be reproduced: e.1"e# . (51) J J 2. The pseudo-wave-functions must be nodeless and have to match the all-electron wave function outside a predetermined radius r : J R.1(r)"R#(r) for r4r . (52) J J J 3. Norm conservation The charge integrated in a sphere of radius r5r has to be the same for both, the all-electron J and pseudo-wave-function:



P



dr r"R.1(r)"" J

P

dr r"R#(r)" for r5r . J J

(53)



(54)

  4. ¹ransferability To guarantee the transferability of the pseudopotential to different chemical environments, equality of the following derivatives has to be satisfied:







j j j j ln(R.1(r)) " ln(R#(r)) . J PPJ PPJJ J J je jr je jr CC CC

The radial logarithmic derivative of R is related to the scattering phase shift. Hence, Eq. (54) J ensures that the original potential and the pseudopotential have the same scattering properties and therefore the transferability of the pseudopotential is guaranteed. An identity for radial wave functions (see for example Eq. (1.1) of Ref. [26]), that relates derivatives in the form of Eq. (54) with the charge included in a sphere of radius r, shows that the condition of norm conservation already implies satisfaction of the transferability criterion. Nodeless pseudo-wave-functions in general require a significantly smaller number of basis functions for their expansion in plane waves. Moreover, the radial Schro¨dinger equation





 j

l(l#1) ! # #v.1 rR.1(r)"e rR.1(r) J J J J 2m jr 2mr

(55)

can be easily inverted to determine the screened pseudopotential v.1 as a function of the atomic J eigenenergy e , the potential related to the angular momentum, and the radial part R.1(r) of the J J pseudo-wave-function. This has to be done separately for each angular momentum leading to l-dependent potentials of the form

l(l#1)  1 j v.1 "e ! # (rR.1(r)) . (56) J J J 2mr 2m rR.1(r) jr J To obtain a potential that can be transfered to different chemical environments, the contribution of the valence electron charge density n.1 to the effective Kohn—Sham potential v.1 has to be  J subtracted (unscreening). The resulting ionic potential v.1 "v.1 !v (r ; n.1)!v (n.1) J J &   

(57)

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is dependent on the angular momentum. By introducing the projectors "lm21lm", the full ionic pseudopotential is given by v.1 " "lm2v.1 1lm" .  J JK

(58)

The projector "lm21lm" leads to a non-local real-space representation of v.1 . Since Eq. (53) has to be  satisfied, v.1 is in general a norm-conserving non-local pseudopotential.  Different schemes have been proposed to construct efficient norm-conserving pseudopotentials. The method suggested by Bachelet, Hamann, and Schlu¨ter [26] allows one to generate relativistic pseudopotentials that include spin—orbit coupling. Softer potentials can be constructed with the help of the procedure introduced by Kerker [196]. By extending Kerkers parametrization, Troullier and Martins have developed an efficient method [366] which produces soft pseudopotentials that yield converged physical properties at significantly lower cutoff energies. Ultrasoft pseudopotentials can be obtained, using the formalism suggested by Vanderbilt [377] in which norm-conservation is given up during the generation of the potentials. Norm-conservation, however, has to be reintroduced in any subsequent electronic structure calculation. On the basis of a suitable representation of the non-local part, Kleinman and Bylander have found a way to introduce a separable form of the pseudopotential [143,144,199], which substantially increases the efficiency to compute the external potential matrix elements. This is important for large systems and especially for ab initio molecular dynamics simulations [111,336]. Norm-conserving pseudopotentials are particularly advantageous in plane-wave calculations, because of the much faster convergence of the Fourier sums. They are also widely used in conjunction with other approaches based on localized Gaussian orbitals [207,208,309], the mixed basis formalism [45,165], or a tight-binding like linear combination of atomic orbitals [10,90,314]. 3.5. Other minimization techniques The self-consistent solution of the Kohn—Sham equations (39)—(41) requires to determine the N lowest eigenstates. In conventional approaches, the Hamilton matrix such as defined by Eq. (48) is diagonalized in order to obtain the Kohn—Sham eigenstates. Since the eigenvectors and the effective potential are related to each other, the Kohn—Sham equations have to be solved iteratively. The self-consistency procedure is normally stopped at a certain degree of convergence which is monitored after each step. The accuracy of the self-consistent results has also to be examined in dependence on the number of special points and the size of the basis set. Hence, a large number of diagonalizations is necessary for this kind of computations. For a converged expansion in momentum space, typically 100—200 plane waves are necessary for each atom in the system. Since the numerical labor for conventional matrix diagonalization scales with the third power of the number of basis set states, such calculations are restricted to about 10—20 atoms per unit cell. Moreover, conventional diagonalization yields all occupied and empty eigenstates up to maximal accuracy even in the early stages of the self-consistency procedure, although high precision for computed eigenvectors is not needed far from selfconsistency.

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3.5.1. Car—Parrinello method By treating the electronic and atomic coordinates simultaneously as dynamical variables, the Car—Parrinello scheme provides the possibility to perform ab initio molecular-dynamics simulations [60]. The approach can also be used to solely determine Kohn—Sham eigenstates. In the spirit of the variational principle, trial wave functions have to be improved until they converge to the Kohn—Sham eigenstates. According to the Kohn—Sham formalism, the Car—Parrinello method starts from the following Lagrangean:



1 1 1 (59) ¸" k dr"tQ "# M RQ lY# k aR !E [+t ,, +Rl ,, +a ,] , ? ?  J ? G J 2 G G 2 l 2 4 ? G J where E [+t ,, +Rl ,, +a ,] is the total energy of the system including the Coulomb repulsion of the ? G  J ionic cores and the Kohn—Sham energy (Eq. (37)) given as a functional of single-particle wave functions +t ,, time dependent atomic positions +Rl ,, and the size and shape of the unit cell +a ,. ? G J The parameters k and k are arbitrary mass-like parameters of appropriate units. G The molecular dynamics equations of motion are derived from the Lagrange dynamics of the variables q"(+t ,, +Rl ,, +a ,) under the constraint J ? G



drt*(r, t)t (r, t)"d . J G JG 4 We obtain the following set of equations:

(60)

kt$ (r, t)"!H t (r, t)# K t (r, t) , (61) J  J JG G G jE jE M R$ l "!  and k a( "!  , (62) ? ? G G jRl ja ? G where H is the Kohn—Sham Hamiltonian of Eq. (39). The parameters K are the Lagrange  JG multipliers introduced by the constraint (Eq. (60)). In order to determine the Kohn—Sham eigenstates for a given configuration +Rl ,, +a , only ? G Eq. (61) needs to be solved. The Lagrange multipliers K ensure that the electronic wave functions IG remain orthonormal during the molecular-dynamics simulations. At equilibrium, the condition t$ "0 is reached. In this case, Eq. (61) is identical to a unitary transformation of the Kohn—Sham J equations. In practical applications, the Lagrange multipliers are replaced by the expectation values j "1t "H "t 2 . J J  J With this, the equations of motion have the form

(63)

kt$ "!(H !j )t . (64) J  J J To solve Eq. (64) Car and Parrinello used the Verlet algorithm [378]. The integration of the equations of motion leads to new wave functions. Since the full constraint of orthonormality has been removed, a separate orthonormalization step is necessary after each single time step. To ensure that the wave functions t (r, t) evolve to stationary orbitals, a damping term in form of J

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!ctQ can be added to Eq. (64). Alternatively, the velocities related to t (r, t) can be gradually J J reduced. In order to guarantee that the Car—Parrinello method yields Kohn—Sham eigenstates, the Gram—Schmidt orthogonalization scheme has to be applied [227]. With this, each state will be forced to yield the lowest possible energy under the constraint that it is orthogonal to all states below [285]. 3.5.2. Direct minimization: conjugate gradient method Beside the diagonalization of the Hamilton matrix (Eq. (48)) or the use of Lagrange-like dynamics, the Kohn—Sham equations can also be solved by direct minimization of the total energy functional (Eq. (37)) according to the variational principle. In a more general and more abstract description, direct minimization aims to locate the minimum of a function F(x), where x is a vector in a multidimensional space. It is assumed that the function F(x) has a single minimum. The Kohn—Sham energy functional E[+t ,] depends on the single-particle states +t ,. Hence, the J J functional E corresponds to F and the wave functions +t , take the place of the vector x. J The general situation is to start the minimization at a point x. To find the minimum of F(x), an optimized direction for the move from x has to be determined. The negative gradient



jF u"! jx x

(65)

 represents the direction of the steepest descent of the function F at x"x. It is assumed that the direction of the steepest descent can be determined with the help of a gradient operator G according to x

u"!Gx .

(66)

In the method of steepest descents a scalar b is determined which minimizes the function F(x#bu). This can be achieved by sampling F(x#bu) along the line x#bu for several values of b. If the gradient operator G is known, the minimizing value b is determined by u ) G(x#bu)"0 .

(67)

At the point x"x#bu, the gradient is perpendicular to u which is the gradient at the point x. Since b minimizes the function F only along a particular line of the multidimensional space for x, the steepest descent algorithm has to be repeated starting from the point x. In general, this method requires several iterations to converge to the minimum of F(x). One disadvantage is that each steepest descent vector is orthogonal to that of the previous iteration. Hence, each minimization step is dependent on the previous one. In certain cases, the number of iterations required for convergence is very large. Substantial improvement can be achieved if consecutive minimization steps are carried out by means of conjugate gradients. Instead of using the nth gradient uL for the nth iteration, more suitable directions d L are chosen. The simple example illustrated in Ref. [285] can be generalized to the conjugate-gradient condition (see Ref. [136]) d L ) G ) d K"0 for nOm .

(68)

In the Kohn—Sham formalism, the relevant gradient operator G essentially is the Hamiltonian H as defined by Eqs. (39)—(41). The initial direction of the minimization d is taken as the negative 

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gradient at the starting point, while all other directions are linear combinations of the new and the previous gradient. The search direction of the nth iteration is defined in the following way: d L"uL#cLd L\ ,

(69)

uL ) uL for n52 cL" uL\ ) uL\

(70)

with

and c"0. The advantage of the conjugate-gradients method is that information from all previous minimization steps is used to generate the new search direction, while the steepest-descents scheme only relies on the gradient of each current sampling point. Hence, the conjugate-gradients formalism is a faster method for determining the minimum of a function F of vectors in a multidimensional space. For the minimization of the Kohn—Sham energy functional, suitable conjugate-gradients schemes have been developed by Gillan [137] and Teter and co-workers [360]. The scheme proposed in Ref. [360] compares in computational labor for each single iteration to alternative techniques and requires only minimal memory [285]. This can be achieved by the following adjustments that are introduced to optimize the conjugate-gradients method for electronic-structure calculations: 1. Instead of treating all electronic wave functions simultaneously, each single band can be updated separately. The restriction to optimize only one band at a time reduces the memory requested during the iterative procedure. The steepest descent vector of an electronic band l is given by uL"!(H !jL)tL , J  J J

(71)

with jL"1tL"H "tL 2 . (72) J J  J 2. To avoid modifications in other bands during the optimization of the lth band, the projection of uL onto all other bands is subtracted from uL. This yields steepest descent directions uL which J J J, are orthogonal to all wave functions t (iOl). G 3. As can be seen from Eq. (71), the steepest descent vectors uL and uL are dominated by the J J, coefficients of a plane-wave expansion corresponding to high kinetic energies. This domination can be compensated by introducing a diagonal preconditioning matrix K which is essentially the inverse of the kinetic energy operator for higher kinetic energy components and unity for the components corresponding to low kinetic energies. The preconditioned steepest descent vectors gL"KuL (73) J J, guarantee a similar rate of convergence for all plane-wave coefficients. The preconditioned vector gL is not orthogonal to all of the bands. A second orthogonalization therefore is required, J yielding the orthogonal and preconditioned steepest descent vector gL . J, 4. The vectors gL are used to construct the conjugate gradient directions DtL in analogy with J, J Eqs. (69) and (70). The directions DtL are orthogonal to all other bands but not necessarily to J the wave function tL which has to be updated. J

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5. By using the normalized direction fL J with fL"DtL!1tL"DtL2tL , mL" J J J J J J 1fL"fL2 J J the updated wave function can be represented by

(74)

tL>"tL cos(h)#mL sin(h) . (75) J J J The determination of the optimal value for h is greatly simplified by the approximation  E(h)"E # [A cos(2mh)#B sin(2mh)]+A cos(2h)#B sin(2h) . (76)  K K   K The three parameters of the approximation (E , A , and B ) are determined by computing the    Kohn—Sham energy E(h) at h"0 (which is already known) and for some small value of h, as well as the derivative jE/jh at h"0 which is inexpensive to calculate [285]. The steps (i)—(v) are performed for each single band starting from the lowest band. After updating one band the next higher band has to be processed. Once all bands have been updated, the procedure has to be started from the lowest band again. The conjugate-gradients method is highly efficient and thus allows one to perform plane-wave calculations for very large systems. By means of separable pseudopotentials [143,144,199], excessive memory usage can be avoided so that computations are not limited by the available memory. The efficiency of the conjugate-gradients scheme has been demonstrated by many examples. Recently, S[ tich has applied this method to perform molecular-dynamics simulations for the Si(1 1 1) (7;7) surface using slab-supercells comprising 400 atoms [343,344]. 3.6. Slab-supercell approach In analogy with the computation of surface phonons with model potentials, electronic structure calculations make widely use of slab geometries to describe crystal surfaces. Appropriate basis functions are necessary to expand the electronic states of a crystal film. For a periodic system like the bulk, the expansion in plane waves is a natural choice since the eigenstates are Bloch states "t k2, which are characterized by a wavevector k and a band index n. Crystal surfaces are periodic L only in two dimensions. According to Bloch’s theorem, the wavevectors kM parallel to the surface are good quantum numbers, while wavevector components along the surface normal (z-direction) cannot characterize a quantum state. Based on the suggestion of Alldredge and Kleinman [13], the infinite barrier model considers a single slab embedded in a potential well. Adequate basis functions are products of the eigenstates of a one dimensional well (in the z-direction) and plane waves parallel to the surface. The infinite barrier model has been successfully applied to the calculation of the electronic structure [384] and the phonon dispersion [298] of metal surfaces. By introducing a slab configuration consisting of identical thin crystal films periodically repeated in the direction perpendicular to the surface (see Fig. 5), we retain a system which is periodic parallel and also perpendicular to its surfaces. In this way, the plane-wave method can be extended to the calculation of surface properties [72,73,406]. The number of atomic layers per crystal film

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Fig. 5. Side view of a configuration of thin crystal films periodically repeated in the direction perpendicular to the surface. Typical dimensions chosen for compound and elemental semiconductors are nine to twelve atomic layers per slab (&16 As ) and a separation between neighboring crystal films of four to six interlayer distances (&8 As ).

has to be large enough, to guarantee that the two surfaces bordering one slab are sufficiently decoupled. The vacuum region between adjacent crystal films has to ensure that also the opposing surfaces of neighboring slabs do not interact. Typical dimensions in slab-supercell calculations for semiconductors are nine to twelve atomic layers per slab (&16 As ) and a separation between two crystal films of four to six interlayer distances (&8 As ) [121,273,329]. The lattice constant of the slab supercell is chosen to be consistent with the theoretical value determined for the bulk. The primitive translations of a periodic-slab configuration are a and a , which lie in the plane of   the surface, and a "¸ eL (77)  X X oriented perpendicular to the surface. The width ¸ of the slab-supercell comprises all layers of X a crystal film and the extension of the vacuum region which separates two neighboring films. The primitive translations a and a are chosen in accord with the reconstruction of the surface. In   analogy with calculations for the bulk, the electronic wave functions are Bloch states t k(r) which L can be expanded in plane waves according to 1 t (k#G) e k>G r . t k(r)" L (» G L

(78)

The reciprocal lattice vectors G are linear combinations G"n b #n b #n b (79)       of the primitive translations b (i"1, 2, 3) of the reciprocal lattice and are defined by G 2n(a ;a ) H I b "e (80) G GHIa ) (a ;a )    with the Levi-Civita tensor e . GHI Basis sets other than plane waves can be applied in slab calculations as well. Conveniently used expansions for the electronic states are based on the mixed-basis approach [45,165], localized Gaussian type orbitals [207,208,309], and pseudo-atomic orbitals [10,314]. A main advantage of

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the slab-supercell approach is the direct transferability of computational methods developed for the bulk to the determination of surface properties. From a technical point of view, the use of the plane-wave based momentum-space formalism [179,260] leads to compact and simple expressions for most of the interaction terms in the electronic structure calculation. Particularly, the computation of first and second-order derivatives of the total energy such as atomic forces and dynamical matrices can be performed in an efficient way by means of the Hellmann—Feynman theorem [159,114] and first-order perturbation theory [32,134]. While first-principle calculations of surface phonons are primarily based on the slab method, some other approaches have been suggested to determine the surface electronic structure. We only briefly mention these methods which apply to semi-infinite systems. In the matching method [21,218] wave functions of the surface region are constructed with the constraint that the vacuum tails converge to zero and that the wave functions match bulk states in the inner region. The scattering-theoretical method uses the Green’s function method [188,206,207,293]. 3.7. Relaxation and reconstruction Because of the covalent nature, semiconductors are characterized by directed bonds which connect nearest neighbors in the bulk. The creation of a surface leads to the appearance of broken bonds. Substantial changes in the atomic equilibrium positions occur for surface near atoms. Two major principles are responsible for the observed relaxations and reconstructions: chemical bonding and autocompensation [96]. Processes like rebonding, the formation of new bonds, rehybridization, and charge transfer between dangling bonds reduce the number of dangling bonds and minimize the energy in the electronic system [341]. Within density-functional calculations, atomic rearrangements and changes in the electronic structure can be treated in a consistent way. The total energy of the system E , which is given by  the sum of the Coulomb repulsion of the ionic cores and the Kohn—Sham energy, is not minimal as long as the atomic positions Rl are different from the equilibrium positions Rl . The negative of the ? ? first-order derivative of E with respect to the position of an atom (Eq. (62)) corresponds to the  force acting on this atom. A local minimum of the total energy can be found by moving the atoms along the forces until all forces are essentially zero. The computation is largely simplified by means of the Hellmann—Feynman theorem [159,114] which is discussed in more detail in Section 4.3.1. In this approach, the calculated forces are physically correct only if the wave functions of the Kohn—Sham formalism are eigenstates. Hence, accurate forces can be obtained in iterative self-consistency schemes only if the wave functions are sufficiently converged to the eigenstates. This is of particular importance for molecular-dynamics simulations since the trajectories sensitively depend on the forces. For the structure optimization of an atomic configuration, small errors are acceptable during the initial steps, while the forces have to be determined with higher accuracy close to the minimum. Additional terms appear in the derivative of the total energy if the basis functions depend on the atomic positions. Such additional contributions are called Pulay forces [297]. They are exactly zero if plane waves are used. Other basis functions, however, generally require a careful treatment of the additional terms. Various methods have been applied to determine atomic zero-force configurations. The traditional diagonalization of the Kohn—Sham Hamiltonian and the conjugate-gradients scheme usually yield forces that are, to a high degree of precision, the exact derivatives of the total energy E . 

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Under such conditions, E can be minimized with respect to the atomic positions by means of  the conjugate-gradients technique in analogy with the solution of the Kohn—Sham equations. The Hellmann—Feynman forces correspond to the steepest-descent vectors. As an example, Umerski and Srivastava used conjugate gradients to determine the relaxation of the (1 1 0) surfaces of GaAs and InP [373]. Convergence in the structure optimization was achieved within 8—12 iterations. Another approach to minimize E as a function of the atomic positions with similar efficiency is  provided by the modified Broyden scheme [375] which is based on the Newton—Raphson procedure to find the zero-force configuration. This method has been applied to determine the relaxation of various semiconductor surfaces, such as the (1 1 0) surfaces of GaAs [121,309], SiC [309], and other III—V compounds, and the Si(0 0 1) surface [124]. Another method to determine atomic equilibrium positions is to integrate the equations of motion by means of molecular-dynamics simulations. The incorporation of damping into the dynamics gradually reduces the kinetic energy of the ions which move about their equilibrium positions with steadily decreasing displacements from the zero-force positions [285]. This approach has been successfully applied in tight-binding like ab initio molecular-dynamics simulations for complex systems such as the Si(1 1 1) (5;5) surface [1], clusters, fullerenes, and amorphous silicon (see references in [1]). By combining the conjugate-gradients technique for minimizing the electronic energy with a Car—Parrinello like scheme for relaxing the ionic degrees of freedom [342], the molecular-dynamics formalism could be applied to the geometry optimization of the higherorder reconstructions of Si(0 0 1) [299]. In the Car—Parrinello approach, the equations of motion associated with the electronic degrees of freedom are integrated at the same time as the equations of motion for the dynamics of the ions. For physically correct forces, the total energy must be minimized up to sufficient convergence. Since high accuracy is not necessary in the early steps of the relaxation, convergence of the calculated forces is approximately achieved by the use of different time steps for the two systems. By chosing sufficiently smaller time steps for the ionic system, the electronic configuration relaxes close enough to the ground-state for each ionic arrangement. Damping has to be introduced to gradually reduce the kinetic energy of the ions. To increase the accuracy near the atomic equilibrium positions, the time steps for the ionic system have to be progressively reduced. Many plane-wave calculations have applied the original or a modified Car—Parrinello-like scheme to determine atomic equilibrium positions for low index semiconductor surfaces. Examples are GaAs(1 1 0) [111], Si(0 0 1) [336], Si(1 1 1) (2;1) [18], and antimony-covered III—V(1 1 0) surfaces [330]. The minimization of the total energy by means of the Hellmann—Feynman forces in general converges to the local minimum which is closest to the initial configuration. No scheme can guarantee to locate the global energy minimum. Already the choice of a particular slab-supercell geometry can impose constraints that prevent the atoms from relaxing into the ground-state positions. As an example, the ab initio molecular-dynamics simulations of Shkrebtii and coworkers have shown that the Si(0 0 1) surface relaxes into higher-order reconstructions, while the (2;1) arrangement assumed in many other computations does not correspond to a local minimum [336]. To find the configuration of lowest energy, a series of calculations has to be carried out for sufficiently large unit cells, starting the relaxation from various initial positions. By comparing the total energies computed for different atomic arrangements, the most stable configuration can be identified. It is important to note that the energy differences can depend on the accuracy of the calculation. This has been demonstrated by Ramstad and co-workers for the Si(0 0 1) surface [299].

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Simulated annealing [198] represents an alternative way to search for a low energy minimum. In this approach, the system can escape from local energy minima, provided that small enough energy barriers exist. This increases the probability to find the global minimum. However, simulated annealing also cannot guarantee convergence to the lowest-energy configuration.

4. Density-functional phonon calculations In this section we illustrate three density-functional methods that are widely applied to compute semiconductor surface vibrations. We start with a brief review of ab initio molecular-dynamics simulations, which yield accurate trajectories that reflect the finite-temperature dynamics of particles moving about their equilibrium positions. Forces have to be computed with a sufficiently high degree of precision, in order to obtain realistic dynamics from the solution of the equations of motion. Particularly in the Car—Parrinello scheme, caution needs to be exercised during molecular-dynamics simulations. We also discuss the frozen-phonon approach which is another method solely based on the determination of Hellmann—Feynman forces. Interatomic or interplanar force constants are extracted from first-order derivatives of the total energy computed for off-equilibrium atomic positions. Finally, we focus on density-functional perturbation theory (DFPT) which combines density-functional calculations with linear-response theory. The efficient formalism suggested by Baroni and co-workers [32,134] allows one to calculate dynamical matrices at arbitrary wavevectors, using numerical labor that is comparable to that necessary for electronic structure calculations for the ground-state. 4.1. Ab initio molecular-dynamics simulations By means of molecular-dynamics simulations the finite-temperature dynamics of particles which move about their equilibrium can be investigated. For low enough temperatures, the harmonic approximation can be applied to describe the computed trajectory as a superposition of harmonic vibrations. The identification of eigenfrequencies and eigenvectors relies on spectral analytical methods that have been discussed in Section 2.2. To obtain accurate trajectories and therefore correct frequencies and eigenvectors, the calculated Hellmann-Feynman forces have to be essentially the exact derivatives of the total energy. The two main approaches which are applied in ab initio molecular-dynamics simulations are the Car—Parrinello scheme [60,111,336], and the conjugate-gradients method [285,343,344]. In the Car—Parrinello approach, equations of motions are employed to describe the dynamics of the ions and also used to deal with the electronic degrees of freedom. Since both systems are treated simultaneously, the electronic ground-state is not exactly reached. To ensure that the electrons are close enough to the ground-state, the time steps chosen for the ionic motion have to be smaller than those for the electrons. Two important requirements are essential for a successful simulation: Errors in the computed trajectories have to be small and must not accumulate with progress in time. The Car—Parrinello method can fulfill both of these criteria. Inaccuracies occurring during a simulation remain bounded for all times. This is related to a compensating behavior which is characteristic for Car—Parrinello molecular-dynamics simulations. Errors are corrected in an oscillatory manner. This prevents successive amplification of inaccuracies. A detailed description of

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the mechanism is given in Ref. [285]. The error cancellation is fundamentally related to the fact that the electrons are treated on the basis of second order equations of motion. This leads to an oscillation of the electrons about their ground-state distribution which has to follow the ionic motion. An important prerequisite for an effective compensation of errors is that the time scales in the electronic system are much smaller compared to those chosen for the ion dynamics. Finally, it is crucial to control the delicate separation between the ionic and electronic dynamics during the simulation [285]. Because of the fictitious mass of the electrons, kinetic energy is associated with the electronic system. Hence, the sum of the kinetic energy of the ions and the potential energy is not exactly constant in simulations based on the Car—Parrinello scheme. A conceptionally different approach is provided by conjugate-gradients dynamics. Instead of integrating simultaneously the equations of motion of the electrons and ions, the electronic wave functions are thoroughly converged to the ground-state for each given atomic configuration by means of the minimization procedure described in Section 3.5.2. Hence, the Hellmann—Feynman forces are in general close enough to the physically correct forces in conjugate-gradients dynamics simulations. The evolution of the electronic system with time can be incorporated to increase the efficiency. Arias and co-workers have proposed to extrapolate the wave functions for the new positions from the data available from the current and the previous time step [22]. By this, the number of iterations needed to reach converged wave functions for the new atomic configuration can be effectively reduced. As an example, this method has been successfully applied by S[ tich and co-workers to study the dynamics of the Si(1 1 1) (7;7) surface [343,344]. 4.2. Frozen-phonon approaches Dynamical matrices and phonon-dispersion curves can be determined on a level of complexity that is comparable to that of ground-state calculations and molecular-dynamics simulations. This is achieved by calculating energy differences and forces acting on the atoms for selected atomic displacements. In conjunction with density-functional theory, the frozen-phonon approach was originally introduced to compute vibrational frequencies for eigenmodes known by symmetry [397,398]. By chosing suitable atomic displacements, the method can be modified to determine interplanar force constants [211]. Frozen-phonon-like techniques have been extended to the determination of surface vibrations only for some systems. While molecular-dynamics simulations can be started in principle from arbitrary atomic positions not too far from the equilibrium configuration, the atomic ground-state positions usually serve as reference configuration in frozen-phonon schemes. To determine interatomic force constants, the total energy and Hellmann—Feynman forces are calculated as a function of atomic displacements from the equilibrium positions. By fitting the energies and forces to quadratic equations in the distortion, the harmonic contributions can be extracted. If only energy differences are used, the number of necessary distortion geometries roughly scales with N for a system with N atoms, while the number scales with N if also the forces are computed to determine dynamical matrices. The frozen-phonon method is greatly simplified if the eigenvector patterns of the lattice vibrations are known by symmetry [398]. In this case, effective force constants need to be determined only for phonon polarization vectors according to the normal coordinates. Compared to bulk vibrations, the determination of slab-phonon modes in general requires substantially more numerical labor, because of the large number of atoms per unit cell and the small number of

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symmetry operations which can be used to decompose the coupling constants into their subspaces. Hellmann—Feynman forces are usually determined for configurations of periodically repeated slabs. Since neighboring crystal films do not interact, we may restrict the following discussion to a single slab. We consider a static and simultaneous displacement of all atoms with index a by the amount 1 ul " v e qN  RM lY , (81) ?YH (M ?YH ?Y where lM  indicates the lM th unit cell, j represents the Cartesian coordinate direction, and qN is the wavevector in the surface Brillouin zone. All other atoms of the slab are in their equilibrium positions. The harmonic force acting on the ath atom in the 0th unit cell along the ith direction is (82) F "! (M D (aa, qN )v ?YH ?G ? GH ?YH according to Eqs. (11) and (14). By chosing appropriate displacement patterns, the dynamical matrix D (aa, qN ) can be determined for various wavevectors of the surface Brillouin zone. The GH unit cell used to calculate F and therefore D (aa, qN ) has to match the wavevector of the ?G GH lattice vibration that is considered. If only zone center modes are investigated, it is not necessary to increase the size of the unit cell compared to a ground-state calculation. To describe zone-boundary modes within the frozen-phonon method, however, supercells which are twice as large in the direction of the wavevector have to be used since the displacements are given by 1 v (83) ul "$ Y?YH (M ?YH ?Y along the direction of propagation. With the help of symmetry, the number of total energy calculations which have to be performed to determine Hellmann—Feynman forces can be restricted to a smaller number of atoms and to a smaller number of directions for which the atoms have to be displaced. If Hellmann—Feynman forces are calculated for more than only the necessary displacement patterns, symmetry relations can be used to increase the accuracy of the computed force constants. By going beyond quadratic terms, higher-order force constants can be determined to investigate anharmonic effects. The frozen phonon approach has been widely employed to study the vibrations of adsorbates on semiconductor surfaces like antimony on the (1 1 0) surfaces of III—V compounds [327,330] as well as hydrogen on GaAs(1 1 0) [43], C(0 0 1) [10], Si(1 1 1) [225], and Ge(1 1 1) [192]. Clean surfaces have been systematically investigated by Schmidt and co-workers [329] who performed calculations for the zone center and X point modes of the (1 1 0) surfaces of GaAs, GaP, InAs, and InP. All of these examples were restricted to the determination of harmonic vibrations. Li and Vanderbilt [225] have extended the frozen-phonon approach by fitting energies and forces to a fourth-order potential energy surface to study phonon—phonon interactions and two-phonon bound states for Si(1 1 1) : H.

