GW self-energy calculations for surfaces and interfaces

Transcription

1 Computer Physics Communications 137 (2001) GW self-energy calculations for surfaces and interfaces P. García-González, R.W. Godby Department of Physics, University of York, Heslington, York YO10 5DD, UK Abstract The many-body GW approximation has become the most popular method in ab initio calculations of excited-state properties in real materials. GW overcomes many of the limitations of traditional density functional theories to predict one-electron excitation spectra for a wide variety of materials. In this article we review some applications of GW to real surfaces, from calculations of surface band structures in semiconductors to estimations of the lifetimes of surface states in metals Elsevier Science B.V. All rights reserved. PACS: Ar; Lc; At; y. Keywords: Electronic structure; Quasiparticle energies; Surface states; Self-energy calculations; GW approximation 1. Introduction Although many of the electronic and structural properties of surfaces can be understood in terms of simplified models, a full understanding of the underlying physical processes requires the use of ab-initio calculations of the electronic structure of real surfaces. However, even at this level, the use of approximations to deal with the many-electron problem are needed. In density functional theory (DFT) [1,2] the many-electron problem is mapped onto a fictitious system of non-interacting electrons and solved for the ground-state properties. DFT is an exact formulation of the problem, but the energy functional E XC [n] that accounts for the many-body exchange and correlation (XC) effects is unknown. In practical applications, this functional is approximated by the local density approximation (LDA) of Kohn and Sham (KS) [2], or extensions such as generalized gradient approximations [3] and even other more sophisticated non-local approaches [4 7]. By using these ground-state calculations, the total energy of any electron system can be obtained and, subsequently, it is possible to determine accurately crystal structures, lattice constants, surface geometries, etc. Furthermore, the development of first-principles molecular dynamics methods [8] together with the improvement in the numerical procedures allows the theoretical study and characterization of complex surface structures containing a large number of atoms [9]. Nonetheless, the one-electron wavefunctions and energies of the fictitious non-interacting system (KS system), even in exact DFT, do not have a well-defined physical meaning (except the energy of the highest occupied * Corresponding author. addresses: pgg1@york.ac.uk (P. García-González), rwg3@york.ac.uk (R.W. Godby) /01/$ see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S (01)

2 P. García-González, R.W. Godby / Computer Physics Communications 137 (2001) state which is equal to the chemical potential [10] and is directly related, for instance, to the work function of a surface [11]). As a consequence, there is no rigorous theoretical support for the interpretation of the KS energies as single-particle excitation energies. Therefore, one should not expect to explain photoemission spectroscopic properties, which are related to the removal or addition of an electron, in terms of the one-electron KS electronic structure. In fact, these single-particle removal/addition energies, known as quasiparticle (QP) energies ε qp n k,have to be obtained from the QP equation [12]: 1 [ V ion ( r)+ V H ( r) ] Ψ n k ( r)+ d r Σ ( r, r ; ε qp ) Ψn n k k ( r ) = ε qp Ψ n k n k ( r), (1) where Ψ n k ( r) is the QP wavefunction, V ion( r) is the potential due to the ions, V H ( r) is the Hartree electrostatic potential, and Σ( r, r ; ε qp ) is the self-energy operator in which the XC effects are contained. Note that, in contrast, n k the KS orbitals φ n k ( r) and energies εks are given by n k [ V ion ( r)+ V H ( r)+ V XC ( r) ] φ n k ( r)= εks n k φ n k ( r) (2) with V XC ( r) = δe XC [n]/δn( r) the exchange-correlation potential. Although the physical origin of the presence of Σ and V XC is the same, it is evident that more information is contained in the non-local and energy-dependent self-energy operator than in the local and static XC potential. Anyway, in simple systems such as sp metals the matrix elements of Re Σ(ε) and V XC are quite similar, and the KS energies are not a bad approximation to the actual QP spectra. Nonetheless, as is very well known, this agreement breaks down when considering the band structure in semiconductors, for which DFT-LDA greatly underestimates the band gaps. As a consequence, a theoretical description of the one-electron spectra with enough predictive accuracy requires the use of manybody techniques allowing the determination of the self-energy operator, Σ. Such techniques, even using simple approximations like Hedin s GW method [13], are more difficult to implement in ab-initio calculations than traditional DFT. This explains why the first results using first-principles QP methods appeared only in the mid- 1980s [14 16] and why further improvements and applications to complex systems have been restricted to the last decade [17 19]. In this review, after a brief description of theoretical foundations and computational methods (Section 2), we will focus on some selected applications to surfaces and interfaces. Specifically, in Section 3 we present some calculations of the QP spectra for intrinsic surface states in semiconductors. These states owe their existence to the breaking of symmetry related to the crystal potential termination (i.e. they are surface-induced states), are localized around the outermost atomic layer and, hence, their characteristics deeply depend on the geometrical details of the surface reconstruction. In the fourth section we will describe the attempts to evaluate the shape of the surface barrier at a metal-vacuum interface. Its knowledge is essential for a proper description of the so-called surface image states, because they arise as a result of the binding by the surface barrier in the vacuum region (in this case the geometrical details of the surface are no so important). 2 The most recent developments in the determination of the lifetimes of such image states in metals will be treated in Section 5. Finally, the importance of many-body effects in the study of Schottky barriers at metal/semiconductor interfaces will be the topic of Section 6. Although it is not our aim to present a comprehensive description of the field, we believe that the selection of topics to be treated here will be extensive enough to provide a general overview of the current state-of-the-art in first-principles calculations of quasiparticle properties of surfaces. 1 Unless otherwise noted, Hartree atomic units ( h = e = m e = 1) are used throughout. 2 In both cases, the surface states do not propagate towards the bulk because they lie on a band gap of the bulk band structure projected onto the surface. If that does not happen, the surface states would be resonances.