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4.3. Density-functional perturbation theory As shown below by a simple application of the Hellmann—Feynman theorem, it is possible to calculate the change of the total energy up to the second order (even up to the third order according to the 2n#1 theorem [142]), provided that the linear variation of the electronic charge density caused by a static lattice distortion is known. Because of this, perturbative methods can be applied to the computation of harmonic force constants that combine electronic structure calculations with the linear-response formalism. In conjunction with density-functional theory, several approaches have been suggested [32,106,298]. The main advantage of all perturbative schemes over direct methods like molecular-dynamics simulations or frozen-phonon calculations is that dynamical matrices can be determined for (in principle) arbitrary wavevectors without the need to increase the size of the unit cell. In direct methods the size of the unit cell has to match with the wavelength of the phonon mode. 4.3.1. Hellmann—Feynman theorem We consider a system of interacting electrons which move in the field of an external potential v that is dependent on parameters +j , (i"1, 2,2, p) such as the atomic positions or an external  G electric field. In the adiabatic approximation these parameters are static quantities. Each set of parameters defines one external potential v "v (r; +j ,). The Kohn—Sham energy is given as   G a functional of the single-particle wave functions and the parameters +j ,: G (84) E[n(r)]"E[+t ,, +t*,, +j ,] . J G J The wave functions t are also dependent on the parameters +j ,. This is important for the J G calculation of the first-order derivatives of the Kohn—Sham functional which are given by jE jt jE jt* dE jE J . J# " # (85) jt jj jt* jj dj jj J H J H H H J J The derivatives of the energy functional with respect to the single-particle wave functions yield the Kohn—Sham Hamiltonian H so that Eq. (85) can also be written in the form  jt jt dE jE J # JH t . " # t H (86) J  jj jj  J dj jj H H H H J J If all wave functions are eigenstates according to H t "e t , the second and third part on the right J J J hand side of Eq. (86) cancel each other, because the eigenstates are normalized to 1t "t 2"d . J G JG Hence, in the case of well-converged Kohn—Sham eigenstates, the derivative of the total energy with respect to j is equal to the corresponding partial derivative. It can be represented by H dE jE jv (r; +j ,) G dr , " " n(r; +j ,)  (87) G dj jj jj H H H where n(r; +j ,) is the electronic ground-state density computed for a fixed set of parameters. G With Eq. (87) the second-order derivatives of the Kohn—Sham energy are simply given by

     







jn(r;+j ,) jv (r;+j ,) jE jv (r; +j ,) G  G dr .  G dr# " n(r; +j ,) G jj jj jj jj jj jj I H H I H I

(88)

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This equation describes the electronic contribution to the harmonic force constants for the case that the parameters j represent displacements of the atoms from their equilibrium positions. G Consequently, lattice vibrations can be determined in a formalism that requires to compute only first-order variations of the charge density. This is achieved by means of linear-response methods as already proposed in Refs. [292] and [334]. The 2n#1 theorem [142] generalizes the results obtained from the Hellmann—Feynman theorem. Particularly, it is possible to calculate anharmonic coupling constants and other quantities defined as third-order derivatives of the total energy in the framework of first-order perturbation theory [89]. 4.3.2. Dielectric approach Let us consider a static perturbation Dv (r) associated with a phonon or displacements of  atoms. The resulting first-order variation of the electron charge density Dn(r) can be computed by means of the density-response function s(r, r). In combination with density-functional theory, this approach has been suggested by Eguiluz and co-workers [106] and Quong and co-workers [298] for determining phonon dispersion curves of simple metal surfaces. Using the local-density approximation to incorporate screening effects due to the electron-electron interaction, the linear response of the electrons has the form



Dn(r)" dr s (r, r)Dv (r) , *" 

(89)

where the response function s (r, r) is related to that of noninteracting electrons s (r, r) by *"  s

 

(r, r)"s (r, r)# dr dr s (r, r ) v (r , r ) s (r , r) .     R   *"  *" 

(90)

The interaction operator v (r , r ) comprises the Coulomb potential and exchange correlation R   effects in the local-density approximation:





jv (r ) e #   d(r !r ) . (91) v (r , r )"  R   jn(r )   "r !r "  L   The derivative of the exchange-correlation potential has to be computed for the electronic ground-state density n(r )"n (r ). For periodic systems, the Fourier transform of s can be     represented by a matrix as introduced by Adler and Wiser [3,389]: 4 1t k"e\Gq>G r"t k q 2 1t k q "eGq>GY r"t k2 A >  T A >  T . s (q#G, q#G )"! (92)  » k E k q !E k T A >  AT The summation in Eq. (92) has to be performed for all valence band states "t k2 and a sufficiently T large number of conduction band states "t k q 2. The energy eigenvalues of the valence and A >  conduction band states are E k and E k q . For the calculation of the surface phonon dispersion of T A >  Al(1 1 0) [106], Eguiluz and co-workers have used the jellium model [105] to determine the response-function for a thin metallic crystal film. This approximation is valid for weak pseudopotentials and therefore justified for metals like aluminum. It simplifies the computation of the response function s and hence the solution of the integral equation (90), which defines the  response function of interacting electrons.

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Fig. 6. Schematic representation of the density-functional perturbation formalism used to calculate self-consistently the first-order variation of the charge density Dn in response to the screened perturbing potential Dv "Dv #Dv .   

In semiconductors, the discrete nature of the crystal lattice has to be taken into account. Particularly, effects related to the formation of new bonds and the existence of dangling bonds are not included in the jellium model. A rigorous treatment of the ionic background potential has been realized by Quong and co-workers [298] who used pseudopotentials of the Heine-Abarenkov form to compute s . From that, the response function s was determined using electronic eigenstates  *" and eigenenergies calculated within the single slab model. The calculation of the density-response function requires summations over conduction bands which can be very time consuming. In context with new energy minimizing techniques, which are efficient methods for determining occupied electronic states, it is desirable to treat the electronic response in a formalism, in which summations over conduction band states are entirely avoided. 4.3.3. Self-consistent linear response The density-functional perturbation formalism introduced by Baroni and co-workers [32,134] combines the virtues of the frozen-phonon method with those of the dielectric approach. An independent derivation of the perturbation formalism has been given by Zein [400]. The method is based on the fact that the linear response of the electrons Dn to an external perturbation Dv is related to the screened effective perturbing potential Dv "Dv #Dv     by the independent-electron polarizability s . As the exact screening contribution of the electrons  is not known at the beginning, the procedure shown in Fig. 6 has to be repeated until selfconsistency is achieved between the first-order variation of the charge density Dn and the screened effective potential Dv . We restrict the following considerations to periodic systems  such as the bulk of a crystal solid or surfaces which are described by means of the periodicslab approach. In the momentum space formalism the electronic response to the screened perturbation Dv is represented by  1t k"e\Gq>G r"t k q 21t k q "Dv (r)"t k2 4 A >   T . T A >  Dn(q#G)"! E k q !E k » k T A >  AT

(93)

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To derive Eq. (93) it has been assumed that Dv is a potential with local real-space representation.  Eq. (93) can be directly extended to non-local potentials such as introduced by the use of norm-conserving pseudopotentials. With the help of the single-particle Green’s function 1 G(E)" E!H

(94)  and the projector P over the conduction manifold A (95) P " "t k21t k""1!P "1! "t k21t k" , A A T T T A k k T A it is possible to compute the first-order variation of the electronic charge density exclusively on the basis of ground-state properties. Hence, sums that include conduction band states and energies can be avoided. With the Kohn—Sham eigenstates, the single-particle Green’s function has the representation 1 1 G(E)" " "t k2 1t " . L E!E k Lk E!H k  L L With this, the variation of the charge density is given by 4 Dn(q#G)" 1t k"e\Gq>G rP G(E k)P Dv "t k2 . A T A  T » k T T With the solution "DtTk q2 of the Sternheimer equation > (H !E k)"DtTk q2"P Dv "t k2 , > A  T  T the variation of the charge density can be written as

(96)

(97)

(98)

4 (99) Dn(q#G)"! 1t k"e\Gq>G rP "DtTk q2 . > A » k T T The screened potential comprises the bare perturbing potential Dv and the screening contribu tion Dv resulting from the electronic redistribution  Dn(r)e dv . (100) Dv (r)" dr #Dn(r)   "r!r" dn LLr Beside the variation of the Coulomb potential, modifications of the exchange-correlation contribution are incorporated. Using Eqs. (98)—(100), the procedure illustrated in Fig. 6 is solved iteratively by means of the following system of coupled equations:



 

4 Dn(q#G)"! 1t k"e\Gq>G rP "DtTk q2 , > A » k T T (H !E k)"DtTk q2"P Dv "t k2 , A  T >  T Dv (r, r)"Dv (r, r)#Dv (r)d(r!r) ,    Dn(r)e dv . #Dn(r)  Dv (r)" dr  "r!r" dn r  LL  





(101a) (101b) (101c) (101d)

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Fig. 7. Flow diagram for the iterative solution of the system of equations (101a)—(101d) to determine n(q#G).

The flow diagram shown in Fig. 7 illustrates the iterative procedure which is performed to compute self-consistently the variation of the charge density. In the initial iteration, Eq. (101b) is solved for the external perturbation (Dv "Dv ). The electronic response given by "DtTk q2 and >   Dn(q#G) results in the screening contribution Dv described by Eqs. (101c) and (101d), which  enters into Eq. (101b) in all further iterations. The procedure is repeated until self-consistency is reached between the change of the charge density and the screening contribution to the perturbing potential. Since the valence band states t k are determined exclusively for wavevectors belonging to a set of T special points k ,2, k , the variation of the charge density can be computed only for a set of  P wavevectors q3+q ,2, q , which is compatible to the special points k ,2, k . More precisely,  N  P a symmetry operation ¹ of the lattice, a special point k , and a reciprocal lattice vector G must exist H for each pair k and q so that G J k #q "$¹(k #G) (102) G J H can be fulfilled. The denser is the grid of special points is, the more wavevectors are in the grid q ,2, q . This is important for the computation of dynamical matrices and real-space force  N constants. 4.3.4. Calculation of dynamical matrices Let us consider a periodic system such as the bulk or a repeated slab configuration. With the help of the Hellmann—Feynman theorem, the electronic contribution to the force constant matrix U (la, la) is given according to Eq. (88) by GH jv jt k jv  t k #2 T  t k #cc , (103) U (la, la)"2 t k T jul jul T T GH jul jul k k ?G Y?YH ?G Y?YH T T where ul denotes the displacement of the ath atom in the lth unit cell in the ith Cartesian direction. ?G Spin degeneracy of the Kohn—Sham eigenstates "t k2 is included by the prefactor 2. T The linear-response formalism greatly simplifies the computation of U (la, la) which is only GH dependent on the Kohn—Sham eigenstates t k and their first-order derivatives "jt k/jul 2. To ?G T T determine dynamical matrices, only periodic displacements characterized by a wavevector q need to be considered:

 

ul "u (q) eiq  Rl . ? ?

 

   

(104)

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The derivatives obey the following relation: j

j " eiq  Rl . (105) ju (q) jul l ?G ?G The contributions to the dynamical matrix which result from Eq. (103) are denoted by D(aa, q) GH and D(aa, q). With the remaining contribution D(aa, q) stemming from the Coulomb interacGH GH tion of the ionic point charges, the dynamical matrix is given by the following sum: D (aa, q)"D(aa, q)#D(aa, q)#D(aa, q) . (106) GH GH GH GH For D(aa, q), which represents the part of the dynamical matrix related to the second-order GH derivatives of the external potential, only contributions with l"l and a"a do not vanish. This simplifies D(aa, q) to GH 2d jv  D(aa, q)" ??Y t k t , (107) GH T ju (q"0)ju (q"0) Tk NM k ? T ?G ?H where N is the number of unit cells corresponding to periodic boundary conditions. Using time-reversal symmetry for the Bloch states, the second part of the dynamical matrix D(aa, q) has the form GH 1 jt k q Rl jv 4 T e   e q  R l Y t k . (108) D(aa, q)" T GH jul N (M M k ll jul ?G Y?YH ? ?Y T Y With Eq. (105), we obtain the following expression:

 

 









 

 

4 1 jt k jv T  t k . D(aa, q)" (109) GH N (M M k ju (q) ju (q) T ?YH ?G T ? ?Y Here, " wR?GTkq 2 is the self-consistent solution of the Sternheimer equation wS   jt k jv T "P  "t 2 , (110) (H !E k) Aju (q) Tk  T ju (q) ?G ?G which is solved with the help of the procedure shown in Fig. 7 by replacing the first-order variation of the charge density and the screening contribution by the respective derivatives. The remaining part of the dynamical matrix D(aa, q) results from the ion—ion interaction. In GH the momentum-space formalism, it can be computed by means of Ewald summation techniques [134,235]:





4ne e\q>GE Z Z e q>GR?\R?Y(q #G )(q #G ) D(aa, q)" ? ?Y G G H H GH (q#G) X(M M G ? ?Yq G > $ e\GE 2ne 1 Z Z e G R?\R?YYG G #cc d , (111) ! ? ?YY G H ??Y G X M G ? $ ?YY where X denotes the volume of the unit cell. Divergent terms of the Fourier expansion corresponding to G"0 and q#G"0 are not included in the summation.





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4.3.5. Polar semiconductors In polar materials, the long-range character of the Coulomb interaction causes macroscopic electric fields generated by longitudinal optical displacements of the ions. In the limit qP0, the influence of such macroscopic electric fields has to be incorporated with care. While polar and nonpolar materials can be treated in exactly the same way by means of Eqs. (101a), (101b), (101c) and (101d) for all wavevectors qO0, longitudinal optical zone center phonons give rise to an electric potential of the form Dv "eE ) r that is part of the screened perturbing potential. The  effects of the macroscopic field E have to be investigated separately. This can be achieved in the following way: For qP0, the increase of the total energy per unit cell De, which is associated with a lattice distortion u (qP0) and the related macroscopic electric field E, is given by ? X 1 (112) De" (M M u* (q)D (aa, q)u (q)! euR(q)(Z )RE(q)! ER(q)e E(q) , ? ?Y ?G ? GH ?YH ?  8n 2 ?G ?YH ? where (Z ) is the Born effective charge tensor of the ath atom and e is the high-frequency static ?  dielectric tensor. The second term on the rhs of Eq. (112) describes the energy of a static dipole (Z )u (q) in the electric field. The analytical part of the dynamical matrix D (aa, q) and the Born ? ? GH effective charges are computed for zero macroscopic electric field boundary conditions. The third term on the rhs of Eq. (112) covers the remaining contributions to the increase of the total energy stemming from the electric field and the electronic polarization. The analytical part of the dynamical matrix D (aa, qP0) is computed according to Eqs. (107), GH (109) and (111) by solving Eqs. (101a), (101b), (101c) and (101d) with the help of Eq. (110), setting the contribution from the macroscopic electric field to the effective perturbing potential to zero. In nonpolar materials like elemental semiconductors, no macroscopic electric field is associated with lattice displacements. Therefore, the dynamical matrix and its analytical part are identical. For compound semiconductors, however, the effects of macroscopic electric fields have to be taken into account in form of an additional contribution D (aa, qP0): GH D (aa, qP0)"D (aa, qP0)#D (aa, qP0) . (113) GH GH GH The nonanalytical part of the dynamical matrix has the general form 4ne (Z )RqL qL R(Z ) ? ?Y , (114) D (aa, qL )" qL Re qL XM M  ? ?Y which is given as a function of the direction of the wavevector qL , the high-frequency static dielectric constant, and the Born effective charges. To determine the effective charges (Z ) and the dielectric constant e , matrix elements of the form  ? 1t k"r"t k2 have to be computed. Such matrix elements are not well-defined for the case that T A Born—von Karman boundary conditions are assumed. This problem can be solved by using the relation 1t k"[H , r]"t k2  A , 1t k"r"t k2" T T A E k!E k A T

(115)

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which is an identity for finite systems [134]. With the auxiliary functions 1t k"[H , r ]"t k2  G T "!P G(E k)P [H , r ]"t k2 , : "t k2 A "UG k2 " A A T A  G T T E k!E k A T A the effective charges and the dielectric constant are given by

(116)

     

jtTk 16ne UG k , (117) (e ) "d # T jE  GH GH » k H T jtTk 4 , (118) (Z ) "Z d # UG k ? GH ? GH N k T ju ?H T where "jtTk/jE 2 describes the electronic response to an applied homogeneous screened electric field H and "jtTk/ju 2 is the response to a lattice distortion computed for zero electric field conditions ?H [134]. The charge of the ath ion is denoted by Z . ? This formalism has been developed to compute the nonanalytical part of the dynamical matrix for the bulk of solids [134]. In the case of periodically repeated thin crystal films, the electric field and the dielectric tensor vary in the direction normal to the planes of the slab. However, "jtTk/jE 2 H is computed according to Eqs. (101a), (101b), (101c) and (101d) under the boundary condition that the screened electric field is constant throughout the system. The electronic response is different near the surfaces of the slab compared to the inner region. Therefore, the self-consistent solution for e yields an effective dielectric constant which represents an average of the  bulk, surface, and vacuum values. Because of this, the LO—TO splitting for bulk phonon modes is described properly also for periodically repeated thin crystal films, without modifying the formalism. Moreover, microscopic surface phonon modes are not affected by the nonanalytical part of the dynamical matrix, provided that the crystal film comprises a sufficiently large number of atomic layers. As can be seen from Eq. (114), the increase in energy due to D (aa, qP0), GH which is given by 1 (119) DeL?" (M M u* (qL )D (aa, qL )u (qL ) , GH ?YH ? ?Y ?G 2 ?G ?YH will scale with the inverse of the size of the unit cell (and hence the number of layers), if the displacements are restricted to only a few planes in the surface region. Therefore, DeL? is essentially zero for localized phonon modes in large crystal films. 4.4. Comparison of the methods Molecular-dynamics simulations, frozen-phonon techniques, and the linear-response formalism now are well-established methods that are widely used in the framework of density-functional calculations in order to determine the phonon modes of the bulk and the surfaces of semiconductors. In the frozen-phonon scheme, dynamical matrices of the system are determined by displacing the atoms from their equilibrium positions and calculating the resulting Hellmann—Feynman forces on all atoms of the slab. By fitting the forces to a quadratic equation in the distortion, the harmonic contributions can be extracted [329]. For small displacements (&0.05 A> ), anharmonic

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effects are negligible, although the frozen-phonon method provides also the possibility to extract higher-order force constants [225]. For the investigation of phonon modes not only at the zone center, it is necessary to use larger unit cells for a proper description of the periodicity according to the wavevector. This is the main disadvantage of the frozen-phonon approach and also of molecular-dynamics simulations in contrast with the linear-response formalism presented in the previous section. Density-functional perturbation schemes, like the dielectric approach or the self-consistent linear-response formalism, allow one to calculate dynamical matrices on a fine grid of wavevectors in the Brillouin zone by using the same unit cell as in the ground-state calculations. Therefore, it is possible to determine complete phonon dispersion curves [121,122] with acceptable numerical labor. In the molecular-dynamics approach, phonon frequencies and eigenvectors are found by analyzing particle trajectories obtained in simulations covering a sufficiently large period of time. The frequency spectra can be determined from the Fourier transform of the velocity auto-correlation function [315]. Moreover, by combining the multiple signal-classification method (MUSIC) [267] with the self-consistent algorithm proposed in Ref. [201], one obtains an efficient procedure to extract not only phonon frequencies but also phonon eigenvectors from the trajectories [111,336]. An advantage of surface phonon calculations based on the determination of force constants above molecular-dynamics simulations is the usually smaller number of self-consistent calculations that are necessary. While the computation of a dynamical matrix typically requires to consider a number of displacement geometries which is proportional to the number of atoms in the unit cell, molecular-dynamics simulations generally have to cover over 1000 time steps, in order to resolve also low-frequency vibrational modes. For very large systems like the Si(111) (7;7) surface, however, the simulation of particle trajectories becomes comparable in numerical labor. Nevertheless, for a precise determination of low-frequency modes, the simulations have to be carried out for a very large number of time steps. All commonly used approaches are based on the slab method. Phonon modes of the bulk and the surface are obtained in the same step of the calculation. Most of the eigenvectors and eigenfrequencies correspond to bulk subbands. The determination of the surface phonon dispersion therefore requires to analyze the eigenvectors of a crystal film with respect to the localization. While bulk derived eigenvectors show significant atomic displacements throughout the crystal film, surface phonon modes are identified because of the restriction of atomic displacements predominantly to the outermost layers of a slab. Beside strongly localized true surface phonon modes, resonant states and macroscopic features characterized by large penetration depths can be observed. The most prominent macroscopic surface vibrations are the Rayleigh wave [300] and the Fuchs—Kliewer phonon [128]. Near the CM point, the dispersion of macroscopic surface phonon modes is strongly affected by the finite size of the crystal films. This reflects the small decrease of the vibrational amplitudes which decay exponentially with a penetration depth roughly proportional to the wavelength. Within force-constant methods, phonon modes can be investigated for much larger crystal films. Interatomic coupling parameters computed for the bulk and the surface can be used in the framework of the Green’s function approach or to model dynamical matrices of very large crystal films [121,122] as discussed in Section 2.1.2. The analysis of surface phonon modes within molecular-dynamics simulations, however, is generally restricted to only comparably thin crystal films [111,336,343].

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5. Phonon dynamics of III—V(1 1 0) semiconductor surfaces The (1 1 0) surfaces of III—V compound semiconductors have been intensively studied by many different approaches using various experimental and theoretical techniques. It is now well established, that the (1 1 0) surfaces of these materials are characterized by an inward relaxation of the surface cations, while the first-layer anions are shifted above the surface [17,223]. In their equilibrium positions, the top-layer anions of the III—V (1 1 0) surfaces are situated in a pyramidal configuration, while the first-layer cations are bonded to their nearest neighbors in a nearly planar sp-like configuration. This has a strong influence on the force constants at the surface. Localized modes that are closely related to the surface relaxation are expected to appear. Surface phonon dispersion curves have been computed for a large number of III—V compounds. Most of the calculations are based on density-functional perturbation theory (Refs. [101,121,122,126,255]) and the bond-charge model (Refs. [316,317,368—370]). In addition, the tight-binding approach [138,194,385] and force-constant models [87,141] have been applied to study the phonon dispersion of GaAs(1 1 0) and InP(1 1 0). Some calculations have been restricted to high-symmetry points of the two-dimensional Brillouin zone. By means of the frozen-phonon method [329] and Car—Parrinello molecular-dynamics simulations [111], density-functional theory has been used to determine localized phonon modes at the zone center and zone boundary points of the (1 1 0) surfaces of GaAs, GaP, InAs, and InP. The tight-binding total-energy calculations of Refs. [138,385] were focused on exploring rotational low-frequency phonon modes [138,385], which are related to the characteristic bond-angle relaxation of the (1 1 0) surfaces of III—V and also II—VI compounds. Experimentally, the phonon modes of the (1 1 0) surfaces have been investigated with the help of inelastic He-atom scattering (HAS), high-resolution electron-energy-loss spectroscopy (HREELS), and Raman spectroscopy. Many HREELS studies provide only a few surface phonon frequencies [78,79,95,287]. Recently, complete dispersion curves along high-symmetry directions have been measured for the (1 1 0) surfaces of GaAs, GaP, InP, and InSb using HREELS [261,262,264] and HAS [92,157,255,364]. In addition, surface vibrational modes of InP(1 1 0) have been detected with high precision by means of Raman spectroscopy [163]. The large number of data now available for the III—V compounds can be used to analyze similarities and chemical trends in the vibrational spectra of the (1 1 0) surfaces of zincblende-phase binary semiconductors. This section summarizes the results obtained within ab initio calculations for the relaxation and the phonon modes of the (1 1 0) surfaces of GaAs, GaP, GaSb, AlAs, AlSb, InP, InAs, and InSb. A detailed description of localized phonon modes is given for GaAs(1 1 0), InAs(1 1 0), GaP(1 1 0), InP(1 1 0), GaSb(1 1 0), and AlAs(1 1 0). These systems cover the three regimes m (m , m +m , and m 'm , where m and m are the anion- and cation masses. The     three different scenarios are particularly helpful to identify clear signatures of the atomic relaxation in the phonon spectra of the III—V (1 1 0) surfaces which are strikingly similar in their atomic geometry. 5.1. Atomic structure of the III—»(1 1 0) surfaces Fig. 8 illustrates the atomic arrangement in a crystal with zincblende structure. Adjacent (1 1 0) planes of the bulk are indicated by shaded areas. Each (1 1 0) plane contains anion—cation chains

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Fig. 8. Bonding structure in a crystal with zincblende structure. Consecutive (1 1 0) planes are indicated by the shaded areas. The anions and cations form zigzag chains in the (1 1 0) planes oriented in the (1M 1 0) direction, neighboring (1 1 0) planes are connected by anion—cation chains. Table 1 Relaxation parameters of the III—V(1 1 0) surfaces. The structural parameters are defined in Fig. 9. The theoretical results (a , D ) are compared with the experimental data (a, D ) reported in Refs. [134,233]. The theoretical data for  ,  , InSb(1 1 0) are taken from Ref. [255]

GaP GaAs AlAs InP InAs GaSb AlSb InSb

a (A> ) 

a (A> ) 

D (A> ) ,

D (A> ) ,

u

E (Ry) 

5.349 5.613 5.633 5.822 5.974 6.054 6.104 6.297

5.451 5.654 5.657 5.869 6.036 6.118 6.130 6.478

0.623 0.655 0.668 0.636 0.754 0.762 0.827 0.869

0.630 0.690

29.4° 29.3° 30.2° 27.5° 31.0° 31.0° 33.5° 33.9°

12.0 10.0 10.0 10.0 10.0 11.0 11.0 8.0

0.730 0.780 0.770 0.880

oriented in the [1M 1 0] direction. Neighboring (1 1 0) planes are connected by anion—cation chains so that the formation of a (1 1 0) surface will lead to one broken bond per surface atom. Related to the different chemical character of the group-III and group-V elements, the dangling bonds of the surface cations are empty, while the dangling bonds of the surface anions are doubly occupied. This leads to the relaxation shown in Fig. 9, which is driven by the fact that the first-layer anions prefer a p-bonding configuration with their group-III neighbors, which lowers the energy of the occupied dangling bonds, while the first-layer cations prefer to be situated in a more sp-like bonding configuration with their group-V neighbors. This increases the energy of the unoccupied dangling bonds [112,309,341]. Compared with the bond angle changes introduced by the relaxation, the variation in the bond lengths is very small. Therefore, the relaxation essentially consists in a nearly bond-length conserving rotation of the surface chains of about 25°—35° and a slight counter-rotation in the second layer of about 3°—5°, depending on the particular compound. Table 1 summarizes our

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Fig. 9. Side view of the first three layers of a relaxed (1 1 0) surface. The first-layer anions are shifted above the surface, while the cations are shifted towards the bulk. All bond lengths are nearly conserved. The relaxation parameters of eight different III—V compounds are listed in Table 1. Fig. 10. Vertical displacement D plotted as a function of the lattice parameter a . The values are taken from Table 1. , 

results for the bulk lattice constant a and the relaxation parameters of the (1 1 0) surfaces of AlAs,  AlSb, GaP, GaAs, GaSb, InP, and InAs, together with the findings of Ref. [255] obtained for InSb(1 1 0). The data are compared with the experimental results reported in Refs. [134,233]. All calculations have been carried out by using a periodic slab configuration of nine-layer crystal films separated by vacuum equal to four interlayer distances. With the exception of InSb(1 1 0) (Ref. [255]), the k-point sampling has been performed with 6 special points. The last row of Table 1 indicates the cutoff energy used in the plane-wave expansion for each compound. The structural parameters D and u are defined in Fig. 9. , Table 1 illustrates that calculations based on DFT in LDA are able to reproduce the measured lattice constant a and vertical displacement D with a slight underestimation of only a few  , percent. The data indicate a pronounced correlation between D and a . In Fig. 10, the vertical ,  displacements obtained from DFT are plotted as a function of the computed lattice parameter a .  The straight line represents the result from a least-squares fit to the data for all compounds to D "0.269a !0.840 A> , (120) ,  with a regression coefficient of 0.974. The data for InP have been excluded from the fit. As in other ab initio calculations [17,373], we observe for InP(1 1 0) a significant underestimation of D by , theory with respect to the experimental findings. From an additional calculation performed for InP(1 1 0) using another set of pseudopotentials (Ref. [122]), we obtain an increase of the perpendicular displacement to D "0.680 A> and a slight decrease in the calculated lattice constant to , a "5.807 A> .  As in density-functional calculations, the tight-binding total-energy scheme was used to determine the surface relaxation angle prior to the computation of surface phonon modes

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[138,194,233,385]. The total energy was described using the model developed by Vogl [379] and Chadi [70], which contains contributions from the one-electron band structure and elastic-energy terms from each of the bonds.

5.2. Density-functional phonon calculations for III—»(1 1 0) surfaces In the framework of density-functional theory, the phonon modes of III—V(1 1 0) surfaces were explored by ab initio linear-response theory [101,121,122,126,255], the frozen-phonon approach [329], and Car—Parrinello molecular-dynamics simulations [111]. Prior to summarizing the main results of these calculations in the following sections, a brief outline of the computational details will be given. In the DFPT scheme, the surface dynamics of all compounds listed in Table 1 were treated by calculating the harmonic force constants of a fully relaxed nine-layer slab [101,121,122,126,255], using special points and a plane-wave expansion as illustrated in the previous section. The computed interatomic force constants decrease at least two orders of magnitude for atoms separated by more than three layers when compared with the nearest neighbor interactions. Therefore, the force constants of a nine-layer slab were used to model the dynamical matrices of a much larger slab comprising 25 atomic layers. We used the slab filling procedure described in Section 2.1.2. By this, resonances and deeply penetrating surface states can be identified. To obtain the full dispersion parallel and perpendicular to the surface chains, we calculate four dynamical matrices in the CX and three dynamical matrices in the CX direction. The one-dimensional Fourier transformation of these matrices along each of the two directions yields the corresponding interplanar force constants, which are used to compute the dynamical matrices at arbitrary wavevectors parallel and perpendicular to the surface chains. The computations of Schmidt, Bechstedt, and Srivastava [329] have been carried out for periodic arrangements of crystal films with eight atomic layers and a vacuum region equivalent in thickness to six atomic layers. Brillouin zone integrals are replaced in their calculation by a sum over four special points. The plane-wave expansion includes basis functions up to an energy cutoff of 15 Ry. The relaxed atomic positions determined in the calculations of Schmidt and co-workers for the (1 1 0) surfaces of GaAs, InAs, InP, and GaP are in excellent agreement with the data listed in Table 1 and the results of Refs. [17,373]. To investigate surface vibrational modes, the atoms of the three outermost surface layers of the eight-layer crystal films were displaced by 0.053 A> away from their equilibrium positions. The harmonic force constants were extracted by fitting the resulting Hellmann—Feynman forces to a quadratic equation in the distortion. For GaAs(1 1 0), a (2;1) surface unit cell has been used to calculate both, the C and X point phonon modes, while the computations were restricted to the C point for InAs, InP, and GaP(1 1 0). The molecular-dynamics simulations of Di Felice and co-workers [111] have been performed using periodically repeated slabs to describe the GaAs(1 1 0) surface. Each crystal film comprises 5 atomic layers, neighboring slabs are separated by vacuum equal to six interlayer distances. The dangling bonds on one side of the films were saturated by hydrogen. By imposing a periodicity of (2;2) in the plane of the slabs, the simulations allow one to extract phonon modes at the CM , XM , XM , and M point of the surface Brillouin zone. Only the point k "0 was used in the Brillouin zone sampling. The relaxation geometry determined by means of a combined steepest descent of the

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Fig. 11. Phonon dispersion of the GaAs(1 1 0) surface. The large shaded area represents the surface projected bulk band structure. Surface-localized modes are indicated by the solid lines. Triangles: experimental data from HAS [157,92]. Squares: experimental data from HREELS [261]. The dashed areas at about 10 meV indicate regions where the A -peak  is present in the calculated scattering cross-section. The irreducible wedge of the surface Brillouin zone is shown in the inset (see also Fig. 1).

atomic and electronic degrees of freedom agrees very well with the results shown in Table 1 and all other ab initio calculations [17,373]. To examine the vibrational properties of the GaAs(1 1 0) surface, free molecular-dynamics simulations have been performed for a temperature of 300 K, covering a time period of 4.74 ps with a time step of about 3 fs. The fifth atomic layer was fixed during the simulations. This can affect bulk states and less localized surface states. Well localized surface states, however, are not influenced by this particular choice of boundary conditions. To extract eigenfrequencies and eigenmodes, the MUSIC algorithm [267,325] was applied in combination with the self-consistent procedure [344] which is illustrated in Section 2.2. 5.3. Phonons in GaAs(1 1 0) In this section, we focus on the most prominent surface-localized phonon modes obtained from DFPT for GaAs(1 1 0) and compare the results with those of the frozen-phonon calculation of Ref. [329] and the molecular-dynamics simulations of Ref. [111], as well as with the experimental data from HAS [157,92] and HREELS [261,262]. Fig. 11 illustrates the phonon dispersion computed for GaAs(1 1 0) together with the data from HAS [157,92] (indicated by triangles), and those from HREELS [261] (represented by squares). As can be seen, the agreement with the experimental data is excellent in the whole range of the frequency spectrum and in both directions of the dispersion. The two lowest modes in the CX direction are the Rayleigh wave (RW) and a surface vibration which starts at the C point as a shear-horizontal mode. Close to the zone boundary, it acquires a strong vertical component in

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the displacement pattern and can be detected by HREELS measurements [261]. The RW, which is not prominent in the HREEL spectra, has been resolved by HAS [157]. Two further distinct features in the CX direction have been observed by HAS: A flat branch at about 10 meV referred to as A and a nearly dispersionless surface mode (A ) at about 13 meV near the zone boundary [157].   In the CX direction, the eigenvectors can be classified as A- and A-modes, according to the irreducible representation of C . In A-modes atomic displacements exclusively in the [1M 1 0] 1 direction are allowed (SH-modes), while A-modes are characterized by atomic motions perpendicular to the surface chain direction (SP-modes). The calculation yields three surface acoustical waves. The RW and the third phonon mode are visible in the HAS time-of-flight spectra of Ref. [92], while HREELS measurements only resolve the higher surface phonon mode. The intermediate branch cannot be detected as the corresponding eigenvectors are characterized by a pure shear-horizontal displacement pattern. The flat branch of the A mode is observed by HREELS in  both directions of the dispersion. In the frequency spectrum above 13 meV, experimental data are available only from HREELS. As can be seen from Fig. 11, three more flat branches at about 16, 21, and 36 meV have been identified in the energy-loss spectra in good agreement with the dispersion predicted by DFPT. A detailed comparison shows that the results obtained from DFPT are nicely confirmed by the findings of the frozen-phonon calculation of Ref. [329], with an excellent agreement in the calculated eigenvectors and frequencies for all well-defined surface vibrational states. In particular, at the C point, the frozen-phonon approach yields well-defined vibrational states at the energies 14.6, 23.2, and 35.8 meV with A character and pronounced A-modes at the energies 31.1 and 31.7 meV. At the X point, the eigenvectors cannot be strictly separated into A- and A-modes. Localized surface states are found in Ref. [329] at 8.7, 8.9, 13.0, 14.2, 22.7, 28.2, 32.0, and 35.5 meV. The agreement with the results obtained from ab initio molecular-dynamics simulations performed by Di Felice and co-workers [111] is similar. Pronounced phonon modes can be identified from the particle trajectories by means of the signal processing algorithm: In accord with the results from the DFPT approach and the frozen-phonon calculation, a characteristic surface optical vibration is found above the bulk continuum, the phonon energies being 34.1, 33.5, and 35.2 meV at the CM , X , and XM  point. A mode mainly characterized by a vibration of the first-layer cations is resolved with an energy of 13.3, 13.6, 13.4, and 13.0 meV at the CM , XM , M , and X  point. Its eigenvector at the X point compares very well with that of the 14.5 meV mode found by DFPT, which is illustrated in Fig. 12. The Rayleigh phonon mode is clearly identified by the MUSIC scheme with zone boundary energies of 8.8, 7.7, and 8.6 meV at the X , X , and M  point. However, two differences have to be pointed out. In Ref. [111] low-frequency vibrations are observed and related to the use of only one kM -point for the sampling. Moreover, only one mode has been reported in Ref. [111] at the X point with an energy of about 13 meV, while two distinct vibrational features, the (A ) mode at about 13 meV and the cation mode above at about 14 meV (see Fig. 12), are  obtained from DFPT and the frozen-phonon method in agreement with the experimental results from HAS and HREELS. The A mode has been discussed with particular interest. The tight-binding model of Wang and  Duke suggests that the flat branch measured at about 10 meV is related to a bond-length conserving rotation of the top-layer atoms with an energy of 10.7 meV [385], while the calculations of Godin, LaFemina, and Duke [138] relate a chain-bouncing mode at 9.3 meV to the experimentally observed feature. The bond-charge model yields a true surface state at the zone boundary,

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Fig. 12. Displacement pattern of two surface-localized phonon modes in GaAs(1 1 0) in a side view of the first five atomic layers. (a) Cation-localized mode in GaAs(1 1 0) at the X point. (b) A -mode in GaAs(1 1 0) at the X point. Arrows with  a kink indicate atomic displacements in the [1 10] direction.

which turns into a broad resonance for small wavevectors [317]. In the ab initio calculations of Refs. [329,111], a resonant vibration has been observed at about 11 meV. It can be described as a rocking motion of the atoms in the surface chains with considerable displacements of the atoms in the deeper layers. Our calculation reveals a similar phonon mode at the C and X point at about 10.5 meV. In the tight-binding calculation of Wang and Duke, the mode has been expected to couple only weakly to atomic motions in the deeper layers. In contrast with that and in agreement with results of the other LDA calculations, we find that the rocking mode strongly mixes with bulk states and cannot be resolved throughout the surface Brillouin zone. For a further clarification of the flat branch seen in the HAS experiment at about 10 meV [157], we calculated the differential reflection coefficient of inelastically scattered helium atoms, using all eigenfrequencies and eigenvectors determined for the crystal film. We used the Born distorted wave approximation (Eq. (6.37) of Ref. [205]) to compute the scattering-cross section for constant-Q scans. The differential reflection coefficient is dominated by the displacements of the surface atoms normal to the surface with smaller contributions from the first-layer cations which are 0.655 A> below the outermost anions [125]. Low-frequency surface phonon modes with significant firstlayer displacements in the direction of the surface normal appear as pronounced peaks in the spectrum, while bulk vibrations lead to a broad background. Fig. 13 shows three typical constantQ scan curves for scans between the C point (m"0) and the X point (m"1). Besides the distinct peak of the RW, we observe a broad feature at 10 meV in large parts of the dispersion in the CX direction. It is present in the regions indicated as dashed areas in Fig. 11. This feature arises from states which penetrate deeply into the bulk. At least partial contributions to the calculated maximum seen at 10 meV [121] originate in the rocking mode. The scattering-cross section computed for constant-Q scans along the CX direction shows no clear feature at about 10 meV for wavevectors larger than zero. 5.4. Phonons in InP(1 1 0) Fig. 14 summarizes the results from DFPT for the surface phonon modes in InP(1 1 0). The computed dispersion is compared with experimental data from HREELS [262] and Raman spectroscopy [163]. As for GaAs(1 1 0), two acoustical phonon modes and four flat branches are

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Fig. 13. Differential reflection coefficient calculated for constant-Q scans at three different points between CM (m"0) and X (m"1).

Fig. 14. Phonon dispersion of the InP(1 1 0) surface. Squares: experimental data from HREELS [262]. Dots: experimental data from Raman spectroscopy [163].

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Fig. 15. Displacement pattern of the two gap modes at the C point. The modes correspond to the experimentally resolved features at 31.5 and 33.5 meV. The eigenvectors are shown in a side view of the first five atomic layers. Indium atoms are represented by the open circles; closed circles refer to phosphorus atoms.

determined by the HREELS experiment. The agreement between theory and the measured data is comparable with the case of GaAs(1 1 0). Most of the phonon modes in InP(1 1 0) are similar to the vibrations of the GaAs(1 1 0) surface. One important difference, however, arises from the large gap between the acoustical and optical phonons of the bulk, which is resulting from the large difference in mass between indium and phosphorus. The two flat branches placed in the gap at 32.0 and 33.9 meV correspond to the probably most significant phonons of the InP(1 1 0) surface. The eigenvectors of both modes are dominated by a vibration of the first-layer anions in the plane perpendicular to the surface anion—cation chains. They are shown in Fig. 15. Because of their location in the frequency spectrum, these modes do not mix with bulk states and can be observed as true surface vibrations throughout the two dimensional Brillouin zone. This is in very good agreement with the vibrational features detected by Raman spectroscopy [163] at 31.5 and 33.5 meV and the data from HREELS [262]. The gap modes show some similarities to the phonon of GaAs(1 1 0) at about 23 meV (Fig. 11). In GaAs(1 1 0), however, the vibration is not restricted to solely the first-layer anions. Therefore and because of the large mass ratio between indium and phosphorus, a direct comparison is not possible. The frozen-phonon calculation of Schmidt and co-workers gives well-defined surface phonon modes with zone-center energies of 12.1, 34.2, 36.0, 43.6 and 46.7 meV [329]. All modes have A-character with the exception of the state at 43.6 meV, which has A-symmetry. As pointed out in Ref. [329], their numbers have to be rescaled according to the overestimation of the bulk TO(C)

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frequency, which is 10% for InP. The corresponding phonon states determined by DFPT have CM point energies of 13.3, 32.0, 33.9, 40.6, and 43.8 meV (see Fig. 14). 5.5. Comparison of the III—»(1 1 0) phonon spectra The (1 1 0) surfaces of all III—V compounds considered in Section 5.1 undergo essentially the same relaxation. Also, the electronic structure is similar in all cases. Therefore, the force constants computed for GaAs(1 1 0), InP(1 1 0), and all other compounds are different to only a small extent, just reflecting minor trends in the bonding properties of the bulk materials. Hence, a series of similarities has to appear in the phonon dispersion curves. Typical features present in GaAs(1 1 0), InP(1 1 0), and the (1 1 0) surfaces of all other III—V compounds are two acoustical surface modes in the CX and three in the CX direction as well as the surface optical vibration above the bulk continuum. Pronounced differences expected for the eigenvectors have to be related to the individual constituent masses in each compound. The anion and cation masses and their ratio are relevant, because of the bonding configuration, which is different for the group-III atoms in the surface compared to that of the first-layer group-V atoms. From calculations of the bulk phonon spectra, it is known that the optical bulk vibrations are separated from the acoustical phonons by a gap, if m (m or m 'm , (121)   as long as the difference between the anion mass m and that of the cations m is large enough. With  the surface phonon dispersion calculated for the (1 1 0) surfaces of GaP, GaSb, and other III—V compounds, we are able to investigate the conditions under which true surface states appear in the gap of the projected bulk bands. Because of the different nearest-neighbor constellation of the surface anions and cations, the two scenarios m (m and m 'm have to be considered   separately. 5.5.1. Compounds with m (m  We first focus on compounds in which the anions have a smaller mass than the cations. Besides InP(1 1 0), calculations based on DFPT have been performed for GaP(1 1 0) and InAs(1 1 0) [101]. In Fig. 16, the phonon dispersion of GaP(1 1 0) is shown together with the results obtained from HREELS [78,95], the tight-binding calculation of Wang and Duke [385], and the frozen-phonon method [329] which gives well-defined surface states at 16.8, 36.4, 37.2, 45.9, and 49.8 meV (indicated by triangles in Fig. 16). Analogously to InP(1 1 0), two surface-localized phonon modes appear in the middle of the gap of the surface-projected bulk bands. The displacement pattern of these surface vibrations is also dominated by a motion of the first-layer anions perpendicular to the chain direction and can be compared to the respective eigenvectors of InP(1 1 0) shown in Fig. 15. Atomic motions in the outermost three layers are involved. Because of their particular displacement pattern, these modes are sensitive to even small variations in the details of the surface structure as has been demonstrated for InP(1 1 0) [122]. A third example for the appearance of such gap modes is InAs(1 1 0). Fig. 17 illustrates the phonon dispersion of this surface. The acoustical and optical bulk phonon bands are separated by a relatively small gap because of the small difference between the masses. Particularly, the lowest gap mode is close to the upper edge of the surface projected acoustical bulk frequencies. Near the

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Fig. 16. Phonon dispersion of GaP(1 1 0). The LDA-results of Ref. [329] are indicated by the triangles. Data from HREELS [78,95] are represented by squares. The diamond indicates the frequency of the bond-length conserving rotation determined in Ref. [385].

zone center and the X point, it lies in the energy range of the acoustical bulk frequencies. The displacement vector of this vibrational state is also characterized by motions of the first- and second-layer anions. In contrast with GaP(1 1 0) and InP(1 1 0), vibrations of the surface anions are less dominant in InAs(1 1 0). The frozen-phonon calculation of Ref. [329] yields for InAs(1 1 0) one clear and one less defined surface state at the CM point in the near of the gap at 21.8 and 24.3 meV and, in addition, two other pronounced surface vibrations at 10.6 and 31.1 meV. All surface-localized features found in Ref. [329], including also those with a more complex displacement pattern, are indicated in Fig. 17 as triangles. From Figs. 14 and 16 it can be seen, that for InP and GaP, a third mode is placed in the gap slightly below the upper edge of the optical bulk phonon bands. It consists mainly in a displacement of the second-layer anions perpendicular to the surface and smaller vibrations of the anions in the first layer. In InP(1 1 0) further modes appear slightly below the optical bulk bands in the near of the zone boundary points. For InAs(1 1 0), the corresponding branches lie above the lower rim of the optical bulk bands. Related to that, the displacement patterns of these modes display pronounced contributions to the vibration from all first- and second-layer atom vibrations. 5.5.2. Compounds with m 'm  As seen from the comparison of Figs. 14, 16 and 17, the gap modes obtained for InAs(1 1 0) are closer to the upper edge of the acoustical bulk modes than those computed for InP and GaP(1 1 0). This trend indicates that the corresponding modes of GaSb, AlAs, and AlSb(1 1 0) should have energies in the range of the acoustical bulk phonons. The phonon dispersion curves computed for GaSb and AlAs(1 1 0), which we discuss in this paragraph, confirm this assumption.

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Fig. 17. Phonon dispersion of InAs(1 1 0). The LDA-results for Ref. [329] are indicated by the triangles. Data from HREELS [79] are represented by squares. The diamond indicates the frequency of the bond-length conserving rotation determined in Ref. [385].

For the determination of the complete phonon dispersion of GaSb(1 1 0), which is illustrated in Fig. 18, we have used a generalized mass approximation (GMA). It combines the mass approximation suggested in Ref. [134] with a rescaling of all frequencies by the factor a"0.926, in order to reproduce the frequency of the longitudinal optical bulk phonon determined by DFPT. Interestingly, this factor is close to the ratio of the lattice constants of bulk GaAs and GaSb, which amounts to a% /a% 1
"0.927. The circles in Fig. 18 indicate the results from a self-consistent   calculation applying the linear-response approach to GaSb(1 1 0) at the CM , X , and X  point of the surface Brillouin zone. The results obtained within the GMA compare very well with the frequencies and eigenvectors determined by means of DFPT. Only small differences are seen in form of a slight overestimation of the acoustical phonon frequencies by the GMA and some deviations in the dispersion at about 25 meV near the C point. The possibility to use the force constants of one system to estimate the phonon dispersion of a surface which has similar effective coupling parameters, is an advantage of the linear-response formalism or the frozen-phonon approach over molecular-dynamics simulations. Particle trajectories such as obtained within Car—Parrinello molecular-dynamics simulations for GaAs(1 1 0) cannot be used to extract the frequency spectrum of another compound. The phonon dispersion of GaSb(1 1 0) illustrates representatively the surface phonon bands of the III—V compounds with m 'm . The most important difference to the phonon spectrum of  InP(1 1 0) or GaP(1 1 0) is the removal of the gap modes which are related to the first-layer anion vibrations. Because of the larger mass of the group-V elements compared to that of the cations, the corresponding phonon branches are shifted into the continuum of the acoustical bulk phonons. Consequently, the phonon modes mix with bulk vibrations and cannot be identified as surfacelocalized states. However, the gap is not free from surface phonon bands. Analogously to InP(1 1 0)

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Fig. 18. Phonon dispersion of GaSb(1 1 0). The full lines and the shaded area have been obtained within the generalized mass approximation using the force constants of GaAs(1 1 0). The circles indicate the results from DFPT applied to GaSb(1 1 0).

and GaP(1 1 0), several vibrational features occur close to the lower rim of the optical bulk phonons. They are dominated by atomic motions in the second and third layer and smaller displacements of the first-layer atoms mainly in the (1 1 0) plane. The same behavior is found for the (1 1 0) surfaces of AlAs and AlSb. The phonon dispersion of AlAs(1 1 0) is shown in Fig. 19. A comparison with the phonon spectra of GaP(1 1 0) (Fig. 16), InAs(1 1 0) (Fig. 17), and GaSb(1 1 0) (Fig. 18) shows that for the compounds with m 'm , typically three surface localized features are present at the X point with nearly the  same energy. The lowest gap mode is always dominated by displacements of the second- and third-layer cations in the plane normal to the surface chains. Two of the three vibrational states have essentially the same energy at the X  point. Their displacement pattern is similar to that of the lowest of the three gap modes at the X point. The third mode has A symmetry and appears at the X  point with a higher energy in AlAs(1 1 0). For AlSb(1 1 0), the lowest gap state is found at about 1 meV below the optic bulk phonons in good agreement with preliminary results from HREELS [263,265]. It is characterized by displacements of the second and third-layer cations mainly in the z-direction and appears with an energy of 35.7, 35.1, and 36.3 meV at the CM , X , and X  point, respectively. The modes lying in the middle and lower part of the gap of the phonon dispersion of GaP(1 1 0), InP(1 1 0), and InAs(1 1 0) are characterized by large displacements of the first-layer anions. In III—V compounds with m 'm , no similar modes are seen. We therefore conclude, that pro nounced surface phonon modes such as illustrated in Fig. 15 result from the interplay between two important factors: The particular bonding configuration of the first-layer anions, which originates from the surface relaxation, is manifested by the appearance of characteristic modes in the gap

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Fig. 19. Phonon dispersion of AlAs(1 1 0) computed by means of DFPT.

between the optical and acoustical bulk vibrations only if the anion mass is smaller than the cation mass (m (m ).  5.5.3. Similarities and chemical trends Apart from the fact that the pair of characteristic gap modes is present only if the anion mass is smaller than the cation mass, the phonon dispersion and the related eigenvectors are similar for the (1 1 0) surfaces of all binary semiconductors under consideration. All (1 1 0) surfaces exhibit two acoustical phonon modes in the CX and three surface acoustic modes in the CX direction. The eigenvectors of the acoustical modes are strikingly similar for all systems. In addition to the detailed information obtained from theory, this has been verified experimentally for GaAs(1 1 0), InSb(1 1 0), and GaP(1 1 0) by means of HAS [92,255,364]. In the CX direction, the intermediate acoustical phonon branch is polarized in the chain direction. Therefore, it cannot be detected by the He-scattering method. The other two branches, however, show atomic displacements normal to the surface chains. Consequently, they can be resolved by HAS. The upper vibrational state starts at the zone center with atomic motions mainly parallel to the surface. However, approaching the X point, the first-layer atoms are vibrating normal to the surface, while the second-layer atoms move in the (1 1 0) plane. The lowest acoustical mode (the RW) is characterized by a complementary displacement pattern. It starts as a vibration with predominantly normal components for small wavevectors. Approaching the zone boundary, the first-layer atoms move parallel to the surface, while the second-layer atoms vibrate mainly in the direction of the surface normal. Consequently, the third phonon branch leads to higher intensities in the HAS-TOF spectra near the X point compared to those of the lowest acoustical mode. Only in the first half of the dispersion, larger intensities are attributed to the Rayleigh mode. This

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behavior has been observed experimentally in complete agreement for GaAs(1 1 0) [92], InSb(1 1 0) [255], and GaP(1 1 0) [364]. The rocking mode predicted by Wang and Duke [385] can be identified for all compounds at the C point. As it mixes strongly with bulk states, its dispersion cannot be resolved in the entire Brillouin zone. Another vibrational feature prominent in all systems is related to the planar bonding configuration of the group-III elements in the first layer. Fig. 12a illustrates the eigenvector of this mode at the X point, using GaAs(1 1 0) as a representative example. As the mode mainly consists in a vibration of the surface cations, its frequency is shifted towards the upper edge of the acoustic bulk phonon bands for the compounds with m 'm (Figs. 18 and 19), while it is shifted  downwards in energy for m (m , touching in this case the lower rim of the stomach gap near the X  point (Figs. 14, 16 and 17). The eigenvector of the A -mode resolved by HAS for GaAs near the X point with an energy of  13 meV [157] is shown in Fig. 12b. For the compounds with m <m , this mode cannot be  identified in the phonon spectra as it mixes strongly with bulk states. In the case of InP(1 1 0), GaP(1 1 0), and InAs(1 1 0), however, the mode is shifted to higher frequencies appearing at the X point as the mode placed in the stomach gap (Figs. 14, 16 and 7). One of the most prominent surface vibrations is the optical phonon mode appearing above the projected bulk bands. Its location and dispersion is similar for all III—V compounds. In Fig. 20 the eigenvector of this vibrational state is shown for the example of GaAs(1 1 0). It is characterized by an opposing motion of the first-layer cations and the second-layer anions in the plane defined by the surface group-III atoms and their nearest neighbors. Depending on the ratio between m and m , this mode is characterized by larger displacements of the first-layer cations (m 'm ) or the   second-layer anions (m (m ). As can be seen from Fig. 8 of Ref. [122], the anions of the second  layer are displaced nearly parallel to the (1 1 0) plane in the case m ;m . Related to the  displacement pattern, the frequency of this optical mode scales with the square root of the inverse

Fig. 20. Displacement pattern of the surface optical phonon mode in GaAs(1 1 0) at the X  point. Fig. 21. Mass trend for the high-frequency optical surface phonon mode at the C (circles), X (crosses), and X  point (squares). Plotted is the phonon energy multiplied by the lattice constant as a function of one over the square root of the reduced mass.