3 110 P. García-González, R.W. Godby / Computer Physics Communications 137 (2001) Theory 2.1. The GW approximation The core idea of Hedin s GW method [13] is to approximate the self-energy operator Σ by Σ( r, r,ω) i 2π + dω W( r, r ; ω )G( r, r ; ω + ω )e iω δ, (3) where G is the one-particle Green s function, W is the dynamically screened Coulomb interaction, and δ is a positive infinitesimal. Eq. (3) is the lowest (first-order) approximation in the expansion of the self-energy operator in terms of G and W. Since in the Hartree Fock approximation the self-energy is obtained as in (3) but with W replaced by the static bare Coulomb interaction v( r, r ) = 1/ r r, GW can be considered as a well motivated generalization of the Hartree Fock method in which the Coulomb interaction is dynamically screened. W is exactly given in terms of the inverse dielectric function ɛ 1 by means of W( r, r ; ω) = d r ɛ 1 ( r, r ; ω)v( r, r ), (4) so the GW evaluation of Σ relies on the calculation of the Green s function and the dielectric function. Besides, G depends on Σ through a Dyson equation and may be determined in a self-consistent fashion [20]. However, in most of the applications we will describe here, G has been approximated by the non-interacting Green s function of the fictitious KS system under the local approximation: G( r, r ; ω) G 0 ( r, r ; ω) = φ n k ( r)φ n k ( r ) ω ε KS, (5) iη n k n k where η is a positive (negative) infinitesimal for occupied (unoccupied) one-particle KS states. Finally, the evaluation of the dielectric function ɛ requires the knowledge of the polarization propagator P : ɛ( r, r ; ω) = δ( r r ) d r P( r, r ; ω)v( r, r ), (6) and P is usually approximated in the RPA: P( r, r ; ω) χ 0 ( r, r ; ω) = i 2π + dω G 0 ( r, r ; ω )G 0 ( r, r; ω ω). (7) Once the self-energy operator has been obtained, the QP energies can be calculated by solving (1). Because Σ is a complex non-hermitian operator, the energies ε qp are complex. The real part of εqp is the QP energy itself n k n k whereas the imaginary part gives the inverse lifetime of the QP. In many cases, there is an almost complete overlap between the QP and LDA wavefunctions, and the full resolution of the QP equation can be circumvented (although, as we will see later, this assumption is not always true). Thus, ε qp may be obtained as a first order perturbation of n k the corresponding LDA energy ε KS n k : ) VXC ε s φ n k. (8) ε qp n k εks n k + φ n k Σ( ε qp n k Eq. (8) can be solved self-consistently or, alternatively, after a first-order Taylor expansion of Σ(ω) around ω = ε KS n k. Note that in (8) a constant ε s has been included to align the chemical potential of the system before and after the inclusion of the GW correction, hence compensating some of the inconsistencies arising from the fact that in this scheme Σ and G are not related through a Dyson equation. For a more detailed account on the foundations of the GW method as well as on the many-body theory we refer the reader to further references [12, 18,19,21,22].

4 P. García-González, R.W. Godby / Computer Physics Communications 137 (2001) Numerical procedures As commented above, the evaluation of the self-energy in the GW approximation starts with a standard selfconsistent LDA ground-state calculation using ab-initio pseudopotentials. In surfaces, to mimic the underlying breaking of the symmetry, the ground-state calculation is carried out using a periodically repeated supercell comprising typically about 10 layers of the crystal together with a vacuum region (obviously the number of crystal/vacuum layers in the supercell is a convergence parameter). Usually, a plane wave representation of the one-electron orbitals is used. However, using a representation based on localized wavefunctions such as Gaussians, faster convergence may be obtained in surface calculations [23]. The most demanding step in a GW calculation is the evaluation of the dynamical dielectric function (Eqs. (6) and (7)). Due to this, the full dynamical dependence of ɛ 1 is often not calculated, but approximated using a plasmon-pole Ansatz [24]. Essentially, the underlying idea is to assume that all the spectral weight of the dielectric function is concentrated at a plasmon excitation. Therefore, the matrix elements of Im ɛ 1 (ω) are simply Dirac delta functions [15]. The positions and weights of such delta functions are typically determined from the static limit (ω = 0) of ɛ 1 and from the f -sum rules. As long as we are interested in QP energies, the plasmon-pole approximation seems to be accurate enough, although it has to be borne in mind that in systems other than sp metals, plasmons cannot be viewed as isolated excitations, and this approximation can hardly be justified. More importantly, the imaginary part of the self-energy is not well reproduced at all because Im Σ(ω) is then equal to zero except at isolated frequencies. Therefore, the evaluation of QP lifetimes is not possible using a plasmon-pole approximation. An effective way to find out the entire spectral representation of the self-energy is the so called space-time method by Rojas et al. [25]. In this procedure, rather than solving all of the preceding equations in reciprocal space, each one is solved in the most favorable spatial representation changing from real space to reciprocal space using fast Fourier transforms. Further, the dynamical dependences are not described in terms of real time or frequency but using their imaginary counterparts. By doing so, the time-dependence in G 0 decays exponentially and is preferable to evaluate G 0 ( r, r,iτ) instead of G 0 ( r, r,ω). Besides, the use of an imaginary axis representation avoids the sharp polar structures in G 0 (ω) and W(ω), allowing the use of coarser meshes. In the final evaluation of the QP energies, the self-energy for real frequencies can be obtained by means of analytical continuation from its values on the imaginary axis after a fitting to a suitable analytical function. An example of the method applied to surface systems can be seen in Ref. [26]. Further details as well as a full description of the computational procedures can be found in [27,28]. 3. Semiconductor surfaces In this section we will pay especial attention to GW calculations of band structures of intrinsic surface states in the prototypical semiconductors Si and Ge, for which the number of well resolved experimental studies allows a good discussion of the theoretical results. As in bulk semiconductors, DFT is unable to provide an accurate description of the band structure of these surface states. Only the inclusion of a full many-body treatment of the electron problem by using the GW approximation allows the reproduction of the experimental electronic structure. In fact, usually GW corrections are even more important than in bulk semiconductors [29] Si(100)-(2 1) and Si(100)-c(4 2) Because of their technological importance, Si surfaces have been studied thoroughly, both theoretical and experimentally. For Si(100), DFT total energies calculations [30 32] show the surface to be reconstructed forming buckled dimers, where one atom moves up and the other down with respect to the ideal (unreconstructed) surface plane. At room temperature,this reconstructionshows a 2 1 pattern, and there are two different types of dangling