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of the reduced mass which is defined by the anion and cation mass. By multiplying each vibrational energy with the respective bulk lattice constant a , a nearly linear function can be obtained in  analogy to the findings of Ref. [265]. This is illustrated in Fig. 21 for the phonon frequencies computed for the zone boundary points and the zone center. Evidence for the scaling of the frequencies with 1/a was already provided in context with the GMA, which was applied in  Section 5.5.2 to determine the phonon dispersion of GaSb(1 1 0). In addition, the frequencies of the lower pair of gap modes appearing for compounds with m (m show a nearly linear scaling with  1/a (m . This clearly indicates that the effective force constants for optical vibrations are rather  similar for all systems, scaling approximately with 1/a when comparing compounds with different  bulk lattice constants.

6. Phonon dynamics of diamond-structure (0 0 1) surfaces In semiconductors with diamond or zincblende structure, neighboring (0 0 1) planes are connected by two bonds per atom. Therefore, ideal diamond-structure (0 0 1) surfaces have two dangling bonds per first-layer atom. To reduce the number of broken bonds, pairs of neighboring surface atoms move closer together and form dimers which are usually arranged in rows perpendicular to the dimer bonding direction. This leads to a p(2;1) structure that was clearly resolved for C, Ge, and Si(0 0 1) by low-energy electron diffraction (LEED) [84,153,231,323,394], scanning tunneling microscopy (STM) [28,151,254,391], and other experiments. Clear signatures of the dimerization are also present in the phonon spectra for all of these surfaces. This has been shown by many calculations based on the tight-binding method [8,11,12,147,238,241,293] and model potentials [174,387,402]. Only recently, density-functional theory was applied to determine the surface phonon modes in C, Ge, and Si(0 0 1) [10,56,124,336,346]. Because of its importance for the fabrication of semiconductor devices, particularly the Si(0 0 1) surface has been subject of intensive investigations. Some computations support the symmetric dimer model [24,34,284], while most of the investigations favor buckled dimers on the (0 0 1) surfaces of Si and Ge [70,85,208,207,273,301,335,358,403]. Highly converged ab initio calculations [85,208,207,299] yield an energy gain per dimer of about 0.1—0.2 eV for the asymmetric configuration when compared with the symmetric case. Experimentally, the controversial discussion about the preference of either symmetric or tilted dimers has definitely been clarified by STM [254] and core level spectroscopy [57,214] for Si(0 0 1). Both experiments resolve clearly the charge transfer from the down atoms to the up atoms as predicted by theory [70], thus discarding unambiguously the symmetric reconstruction in favor of the tilted dimer configuration. Also for Ge(0 0 1), experimental data indicate the presence of buckled dimers [84,113,197]. In contrast with Si(0 0 1) and Ge(0 0 1), where asymmetric dimers give a lower total energy than untilted dimers [70,299], the diamond-structure C(0 0 1) surface is characterized by symmetric dimers [10,208]. The different behavior of C(0 0 1) relative to Si and Ge(0 0 1) can be explained by the n-bonding which is much stronger for carbon dimers. Hence, the electronic surface bands originating in the bonding n and antibonding n* dimer orbitals are separated by a gap already for untilted dimers in C(0 0 1), while the Si and Ge(0 0 1) surfaces are semiconducting through a JahnTeller-like distortion which leads to buckled dimers [208].

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In scanning tunneling microscopy [28,209,391], low-energy electron diffraction [84,197,354], X-ray diffraction [113], elastic He-atom scattering [61,62,213], photoemission spectroscopy [184,185,103], and other experiments at least three different reconstructions have been found for Si and Ge(0 0 1), namely the “ferromagnetic” ordering in the p(2;1) phase and the “antiferromagnetic” ordering in the higher-order reconstructions (HOR) p(2;2) and c(4;2). With tilted dimers as the main building blocks prevalent in Si and Ge(0 0 1), these reconstructions can simply be explained in terms of an in- or out-of-phase arrangement of the dimer buckling along and perpendicular to the rows as illustrated in Fig. 22. The calculations of Refs. [70,180,257,258,301,403] show that an alternating buckling along the dimer rows lowers the total energy with respect to the asymmetric p(2;1) reconstruction, favoring the c(4;2) geometry proposed by Chadi [70] slightly above the p(2;2) reconstruction [403,180]. Since symmetric carbon dimers are stable in contrast with the situation on Ge or Si(0 0 1), HORs of this kind do not appear on C(0 0 1) [153,231].

Fig. 22. Surface geometries investigated for Si and Ge(0 0 1). Top view of the first two atomic layers of (a) the symmetric p(2;1), (b) asymmetric p(2;1), (c) p(2;2), and (d) c(4;2) reconstruction. In the symmetric p(2;1) structure all surface atoms are represented by open circles. For tilted dimers, the down atoms are indicated by the grey circles, while the open circles symbolize the up atoms. The second layer atoms are depicted as black circles. The corresponding unit cells are illustrated by the dashed lines.

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It is now generally accepted, that the ground-state configuration of Si(0 0 1) at low temperatures is the c(4;2) arrangement of tilted dimers. This has been clearly shown by highly converged ab initio calculations [299], scanning-tunneling microscopy [28], and LEED [354]. In the case of Ge(0 0 1), the two HORs, namely the c(4;2) and p(2;2) arrangement, are almost identical in energy and about 0.08 eV per dimer lower in energy than the p(2;1) phase [346]. For Si(0 0 1), STM and LEED indicate a second-order phase transition from the ordered c(4;2) phase to a disordered p(2;1) configuration in a temperature range from about 200 K through 300 K [28,354,391], in agreement with the model calculations of Refs. [180,242]. The phase transition can be explained by means of thermally excited dimer-flipping processes as suggested in Refs. [85,391]. A similar behavior is found for Ge(0 0 1) as demonstrated by LEED [84] and helium-atom diffraction [213]. These investigations indicate that the long-range order of buckled dimers, which is present at low temperatures, is broken at room temperature. However, there is evidence that local order still is prevalent at higher temperatures. For Ge(0 0 1), streaks of quarter order spots were seen in diffraction experiments [84,213]. Local p(2;2) and c(4;2) domains were resolved by STM [209]. For Si(0 0 1), helium-atom diffraction [61,62] and other experiments [110] gave indication for the existence of HORs also above the critical temperature of the order-disorder phase transition. Room-temperature photoemission spectroscopy [103,184,184] resolves pronounced branches in the electronic band structure of Si(0 0 1). Not every branch can be reproduced by theory if the p(2;1) geometry is assumed. However, all experimental features, including the “additional” branches, can be explained by means of the surface bands computed for the c(4;2) reconstruction [124,273]. In analogy, the surface phonon dispersion of Ge(0 0 1) shows branches, that are reproduced by theory only if HORs are assumed. This has been demonstrated by a recent investigation of Ge(0 0 1) based on helium-atom scattering [381] and calculations using DFPT [346]. The appearance of “additional” branches indicates that there still is local order with p(2;2) and c(4;2) symmetry, although dimer flipping is thermally activated. As proposed by Kubby and co-workers [209], the phase transition has to involve collective interactions. Important clarification was given by the ab initio molecular-dynamics simulations of Shkrebtii and co-workers performed for Si(0 0 1) [336]. The simulations show that the low-temperature phase of Si(0 0 1) has c(4;2) symmetry. At higher temperatures, dimers tend to flip simultaneously along a row, thus preserving the alternating sign of buckling along the rows. This kind of correlated flipping along a row switches between local p(2;2) and c(4;2) symmetry and therefore accounts for the observations made in many experiments at room temperature. Here we summarize the main results obtained from density-functional calculations for the vibrations of C(0 0 1) at high symmetry points and the phonon dispersion of the (0 0 1) surfaces of silicon and germanium. In all cases, a dimer-stretch vibration occurs at a frequency which directly reflects the strength of the dimer bond. For Si and Ge(0 0 1), full phonon dispersion curves have been computed for the asymmetrically reconstructed p(2;1) surface. Vibrational features are identified, that are related to the appearance of buckled dimers which are the main building blocks of both surfaces. To investigate the influence of the dimer correlation along and across the rows, additional calculations have been performed for the HORs. For Si(0 0 1), interesting novel vibrational modes appear, that can support correlated dimer flipping. For Ge(0 0 1), phonon dispersion curves measured by HAS can be explained only if higherorder reconstructions are assumed. This shows that the surface phonon spectra of the (0 0 1)

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surfaces exhibit clear signatures which indicate the presence of dimers and their correlation in the case of Si and Ge(0 0 1).

6.1. Application of DFPT to Si and Ge(0 0 1) The (0 0 1) surfaces of Ge and Si are investigated within plane-wave pseudopotential calculations [124,346], using a cutoff energy of 8 Ry for Si and 9 Ry for Ge, which guarantees convergence for the computed bulk lattice constants and phonon frequencies. The calculated lattice constant for Si amounts to a "5.42 A> in good agreement with a "5.43 A> measured at room temperature [216].   For Ge, the respective values are 5.57 and 5.66 A> . To describe the surfaces, periodically repeated thin crystal films are used. For the ideal surface and the p(2;1) reconstruction, the computations are carried out for crystal films spanning ten atomic layers, while the number of layers is reduced to eight for the description of the p(2;2) and c(4;2) configuration. In all calculations, neighboring slabs are separated by a vacuum spacing which is equal to six interlayer distances. Fig. 22 illustrates the different structural phases and the respective unit cells considered for the Si and Ge(0 0 1) surfaces. The three different two-dimensional Brillouin zones corresponding to the p(2;1), p(2;2), and c(4;2) geometry are shown in Fig. 23. Integrations in the reciprocal space are carried out by means of special point summations using k-points according to a (4;6) mesh for the p(2;1) structure, while a (4;4) mesh of k-points is used for the higher-order reconstructions [124,127]. This choice gives 4 special points for the p(2;2) geometry and 6 for the p(2;1) and c(4;2) arrangement. By applying the scheme of Ref. [127], the (4;6) mesh determined for the p(2;1) configuration can be reduced to a set of 3 special points in the Brillouin zone of the c(4;2) geometry. In order to compare the total energy of the p(2;1) phase with the energies of the HORs, the structure optimization for the p(2;1) arrangement has to be repeated using the unit cell and the special points of the p(2;2) and c(4;2) reconstruction, respectively. The atomic equilibrium positions are determined for each geometry starting the relaxation from initial positions that are close enough to the respective zero-force configuration. By means of the modified Broyden scheme [375], the positions of all atoms in the crystal films are varied until all forces are essentially zero and the computed positions are defined with a numerical uncertainty of less than 0.01 A> . For the calculation of the phonon-dispersion curves of Si and Ge(0 0 1) in the tilted-dimer p(2;1) phase, dynamical matrices of fully relaxed ten-layer crystal films (in the configuration of Fig. 22b) are determined by means of DFPT. The force constants in the outermost four layers of the slabs are significantly different from the respective bulk values. Sufficient agreement between bulk and slab force constants is seen only for the central two layers of the crystal films. This underlines the deep penetration of the structural perturbations appearing in Si and Ge(0 0 1) p(2;1) through the dimerization. With the help of the slab-filling scheme discussed in Section 2.1.2, dynamical matrices of larger slabs consisting of 26 atomic layers are calculated using for the inner region force constants obtained from a separate computation performed for the bulk. Phonon dispersion curves are obtained by means of interplanar force constants calculated from four dynamical matrices in the CJ and JK direction, while three dynamical matrices are used in the CJ and JK direction.

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Fig. 23. Surface Brillouin zones (SBZ) of the considered surface structures. (a) SBZ of the p(2;1) and the p(2;2) geometry (solid and dashed lines, respectively) illustrated together with the corresponding irreducible wedges (heavy lines). (b) SBZ of the p(2;1) and c(4;2) structures shown together with the corresponding irreducible parts (heavy lines).

6.2. Surface reconstruction in Si(0 0 1) and Ge(0 0 1) The two dominating structure forming principles in Si and Ge(0 0 1) are rebonding, which reduces the number of dangling bonds, and a Jahn-Teller like distortion, which lowers the total energy by tilting the surface dimers. Because of the dimer-dimer correlation along and across the rows, the buckled-dimer p(2;1), p(2;2), and c(4;2) arrangements are characterized by slightly different energies and equilibrium positions. For a comprehensive investigation of the physical principles that cause the reconstruction in Si and Ge(0 0 1), we have to consider all of these surface geometries. By using a unit cell with p(2;1) periodicity, the formation of symmetric and buckled dimers can be described. When the structure optimization is started from an asymmetric configuration, tilted dimers result. To obtain unbuckled dimers, the symmetry of the forces and atomic positions has to be controlled during the relaxation. The change of bond lengths due to the relaxation is less than 4 percent for the p(2;1) geometries and the HORs. The convergence of the structure optimization can be accelerated by using initial positions, where nearest neighbors, including the pair of surface atoms forming a dimer, are separated by approximately the same distance as nearest neighbors in the bulk. In crystal films with p(2;1) symmetry, atomic forces are zero in the [1 1 0] direction (along the dimer rows as illustrated in Fig. 22). Hence, the symmetric and tilted dimer p(2;1)

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Table 2 Total energy per dimer (in eV), dimer bond length (in A> ), and dimer buckling angle of the p(2;1), p(2;2), and c(4;2) surface geometries calculated for Si(0 0 1), using slab supercells as indicated in Fig. 22. In the p(2;2) structure, the two individual tilt angles per surface unit cell are u "21° and u "!16°, while for the c(4;2) geometry the buckling   angles are u "20° and u "!16°. The value of u listed in the table is defined as u"1/2(u !u ) for the     higher-order reconstructions Si(0 0 1)

Ideal p(1;1)

Symmetric p(2;1)

Asymmetric p(2;1)

p(2;2)

c(4;2)

E  d u

0.00

!1.51 2.26 0.0°

!1.64 2.28 17°

!1.71 2.33 19°

!1.71 2.32 18°

Table 3 Same as Table 2, but for the Ge(0 0 1) surface. In the p(2;2) and the c(4;2) structure the two individual tilt angles per surface unit cell are approximately u "20° and u "!17°   Ge(0 0 1)

Ideal p(1;1)

Symmetric p(2;1)

Asymmetric p(2;1)

p(2;2)

c(4;2)

E  d u

0.00

!1.22 2.38 0.0°

!1.41 2.39 17°

!1.49 2.45 18°

!1.49 2.44 18°

configurations show atomic relaxations exclusively in the plane defined by the surface normal and the dimer bond. To determine the relaxed positions of the p(2;2) and c(4;2) arrangement of tilted dimers, the structure optimization is started from initial positions obtained by transferring the results from the tilted dimer p(2;1) reconstruction. Apart from the alternating dimer bucking along the rows, the most significant difference between the HORs and the p(2;1) reconstructions is that atomic relaxations along the dimer rows are not forbidden by symmetry in the p(2;2) and c(4;2) structures. In both cases, zero-force configurations are determined for Si and Ge(0 0 1) with the subsurface atoms displaced not only in the (1 1 0) plane, but also by about 0.1 A> along the dimer rows. The upper first-layer atoms pull their nearest neighbors in the second layer closer to the dimers, while the lower first-layer atoms push their nearest neighbors of the second layer in the opposite direction. This lowers the total energy of the HORs by about 0.07—0.08 eV relative to the p(2;1) tilted-dimer arrangement. In addition, the alternating buckling along the dimer rows leads to a significant relief of surface stress relative to the p(2;1) reconstruction as demonstrated for Si(0 0 1) [86,133]. Tables 2 and 3 summarize the total energies and the structural parameters of Si and Ge(0 0 1) determined for the geometries illustrated in Fig. 22. The ideal surface, in which all atoms reside at bulk-derived positions, serves as reference configuration. The total energy of the p(2;2) and c(4;2) phase is defined relative to the asymmetric p(2;1) structure. Deeply penetrating

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relaxations result in different magnitudes of the two individual tilt angles per unit cell, because of the small size of the eight-layer slabs used to describe the p(2;2) and c(4;2) phase. Due to symmetry, the splitting is suppressed in crystal films spanning six or ten atomic layers [124]. The total energies determined for Si and Ge(0 0 1) show that the dominating reconstruction mechanism in both surfaces consists in the formation of buckled dimers which are correlated along the dimer rows, while the interaction is weak between neighboring rows of dimers. 6.3. Surface phonons in Si(0 0 1) In the phonon spectrum of Si(0 0 1), vibrational modes are expected which originate in the buckled-dimer geometry and thus reflect the microscopic structure of the surface. We have calculated the complete phonon dispersion of Si(0 0 1) in the tilted dimer p(2;1) configuration, which already exhibits the most important features of the surface reconstruction. To examine the effects of the dimer correlation along the rows, the phonons of the Si(0 0 1) c(4;2) surface are determined at the J point of the corresponding surface Brillouin zone. We discuss pronounced surface phonon modes obtained from DFPT and compare the results with those obtained from Car—Parrinello molecular-dynamics simulations [336]. 6.3.1. Phonon dispersion of Si(0 0 1) p(2;1) Fig. 24 illustrates the phonon dispersion of the asymmetrically reconstructed Si(0 0 1) p(2;1) surface. Because of the large structural changes in the surface region, there are many different

Fig. 24. Phonon dispersion of the tilted dimer Si(0 0 1) p(2;1) surface. The solid lines represent surface-localized features, the shaded area illustrates the bulk phonon bands projected onto the SBZ. The broken lines indicate the velocity of the Rayleigh wave (RW) measured by Brillouin light scattering [99]. Fig. 25. Eigenvector of the dimer back bond mode at the J point of the SBZ. The displacement pattern is shown in a side view of the first six atomic layers.

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modes localized in the outermost layers. Surface vibrations of particular interest are the modes labeled r (rocking), ds (dimer stretching), and sb (dimer back bond). Most of the features identified in the DFPT calculation for the p(2;1) phase are similar to the results found by previous investigations based on the tight-binding method [8,11,12,238,241,293]. The mode r, which is placed in an energy range from 20 to 23 meV, attracts much attention as it is closely related to the buckling of the dimers. It is characterized by a rocking motion of the dimer atoms with an opposing up and down displacement of the first-layer atoms leading to a periodic increase and decrease of the tilt angle. A dimer-stretching mode, denoted ds, is present near the K point of the SBZ with an energy of about 43 meV. Additionally, another bond-length oscillation of the dimers appears with an energy of about 58—59 meV in the KJ direction just below the mode sb. The frequencies of such stretching vibrations provide information about the dimer bonding structure. This is underlined by the phonon dispersion computed for Ge(0 0 1) [345], which shows similar dimer bond-length vibrations. Relative to the corresponding bulk-phonon spectrum, the dimer stretching oscillations are 1—2 meV lower in energy in Ge(0 0 1) compared with Si(0 0 1). This is in agreement with the results of Ref. [208], which show that the dimer bond in Si(0 0 1) is slightly stronger than in Ge(0 0 1). Above the bulk continuum, we observe the optical surface phonon mode sb. In the CJ direction it is dominated by an opposing motion of the down shifted dimer atoms and their nearest neighbors in the second layer in the plane defined by the lower dimer atoms and their bonding partners. Because of this, the mode has the character of a dimer back bond mode. Fig. 25 illustrates the eigenvector of this vibration at the J point of the SBZ. The lowest mode in the surface phonon spectrum is the Rayleigh wave (RW). From the dispersion curves in the CJ direction, we estimate the velocity of the RW to be  v ([1 1 0])"4860 m/s. This compares very well with the experimental result of Ref. [99] which is 05 indicated by the broken lines in Fig. 24. The velocity of the shear-horizontal acoustical bulk wave for propagation in the [1 1 0] direction is v "4674 m/s [216]. Hence, the RW is not pushed below 1& the bulk continuum in the CJ and CJ direction. The preparation of the Si(0 0 1) surface usually leads to a two-domain structure. The dimer row orientation in terraces being separated by monoatomic steps is rotated by 90°. Under such conditions the dispersion along the CJ and CJ direction is recorded simultaneously. This problem can be solved in an elegant way. Fig. 26 illustrates the SBZ of a p(2;1) and p(1;2) domain. The J  point of the second Brillouin zone and also the CJ direction are identical for both   domains. Hence, the dispersion measured along the [1 0 0] direction is the same for the p(2;1) and p(1;2) orientation. Pronounced features appearing in the phonon dispersion along the [1 0 0] direction are the rocking mode (r), the dimer back bond mode (sb), and the RW (see Fig. 27). The velocity of the RW can be estimated to v ([1 0 0])"4750 m/s. In contrast with 05 the CJ and CJ direction, the RW now is found below the bulk continuum also for small wavevectors. This compares very well with the experimental results of Refs. [99,294] and the findings of continuum theory [374]. 6.3.2. Vibrational states of the c(4;2) phase In the SBZ of the c(4;2) geometry, the J  point is equivalent to the J point. This can be seen from Fig. 23. Hence, the phonons calculated at the J point of the c(4;2) reconstruction have to be compared with the respective surface-vibrations determined for the J and J  points of the p(2;1)

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Fig. 26. Surface Brillouin zones for p(2;1) and p(1;2) domains on the Si(0 0 1) surface. The [1 0 0] direction is defined by the zone center point and the J  point of the second Brillouin zone, and is oriented diagonal to the dimer rows in both  domains. Fig. 27. Phonon dispersion of the tilted dimer Si(0 0 1) p(2;1) surface from the zone center to the J  point of the second  Brillouin zone.

phase. The majority of the phonon eigenvectors and frequencies are similar in both reconstructions. Representatively, the energies of the modes RW, r, and sb calculated for the c(4;2) as well as the p(2;1) structure are summarized in Table 4. Especially the energies of the rocking mode r are in excellent agreement comparing the results of the two different surface reconstructions. The most significant variations are seen for the mode sb. While this mode is found above the bulk continuum for the p(2;1) geometry, the two corresponding frequencies in the c(4;2) structure are 57.2 and 59.8 meV. Therefore, by measuring the mode’s location with respect to the upper rim of the bulk continuum, information could be obtained about the in- or out-of-phase arrangement of the dimers in Si(0 0 1). An interesting surface vibration not seen in the p(2;1) geometry occurs in the c(4;2) surface with an energy of 7.1 meV at the J point. The eigenvector of this low-frequency mode is illustrated in Fig. 28. It is dominated by a twisting motion of the dimer atoms originating in first-layer displacements parallel to the dimer rows. While a similar mode has been found in the ab initio molecular-dynamics simulations of Ref. [336], previous model calculations did not reveal any similar low-energy vibrations [8,238].

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Table 4 Energies (in meV) calculated for the modes RW, r, and sb at the J point of the c(4;2) structure compared with the energies of the corresponding modes at the J and J point of the p(2;1) surface Mode

RW r sb

c(4;2) J 8.5 23.0 59.8

10.8 20.4 57.2

p(2;1) J

J 

9.0 22.6 63.9

9.2 20.6 58.9

Fig. 28. Eigenvector of the low-frequency dimer twisting mode at the J point of the c(4;2) structure. (a) Side view of the atomic displacements in the first three layers projected onto the plane which is defined by the broken line shown in (b) and the surface normal. (b) Top view of the displacements in the first layer.

6.3.3. Car—Parrinello molecular-dynamics simulations Besides the linear-response formalism, the only other ab initio approach applied to study the vibrational modes of the tilted dimer reconstruction on Si(0 0 1) is the Car—Parrinello moleculardynamics simulations scheme used by Shkrebtii and co-workers [336]. Periodically repeated crystal films made up of six atomic layers were employed, with neighboring slabs being separated by 8 A> . The bottom plane of each slab was covered by hydrogen at a full monolayer completion. The molecular-dynamics simulations were carried out for p(4;2) supercells which contain eight silicon atoms per layer. This allows one to investigate all reconstructions shown in Fig. 22. Optimized atomic positions were determined by simultaneously relaxing the atomic and electronic degrees of freedom. Vibrational states and frequencies were extracted by means of the MUSIC algorithm illustrated in Section 2.2. Atomic trajectories were recorded for molecular-dynamics runs at 300 and 900 K. In the simulations, the ideal surface and the p(2;1) arrangement of tilted or untilted dimers turned out to be unstable in favor of the p(2;2) and c(4;2) reconstructions which were formed

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spontaneously. The energy difference between the structural phases is similar to those summarized in Table 2. One important result obtained for the free dynamics is the observation that dimers along a row tend to flip simultaneously. This originates in the pronounced dimer correlation within the rows. Across dimer rows, the interaction is weak. Correlated dimer flipping leads to an alternation between the p(2;2) and c(4;2) phase. The change is more frequent at higher temperatures. In their switching positions the dimers show significant amounts of twisting and shifts along the rows but only a very small buckling. The energy surface for this path is almost flat. Vibrational modes have been extracted from atomic trajectories obtained during a period in time, where the system stayed sufficiently long in the p(2;2) phase. Because of the use of a p(4;2) unit cell, the atomic motions comprise contributions from phonon modes at the C and the J point of the p(2;2) SBZ (see Fig. 23). For a comparison with the phonon modes of the tilted-dimer p(2;1) reconstruction, we have to consider the vibrations of the p(2;1) phase at the C and J  point as well as those at the J and K point. The following pronounced vibrational states can be identified by means of spectral analysis: A mode is present at the J point with an energy of 14 meV. This vibration corresponds to one of the modes near 14 meV folded from the K point of the p(2;1) structure (see Fig. 24). The RW at the J point and the RW folded from the K point have energies of 9 and 12 meV, which correspond to the respective states of the p(2;1) phase seen at 8.5 and 10.3 meV. In agreement with the results from DFPT for the c(4;2) phase, a low-frequency mode is found below the bulk continuum with a J point energy of 5.1 meV in the p(2;2) reconstruction. Its character mainly consists in a twisting motion of the surface dimers, which is similar to that illustrated in Fig. 28. The displacements of the twisting mode bring the surface atoms close to the switching position of the dimers. It is expected that the thermal excitation of this low-frequency vibration contributes significantly to a correlated change in the sign of the tilt of all dimers along a row, thus supporting the experimentally observed phase transition. 6.4. Phonon dispersion in Ge(0 0 1) In contrast with Si(0 0 1), only a few studies have been performed to explore the surface dynamics of Ge(0 0 1). Besides the molecular-dynamics simulations of Ref. [387], which are based on model potentials, only DFPT [346] has been applied to investigate the phonon modes of this system. The Ge(0 0 1) surface is very similar to Si(0 0 1). In particular, the HORs are slightly favored above the tilted-dimer p(2;1) arrangement in analogy with Si(0 0 1). From the surface energies listed in Table 3, it can be seen that the germanium dimers are correlated along the rows, while the interaction is weak between neighboring rows. The investigation of the surface phonon modes in Ge(0 0 1) becomes particularly interesting, because it was possible to measure phonon dispersion curves in the low-frequency spectrum by means of HAS [346,381]. Hence, computed phonon frequencies and eigenvectors can be checked by comparing them with experimental data. The results and insights obtained for Ge(0 0 1) are directly transferable to Si(0 0 1). 6.4.1. Phonon dispersion of the p(2;1) phase The main features of the surface vibrational spectrum of Ge(0 0 1) are already present in the phonon dispersion of the buckled-dimer p(2;1) reconstruction. As for Si(0 0 1), the rocking mode (r), dimer stretching vibrations (ds), and a dimer back bond mode (sb) are pronounced vibrational

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Fig. 29. Phonon dispersion of Ge(0 0 1) along the [1 0 0] direction measured by HAS (crosses) and computed for the p(2;1) phase by DFPT (solid lines). Different gray scales reflect the magnitude of the calculated scattering intensities.

features of the p(2;1) phase. The phonon dispersion of Ge(0 0 1) is essentially the same as that shown in Figs. 24 and 27, because of the structural similarity of Si and Ge(0 0 1). The larger atomic mass and the slightly smaller force constants in germanium, however, give vibrational frequencies for the bulk and the surface of Ge, which are smaller by one common factor when compared to those of Si [345,347]. For individual surface phonon modes, like the dimer-stretching oscillation, small deviations from this behavior are found. Important new information about the vibrational states in Ge(0 0 1) can be obtained by comparing the results from theory [346] with those recently obtained by means of inelastic helium-atom scattering [381]. In the [1 0 0] direction (diagonal to the dimer rows), domains with p(2;1) and p(1;2) orientation give the same dispersion curves. We therefore restrict a detailed  is the zone boundary point of the (1;1) discussion of our results to the CM direction, where M structure, which is equivalent to the J  point of the second Brillouin zone of the p(2;1) phase. Since  HAS is generally restricted to the measurement of phonon modes in the low-frequency range, we focus on vibrational modes with energies from 0 to 10 meV. Fig. 29 illustrates the measured phonon dispersion of Ge(0 0 1) along the [1 0 0] direction and compares the data with phonon branches computed for the p(2;1) reconstruction by means of DFPT. The experimental results are indicated by the crosses, while localized surface states obtained by theory are represented by solid lines. The RW close to the zone center and its branch at 6 meV in the near of the M  point are explained very well by DFPT. However, the acoustical mode close to the M  point and the flat branch at about 6 meV near the zone center are not seen in the dispersion curves computed for the p(2;1) phase. The A mode present in the calculated dispersion at about 7 meV has shear-horizontal  polarization. Therefore, the A mode does not give any significant contribution to the differential  reflection coefficient as can be seen from the small magnitude of the computed scattering intensities indicated by the gray scale in Fig. 29. This deviation is crucial since the experimental data giving the mode at about 7 meV are related to a strong feature present in the time-of-flight spectra. Moreover, the linear dispersion of the lowest branch close to the M  point indicates, that this zone boundary point is equivalent to the zone center point of a probably prevalent higher-order

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reconstruction. As can be seen from Fig. 23, the J  and J  point of the p(2;1) structure are folded  onto the C point of the p(2;2) phase. Hence, the appearance of additional branches resolved by experiment underlines that the p(2;1) structure cannot describe conclusively the phonon dispersion of Ge(0 0 1) at room temperature. Important contributions to the phonon spectra have to come from HORs. The situation is similar to the electronic dispersion of the Si(0 0 1) surface: While the p(2;1) reconstruction model is not able to account for all electronic states resolved by photo-emission spectroscopy [103,184,185], the surface states computed for the p(2;2) and c(4;2) phase are in excellent agreement with the experimental data [124,273]. 6.4.2. Local coupling transfer A simple method to estimate the dispersion for the HORs is to fold the phonon branches computed for the p(2;1) reconstruction back to the smaller Brillouin zones of the p(2;2) and c(4;2) phase. This approach, however, does not include the correlation along the dimer rows. The coupling of dimers leads to significant changes in the electronic and vibrational structure. An example is the frequency of the RW determined for Si(0 0 1) at the J and J  point of the p(2;1) structure. Table 4 shows that the correlation of the dimers separates the corresponding modes determined for the c(4;2) phase by more than 2 meV, while the vibrational states are nearly degenerate in the p(2;1) reconstruction. Therefore, it is crucial to calculate the dynamical matrices for the HORs. In particular, nearest neighbor and second-nearest neighbor force constants are sensitively dependent on the alternating buckling along the rows of dimers. The direct ab initio calculation of the surface phonon dispersion of Ge(0 0 1) p(2;2) and Ge(0 0 1) c(4;2) is computationally very demanding. Since buckled dimers are the basic building blocks of all reconstructions, however, the force constants determined for the tilted-dimer p(2;1) geometry can be used to obtain the dynamical matrices for the HORs by adapting the coupling constants to the local geometry. By this, no additional numerical labor beyond that required for the p(2;1) geometry is necessary. As input for the recently developed local coupling transfer (LCT) approach [346], only the atomic equilibrium positions must be calculated for all reconstructions. In addition, the dependence of the local coupling constants as a function of the nearest-neighbor distances and the relative orientation of the bonds has to be determined. The LCT method is quite general and can be applied to the computation of the phonon dispersion of large-size superstructures, provided that realistic force constant matrices of similar smaller systems are known. A detailed description of the approach is given in Ref. [346,347]. For any pair of atoms at RlM and RlM , the force constants between the two atoms U (lM a, lM a) in general ? Y?Y GH give a non-symmetric 3;3 matrix. However, for nearest-neighbor pairs, the matrices UL L are GH found to be almost symmetric. By representing UL L in terms of the real parts of the eigenvalues and GH eigenvectors, one can identify one stretch force constant U and two transverse force constants 1 U  and U , which correspond to eigenvectors V , V , and V . While the transverse force 2 1 2 2 2 constants U K (m"1, 2) are nearly identical throughout the crystal film, the longitudinal coupling 2 parameters U change linearly with the bond length. Using this information together with the 1 knowledge of the direction of the bonds in the HORs, the respective coupling constants and eigenvectors can be determined for all nearest-neighbor pairs in the p(2;2) and c(4;2) phase. This provides all data needed to determine a complete set of nearest-neighbor force-constant matrices for the HORs.

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Fig. 30. Phonon dispersion of the Ge(0 0 1) surface along the [1 0 0] direction measured by HAS (crosses) and computed for the c(4;2) and p(2;2) phase (solid lines). The M  point is a zone edge point of the basic (1;1) geometry and is equivalent to the J (C ) point of the c(4;2) (p(2;2)) structure. For a comparison with the scattering data, the magnitude of the calculated differential reflection coefficient is illustrated by the gray scale. In the sketched window, 6 to 8 resonant bands with complicated displacement patterns are observed. Therefore, we refrain from continuing the calculated dispersion curves within these areas.

The coupling constants between atoms at greater distance, which have only a small effect on the phonon dispersion, are adapted from the p(2;1) structure to the antiferromagnetic ordering and subsequently symmetrized with respect to the mirror plane normal to the [1M 1 0] direction. The on-site matrices U (lM a, lM a) are determined by using the translational invariance of the slab. Finally, GH the dynamical matrices are corrected to assure also rotational invariance of the system. The accuracy of the LCT method has been tested for Si(0 0 1) by constructing the dynamical matrix at the J point of the c(4;2) phase from the real-space force constants determined for the tilted-dimer p(2;1) reconstruction. This matrix compares very well with the one directly calculated by means of DFPT. Particularly, the splitting between the RW and the corresponding folded branch by more than 2 meV at the J point is reproduced using the dynamical matrix obtained from LCT. 6.4.3. Phonon dispersion of the higher-order reconstructions Fig. 30 illustrates the measured phonon frequencies of the Ge(0 0 1) surface together with the dispersion computed for the c(4;2) and p(2;2) phase along the [1 0 0] direction. Because of the doubling of the unit cell relative to the p(2;1) structure, folded phonon branches corresponding to

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the modes illustrated in Fig. 29 are prevalent besides the modes RW, A , and A . In particular, the   flat branch measured at about 7 meV close to the C point can be related to the folded Rayleigh wave RW in both, the c(4;2) and p(2;2) reconstruction, while the 7 meV feature is completely D missing in the spectrum of the p(2;1) geometry. This demonstrates that clear fingerprints of the HORs are present in the phonon dispersion of Ge(0 0 1) measured at room temperature. A vibrational state that is also characteristic for both HORs is the twisting mode t. It has been found for Si(0 0 1) c(4;2) at 7.1 meV. In the phonon dispersion of Ge(0 0 1), it is present near the zone center and the M  point with an energy between 6 and 8 meV. Not displayed in Fig. 30 is the rocking mode. It has energies ranging from 13 to 14.5 meV. In agreement between experiment and theory, the rocking mode produces only weak peaks in the scattering intensities. Interestingly, signatures of both, the p(2;2) and c(4;2) structure are found in the phonon spectrum. The M  point of the (1;1) geometry is equivalent to the zone center of the p(2;2) phase and corresponds to J in the c(4;2) structure. Therefore, the low-frequency branch appearing in the measured spectrum with linear dispersion close to the M  point can only be related to the RW in domains with p(2;2) symmetry. The data points seen in the experiment close to M  at about 5 meV, however, are not reproduced by any phonon mode computed for the p(2;2) arrangement. These points can be explained very well by the RW calculated for the c(4;2) geometry. These observations can be interpreted in the following way: The decreasing intensity of higherorder peaks observed in diffraction experiments indicates thermally excited dimer flipping processes. Since the phonon dispersion computed for the p(2;1) phase cannot account for all features seen by HAS, only the long range order is destroyed by dimer flipping, while local order with p(2;2) or c(4;2) still dominates the vibrational spectrum. Both structures have to be prevalent on the Ge(0 0 1) surface at room temperature. Because of the correlation along the rows, dimers are likely to flip simultaneously as observed for Si(0 0 1) in the molecular-dynamics simulations by Shkrebtii and co-workers. Correlated flipping induces a change from local c(4;2) to local p(2;2) symmetry and vice versa. 6.5. Surface vibrations in C(0 0 1) p(2;1) Because of the much stronger bond in the carbon dimer, the diamond structure C(0 0 1) surface is reconstructed into a symmetric p(2;1) configuration of non-buckled dimers [153,208]. We therefore discuss the surface dynamics of this system separately from Si and Ge(0 0 1). The calculations were carried out by Alfonso and co-workers [10] on the basis of the Harris functional approach, using the local-orbital method of Ref. [314]. In the computations crystal films spanning 10 atomic layers were used. Per layer, the slabs contain 8 atoms. The dangling bonds of the bottom layer are terminated by hydrogen, while the top layer atoms are forming dimers. For the k-point sampling, two special points were used. All the atoms except those of the bottom layer are allowed to relax. Optimized atomic positions were determined with the help of ab initio moleculardynamics simulations by gradually reducing the kinetic energy of the system. The clean surface forms symmetric dimers with a bond length of 1.36 A> . This is in good agreement with the density-functional calculation of Kru¨ger and Pollmann, who found that tilted dimers always relax back into the same symmetric configuration [208]. Due to the dimerization, the first interplanar spacing is reduced by 24% with respect to that of the bulk [10]. Moreover, atomic relaxations in subsurface layers are observed.

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The frozen-phonon approach was applied to compute the dynamical matrix at the zone center point of the supercell. Since there are eight atoms per layer, in-phase and out-of phase vibrations of adjacent dimers along and across the rows can be investigated. Pronounced surface phonon modes of C(0 0 1) p(2;1) involve rocking, swinging, twisting, bouncing, and stretching motions of the dimers coupled with subsurface layer vibrations. With the exception of the stretching vibration, all of these modes have the character of hindered translations (swinging and bouncing modes) or rotations (rocking and twisting vibrations) of the dimers [10]. The highest-frequency mode on C(0 0 1) is a dimer-stretching vibration with an energy of 184.9 meV which is about 20 meV above the bulk continuum. Because of this, the stretching oscillations of adjacent dimers couple only weakly to atomic subsurface vibrations. Slightly below the upper rim of the projected bulk bands, which is at 165 meV, the frozen-phonon approach gives a dimer swinging mode with an energy of 153.3 and 126.5 meV for the in- and out-of phase motion of adjacent dimers. The corresponding vibrational energies of the twisting mode are 163.0 and 157.9 meV. The rocking mode is found at 151.1 and 75.0 meV, while the dimer-bouncing vibration occurs at 87.9 and 62.4 meV. In all cases, in-phase vibrations of neighboring dimers are characterized by higher frequencies than out-of phase oscillations. The larger frequencies characterizing the in-phase dimer oscillations are related to larger contributions from atomic motions in the second and deeper layers. A similar behavior was found for Si(0 0 1). As can be seen from the phonon dispersion of the p(2;1) phase illustrated in Fig. 24, the vibrational frequencies are overall lower along the KJ direction. Only a few experimental data for the surface dynamics in C(0 0 1) are available. Lee and Apai applied HREELS to measure surface phonon frequencies in C(0 0 1) p(2;1). Two distinct features at 87 and 152 meV as well as a weak peak at about 126 meV have been identified in the energy-loss spectra. For the interpretation of their experimental data Lee and Apai used scaled frequencies of Si(0 0 1) p(2;1) surface phonon modes. With the exception of the dimer-stretching vibration, the scaling method is useful for a qualitative interpretation and shows that many vibrational states are related to the existence of dimers on these surfaces. Dimer-stretching modes give a more detailed insight. In Si and Ge(0 0 1), these phonon states have lower energies than the highest optical modes. The corresponding mode in C(0 0 1), however, lies above the bulk continuum. This directly reflects the much stronger bond in the carbon dimers, which indicates a pronounced tendency of n bonding in C—C pairs in contrast with the much weaker dimer bond in Si and Ge(0 0 1). Another important difference between C(0 0 1) and Si or Ge(0 0 1) is the appearance of a dimer back-bond mode (sb) which is characteristic for the tilted-dimer reconstruction. In C(0 0 1), no similar state is present in the phonon spectrum. Hence, the mode sb is a clear fingerprint of asymmetric dimers on the (0 0 1) surface of Si and Ge.

7. Structure and dynamics of the Si(1 1 1) surface A pioneering experiment, which stimulated the detection of microscopic vibrations in semiconductor surfaces by means of electron-energy loss spectroscopy, was the measurement of a pronounced 56-meV optical mode observed by Ibach [178] for Si(1 1 1) (2;1). Motivated by this discovery, several theoretical investigations aimed to provide a microscopic picture for the dipole-active mode. However, model calculations based on the tight-binding approach and the bond-charge model gave conflicting interpretations for the excitation seen at 56 meV [9,244].

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Driven by the fundamental interest in this controversially discussed vibrational feature, the Si(1 1 1) (2;1) surface was one of the first examples for the application of DFT to the calculation of semiconductor surface phonon modes. With the help of Car—Parrinello molecular-dynamics simulations [18] and DFPT [19], the origin of the well-known 56-meV mode was successfully clarified. The results of these investigations are in good agreement with the energy-loss spectra of Ref. [178] and those from newer measurements based on HREELS [256,286] and HAS [156]. One important observation of the HAS experiment is that the branch associated with the Rayleigh mode rises above the flat branch of a mode seen at 10.5 meV for about half of the samples, while this is not the case for the other half of the cleaves. This “sample-dependent behavior” has not been explained by any of the theoretical approaches discussed in Refs. [9,18,19,244]. 7.1. Atomic surface geometry of Si(1 1 1)(2;1) The (2;1) reconstruction of Si(1 1 1) results from cleaving the material at low temperatures. As illustrated in Fig. 31, silicon and other elemental semiconductors are characterized by a doublelayer structure along the [1 1 1] direction. The two individual planes which form a bi-layer are connected by three bonds per atom and are separated by a distance of a /4(3, where a is the   cubic lattice constant of the bulk. Adjacent bi-layers are connected by only one bond per atom along the [1 1 1] direction and are separated by a distance of 3a /4(3. In each of the (1 1 1) planes,  the atoms are arranged in a hexagonal two-dimensional lattice with a lattice constant a "a /(2. ,  The number of bonds between neighboring bi-layers is three times smaller than that within a double-layer. Therefore, cleavage leads to (1 1 1) surfaces which are terminated by a complete bi-layer with one dangling bond per surface atom as illustrated in Fig. 31. Each dangling bond of the ideal surface is occupied by one electron. Through rebonding and rehybridization, the silicon surface undergoes a (2;1) reconstruction for which several models have been suggested. A conclusive explanation for all existing experimental data is given by Pandey’s n-bonded chain model [279,280] which is illustrated in Fig. 32. Starting from the ideal surface geometry (Fig. 31), the n-bonded surface chain structure is obtained by replacing the bonds between the atoms labeled as 2 and 3 by bonds between the atoms 1 and 3. As a consequence of the rebonding, the atoms indicated by 1 increase their coordination from three to four, while the atoms labeled as 2 reduce the number of bonding partners to three. To establish bond lengths that are close to the nearest-neighbor distance in bulk silicon, the atoms 1 are shifted downwards, while the atoms 2 are shifted upwards. The rebonding and the vertical dislacements lead to zigzag chains along the [1 0 1] direction formed by the atoms 2 and 4 in the first layer and by the atoms 1 and 5 in the subsurface layer. The zigzag chains are joined to the underlying bulk by fivefold and sevenfold rings. In the (2;1) unit cell there are two dangling bonds located at the threefold coordinated first-layer atoms. Through a Jahn-Teller like mechanism, the first-layer chains of the cleaved Si(1 1 1) surface are tilted. A positive tilt angle leads to the “chain-low” configuration, with the atoms labeled as 2 shifted above the atoms labeled as 4. In the “chain-high” arrangement the surface chains are tilted in the opposite direction. In a recent computation of the structural and dynamical properties of Si(1 1 1) (2;1) on the basis of DFPT [405], crystal films spanning ten atomic layers were used with a total number of 20 silicon atoms in the (2;1) unit cell. The dangling bonds of the bottom layer of the slabs were saturated by

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Fig. 31. Side and top view of the (1 1 1) surface of an elemental semiconductor. Corresponding atoms in the side and top views are numbered. The dangling bonds on top of two first-layer atoms are indicated.

hydrogen, while the top layer atoms were allowed to reconstruct according to the chain-low and chain-high geometries. The electronic wave functions were expanded in plane waves with a cutoff energy of 9 Ry. Brillouin zone integrals were replaced by summations over six special points. The atomic equilibrium positions for positive and negative buckling of the surface chains were determined by means of the Hellmann—Feynman forces, starting the relaxation from configurations which are close to the respective local minima. The calculations of Ref. [405] gave the following relaxation parameters: In the positions of minimal energy, all of the bond lengths do not deviate by more than 3.6% from the nearestneighbor distance in bulk silicon. As illustrated in Fig. 32, the vertical displacement between the atoms 2 and 4 is 0.44 A> for the configuration with positive buckling of the chains, while it is 0.50 A> for the opposite sign of the tilt. These numbers compare very well with the measured data reported in Refs. [162] and [365] and the results of other ab initio calculations [18,83,271,272]. In agreement with the molecular-dynamics simulations of Ref. [18], the energy difference between the chain-high

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Fig. 32. Surface geometry of the Si(1 1 1) (2;1) reconstruction. Corresponding atoms in the side and top views of this figure and of Fig. 31 have the same numbers. The two possible tilt angle orientations of the surface chains are illustrated. For positive buckling (chain-low), the vertical displacements of the first-layer atoms is 0.44 A> , while it is 0.50 A> for negative buckling (chain-high). Also illustrated are the SBZs of the (1;1) surface (dotted lines) and the (2;1) reconstruction (solid lines).

and the chain-low arrangement is very small. While the final state found in Ref. [18] is the configuration characterized by a positive buckling of the chains, the computations of Ref. [405] slightly favor the chain-high arrangement by 2.7 meV per surface atom. Such energy differences have the same order of magnitude as the numerical uncertainties and depend on the numerical details of the computations. Therefore, some ab initio approaches favor the chain-high configuration [2,83,405], while other calculations yield a lower surface energy for a positive buckling of the chains [18]. 7.2. Phonons in Si(1 1 1)(2;1) We now focus on the results obtained from the DFPT calculations of Ref. [405] for the phonon dispersion of Si(1 1 1) (2;1) along the chain direction. The computations were carried out for the positive (“chain-low”) and the negative (“chain-high”) tilt angle of the surface chains. Fig. 33 illustrates the phonon dispersion for wavevectors parallel to the chain direction. The calculations were carried out for both orientations of the buckling. As can be seen from the figure, the dispersion curves of the two systems are very similar. Corresponding to that, also the eigenvectors are nearly

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Fig. 33. Phonon dispersion of the Si(1 1 1) (2;1) surface computed by means of DFPT [405] for the chain-low (a) and chain-high (b) configuration. Experimental data are indicated by the square (HREELS, Ref. [178]) and the circles (HAS, Ref. [156]).

the same for the chain-high and the chain-low configuration. The main results can be summarized as follows: In agreement with the calculations of Refs. [18,19,244], several phonon modes were found above the continuum of the bulk bands. At about 56 meV, the phonon dispersion exhibits three vibrational features close to the zone center. While the state at 56.7 meV is a resonance with longitudinal optical polarization, two localized eigenmodes are found at 55.6 and 58.4 meV for the chain-high arrangement. For the opposite buckling of the chains, DFPT yielded the energies 55.6 and 58.6 meV for the localized modes and 56.7 meV for the longitudinal optical resonance. The mode appearing at 58.4 meV (58.6 meV) has the character of a longitudinal optical surface phonon. It mainly consists of an opposing motion of the atoms in the first-layer atoms and smaller oscillations of the second-layer atoms parallel to the chain direction. The atoms of the deeper layers are essentially at rest. In the tight-binding calculations of Ref. [9], this phonon mode was found with an energy of 51.0 meV, while the bond-charge model revealed a similar state at 52.5 meV. The other surface-localized mode found by DFPT has a C point energy of 55.6 meV and shows atomic motions mainly in the third and forth layer and therefore corresponds to the “interface mode” reported in the literature [18,19,244]. It exhibits large atomic displacements normal and parallel to the surface. Finally, the resonance determined at 56.7 meV within DFPT is similar to a vibrational state found at about 56 meV by means of the partial-density formalism [56]. The phonon dispersion illustrated in Fig. 33 was determined for crystal films spanning a large number of atomic layers, using the slab-filling method described in Section 2.1.2. Although slabs with only six to eight (1 1 1) layers were used in the investigations of Refs. [18,19], these calculations revealed similar features. A longitudinal optical mode with large dipole activity was found at

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57.2 meV, while an interface vibration with normal polarization but considerably small dipole moment was revealed at 55.8 meV. As discussed in Refs. [18,19], both of the modes are necessary to provide a conclusive explanation of the available experimental data. The dipole-active longitudinal mode accounts for the anisotropy of the loss peak seen in the electron-scattering experiments of Ref. [256]. The isotropic contribution to the scattering cross section, which was resolved in the experiments of Ref. [286], can be attributed to the interface vibrations. In the tight-binding calculations of Ref. [9], only the dipole active longitudinal mode was found. The bond-charge model revealed a longitudinal vibration and an interface mode with displacements of the atoms normal to the surface [244]. However, the dipole moment computed in the BCM for the longitudinal mode was comparably small. Hence, none of the model calculations could entirely explain the details of the experimental data. The results obtained in the ab initio calculations of Refs. [18,19,405], however, combine the findings of the previous model calculations and therefore account for all of the details in the measured spectra. In the low-frequency region, the surface phonon dispersion was measured by means of HAS [156]. An interesting feature resolved in the TOF spectra beside the RW is a flat branch, with a zone boundary energy at the J point of about 10.2 meV. For about half of the samples, the branch associated with the RW was observed to rise above the 10.2 meV feature, while for the other half of the cleaves, this branch seemed to merge into the flat mode. While the phonon spectra calculated in Refs. [9,18,19,244] provide an explanation for the flat mode, which is associated with a broad band of resonances near the C point, no interpretation was given for the behavior of the RW. In the dispersion illustrated in Fig. 33, resonant states can be identified in the region indicated by the dotted lines, in agreement with the previous studies. Moreover, it is interesting to note that the zone boundary frequency computed for the RW within DFPT is 12.3 meV for the chain-high configuration, while it is 11.1 meV in the case of the chain-low arrangement (see Fig. 33). The energy difference of 1.2 meV compares very well with the splitting of the two distinct branches seen in the HAS experiments, which have an energy of 11.2 and 10.2 meV when approaching the J point. The partial-density calculations of Ref. [56] revealed two modes below the bulk continuum. However, this cannot account for the observation that the higher branch is present in the TOF spectra only for half of the samples. One possible interpretation is, that for the cases where the RW was reported to rise above the flat mode, the HAS experiment probably detected the vibrational modes of the chain-high configuration, while for the other half of the samples, the phonons of the chain-low arrangement were resolved. Because of the small energy difference between the two geometries, it can be assumed that domains with positive and negative buckling of the zigzag chains coexist on the surface or that the cleavage of the surface results in either of the tilt orientations. 7.3. Phonons in the Si(1 1 1)(7;7) surface As a result of temperature treatment, the (2;1)-cleavage structure of the Si(1 1 1) surface can convert to the famous (7;7) reconstruction, which was observed for the first time by Schlier and Farnsworth [323]. Because of the large number of atoms in the (7;7) unit cell, density-functional calculations became feasible for this system only recently. By means of the conjugate-gradients formalism [137,285,360], combined with the wave-function extrapolation method [22] and the use of massive parallel computing, S[ tich was able to perform successfully ab initio molecular-dynamics

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simulations for the computation of surface vibrational modes on Si(1 1 1) (7;7) [343,344]. Prior to that, density-functional cluster calculations were carried out only for small systems which were chosen to describe the local atomic structure and the vibrational signatures of a Si-adatom [88]. However, these investigations suffer from finite-size effects and related to that from the incomplete treatment of the charge transfer between the dangling bonds of the (7;7) surface. For the Si(1 1 1) surface reconstructed into the dimer-adatom stacking fault (DAS) configuration [356], electron-energy loss spectroscopy revealed the branch of a split-off vibrational mode at 70.8 meV, about 10% above the bulk continuum, and a broader feature at 29.8 meV [88]. The cluster calculations of Ref. [88] associated the split-off mode and the state at 29.8 meV with vibrations mainly located at the second-layer atoms and the adatoms. To find an explanation for the energy losses, S[ tich performed highly efficient molecular-dynamics simulations for crystal films spanning eight atomic layers, with a total number of 400 silicon atoms. The atomic trajectories were determined for an average temperature of ¹+200 K, using a time step of 1 fs and a total simulation time of 1.2 ps. Surface vibrational states were identified by analyzing the Fourier spectra for individual atomic sites, using the multiple signal classification scheme (MUSIC) [267,325] and its self-consistent extension [201,344] (see Section 2.2). The most prominent features seen in the spectra of the conjugate-gradient molecular-dynamics simulations of S[ tich are four main peaks located at 14.9, 29.8, 47.2, and 67.0 meV. Because of its high frequency, the split-off mode at 67.0 meV is particularly interesting. The peak at 67.0 meV corresponds to the energy-loss feature seen at 70.8 meV. It is associated with an opposing motion of atoms in the second and third substrate layer, which are directly below silicons atom residing at ¹ adsorption sites. The presence of the silicon-adatoms leads to a compression of the respective  bonds between the second and third layer and therefore to a slightly increased bond-stretching frequency. The vibrational mode identified in the molecular-dynamics simulations at 29.8 meV is dominated by in-plane and out-of plane motions of the adatoms. Its computed frequency is in excellent agreement with the results from electron-energy loss spectroscopy. The results reported by S[ tich are different from those obtained in the cluster calculations of Ref. [88], in which the adatom configuration was described by means of a Si cluster. Two additional features were  obtained within the ab initio molecular-dynamics: The surface vibration at 14.9 meV is a wagging mode which involves motions of the adsorbed atoms and the second-layer atoms directly below the adatoms. At 47.2 meV, a vibrational state located at rest atoms was identified. Particularly, the conjugate-gradients molecular-dynamics simulations of S[ tich demonstrate that the density-functional schemes developed for the calculation of semiconductor surface vibrations became highly efficient in the course of the last decade.

8. Adsorbate covered semiconductor surfaces The adsorption of atoms or molecules can lead to fundamental changes in the atomic structure and the electronic properties of a semiconductor surface. In many cases, the dangling bonds of the clean surface are active sites for the formation of new bonds, which are then generally involved in the chemisorption configuration. Related to the introduction of additional atoms into the surface region, a significant redistribution of electronic charge can occur. For the formation of a chemical bond, two electrons are needed, which often stem only partially from the adsorbed atom or

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molecule. For the example of hydrogen adsorption, also the surface has to provide one electron per bond. Moreover, chemisorbed atoms like group-III or group-V elements on GaAs(1 1 0) introduce new dangling bond states into the system. Therefore, the basic principles responsible for the formation of stable surface structures (chemical bonding and autocompensation) can lead to completely different atomic arrangements, when comparing a clean surface with an adsorption configuration. A well-known example is the hydrogen coverage on the III—V(1 1 0) surfaces. The atomic structure of the clean surfaces results from a relaxation-related rehybridization of the anion-derived and cation-derived dangling bonds. The sign of the tilt angle and hence the kind of rehybridization is predetermined by the autocompensation principle, which requires that the anion-dangling bonds are occupied and those of the cations are empty. With the adsorption of hydrogen, the situation is fundamentally different, as electrons have to occupy at least partially cation-derived dangling bond states. Since the energy of these states is increased by the relaxation of the clean surface, the chemisorption of hydrogen has to induce a removal of the bond-angle relaxation. Similar processes occur in a large variety of other adsorption systems. The exploration of surface phonon modes in chemisorption systems therefore is interesting under many aspects. One important application of probing adsorbate vibrations is based on the fact, that chemical bonds can be identified by characteristic phonon frequencies appearing in the vibrational spectra. This allows one to monitor active adsorption sites on a surface during its exposure to the chemisorbing material. A commonly used experimental technique is HREELS, which is primarily sensitive to dipole active surface excitations, such as the Fuchs—Kliewer phonon and stretching vibrations of adsorbed atoms or molecules along the bonding direction [95,230,263,264,288,289,320]. Moreover, the frequencies of adsorbate vibrations directly characterize the bonding site. Hydrogen, for instance, generally chemisorbs on dangling bonds forming an overlayer above the surface. In contrast, boron is known to occupy second-layer substitutional positions in Si(1 1 1) [38,158,193,232]. Therefore, the vibrational spectrum of the B : Si(1 1 1) adsorption system has to show clear indications for the cage-like bonding configuration of boron with a phonon frequency that is comparable to that of the impurity mode of substitutional boron in bulk silicon. In the case where the clean surface geometry is modified only slightly by the adsorption, many vibrational states of the uncapped system are expected to be still present after the chemisorption. Under such conditions, only vibrational states are affected, which are particularly related to the structural details of the clean system. By comparing the vibrational states of the uncovered surface and those of the chemisorption geometry, clear fingerprints of the relaxation or reconstruction can be identified. Macroscopic modes (like the RW) should not be disturbed by the adsorption. Microscopic modes and the modification of their frequencies and eigenvectors can provide detailed information about the change in the local symmetry and configuration introduced by the adsorption. In this section, we summarize the results from density-functional calculations (Refs. [43,100,102,111,123,167,330]) and experiments (Refs. [78,95,230,263,264,288,289,320]) for the adsorption of hydrogen and other elements on the (1 1 0) surfaces of III—V compounds. Other systems discussed in this section are hydrogen coverages on the (0 0 1) and (1 1 1) surfaces of silicon, germanium, and diamond (Refs. [10,23,129,168,192,225,404]), the adsorption of As on Si(0 0 1) [23], as well as the chemisorption of group-III elements on Si(1 1 1) [102,170,270].

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8.1. Hydrogen coverage on III—»(1 1 0) surfaces The adsorption of hydrogen on the III—V(1 1 0) surfaces gives prototype systems for the research of chemisorption processes on semiconductors. In recent years, detailed information about the structure and the vibrational properties of hydrogen-covered (1 1 0) surfaces of binary compounds like GaAs, GaP, InP, and InAs has become available by density-functional calculations (Refs. [43,44,100,102,111,123,126]) and HREELS (Refs. [78,95,230,263,264,288,289,320]). By means of the HREELS experiments and other techniques like core-level photoemission spectroscopy [319] and ultraviolet spectroscopy [295], it has been demonstrated for GaAs, GaP, and InP(1 1 0), that hydrogen atoms bind simultaneously to the surface anions and cations in each stage of the adsorption. The data of Refs. [78,319] for GaAs and GaP(1 1 0) indicate that the hydrogenation saturates at one monolayer (ML) with two hydrogen atoms per surface unit cell. For InP(1 1 0), however, Auger electron spectroscopy and LEED [243] show, that the adsorption of hydrogen on this surface saturates at the stage of 0.5 ML with only one hydrogen atom per surface unit cell. An important effect of the hydrogenation is, that the characteristic relaxation of the III—V(1 1 0) surfaces is removed as a result of the saturation of the dangling bonds. Evidence for this is given by the experimentally observed decrease of the ionization energy [189,243] and the data from photoemission spectroscopy [25,243]. Atomically resolved measurements of the hydrogen covered GaAs(1 1 0) surface have been made by means of surface-shift photoelectron diffraction [308] and grazing incidence X-ray diffraction (GIXD) [307]. The removal of the relaxation seen in the experiments is confirmed by the computations of Refs. [43,100,102,111,123,126,234,276,392] which predict a slight counterrotation in the first substrate plane at coverages of 0.5 and 1 ML. Interesting vibrational states of the chemisorption systems are the stretching and bending modes of the hydrogen atoms bonded to the surface anions and cations. While stretching modes are seen as pronounced features in the energy-loss spectra of inelastically scattered electrons (Refs. [78,95,230,264,288,289,320]), bond-bending oscillations have been detected only recently with the help of HREELS [263]. Since the phonon dispersion curves have been computed using DFPT for both, the clean and the covered surfaces of all III—V compounds considered here, clear fingerprints of the bond-angle relaxation can be identified by comparing the vibrational spectra of the free surfaces with those of the hydrogen covered surfaces. The dynamical properties of the hydrogenated III—V(1 1 0) surfaces have been investigated by means of DFPT [100,102,123,126], the frozen-phonon approach [43], and Car—Parrinello molecular-dynamics simulations [44,111]. In the linear-response approach, periodically repeated slabs have been used. The slabs comprise seven substrate layers which form thin films covered on both surfaces with hydrogen. Two neighboring slabs are separated by a distance which is equal to six interlayer distances prior to hydrogenation. The adsorbed atoms are placed in the dangling bond direction of the substrate. For a coverage of 0.5 ML, hydrogen is introduced into the system either only above surface anions or cations, while all dangling bonds are saturated in the 1 ML case. Brillouin zone integrations are carried out using six special points. The electronic wave functions are expanded in plane waves up to a kinetic energy of 10 Ry. For H : GaP(1 1 0), the cutoff energy is 12 Ry. Similar geometries have been considered in the other theoretical investigations. In the frozenphonon approach applied to H : GaAs(1 1 0) [43], the same number of atomic layers and the same

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Fig. 34. Equilibrium geometry of a III—V(1 1 0) surface covered by 1 ML of hydrogen illustrated in a side view of the first four substrate layers. Black circles: anions; open circles: cations; grey circles: hydrogen. The relaxation parameters ½ , ½ ,  Z , and Z indicate the atomic shifts in the first layer with respect to an ideal (bulk-like) geometry. The bond lengths of the  hydrogen atoms adsorbed on anions and cations are denoted by d and d . 

cutoff energy were used as in the DFPT calculations. However, the special-points summation was restricted to only one k-point. The ab initio molecular-dynamics simulations of Refs. [44,111] performed for H : GaAs(1 1 0) were carried out for periodically repeated slabs comprising five substrate layers. The complete saturation of the bottom-layer dangling bonds of the thin films was used to study the vibrational modes of 1 ML hydrogen chemisorbed on GaAs(1 1 0). In addition to that, a 0.25 ML configuration with two hydrogen atoms per p(2;2) unit cell was considered in Refs. [44,111]. More details of these computations are summarized in Section 5.2. 8.1.1. Structure of the hydrogenated surfaces Fig. 34 illustrates the (1 1 0) surface of a III—V compound covered by 1 ML of hydrogen. In the equilibrium geometry, the substrate atoms are close to their ideal (bulk-derived) positions. Table 5 summarizes the results obtained within our computations for the chemisorption of 1 ML. The numbers are in good agreement with those found in the other ab initio approaches [43,111,234,276,392] and confirm the experimentally observed removal of the substrate relaxation [25,189,243]. For H : GaAs(1 1 0) experimental data are available. The computed surface geometry compares very well with structural parameters measured by GIXD [307]. For both coverages (1 and 0.5 ML), hydrogen adsorption eliminates the bond-angle relaxation of the clean surface, which is replaced by a slight counter rotation in the first layer. In the 1 ML case, the magnitude of the angle "u", which characterizes the counter relaxation, increases with the lattice constant of the substrate. Lower coverages, however, do not entirely remove the relaxation of the clean surfaces.

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Table 5 Optimized structure of the (1 1 0) surfaces of GaP, InP, GaAs, and InAs covered by 1 ML of hydrogen. The atomic shifts and bond lengths are given in A> . The quantities are defined in Fig. 34. The data obtained for H : GaAs(1 1 0) from gracing incidence X-ray diffraction (GIXD) are taken from Ref. [307] ½

½ 

Z

Z 

d

d 

u

H : GaP(1 1 0) H : GaAs(1 1 0) H : InP(1 1 0) H : InAs(1 1 0)

0.069 0.078 0.088 0.096

0.078 0.084 0.102 0.115

!0.078 !0.089 !0.101 !0.099

0.079 0.086 0.092 0.113

1.447 1.535 1.441 1.531

1.552 1.571 1.734 1.732

!6.7° !7.1° !7.8° !8.0°

GIXD (GaAs)

0.110

0.055

!0.075

0.040

!5.0°

This has been shown by means of Car—Parrinello molecular-dynamics simulations for 0.25 ML of hydrogen chemisorbed on GaAs(1 1 0) [111]. The relaxation parameters listed in Table 5 show, that the bonding distance of hydrogen adsorbed on an anion or cation depends only slightly on the particular compound. The largest differences are seen for the H-Ga bond length, which amounts to 1.552 A> in H : GaP(1 1 0) and 1.571 A> in H : GaAs(1 1 0). The variation of the bond lengths is similarly small when comparing the results obtained for a coverage of 1 and 0.5 ML [123]. Also the stretching-mode frequencies of the hydrogen-surface bonds change only by small amounts as a function of the coverage and in dependence on the III—V compound. Therefore, the formation of a particular chemical bond can be detected by probing the corresponding phonon mode frequencies. 8.1.2. Bond-stretching and bond-bending vibrations The HREELS experiments of Refs. [78,95,230,263,264,288,289,320] clearly resolve the stretching modes of hydrogen adsorbed on the (1 1 0) surfaces of GaAs, InP, and GaP. The associated energy losses are the most prominent features seen in the HREEL spectra besides the Fuchs—Kliewer phonon mode. Table 6 summarizes the C point frequencies obtained within DFPT [102,123] for a coverage of 1 ML and compares the vibrational energies with the HREELS data and the results from the ab initio calculations of Refs. [43,111]. The respective frequencies computed for a coverage of 0.5 ML are close to those of the 1 ML case. They are summarized for H : GaAs(1 1 0) and H : InP(1 1 0) in Ref. [123]. The dispersion of the stretching modes is negligible (41 meV) along and perpendicular to the first-layer anion-cation chains. This can be seen from the upper part of Fig. 36. For each atom adsorbed on the (1 1 0) surface of a III—V compound two bending vibrations are observed. One mode is polarized in the [1 1 0] direction, while the other one is characterized by displacements in the plane perpendicular to the anion-cation chains of the first substrate layer. The phonon branches associated with the bending modes are nearly flat showing a dispersion of less than 3 meV. They are illustrated in Fig. 36 for 1 ML hydrogen chemisorbed on GaP and InAs(1 1 0). The two lower bending modes exchange their vibrational character along the C X direction close to the zone center. For H : GaAs(1 1 0), we obtain four narrow bands between 62 and

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Table 6 Bond stretching frequencies l and l (in meV) computed at the C point for 1 ML of hydrogen adsorbed on GaAs and  InP(1 1 0) (first line) and 1 ML chemisorbed on GaP and InAs(1 1 0) (second line). The frequencies are compared with the experimental data from HREELS (Refs. [95,230,288,289,320]) and the LDA results of former calculations (Refs. [43,111])

Source of data

l  H—Ga

l H—As

l  H—In

l H—P

H on GaAs, InP H on GaP, InAs

222 224

259 264

204 205

278 272

[288,289] [320] [95] [230]

229 233 233 235

262 264 262 267

208 211 211

281 282 292

[111] [43]

219 218

246 242

Table 7 Bond bending frequencies (in meV) of 1 ML hydrogen adsorbed on GaP and InAs(1 1 0) (first line), 1 ML hydrogen on GaAs and InP(1 1 0) (second line), and 1 ML deuterium on GaAs and InP(1 1 0) (third line) calculated at the C point of the SBZ. The vibrations polarized in the [1 1 0] direction are labeled as (1), while the modes polarized perpendicular to the chain direction are labeled as (2) System

Ga(1)

Ga(2)

As(1)

As(2)

In(1)

In(2)

P(1)

P(2)

H on GaP, InAs H on GaAs, InP D on GaAs, InP

65.5 62.0 44.3

66.8 62.7 44.8

62.5 64.8 46.6

67.0 68.7 49.8

53.4 54.5 40.0

54.8 56.8 44.2

70.4 67.1 49.9

75.1 72.0 54.0

68 meV in agreement with the results from the frozen-phonon approach of Ref. [43], while molecular-dynamics simulations yield slightly lower bending mode frequencies [111]. For H : InP(1 1 0) and H : InAs(1 1 0), we observe a gap of at least 5 meV between the H—In and the H-anion vibrations. By means of HREELS, Nienhaus and co-workers [263] succeeded in measuring bending mode frequencies at a hydrogen-exposed InP(1 1 0) surface. The energy-loss spectra show a weak feature at about 54.3 meV. This compares with 56.8 meV computed for the H—In vibration labeled as In(2) in Table 7. The bending mode at the surface anions was detected by means of deuterium adsorption, which gives an energy loss at about 57.7 meV. The calculated frequency of the corresponding D—P vibration is 54.0 meV. Interestingly, the ratio of the hydrogen and deuterium bending mode frequencies is only slightly smaller than (2 for the vibrations on GaAs(1 1 0), while the ratio of corresponding hydrogen and deuterium bending oscillations is much smaller than (2 for the case of InP(1 1 0). This results from the fact that the bending mode frequencies for deuterium are close to the upper rim of the optical bulk modes particularly in the

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Fig. 35. Phonon dispersion of the (1 1 0) surfaces of GaAs and InP covered by 1 ML of hydrogen. For InP(1 1 0), the dotted lines indicate the gap phonon frequencies calculated for the relaxed clean surface, while the diamonds show the corresponding phonon frequencies of the ideal clean surface.

case of InP(1 1 0). Because of this, the bending oscillations of deuterium couple stronger to vibrations in the substrate layers than the corresponding hydrogen modes. 8.1.3. Substrate vibrations We now analyze the effects of the hydrogen induced derelaxation upon the vibrational modes in the (1 1 0) surfaces of GaAs, InP, GaP, and InAs. The phonon dispersion curves computed for a 1 ML coverage of hydrogen is illustrated in Figs. 35 and 36. From the comparison with the phonon spectra of the clean surfaces shown in Figs. 11, 14, 16 and 17, it is seen that most of the localized phonon modes are still present with similar dispersion and similar eigenvectors after the hydrogenation. Especially the two lowest branches in the CX and CX direction are not affected by the adsorption of hydrogen. However, the third acoustical branch observed in the CX direction is shifted slightly above the lower rim of the continuum close to the X  point in all cases. In H : GaAs(1 1 0) and H : InP(1 1 0) its dispersion cannot be resolved due to the strong mixing with bulk bands. For the clean GaAs(1 1 0) surface, this mode was detected close to the zone boundary as a strong feature in TOF spectra of inelastically scattered helium atoms [92]. A vibrational state of particular interest is the rocking mode of the clean surfaces. As pointed out for GaAs(1 1 0), this mode strongly mixes with bulk states and cannot be resolved throughout the Brillouin zone as a localized surface state. It appears at the C and X point with an energy of about 10.5 meV and contributes at least partially to the flat branch at about 10 meV observed in the HAS experiments (see also Section 5.3). Interestingly, the frequency of the rocking mode is nearly the same after hydrogenation. In H : GaAs(1 1 0), the flat branch observed in the dispersion close to the zone center at about 11 meV (see Fig. 35) originates in a rocking motion of the substrate atoms. The hydrogen atoms bonded to the anions and cations follow instantaneously the motion of the substrate atoms with slightly larger amplitudes. In analogy with H : GaAs(1 1 0), the rocking mode was found for the uncovered InP(1 1 0) surface with nearly the same energy for the ideal and relaxed geometry. Compared with the case of the relaxed surface, the mode is characterized by a stronger localization in the ideal structure [122]. For the hydrogen covered InP(1 1 0) surface, however, the

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Fig. 36. Phonon dispersion of the (1 1 0) surfaces of GaP and InAs covered by 1 ML of hydrogen. The dotted lines indicate the gap phonon frequencies calculated for the relaxed clean surfaces, while the filled circles show the corresponding phonon frequencies of the ideal clean surfaces.

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rocking mode strongly mixes with bulk phonon bands. This results from the large density of bulk states in this energy region. For all systems, we observe that the phonon mode above the bulk continuum obtained by theory and resolved by HREELS for the clean surfaces is shifted downwards into the optical bulk bands. It mixes strongly with bulk phonons and cannot be identified as a surface localized state. The moleculardynamics simulations of Refs. [44,111] show the same behavior for this surface optical mode. A common feature of the clean InP, GaP, and InAs(1 1 0) surfaces is the pair of modes that appears in the gap between the acoustical and optical bulk vibrations. As discussed in Section 5.5.1 and in Ref. [122], the gap modes are sensitive to even small variations in the details of the surface structure. The comparison of the phonon dispersion of the clean and hydrogen covered surfaces clearly underlines this conclusion. In addition to the phonon branches of the hydrogenated surfaces, Figs. 35 and 36 illustrate the gap mode frequencies of the clean relaxed surfaces by the dotted lines and the respective energies of the clean ideal surfaces, which are represented by the data points. In agreement with the observation for the ideal clean surfaces, hydrogenation results in a larger separation of the two gap modes. In all cases, the removal of the surface relaxation reduces the frequency of the lower branch and increases the vibrational energy of the upper gap mode. The splitting should be observable for H : InP(1 1 0) and H : GaP(1 1 0), where the two branches are placed far apart from the features resolved by HREELS for the clean surfaces [262,266]. 8.2. Other adsorbates on III—»(1 1 0) surfaces Besides the deposition of hydrogen, the adsorption of a large variety of other elements on the (1 1 0) surfaces of III—V compounds has been investigated by experiment and theory. The chemisorption of group-V elements like Bi or Sb leads to ordered overlayers with a (1;1) structure as observed by LEED [187,240,337,350,367]. Particularly in the case of Sb deposition, the adsorbed atoms are proposed to reside in positions, which correspond to a continuation of the underlying material [249]. This results in a zigzag-chain arrangement of the Sb atoms covering the (1 1 0) surfaces over the valleys between the anion-cation zigzag chains of the substrate. This so-called epitaxial continued layer structure (ECLS) has also been suggested for Ge adsorption on GaAs(1 1 0) [167,188]. In contrast, the deposition of metal atoms on semiconductor surfaces often generates continuous metal films. In dependence of the respective metal-semiconductor combination and the experimental conditions, different growth modes can be observed [249]. Only a few theoretical investigations based on DFT have been dedicated to compute the vibrational modes of adsorbates other than hydrogen on the (1 1 0) surfaces of III—V compounds. The calculations were restricted to systems, for which the formation of an ECLS can be assumed. Results from ab initio calculations exist for Ge : GaAs(1 1 0) [167] and the adsorption of Sb on the (1 1 0) surfaces of GaAs, GaP, InP, and InAs [327,330]. The computations of Refs. [327,330] were focused on analyzing the vibrational modes of the Sb-III—V(1 1 0) interface, in order to provide an explanation for the structures seen in Raman spectroscopy [108,109,175,304]. 8.2.1. Ge on GaAs(1 1 0) Because of the nearly perfectly matching lattice constants and thermal expansion coefficients, layers of Ge on GaAs(1 1 0) appear to be an ideal system to study the interface between an elemental and a binary semiconductor. Ordered overlayers of Ge can be produced over a wide

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temperature range. STM topographs clearly show, that Ge deposition at elevated temperatures results in the nucleation of isotropic islands as high as 10 ML [395]. Nevertheless, there are indications from experiment, that under appropriate conditions, regions with epitaxial growth might be obtainable [77]. However, because of the low temperature of 330 K used in Ref. [77], the crystal films were far from being perfect. The atomic structure of an ordered Ge monolayer on GaAs(1 1 0) has been investigated by theory using a second-nearest neighbor empirical tight-binding model [71,188] and DFT [167]. The plane-wave pseudopotential computations were carried out for periodically repeated crystal films comprising 9 substrate layers, which are covered on both surfaces of each slab by 1 ML of Ge. For the surface phonon modes, dynamical matrices have been calculated using crystal films with 7 substrate layers. Brillouin zone integrations were performed by summations over six special points. More details of the calculations are summarized in Ref. [167]. Fig. 37 illustrates the lowest-energy configuration determined by DFT. As can be seen, the cation bonded Ge atoms are shifted slightly above the surface, while the anion bonded adatoms are shifted downwards. This kind of relaxation can be explained in terms of a substrate induced pseudo-anionic and pseudo-cationic character of the adsorbed Ge atoms. The adatoms bonded to second-layer group-III neighbors are more likely to acquire pseudo-anionic character, while the anion bonded Ge atoms tend to be pseudo-cationic. Under such conditions, the relaxation turns out to be the analogue of the clean surface. The Ge atoms bonded to cations of the substrate material are shifted outwards, adatoms bonded anions are shifted downwards. Subsequently, the pseudo-ionic character is enhanced by the transfer of electronic charge from the dangling bonds of the lower to the upper atoms in the tilted Ge zigzag chains. The bond lengths of the relaxed system do not change by more than 3 percent with respect to the bulk value of GaAs. Table 8 summarizes the structural parameters characterizing the lowest-energy configuration. They are compared with the parameters of a second zero-force geometry found in the densityfunctional calculations [167]. The equilibrium positions of the second, metastable structure, which

Fig. 37. Side view of one ordered overlayer of Ge adsorbed on GaAs(1 1 0). The Ge atoms reside in positions close to the corresponding relaxed atomic configuration of the clean surface.

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Table 8 Relaxation parameters of 1 ML Ge adsorbed on GaAs(1 1 0). The angles u, u, and 0 are defined in Fig. 37. The energy DE gives the energy per unit cell with respect to the ideal surface geometry System

u

u

0

DE (eV)

1 ML (stable) 1 ML (metastable)

36.1° !29.9°

!5.9° !1.9°

2.0° 0.0°

!0.56 !0.26

is about 0.3 eV per unit cell higher in energy, are similar to those of the ground-state configuration. However, the adsorbate-layer zigzag chains are tilted in the opposite direction. Again, the upper Ge atoms in the adsorbed layer have anionic character. This shows that the relaxation induced charge transfer is primarily responsible for the pseudo-ionicity in the adsorption layer. The larger surface energy of the second structure, however, indicates that the trends induced by the polarity of the substrate and the charge transfer caused by the relaxation and rehybridization compete in the second structure, while they support each other in the first case. In contrast with the result from DFPT, the tight-binding calculations of Refs. [71] and [188] give a local minimum only for negative buckling, while the total energy is found to increase for positive buckling angles. The lowest-energy adsorption configuration and the clean GaAs(1 1 0) surface are very similar with respect to the atomic structure and the electronic charge density. The interatomic force constants of the two systems therefore have to be nearly the same. Since there is almost no difference between the atomic masses of Ga, Ge, and As, the phonon dispersion of the adsorption system and that of the free surface are similar to a large extent. This is obvious by comparing the vibrational spectra of the two systems, which are illustrated in Figs. 11 and 38. In the upper and intermediate energy regime, the phonon modes of the Ge covered surface correspond to similar states of the clean surface. However, most of the vibrational features of the adsorption configuration are approximately 2 meV lower in energy. Significant changes only occur in the low-energy part of the vibrational spectrum. In both directions of the dispersion, the chemisorption system shows one more surface acoustical branch than the clean surface. In the CX direction, a flat branch at about 7.7 meV appears close to the zone boundary in addition to the two acoustical modes shown in Fig. 11. This flat branch crosses the upper acoustical mode and mixes with the lower branch near to the X point. The associated phonon mode is dominated by a vibration of the uppermost As ions normal to the surface and displacements of the outward relaxed Ge atoms parallel to the surface perpendicular to the chain direction. In the CX direction, the adsorption of Ge does not influence the two acoustical modes with sagittal polarization. However, the intermediate phonon branch of the clean surface, which originates in a pure shear-horizontal vibration, is drastically changed. We observe two modes with displacements exclusively along the chain direction. One branch lies clearly below the pair of acoustical SP modes with a zone boundary energy of 3.20 meV, while the other vibrational states has a significantly larger energy than the two SP modes. Both states involve a shear-horizontal displacement of the outermost anion—cation pairs of the substrate similarly to the intermediate branch of the clean surface. The Ge atoms bonded to the anion—cation chains of the substrate can vibrate either in-phase or out-of-phase with the atoms of the first substrate plane. This explains the

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Fig. 38. Phonon dispersion of GaAs(1 1 0) covered by 1 ML of Ge in the epitaxial continued layer structure relaxed into the lowest-energy configuration. Localized surface phonon modes are indicated by heavy solid lines. The small hatched regions represent rather deeply penetrating surface states.

large splitting of the intermediate branch of the free surface into the pair of shear-horizontal acoustical modes.

8.2.2. Sb on III—»(1 1 0) surfaces Antimony has been found to form an epitaxial continued one-monolayer structure on the (1 1 0) surfaces of III—V compounds like GaAs, GaP, InP, and InAs. The deposited layers are arranged in a (1;1) symmetry with two adatoms per unit cell. The chemisorbed atoms form zigzag chains above the valleys between the anion—cation chains of the substrate [117,248,337,350,367]. Similarly to the first-layer atoms of the clean surface, each deposited atom has one dangling bond. In contrast with germanium, the antimony atoms have five electrons in the valence shell. Consequently, the dangling bonds of both, the anion- and cation-bonded adatoms have to be doubly occupied. Therefore, the surface relaxation is determined only by the inequivalence of the substrate atoms. The geometry of the antimony-covered (1 1 0) surfaces of GaAs, GaP, InP, and InAs has been studied in detail by means of LEED and X-ray standing wave (XSW) measurements [117,248,337,350,367] as well as with the help of first-principles total-energy calculations. Only some investigations have been made to explore the vibrational properties of the interface. Experimental data are available from Raman spectroscopy [108,109,175,304]. Theoretical results were obtained within a tight-binding total-energy model [138] and density-functional plane-wave

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Fig. 39. Side view of one ordered overlayer of Sb adsorbed on the (1 1 0) surface of a III—V compound. All atoms reside in positions close to corresponding sites of the ideal unrelaxed surface. Table 9 Relaxation parameters of 1 ML Sb adsorbed on the (1 1 0) surfaces of GaAs, GaP, InAs, and InP. The displacements D and D are indicated in Fig. 39 and represent the vertical separation (in A> ) between the adsorbed atoms and the , , first substrate-layer anions and cations, respectively

D , D ,

GaAs

GaP

InAs

InP

0.05 0.09

0.10 0.07

0.12 0.08

0.16 0.07

calculations [327,330]. The measured phonon frequencies were assigned to characteristic eigenmodes computed for the adsorption system on the basis of a restricted dynamical model. Here, we summarize the results obtained from the frozen-phonon approach applied by Schmidt and Srivastava [330]. The calculations were carried out within the density-functional theory using periodically repeated crystal films composed of eight (1 1 0) substrate layers, both surfaces of the slabs being covered by 1 ML of antimony. For the expansion of the electronic wave functions, plane waves up to a cutoff energy of 15 Ry were used. Brillouin zone integrals were approximated by summations over four special points. Among the structural models discussed for the adsorption of Sb on the (1 1 0) surfaces of GaAs, GaP, InP, and InAs [212], the ECLS has been found to be the most favorable 1 ML configuration. The lowest total energy found for the ECLS model is consistent with the good agreement between experimental and theoretical results obtained for the phonon frequencies [108,109,175,304,330] and geometrical parameters [328,339,340]. Similarly to the chemisorption of Ge on the (1 1 0) surface of GaAs, the Sb zigzag chains of the ECLS are tilted in their relaxed positions. As illustrated in Fig. 39, the cation-bonded Sb atoms are shifted upwards, while the anion-bonded adatoms are shifted downwards. The structural parameters summarized in Table 9 show that the vertical separation D between , the adsorbed atoms steadily increases with the inequivalence of the substrate atoms. As pointed out

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Table 10 Vibrational modes of 1 ML of Sb adsorbed on the (1 1 0) surfaces of InAs, GaAs, InP, GaP computed by the frozen-phonon method [330]. All frequencies are in meV. Experimental results obtained from Raman spectroscopy are in parenthesis Mode

InAs/Sb

GaAs/Sb

InP/Sb

GaP/Sb

A  A  A  A  A  A  A  A  A 

7.5 10.3 20.2 22.5 25.1 31.3 6.9 19.0 27.4

9.8 11.1 23.2 24.1 27.2 34.6 7.8 19.9 30.5

11.0 17.7 20.9 24.0 37.5 47.6 8.4 19.6 40.3

11.1 13.1 21.9 26.6 39.7 51.3 8.4 20.8 (20.7) 44.0

(11.9) (21.6) (24.5)

(19.1)

(9.2) (11.0) (22.3)

(20.6)

(11.9) (19.5) (22.9) (35.8) (39.8) (43.9) (20.0) (36.0)

in Ref. [330], the degree of inequivalence for a given pair of group-III and group-V atoms can be expressed in terms of the difference of the tetrahedral radii [59] or the charge asymmetry [132]. In contrast with Ge : GaAs(1 1 0), the dangling bonds of the anion- and cation-bonded Sb atoms have to be doubly occupied. Therefore, the energy of the Sb adsorption system cannot be lowered by an additional rehybridization induced charge transfer as seen for the Ge covered GaAs(1 1 0) surface. The vibrational modes of the interface were determined by treating the particles in the three outermost layers of the crystal films as coupled harmonic oscillators [330]. By means of the Hellmann—Feynman forces computed for off-equilibrium geometries, dynamical matrices were calculated at the C point of the surface Brillouin zone. As two Sb atoms and four substrate atoms per unit cell are included in the dynamical problem, there are nine normal modes with non-zero frequency. Three of these modes (A modes) have atomic displacements exclusively in the direction of the adsorbed Sb zigzag chains, while six vibrational eigenstates (A modes) are polarized in the plane normal to the surface chain direction. Table 10 summarizes the corresponding frequencies and assigns the vibrational states to features resolved in the Raman scattering experiments of Refs. [108,109,175,304]. Not all modes listed in Table 10 have been detected by the experiments. This is not surprising since not all of the eigenstates determined in the restricted dynamical models have to be necessarily surface states. Moreover, some of the computed modes probably have insufficient scattering intensity. The frequencies computed for the A modes are in good agreement with the results from Raman spectroscopy. The vibrational energies of some of the A modes, however, cannot be reproduced with the same accuracy by the restricted dynamical model. Most of the modes reflect chemical trends linked to the atomic masses of the substrate material and the strength of the chemical bond [330]. The A mode, however, has nearly the same vibrational energy irrespective of the substrate.  This originates in the fact that the A mode is dominated by a shear vibration which is restricted to  the zigzag chains of the adsorbed atoms. Hence, the frequency is mainly predetermined by the strength of the Sb—Sb bond and the atomic mass of Sb. The larger variation seen for the vibrational

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Fig. 40. Side view of the (a) (2;1)-monohydride, (b) (1;1)-dihydride, and (c) (3;1)-monohydride/dihydride reconstructions of the H : Si(0 0 1) surface. The H : Si(0 0 1) (1;1)-dihydride structure, which is stabilized by the formation of canted dimer rows [271], is illustrated only schematically.

energies of all other modes in dependence on the substrate indicates, that most of the vibrational features involve atomic motions of the adatoms and substrate atoms. 8.3. Hydrogen coverage on Si(0 0 1) and C(0 0 1) The adsorption of hydrogen on the Si(0 0 1) surface leads to at least three different reconstructions depending on the experimental conditions. The most commonly observed geometry is the (2;1) monohydride phase [310]. It is illustrated in Fig. 40, together with the structural models proposed for two other reconstructions seen in experiments, namely the (1;1) phase [310] and a (3;1) arrangement [66]. Monohydrides and dihydrides are the two fundamental structural subunits of the hydrogen-induced reconstructions reported for Si(0 0 1). The (2;1) monohydride structure results from the saturation of all dangling bonds of the dimer-reconstructed surface with atomic hydrogen. To obtain the dihydride structure, the dimer bonds of the free surface and/or the monohydride phase have to be broken, followed by a passivation of the created dangling bonds with hydrogen. By measuring characteristic stretching mode frequencies using high-resolution infrared spectroscopy, it was demonstrated that monohydrides are prevalent already at low coverage, while dihydrides appear after larger exposure of Si(0 0 1) to atomic hydrogen [67]. This is consistent with the results obtained from plane-wave calculations [271], which were carried out to compare the surface formation energy of the three structures shown in Fig. 40 as a function of the hydrogen chemical potential. An interesting result reported in Ref. [271] is that the energy of the H : Si(0 0 1) (1;1) surface is reduced by the formation of canted dihydride rows. Reconstructions with (1;1) and (2;1) symmetry were also reported for the hydrogen-covered diamond-structure C(0 0 1) surface. Experimental and theoretical studies clearly indicate that the

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observed (2;1) phase can be attributed to the formation of monohydrides [5,6]. For the (1;1) reconstruction, again dihydride arrangements are suggested [220,313,401]. However, because of large steric repulsions, the full dihydride coverage is found to be unstable relative to H desorption  in several calculations [396,401]. The measurement of phonon mode frequencies is a powerful method to explore the details of the bonding configurations characterizing the (2;1) and (1;1) reconstructions. By probing its respective vibrational spectrum, the presence of a characteristic structural subunit can be monitored. Commonly used experimental techniques are HREELS [5,220,351,361] and highresolution infrared spectroscopy [64,67]. On the theoretical side, only some ab initio calculations have been performed to investigate the phonon spectra of the hydrogen covered (0 0 1) surfaces of Si and C [10,23,64]. These computations were focused on the monohydride reconstruction of H : C(0 0 1) (2;1) and H : Si(0 0 1) (2;1). In addition, a semiempirical approach [338] and the tight-binding model [313] were used to study the monohydride and dihydride configurations of H : C(0 0 1). Recently, surface phonon modes have been determined by means of DFPT for the canted-row dihydride H : Si(0 0 1) (1;1) configuration [23]. Here we summarize only the results obtained for the monohydride reconstruction of H : Si(0 0 1) [23] and H : C(0 0 1) [10]. 8.3.1. Structure and dynamics of H : Si(0 0 1) (2;1) The ab initio calculations for the hydrogen covered Si(0 0 1) surface were carried out in the framework of the plane-wave pseudopotential method and its extension to DFPT [23]. To describe the surface, periodically repeated thin crystal films comprising ten (0 0 1) substrate layers were used. The distance between neighboring slabs was 8 A> . The two surfaces of each film were terminated by the (2;1) monohydride structure. Plane waves up to 9 Ry were used for the expansion of the electronic eigenstates. Brillouin zone integrations were approximated by summations over six special points. The structure optimization of the monohydride geometry resulted in symmetric silicon dimers in the first substrate layer. Relative to the ideal bulk-like positions, the first substrate-layer atoms are shifted by $0.84 A> in the [1M 10] direction giving a dimer bond length of 2.38 A> . The hydrogen atoms are bonded to the silicon dimers at a distance of 1.49 A> . These numbers compare very well with 2.40 and 1.50 A> determined by the density-functional calculations of Ref. [271]. Symmetric dimers are stabilized by the adsorption of hydrogen which saturates both of the dangling bonds per unit cell. As a consequence of this, the two atoms in the silicon dimers are equivalent in the lowest-energy configuration. This is not the case for the clean surface, where one of the two dangling bonds per unit cell can be empty. Therefore, the creation of an energetically favorable doubly occupied and one empty dangling bond is supported in the absence of adsorbed atoms. This is achieved by a JahnTeller-like distortion which lowers the energy of one dangling bond to the expense of the other. Characteristic modes of the monohydride complex are the stretching and bending vibrations of the adsorbed hydrogen atoms as well as hindered translations and rotations of the monohydride unit which is bonded to the second substrate layer. By means of DFPT [23], the following vibrational frequencies were obtained at the C point of the SBZ: The individual stretching oscillations of the two hydrogen atoms are coupled to an antisymmetric vibration at 269.7 meV and a symmetric stretching mode at 270.7 meV as illustrated in Fig. 41. The computed frequencies are in good agreement with 259.1 and 260.6 meV measured by infrared spectroscopy [67]. The splitting of 1.5 meV is slightly underestimated by theory.

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Fig. 41. Symmetric and antisymmetric stretching and bending modes of the (2;1) monohydride structure polarized in the (1 1 0) plane.

For the bending oscillations, DFPT yields four states with C point energies of about 71.4,72.6,73.1, and 74.9 meV. The modes at 73.1 and 74.9 meV involve atomic motions of the adsorbed hydrogen atoms parallel to the dimer rows. The two other eigenstates are illustrated in the lower portion of Fig. 41. In contrast with the stretching modes, the lower vibrational energy is associated here with the symmetric bending mode. With the help of electron-energy loss spectroscopy, a bond-bending oscillation was resolved at about 78.5 meV [351]. The hydrogen-induced formation of the Si(0 0 1) (2;1) monohydride geometry changes two fundamental structural details of the pristine surface. As a result of the hydrogen adsorption, tilted dimers are no longer favored over symmetric dimers. The dimer-bond length is increased from 2.28 A> to about 2.38 A> . Therefore, it is interesting to analyze the effect of the hydrogenation on vibrations like the modes r and sb, which are characteristic features of the buckled dimer geometry as discussed in Section 6.3.1. The dimer-rocking mode r is shifted to higher frequencies in the monohydride configuration. Along the CJ direction, it occurs at about 26—27 meV. The respective frequencies computed for the p(2;1) and c(4;2) reconstruction of the clean Si(0 0 1) surface lie in the range 20—23 meV. The lower frequencies obtained for buckled dimers is probably related to the smaller force constant associated with the planar bonding configuration of the down shifted first-layer atoms in the clean surface. For the asymmetrically reconstructed Si(0 0 1) p(2;1) surface, DFPT yielded a mode sb slightly above the bulk continuum. At the C point it is characterized by a bond-stretching motion of the second-layer atoms bonded to the down shifted dimer atoms against their nearest neighbors in the first and third atomic layer as illustrated in Fig. 25 of Section 6.3.1. The appearance of this mode is a direct consequence of the buckled-dimer geometry of Si(0 0 1) p(2;1). It is not prevalent in the symmetrically reconstructed clean C(0 0 1) p(2;1) surface. Also for the Si(0 0 1) p(2;1)

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monohydride structure, no similar mode was obtained. However, by means of tight-binding calculations, a mode predominantly localized in the second and third substrate layer was found near to the K point [12]. This vibrational state is related to the fivefold rings defined by the dimer atoms, their second-layer nearest neighbors, and the third-layer atoms below the dimers. Since the mode sb of the clean surface acquires a similar character along the CJ direction, it was also attributed to the fivefold rings. 8.3.2. Structure and dynamics of H : C(0 0 1) (2;1) The vibrational states of hydrogen adsorbed on the C(0 0 1) surface have been computed by means of ab initio frozen-phonon calculations [10] and the tight-binding approach [313]. Experimental results are available from HREELS [5,361]. While general agreement was achieved concerning the atomic configuration of the hydrogen-terminated C(0 0 1) (2;1) phase, no consensus has been reached for the (1;1) structure [401]. Hence, only some investigations were focused on the vibrations of the dihydride structure [220,313] of H : C(0 0 1). Here we summarize the results from the local-orbital calculations of Ref. [10], which were performed to investigate the surface vibrations of the monohydride configuration. The computational details are the same as those used for the clean C(0 0 1) surface (see Section 6.5). The C(0 0 1) (2;1) monohydride structure is obtained by attaching hydrogen to the dangling bonds of the dimer configuration which was determined for the clean surface. The relaxation of the hydrogen-terminated symmetric dimers resulted in an increase of the dimer-bond length from 1.36 to 1.62 A> . The angle defined by the H—C bond and the dimer bond is 114°. Compared with the bulk, the interplanar spacing between the first and second substrate layer is reduced by 5%. These numbers are in good agreement with the Car—Parrinello moleculardynamics simulations of Ref. [401]. Fig. 42 illustrates the vibrational features obtained for the relaxed monohydride geometry within the frozen-phonon approach. Similar to the hydrogen-covered Si(0 0 1) surface, a symmetric CH-stretching mode is found at 341.4 meV. Its antisymmetric counterpart has an energy of 336.8 meV. Four bending modes in the energy range from 124 to 186 meV are present in the vibrational spectrum. These modes are dominated by opposing motions of the atoms in each of the CH-units perpendicular to their bond. At the C point, the bending modes are polarized either only along the dimer row direction or in the plane perpendicular to the rows and have a strictly symmetric or antisymmetric vibrational pattern. The bending oscillations involve twisting, rocking, stretching, and swinging motions of the C—C dimers as illustrated in Fig. 42. The dimer-stretching mode has the character of a H-scissoring motion, with an energy of 163 meV. The corresponding stretching vibration of the clean surface occurs at 185 meV. Its lower energy in the monohydride is consistent with the hydrogen-induced increase of the C—C bond length. Analogous to the clean surface, a rocking motion of the dimers is present in the vibrational spectrum. Its vibrational energies are 155 meV at the C point and 166 meV for out-of-phase oscillations of adjacent dimers. In addition to the CH-bending and CH-stretching modes, vibrational features appear, where the two atoms in the CH-unit are vibrating in phase. A CH—CH-stretching oscillation is found in the energy range from 112—122 meV. Unlike the corresponding C—C stretching mode of the clean surface, it mixes with optical bulk phonons and therefore has a dispersion of about 10 meV. Finally, vibrational features appear which resemble hindered normal translations of the dimers. These

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Fig. 42. Phonon modes of the C(0 0 1) (2;1) monohydride structure computed by means of the frozen-phonon approach [10].

CH—CH-bouncing modes couple to bulk modes and are found with energies between 65 and 80 meV. Experimental data for the monohydride structure are available from HREELS [5,361]. The spectra of Ref. [5] exhibit a narrow symmetric peak at about 363 meV that is assigned to a CH-stretching vibration. In addition, a band of bending modes was detected in the range from 124 to 180 meV, with maxima at 137 and 152 meV. The measurements of Ref. [361] resolved vibrational features at 363 and 155 meV. The intensity of these energy losses is due to dipole scattering and impact scattering. It is therefore difficult to obtain information about the orientation of the surface vibrations associated with the spectral peaks [361]. Nevertheless, the measured frequencies are in good agreement with the theoretical results obtained from the frozen-phonon approach [10] and with the findings of the tight-binding calculations of Ref. [313]. 8.4. Arsenic on Si(0 0 1) The adsorption of arsenic on the Si(0 0 1) p(2;1) surface leads to an ordered monolayer with the same reconstruction pattern. This has been demonstrated in several experiments, such as LEED [183], STM [35], and grazing incidence X-ray diffraction (GIXD) [183]. A simple picture of the adsorption process is that the chemisorbed As atoms break the Si—Si dimers of the substrate. The adsorbed As atoms form As—As dimers arranged in rows perpendicular to the dimer rows of the pristine surface [35]. This explains the rotation of the diffraction pattern from p(2;1) for the clean surface to p(1;2) for the As-covered surface as observed by LEED [50]. Density-functional calculations [207] performed to investigate the atomic and electronic structure of the adsorption configuration show that the As : Si(0 0 1) p(2;1) adsorption model is consistent with the observed

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Fig. 43. Side view of the As : Si(0 0 1) p(2;1) adsorption geometry. The dimer bond length and the back bond length obtained by the plane-wave calculation of Ref. [23] are indicated.

LEED intensities [50,183]. STM topographs [35], and electronic surface bands measured by angle-resolved photoemission spectroscopy [371]. In terms of a simple electron-counting model, the As : Si(0 0 1) p(2;1) reconstruction is similar to the hydrogen-induced monohydride configuration. Since each arsenic atom introduces five valence electrons into the system, four electrons have to be incorporated into the two dangling bonds of each As—As dimer. Therefore, buckled dimers are unstable. In the equilibrium configuration, the adsorbed layer consists of symmetric dimers which are composed of three-fold coordinated As atoms, each having a doubly occupied lone-pair state. Hence, the Si—H-units of the monohydride structure on the one hand, and the As—As dimers of the As : Si(0 0 1) adsorption system on the other, correspond to each other. While several investigations were focused on the structural and electronic properties of As adsorbed on Si(0 0 1), only some attention was paid to the vibrational excitations of this system. Recently, the complete phonon dispersion of As : Si(0 0 1) p(2;1) was determined in the framework of DFPT [23]. The numerical details of the computations are the same as those summarized in Section 8.3.1. The adsorption system was described in the slab-supercell formalism using thin crystal films with ten (0 0 1) substrate layers. Both surfaces of the slabs were covered by one monolayer of As, reconstructed into the p(2;1) dimer configuration. Fig. 43 illustrates the equilibrium positions of the As : Si(0 0 1) p(2;1) surface. For the relaxed geometry, the DFPT calculations yielded 2.50 A> for the As—As dimer bond length and a distance of 2.46 A> for the adsorbed atoms from their nearest neighbors in the first substrate layer. These numbers are in good agreement with 2.52 and 2.42 A> found in the density-functional calculations of Ref. [207] and the experimentally determined As—As bond length of 2.55 A> [183]. The phonon dispersion of the As-covered Si(0 0 1) surface is illustrated in Fig. 44. The lowest mode in the spectrum is the RW which has a dispersion different from that of the corresponding mode in the clean surface (see Fig. 24 in Section 6.3.1). At the zone boundary points J , K , and JM , the RW has the energies 7.0, 10.5, and 11.3 meV. For the clean Si(0 0 1) surface, the corresponding energies of the tilted-dimer p(2;1) structure are 9.0, 10.3, and 9.2 meV, while the J point frequencies are 8.5 and 10.8 meV for the c(4;2) geometry. Hence, the J point frequency of the RW is significantly lower for the As covered Si(0 0 1) surface. This is related to the larger mass of the adatoms. Close to the K point, there are four additional branches besides the RW below the continuum of the bulk bands, while only three surface-localized features (RW, A , and A ) are   present in the case of the clean surface.

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Fig. 44. Phonon dispersion of the As : Si(0 0 1) p(2;1) surface computed by means of DFPT [23].

As for the pristine surface, several vibrational states are observed, which have the character of rotational and translational motions of the As—As dimer units. A dimer-rocking mode (r) is found between 25 and 26 meV along the KJ direction. Similarly to the monohydride structure, the rocking mode frequency is higher than for the clean surface. Two localized states appear in the window gap at the K point with vibrational energies of about 29 and 31 meV. The lower mode (S )  is a dimer-twisting oscillation which is dominated by an opposing in-plane motion of the two dimer atoms perpendicular to their bonding direction. The vibrational feature S is characterized by an  in-phase displacement of the dimer atoms along the rows leading to a swinging motion of the As—As dimers. The first-layer oscillations of these modes are coupled to substrate-atom vibrations. Hence, there are some more states with similar first-layer displacements appearing in the spectrum above the S and S features. An example is the pair of modes occurring at 44 and 46 meV close to   the K point. Again, these states are dimer-twisting and dimer-swinging oscillations. The mode sb found above the continuum of the optical bulk modes is related to the interface. It is characterized by displacements of the atoms in the outermost three substrate layers mainly in the [0 0 1] direction. The adatom-layer remains essentially at rest, due to the larger atomic mass of arsenic. 8.5. Hydrogen coverage on Si(1 1 1), Ge(1 1 1) and C(1 1 1) The (1 1 1) surfaces of diamond-structure semiconductors are ideal systems to study the passivation of surface dangling bonds by the adsorption of hydrogen. The hydrogenation of the dangling bonds results in a nearly perfect bulk-terminated plane of surface atoms with well-defined properties. Recently, it was demonstrated that hydrogen-covered, unreconstructed H : Si(1 1 1)(1;1) surfaces can easily be produced by means of wet chemical preparation [65,97,161]. The high

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degree of structural perfection that can be obtained with this and other experimental techniques has made the H : Si(1 1 1)-(1;1) surface to a prototype system for many experimental studies. Also the hydrogen-covered (1 1 1) surfaces of germanium [181] and diamond [80] are known to adapt a (1;1) geometry which is close to the ideal, bulk-truncated configuration. Besides the fundamental interest in the physical properties of such systems, passivation of surface dangling bonds is of technical importance. Particularly for the fabrication of thin diamond films, hydrogen-termination is known to play a crucial role in the achievement of a successful growth [380]. The simplicity of the hydrogen-adsorption systems is related to the bonding symmetry of the bulk material (see Fig. 31 of Section 7). The singly occupied dangling bonds of the bulk-terminated surface are ideal adsorption sites for hydrogen atoms which form a flat overlayer. There are no in-plane relaxations in the adsorption system, because of the hexagonal symmetry of the substrate and the equivalence of all of the first substrate-layer atoms. Several ab initio calculations have been carried out to study the structural and the vibrational properties of the hydrogen-covered (1 1 1) surfaces of diamond [404], silicon [20,56,129,168,192,225], and germanium [192]. The results of these investigations are in good agreement with the experimental data from LEED [181], core-level-spectroscopy [190], STM [37], helium-atom scattering [94,219], HREELS [5,6,98,220,352,382] and other techniques [80,303]. Besides the approaches of Refs. [20,56,129,168,192,225,404], some tight-binding calculations were used to study the surface phonon modes of H : Si(1 1 1) [239,246,311], H : C(1 1 1) [239,312], and D : Si(1 1 1) [146]. For hydrogenated germanium, additional theoretical data are available from cluster calculations [31,160]. Here we summarize the results obtained from DFPT for the vibrational modes of H : Si(1 1 1) [168] and the findings of the total-energy calculations of Refs. [192] and [404] for the hydrogencovered (1 1 1) surfaces of germanium and diamond. The DFPT calculations of Ref. [168] were carried out for periodically repeated slabs spanning ten substrate planes and one adsorbate layer on the two surfaces of each crystal film. Neighboring slabs were separated by about 8 A> . Special points summations over seven points were used to approximate Brillouin zone integrals. The Kohn—Sham eigenstates were expanded in plane waves with a cutoff energy of 13 Ry, using a pseudopotential for Si of the Kerker type [168,196]. The corresponding numbers used in the plane-wave calculations of Ref. [192] are eight layers per crystal film prior to hydrogenation, four special points, and a cutoff energy of 10 Ry. The frequencies of the hydrogen-stretching and bending modes were determined by fitting quadratic polynomials to values of the total energy as a function of the bond length and the bonding angle. The first-principles calculations of Ref. [404] were carried out for crystal films containing six carbon atoms per unit cell. Plane waves with a cutoff energy of 49 Ry were used for the expansion of the electronic wave functions. Table 11 summarizes the results of the ab initio calculations of Refs. [168,192,404] for the relaxation parameters of one monolayer of hydrogen adsorbed on the (1 1 1) surfaces of silicon, germanium, and diamond. The listed data are in good agreement with the findings of other density-functional calculations for H : Si(1 1 1) [20,225], H : C(1 1 1) [353], and the results from LEED for H : Ge(1 1 1) [181]. In all cases, only small relaxations along the [1 1 1] direction are reported. Hence, the hydrogenated surfaces exhibit only minor changes with respect to the ideal bulk-truncated case. While the computations of Refs. [20,129,192,225,404] were restricted to the determination of only some characteristic vibrational modes of the adsorption systems, the linear-response

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Table 11 Relaxation parameters determined in the density functional calculations of Refs. [168,192,404] for 1 ML hydrogen adsorbed on the (1 1 1) surfaces of Si, Ge, and C. The hydrogen-substrate bond length is denoted by d . The distance & d between the ith and jth layer is compared with the respective value for the ideal surface d . All quantities are given in A> GH GH

H : Si(1 1 1) H : Ge(1 1 1) H : C(1 1 1)

a 

d &

d 

d  

d 

d  

5.44 5.66 3.51

1.49 1.57 1.12

0.76 0.73 0.51

0.79 0.81 0.51

2.35 2.38 1.52

2.36 2.45 1.52

Fig. 45. Phonon dispersion of the H : Si(1 1 1) (1;1) surface along the edges of the surface Brillouin zone computed for a slab with 46 substrate layers. The heavy solid lines represent surface-localized modes. In the CM direction, the vibrational states are strictly separated into SP- and SH-modes. The labels used for the other directions are taken from Ref. [311]. Experimental data from HREELS [352] are illustrated by the circles.

formalism was used in Ref. [168] to study the complete phonon dispersion of H : Si(1 1 1) (1;1). Fig. 45 illustrates the results from the DFPT approach. The phonon dispersion was calculated for wavevectors along C !M  !K !C of the SBZ (see Fig. 31). Because of the mirror symmetry of the crystal films, the vibrational states of the slabs can be classified as pure shear-horizontal or sagittally polarized modes for propagation along the CM direction. Characteristic vibrations of the adsorption system are the hydrogen-stretching mode and two bond-bending oscillations. The stretching mode forms a flat band at about 264.8 meV with a dispersion of less than 0.7 meV. The computed frequency is about 2% larger than the vibrational energies resolved by HREELS [352]. As for the hydrogen-covered (0 0 1) surface, bending modes appear above the continuum of the bulk phonons. Since one hydrogen atom is adsorbed per unit cell, only two branches are present in the spectrum. In large parts of the dispersion, the two branches are split by about 2.5 meV. At the zone center, however, the bending modes have

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a common frequency of 72.6 meV which is 4.6 meV below the HREELS data of Ref. [352]. The measured dispersion along the CM and MK directions reveals a splitting between the two bending modes of about 2.5 meV at the M point. This is in good agreement with the calculated splitting of 2.9 meV. The underestimation of the measured bending mode frequencies and the overestimation of the stretching frequencies may partially originate in the particular choice of pseudopotentials and the finite size of the plane-wave basis set [167]. In addition, the results from temperature-dependent infrared-absorption spectroscopy [97] and other experiments [148,149,210] indicate that the deviations are also related to anharmonic effects. Several ab initio calculations were carried out to study non-linear effects in the vibrational spectrum of the hydrogen-covered Si(1 1 1) surface. Gai and Voth [129] examined the vibrational dynamics by means of Car—Parrinello moleculardynamics simulations. Side peaks detected in the Fourier transforms of the particle-displacement autocorrelation function clearly indicated mode coupling. Li and Vanderbilt [225] computed the phonon-interaction strength, the phonon band width, and the binding energy of two-phonon bound states of the Si—H stretching mode. This was achieved by means of a frozen-phonon approach in which the total energies were fitted to a polynomial expansion of the potential up to forth-order terms. Honke and co-workers [169] used a combination of DFPT and the frozenphonon technique to determine the temperature-dependence of the stretching-mode frequency and line width from anharmonic coupling coefficients. The nonlinear terms were represented by derivatives of the dynamical matrices with respect to normal-mode displacements. The anharmonicity of the surface modes may also result in the occurrence of intrinsic localized modes (ILM) [46]. In contrast with harmonic oscillations, ILMs show a strong localization not only in the z-direction, but also in the surface plane. They involve motions of only a few atoms in the surface layer. We now turn to the remaining vibrational features in the hydrogen-covered Si(1 1 1) surface. The lowest mode in the phonon spectrum is the RW. Only in the near of the K point, the RW is below the bulk continuum. Its dispersion computed from DFPT is in good agreement with the HREELS data [352] and the results from HAS [94]. A mode labeled S appears close to the lower rim of the  large stomach gap between 25 and 40 meV. It is dominated by vibrations of the first-substrate-layer atoms and hydrogen atoms normal to the surface. Therefore, it is clearly seen in the energy-loss spectra of inelastically scattered electrons. Other vibrational features seen in the HREELS experiments are the resonances R , and R , and the mode ¸ which mixes strongly with bulk states in    large parts of the dispersion. Hence, all salient features of the HREELS experiment are reproduced by DFPT. In addition to the experimentally detected branches, the computed spectra contain some shear modes, like ¸ and S , which cannot be resolved in the loss spectra. The dispersion of these   shear-horizontal vibrations and all other surface-localized states is in good agreement with the findings of the tight-binding approach used in Refs. [239,311]. While the vibrational spectrum of H : Si(1 1 1) was computed in detail, only some data from ab initio calculations are available for the adsorbate vibrations of H : Ge(1 1 1) [192] and H : C(1 1 1) [404]. Table 12 summarizes the bond-stretching and bond-bending frequencies obtained from density-functional calculations for the three adsorption systems and compares the theoretical results with experimental data. The bond-stretching and bond-bending modes of the hydrogen-covered Ge(1 1 1) surface are found above the continuum of bulk phonons, similarly to H : Si(1 1 1). Being largely decoupled from

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Table 12 Zone-center frequencies (in meV) computed in Refs. [168,192,404] for the bond-stretching and bond-bending modes of the hydrogen-covered (1 1 1) surfaces of Si, Ge, and C. Experimental data are given in parenthesis and are taken from Refs. [80,192,303,352], for H : Si(1 1 1), H : Ge(1 1 1), and H : C(1 1 1), respectively System

u 

u


H : Si(1 1 1) H : Ge(1 1 1) H : C(1 1 1)

264.8 (259.4) 230.9 (245.2) 342.4 (351.0)

72.6 (77.2) 62.7 (65.8) — —

the bulk states, the characteristic adsorbate vibrations give rise to flat branches in the phonon dispersion. In the ab initio calculations of Ref. [404], only the stretching mode of hydrogen adsorbed on C(1 1 1) was investigated in detail. By means of the frozen-phonon approach, total energies were computed for hydrogen and first substrate-layer displacements normal to the surface (z-direction). From that, the stretching mode frequency was determined for C and C isotopes in the substrate. Table 12 gives the fundamental frequency for the excitation from the ground-state to the first excited state, calculated for hydrogen bonded to C. To study anharmonicity-related frequency shifts, the adiabatic potential computed for the hydrogen displacements was fitted to a forth-order polynomial. Using second-order perturbation theory, a difference between the overtone frequency and twice the fundamental energy of 14 meV was found [404]. Finally, the bending modes in H : C(1 1 1) were investigated within the tight-binding approach [239,312]. In marked contrast with the two other systems, these calculations show that the bending-mode branches are not separated from the continuum of the bulk modes and hence mix with the vibrational states of the bulk. 8.6. As on Si(1 1 1) The deposition of arsenic on Si(1 1 1) leads to a well-ordered (1;1) surface geometry. This was demonstrated in several experiments like STM [36] and LEED [275] which gave sharp (1;1) patterns after the chemisorption of arsenic. The adsorption system resembles the bulk-terminated Si(1 1 1) surface. It is obtained by replacing the silicon atoms in the outermost layer of the ideal surface by arsenic. Just as for the hydrogen-covered surface, the remaining substrate atoms are very close to their ideal bulk-derived positions. The structural similarity of H : Si(1 1 1) and As : Si(1 1 1) is not surprising, since the electronic density of As atoms bonded to three silicon atoms is comparable to that of the Si-H units on the H : Si(1 1 1) surface. For the adsorption of arsenic on Si(1 1 1), three valence electrons of the adatom are necessary to establish the chemical bonds to the substrate. Hence, the chemisorbed arsenic atoms have two remaining valence electrons for the occupation of their dangling bonds which are chemically inert for that reason. Several density-functional calculations have been performed to investigate the relaxation of the As : Si(1 1 1) (1;1) surface [48,168,372]. The results of these investigations are in good agreement with the data obtained from STM [36], LEED [275], and X-ray standing-wave measurements [283]. Vibrational modes of As : Si(1 1 1) (1;1) were computed by means of DFPT [168,170], the

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Fig. 46. Phonon dispersion of the As : Si(1 1 1) (1;1) surface along the edges of the surface Brillouin zone computed for a 46-layer slab. The heavy solid lines represent surface-localized modes. In the C M  direction, the vibrational states are strictly separated into SP- and SH-modes. Open circles show the results from helium-atom scattering [318], while full circles represent the data from HREELS [326].

BCM [318], and a Green’s function approach [253]. Here we focus on the plane-wave calculations of Refs. [168,170]. We compare the computed phonon dispersion curves with experimental data from HREELS [326] and HAS [93,318]. In the DFPT calculations of Ref. [168] periodically repeated slabs spanning ten atomic layers were used. The top and bottom layers of the crystal films were composed of arsenic atoms, while the remaining eight planes per slab described the Si-substrate. Neighboring slabs were separated by about 8 A> . Brillouin zone integrals were computed by summations over seven special points. A basis set with a cutoff energy of 10 Ry was chosen for the plane-wave expansion. In good agreement with the results of the LDA calculations of Refs. [48,372], the first and second substrate layers were found to relax from the ideal positions by 0.02 and 0.01 A> towards the inner layers of the slabs. The As ions are located about 0.20 A> above the corresponding Si sites of the bulktruncated structure. With the help of X-ray standing wave measurements, Patel and co-workers determined a value of 0.17 A> [283]. The phonon dispersion computed by means of DFPT is illustrated in Fig. 46, together with the data from HAS [318] and HREELS [326]. The results from the plane-wave calculations are in overall good agreement with the measured dispersion. In contrast, the BCM can reproduce surface vibrational energies with high accuracy only in the lower frequency region [318]. As for the hydrogen-covered Si(1 1 1) surface, the vibrational states of As : Si(1 1 1) (1;1) are strictly characterized as SH- or SP-modes along the C M  direction. Two acoustical surface modes are found in the DFPT calculation below the surface projected bulk phonon dispersion. The lowest mode is the RW which was also resolved in the HREELS and HAS experiments. At the K point, a third mode appears in the computed dispersion below the bulk modes. Since it is dominated by horizontal displacements close to the K point, it cannot be detected by the scattering experiments. The flat branch appearing between 25 and 28 meV can only be identified in the stomach gap. In the other parts of the dispersion, it mixes strongly with bulk states and loses its surface-localized

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character. The two branches between 40 and 45 meV are related to each other: At the C point, both of the modes are dominated by an opposing motion of the As and Si atoms in the outermost bi-layer parallel to the surface. For finite wave vectors, the upper branch acquires appreciable z components and is therefore seen in the HREEL spectra, while the lower mode essentially keeps its horizontal polarization. The highest mode is a resonance which is mainly characterized by displacements of the first and second substrate layer atoms, while the As atoms are essentially at rest. In the hydrogen-covered surface a similar mode was found, with atomic vibrations in the second and third substrate layer. 8.7. Group-III elements on Si(1 1 1) We now turn to the adsorption of group-III elements on the Si(1 1 1) surface. The deposition of Al, Ga, and In on Si(1 1 1) leads to the formation of ((3;(3)R30° superlattices [39,164,215,274]. The particular order of the reconstruction is related to the fact that the adsorbed group-III atoms can saturate three dangling bonds of the free surface. Geometries for which this is achieved are illustrated in Fig. 47. The results of early LEED experiments [215] led to the suggestion that the observed ((3;(3)R30° reconstruction originates in the deposition of the group-III atoms on H adsorption sites, where the adatoms are bonded to three first-layer  atoms in positions above atoms of the forth substrate layer. Newer experiments which used STM, however, clearly show that chemisorbed Al and Ga atoms are bonded to Si(1 1 1) in positions above atoms of the second substrate layer [152,268,399]. The exact position of Al, Ga, and In atoms adsorbed on ¹ sites was determined by means of X-ray standing-wave  measurements [390,399] and LEED [172,191]. The results of these experiments are in good agreement with the relaxation parameters computed on the basis of density-functional theory [170,259,269,399]. The deposition of boron on Si(1 1 1) leads also to a ((3;(3)R30° reconstruction pattern. Because of the small covalent radius of boron, however, the ¹ and H adsorption sites are   unfavorable for the group-III element. Various experiments like X-ray diffraction [158], LEED [173], and STM (combined with ab initio total-energy calculations) [38,232] show that the chemisorption of boron on Si(1 1 1) results in the S configuration, in which boron atoms reside at  atomic positions of the second substrate layer. As in bulk silicon, boron occupies substitutional lattice sites. The silicon atoms which are replaced by boron are shifted above the surface and are bonded to three silicon atoms at ¹ sites.  The vibrational modes of the III : Si(1 1 1) ((3;(3)R30° adsorption geometries have been measured by means of HREELS [7,195,331]. Complete phonon-dispersion curves were determined for the adsorption of Al [7] and Ga [331], while the measurements for the In : Si(1 1 1) ((3;(3)R30° surface were restricted to the zone center [195]. Phonon-dispersion curves were also determined with HAS for Ga and B deposited on Si(1 1 1) [93]. Interestingly, all of the HREELS experiments resolved a surface mode at about 65—69 meV, slightly above the upper edge of the bulk phonon spectrum. The frozen-phonon calculation of Northrup [270] for Al adsorbed on Si(1 1 1) gave an interpretation for this feature by attributing the mode to vibrations in the second and third substrate layer. Recently, DFPT was applied to determine the complete phonon dispersion of Ga : Si(1 1 1) ((3;(3)R30° [102,170] and B : Si(1 1 1) ((3;(3)R30° [170].

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Fig. 47. Top and side views of possible ((3;(3) surface reconstructions for the adsorption of group-III elements on Si(1 1 1). Corresponding atoms in the top and side views are numbered. In the ¹ and H adsorption geometries, the   chemisorbed atoms are bonded to three first-layer atoms. The ¹ positions are above atoms of the second substrate layer,  the H positions are above atoms of the forth substrate layer. The S geometry can be obtained from the ¹ arrangement    by exchanging the adsorbed atoms with the corresponding atoms of the second substrate layer.

Here we focus on the results reported in Refs. [102,170] for the Ga-covered Si(1 1 1) surface. A more detailed analysis and the comparison with the results obtained for the adsorption of boron and other atoms is subject of a future publication [171]. The computations were carried out for periodically repeated thin crystal films spanning eight substrate layers, with a total of 24 Si atoms in the unit cell. The bottom layer of each of the crystal films was terminated by 1 ML of hydrogen, while the top layer was covered in ¹ positions with one Ga atom per unit cell. Neighboring crystal  films are separated by about 9 A> . For the expansion of the electronic states, a cutoff energy of 9 Ry was used to restrict the plane-wave basis set. Brillouin zone integrals were approximated by summations over four special points. The computed relaxation parameters are in good agreement with the results of Ref. [399], and the the data from LEED [191] and X-ray standing wave measurements [399]. In the ((3;(3)R30° adsorption geometry, the atoms relax under the constraint of preserving the C symmetry which is T

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Fig. 48. Phonon dispersion of the Ga : Si(1 1 1) ((3;(3)R30° surface computed by means of DFPT [101,170] in comparison with the results from HREELS (circles) [331] and HAS (triangles) [93]. The figure’s right portion illustrates in a gray scale the projected density of states as defined in Eq. (122).

characterized by a three-fold axis oriented normal to the surface and defined by the atoms labeled by 2 and 3 in Fig. 47. Hence, the atoms in the second and third substrate layer, as well as the adsorbed Ga atoms are allowed to relax only perpendicular to the surface, while relaxations in the first substrate layer are allowed along and radially to the symmetry axis. The first substrate layer atoms indicated by the numbers 1 and 4 in the figure are shifted upwards by 0.03 A> and relax radially towards the adatom by 0.14 A> . The atoms in the second substrate layer are not equivalent. While the atoms directly under the adatoms (labeled as 2 in the figure) are shifted downwards by 0.33 A> , the other two atoms in this layer relax upwards by 0.17 A> . The atoms in the third layer show similar but smaller shifts as their bonding partners above. Therefore, the bonds between the atoms 2 and 3 are compressed as a result of the adsorption. Finally, the chemisorbed Ga atoms are 1.79 A> above the relaxed first substrate layer. Fig. 48 illustrates the phonon dispersion computed by DFPT for the Ga : Si(1 1 1) ((3;(3)R30° adsorption configuration [102,170] and compares the results with the data from HAS [93] and HREELS [331]. Most of the features recorded by means of inelastically scattered electrons and helium atoms can be attributed to localized surface states identified in the densityfunctional calculation. For the explanation of the flat branch at about 40 meV and the other vibrational features seen by HREELS, the projected density of states "n ) €Q (qN )" (122) Z(u, qN ) du" ? Q SZSS>S ? was calculated. Here €Q (qN ) is the displacement of the ath atom given by the normalized eigenvector ? of the sth phonon branch at the wavevector q . The summation over a is restricted to the outermost four atomic layers. To account for atomic displacements normal and parallel to the surface, the vector n was chosen to lie in the sagittal plane at an angle of 30° with respect to the surface normal. As can be seen from the right portion of Fig. 48, all of the branches resolved in the HREELS measurements follow maxima of the projected density of states. However, for a more detailed analysis of the HREELS spectra, multiple scattering theory (see e.g. Ref. [245]) has to be used.

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A prominent feature in the measured and calculated phonon dispersion of Ga : Si(1 1 1) ((3;(3)R30° is a flat branch above the continuum of the bulk states, with a computed C point frequency of about 67.7 meV, which is in good agreement with the experimental value of about 68.0 meV. The corresponding eigenvector is dominated by a bond-stretching oscillation of the atoms 2 and 3, which are in the second and third substrate layers and directly below the adatoms. The gallium atoms and the atoms in the first substrate layer are essentially at rest. As suggested by Northrup [270], this vibrational feature is common for all III : Si(1 1 1) ((3;(3)R30° adsorption systems, in which the chemisorbed atoms reside at the ¹ position on  the surface. The calculations in Ref. [270] were carried out for periodically repeated crystal films, with Al-termination on both sides of an eight-layer slab. For the plane-wave expansion a cutoff energy of 6 Ry was used. For the aluminum-covered Si(1 1 1) surface, Northrup obtained an energy of 69 meV for the subsurface stretching mode. This result is in good agreement with 68.9 meV measured by HREELS [7]. The stretching mode’s separation from the bulk continuum indicates that the bonds between the atoms 2 and 3 are compressed as a consequence of the adsorption of the group-III atoms on the ¹ site. The adsorption on the H position most probably would not lead   to the same compression of the bonds between the atoms 2 and 3. Therefore, the exact frequency of this phonon mode gives unambiguously indication for the ¹ adsorption configuration.  Since the atomic and electronic structure is nearly the same for the adsorption of Al, Ga, and In on Si(1 1 1), the force constants determined for one adatom system can be used to obtain an estimate for the vibrational features of the other III : Si(1 1 1) ((3;(3)R30° configurations. This can be done by replacing the respective atomic masses for the calculation of the dynamical matrices. Using the force constants computed from DFPT for Ga : Si(1 1 1) ((3;(3)R30°, the mass approximation gives a vibrational energy of 68.5 meV for the subsurface stretching mode in the case of Al coverage and 67.1 meV for the coverage of Si(1 1 1) with In. These values agree with the HREELS data of Refs. [7] and [195] which are 68.9 and 65.0 meV, respectively. For In deposited on Si(1 1 11), Northrup obtained a frequency of 68 meV within the mass approximation. Finally, for the boron-induced ((3;(3)R30° reconstruction of the Si(1 1 1) surface, a prominent peak at about 99 meV is present in recently recorded EELS spectra [355]. The results from DFPT show that this feature can be attributed to a stretching mode of the boron atoms and their bonding partners in the third substrate layer [171]. In the force-constant calculations of Ref. [270], a second surface localized feature was determined with a vibrational energy of 33.0 meV for Al and 28.0 meV for In deposited on Si(1 1 1). The mass approximation applied to the results from DFPT for Ga covering Si(1 1 1) yields the energies 33.0 and 29.2 meV for Si(1 1 1) covered by Al and In, respectively. In this mode, the group-III atom oscillates normal to the surface and couples to substrate layer vibrations. For Al : Si(1 1 1) ((3;(3)R30°, the corresponding surface phonon mode was detected by HREELS at about 33 meV [331], in good agreement with the computed frequency.

9. Conclusions and outlook By means of density-functional theory, the phonon dispersion of free and adsorbate-covered semiconductor surfaces can be calculated with high accuracy. The exact knowledge of prominent microscopic vibrational modes is helpful to understand and characterize in great detail the atomic

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arrangement and the bonding structure of many different surface structures. For all of the systems which have been investigated so far, fundamental relations between the particular geometry of a surface and its vibrational spectrum have been revealed. Characteristic surface vibrations were identified which directly reflect the formation, orientation, and strength of chemical bonds, resulting from a surface reconstruction or the adsorption of atoms. One important example is the dimer-stretching mode of the clean (0 0 1) surfaces of silicon, germanium, and diamond (see Section 6 and Refs. [10,124,346]). Its existence is a clear manifestation for the formation of bonds between neighboring surface atoms. Moreover, the vibrational energy of the stretching mode is an indicator for the strength of the bond. The stretching-mode frequency was found to be above the continuum of the bulk modes in C(0 0 1) and below the upper edge of the surface projected bulk bands for Si(0 0 1) and Ge(0 0 1). This is consistent with the computations of Ref. [208] which revealed a pronounced tendency of n bonding in the C—C pairs in contrast with the much weaker dimer bond in Si and Ge(0 0 1). Another example is the flat branch resolved by HREELS experiments (Refs. [7,195,331]) slightly above the continuum of the surface projected bulk band, seen for the adsorption of Al, Ga, or In on Si(1 1 1). Its frequency is nearly independent on the particular choice of adsorbate atoms and lies in the range from 65 to 69 meV. The mode is dominated by a bond-stretching oscillation of the atoms in the second and third substrate layer, which are directly below the adsorbed atoms. Therefore, this vibrational state provides unambiguous information about the adsorption site. While one third of the second and third substrate layer atoms are directly below adatoms in the ¹ chemisorption configuration, the adsorption at H positions would not lead to this kind of   coverage of second and third substrate layer atoms. The splitting of the mode from the bulk continuum indicates that the chemical bonds involved in the stretching vibration are compressed as a result of the adsorption. In the framework of the discussed approximations, ab initio calculations are free from any parameters. Therefore, the computed frequencies and eigenvectors are essentially predetermined by the equilibrium positions found for a chosen reconstruction. In this respect, the calculated phonon dispersion curves will be in agreement with the experimental data only if the right atomic arrangement and the correct symmetry are assumed. An important example for this is the phonon spectrum of the Ge(0 0 1) surface. Only the phonon dispersion curves computed for the higherorder reconstructions [346] are in agreement with the results from HAS [381], while the surface vibrations calculated for the p(2;1) arrangement of tilted dimers do not account for the features seen in the TOF spectra. The high degree of reliability which can be achieved for simple semiconductor surfaces within ab initio calculations is crucial, especially if the results from model calculations give conflicting interpretations. An example for this is the pronounced optical phonon mode at 56 meV resolved for the Si(1 1 1) (2;1) surface by means of HREELS [178,286]. While the tight-binding calculations of Ref. [9] assigned a longitudinal optical vibration with the energy loss at 56 meV, the bond-charge model of Ref. [244] related an interface mode characterized by normal displacements of secondlayer atoms to the feature seen by HREELS. In such an ambiguous situation, it is essential to have reliable results from an unbiased calculation. Clarification about the mode at 56 meV was provided by the Car—Parrinello molecular-dynamics simulations of Ref. [18] and the DFPT approach applied in Ref. [19]. The findings of these investigations are in good agreement with a recent DFPT study of the complete phonon dispersion along the chain direction [405]. The ab initio calculations

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show that the dipole-active longitudinal mode as well as interface vibrations are responsible for the anisotropic and isotropic contributions to the peak observed in the loss spectra. Another example is the surface phonon dispersion of GaSe(0 0 1), measured by means of HAS [55]. For a comparison with the bulk phonon spectrum, surface projected bands were calculated in Ref. [55], using a shell model with parameters fitted to the data from inelastic neutron scattering [182]. The computations revealed a large separation of the surface mode frequencies from those associated with the bulk. Related to this, significant changes in the force constants were necessary to reproduce the experimentally determined surface phonon dispersion. This is surprising since neighboring films of a layered compound semiconductor such as GaSe show only weak Van der Waals like interactions. In Ref. [55], the modifications were interpreted in terms of local changes of the electron susceptibility. A recent ab initio calculation [4], however, shows that the atomic positions and the electronic charge density of the bulk and the (0 0 1) surface of GaSe are very similar. Consistent with this, DFPT yielded vibrational frequencies for the bulk [4], which are in good agreement with the surface phonon dispersion curves measured in the HAS experiments of Refs. [55] and [281]. Because of the large numerical labor, most of the density-functional calculations are feasible only for low-index semiconductor surfaces which are characterized by a simple reconstruction. The (0 0 1) surfaces of Si and Ge are an example for this. While the complete phonon dispersion of the p(2;1) reconstruction was computed for both surfaces by means of DFPT [124,346,347], the vibrational states of the higher-order reconstructions were determined within the linear-response formalism only for the Si(0 0 1) c(4;2) surface, restricting the calculations to the J point of the SBZ [124]. Since tilted dimers represent the basic structural units of the p(2;1), p(2;2), and c(4;2) surface geometries, the phonon dispersion of the higher-order reconstructions can be obtained from DFT by performing self-consistent computations only for the buckled-dimer p(2;1) configuration and subsequently adapting the force-constants according to the in- and out-of-phase arrangement of the dimers along and perpendicular to the rows (local coupling transfer, Section 6.4.2). This method was successfully applied in Ref. [346] to calculate the phonon dispersion of Ge(0 0 1) p(2;2) and Ge(0 0 1) c(4;2). However, the method of local coupling transfer can be successfully applied only to large-size superstructures that are composed of structural subunits which appear with similar symmetry and physical properties in smaller systems. This is not the case for complex systems like the dimer-adatom stacking fault (DAS) (7;7) reconstruction of the Si(1 1 1) surface [356]. Because of the large size of the (7;7) unit cell, this surface still is out of reach for most of the density-functional approaches. By means of the conjugate-gradients formalism [137,285,360], combined with the wave-function extrapolation method [22] and the use of massive parallel computing, S[ tich was able to extend successfully ab initio molecular-dynamics simulations to the computation of surface vibrational modes on Si(1 1 1) (7;7) [343,344]. The results of these simulations are different from the findings of previously performed density-functional cluster calculations [88], which suffer from finite-size effects and, related to that, the incomplete treatment of the charge transfer between the dangling bonds on the (7;7) surface. Particularly, the conjugate-gradients molecular-dynamics simulations of S[ tich demonstrate that the density-functional schemes developed for the calculation of semiconductor surface vibrations became highly efficient in the course of the last decade. The first successful applications of DFT to compute surface phonon modes were based on the frozen-phonon formalism and/or the force-constant method. These approaches, which require the

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same theoretical level as the ground-state calculations, were used to determine the frequencies of adsorbate vibrations and surface-localized phonon modes at high-symmetry points of the surface Brillouin zone [64,192,270,329]. In combination with advanced spectral analytical methods, also Car—Parrinello molecular-dynamics simulations were successfully applied to the investigation of surface phonons [18,111,336]. From the particle trajectories, vibrational states were identified at the zone center and zone boundary points of low-index surfaces of materials like silicon [18,336] and GaAs [111]. An important stimulus for the calculation of semiconductor surface phonons was provided by the introduction of the linear-response formalism into the density-functional method. The application of the efficient density-functional perturbation scheme developed by Baroni and co-workers to semiconductor surfaces allows one to determine dynamical matrices on a dense grid of wavevectors in the Brillouin zone. Therefore, complete phonon dispersion curves can be computed monitoring the vibrational character of the surface-localized states in great detail [101,121,126,169,346]. For several semiconductor surfaces, highly localized phonon states were identified. Particularly in the case where the oscillator has a small reduced mass, large vibrational amplitudes can occur for some of the localized modes. Because of this and the lower symmetry on the surface, anharmonicity of the effective potential can become important for an appropriate description of the vibrational states. The adsorption of light atoms such as hydrogen might give ideal candidate systems which show enhanced anharmonic effects. Current investigations are focused on the determination of the temperature dependence of frequency shifts [169] and phonon line widths [169,225], overtone spectra [404], and the binding energy of two-phonon bound states [225]. Strong anharmonic coupling may also result in the occurrence of intrinsic localized modes (ILM) [46,47]. This new type of vibrational excitations involves motions of only a few atoms in an extended system. Criteria for the stability and the existence of ILMs have been studied intensively for several model cases [47,306]. Adsorbate covered surfaces and clean surfaces with an appropriate vibrational spectrum are promising candidates to study properties of ILMs in realistic systems [46]. Therefore, they are in the focus of future theoretical research. By means of linear-response theory, the phonon dispersion can be determined throughout the entire Brillouin zone. This allows one to calculate mean-square displacements of surface near atoms and Debye—Waller factors [167,170,171], which relate the diffraction intensities of elastical scattering experiments to the thermal motion of the atoms. For an exact interpretation of scattering experiments, the vibrational amplitudes of the surface atoms have to be known as a function of the temperature. The magnitude of the Debye—Waller factors is sensitively dependent on the computed eigenvectors and eigenfrequencies. It is therefore crucial to have a highly accurate description of the surface dynamics, as provided by the density-functional approaches. The knowledge of the complete phonon dispersion is also a prerequisite to compute the free energy of a surface for the investigation of surface phase transitions and thermal expansion. The availability of computational resources and the development of efficient density-functional schemes will intensify theoretical research of temperature dependent phenomena on semiconductor surfaces. The study of anharmonic effects [169,225,404] and the calculation of mean-square displacements [167,170,171] represent first examples for this. The ab initio calculation of semiconductor surface phonons so far was focused on III—V compounds and group-IV materials, for which sufficiently soft pseudopotentials can be generated. In most of the computations, plane waves are used for the expansion of the electronic wave

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functions. Because of the increase of the numerical labor, large systems were studied only to a limited extend. An example is the Si(0 0 1) c(4;2) surface, for which the phonon modes were determined by means of DFPT only at the J point. To study complicated systems, highly efficient algorithms have to be used. Like in the molecular-dynamics simulations of S[ tich for the Si(1 1 1) (7;7) surface [343,344], the conjugate-gradient technique can be successfully implemented into the calculation of higher-order derivatives by means of DFPT [145]. Beside the elemental and compound semiconductors examined so far, new materials like the group-III nitrides have become subject of intensive research. The surface structure of these materials has been examined only recently by means of total-energy calculations. For the expansion of the wave functions in plane waves, considerably high cutoff energies had to be used. To study dynamical phenomena on the surfaces of these materials, ab initio calculations are only practicable if the number of basis functions can be reduced. This can be achieved by the use of other basis sets like the mixed-basis approach [228] as applied in Ref. [247] to compute static and vibrational properties of bulk GaN and bulk AlN. To determine the complete phonon dispersion, an extension of the linear-response formalism to basis sets other than plane waves is required in analogy to Ref. [386], where the linearized augmented plane wave method was used to compute the phonon dispersion of the polar semiconductor CuCl.

Acknowledgements Most of the work on the writing of this review was done during a research fellowship of one of the authors (J.F.) hosted by John B. Page at the Arizona State University. Financial support for this stay was provided by the Alexander von Humboldt-foundation through a Feodor-¸ynenForschungsstipendium, the Office of Naval Research and the National Science Foundation. The review article mainly summarizes the results of several research projects performed at the Institute for Theoretical Physics of the University in Regensburg. We are very much indebted to all the members of the group, for their enthusiasm in performing the calculations and analyzing the data, for numerous discussions, inspiring suggestions, and their kind permission to use their original figures and data, some of which are unpublished. In chronological order, these are P. Pavone, R. Honke, C. Eckl, W. Stigler, M. Zitzlsperger, C. Adler, and M. Arnold. We gratefully acknowledge numerical support by P. Giannozzi, S. de Gironcoli, S. Baroni, and P. Pavone for providing their ab initio linear-response code. Large parts of the computations were carried out on the CraySupercomputers of the HLRZ in Ju¨lich under Contract No. K2710000 and the Leibniz Rechenzentrum in Munich. It is a pleasure to thank all of our colleagues for their lively interest in our results and the wealth of new ideas which evolved in all the discussions. The work was also supported by a grant of the Deutsche Forschungsgemeinschaft through the Graduiertenkolleg “Komplexita¨t in Festko¨rpern: Phononen, Elektronen und Strukturen”.

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