5 112 P. García-González, R.W. Godby / Computer Physics Communications 137 (2001) Fig. 1. Calculated dangling-bond bands of Si(100)-(2 1). Solid lines: GW; dashed lines: LDA; symbols: photoemission experiments (diamonds from Ref. [35] and circles from Ref. [36]). The shaded region corresponds to the bulk-projected band structure. After Rohlfing et al. (Ref. [23]). bonds. The occupied one (D up ) is mainly located at the up atom of the dimer, and the empty one (D down )atthe down atom. The surface band structure of these dangling bonds has been evaluated in the GW approximation by Kress et al. [33] and by Rohlfing et al. [23], both using a plasmon-pole approximation [34] for the dynamical dielectric function. In the calculation by Kress et al. a model static dielectric function was also included. We focus on the calculation by Rohlfing et al., which was performed in a supercell containing eight layers of Si and six layers of vacuum, and using 20 Gaussian basis functions at each atom. To avoid interactions between the dangling bonds of the two surfaces, the dangling bonds of one surface were saturated with hydrogen atoms. The resulting bands are shown in Fig. 1, where they are compared with the LDA energies and the photoemission experiments by Uhrberg et al. [35] and by Johansson et al. [36]. The GW correction raises the occupied band D up by an average of 0.15 ev relative to the top of the bulk valence band, whereas the band D down is raised by an average of 0.50 ev. As a result, the indirect fundamental surface gap increases from 0.20 ev in the LDA to 0.70 ev in the GW approximation. The latter is consistent with experimental estimations that range from 0.44 to 0.9 ev. As we can see in Fig. 1, the GW D up band reproduces fairly well the experimental results. However, there is a series of occupied states (represented in the figure by open circles) that have no correspondence in the theoretical data. These states may be explained as a second dangling-bond band resulting from local c(4 2)-reconstructed regions. In fact, the GW calculation by Northrup [31] for Si(100)-c(4 2) (which is the favored reconstruction at low temperatures) fits relatively well the direct-photoemission data taken at room temperature (see Fig. 2) Si(111)-(2 1) For Si(111)-(2 1), a number of experimental results supports the idea that the reconstructed surface is terminated by a π-bonded chain of Si atoms (Pandey chain [37]). These chains are buckled by about 0.40 Å, and calculations [38,39] indicate the positive sense of buckling to be the most stable (see Fig. 3). The corresponding QP band structure was calculated using the GW by Northrup et al. [39] and, more recently, by Rohlfing and Louie [40] (again using a plasmon-pole approximation to ɛ 1 (ω)). It exhibits one occupied (D up ) and one empty surface band (D down ) that are formed from the p z -like dangling-bond orbitals of the chain atoms relaxed towards the vacuum and towards the bulk, respectively. Because of the surface geometry, the dangling-bonds are strongly coupled along the chain, but the coupling is much weaker between parallel chains due to the large chain chain distance. As a consequence, the surface bands behave as one-dimensional-like bands, which is reflected by the weak dispersions along the JK line (see Fig. 4). This calculated QP band structure agrees very well with the experimental photoemission data [41] as well as with scanning-tunneling spectroscopy results [42]. The physical

6 P. García-González, R.W. Godby / Computer Physics Communications 137 (2001) Fig. 2. Calculated dangling-bond bands of Si(100)-c(4 2). The GW quasiparticle energies (circles) are compared with the experimental results by Johansson et al. [36] (crosses). As before, the shaded region corresponds to the bulk projected band structure. After Northrup (Ref. [31]). Fig. 3. Side view of the two possible bucklings of Pandey chains in Ge(111)-(2 1) and Si(111)-(2 1). Whereas in Si the positive buckling is energetically favorable, in Ge both isomers have almost the same total energy. From Rohlfing et al. [49]. origin of the differences between the theoretical data and those obtained using heavily n-doped Si samples [43] may be understood in terms of the influence that the partial occupation of the D down state could have on its electronic self-energy [40], in close resemblance with the band-gap narrowing in bulk n-doped Si [44]. Although not directly related to the main topic of this review, it should be noted that the surface QP band gap obtained by photoemission spectroscopy differs from the optical gap (determined by, for instance,

7 114 P. García-González, R.W. Godby / Computer Physics Communications 137 (2001) Fig. 4. GW quasiparticle band structure of the Si(111)-(2 1) surface (full lines). The shaded areas denote Si bulk bands. The dots represent experimental data. From Rohlfing and Louie [40]. differential reflectivity spectroscopy) by an amount of ev. Whereas photoemission and scanningtunneling spectroscopies probe single-particle processes (hence well described in terms of the QP electronic structure), in optical spectroscopies electron-hole interactions may play an important role. In fact, the hypothesis of an optical spectrum dominated by a surface exciton were supported by model calculations by Northrup et al. [39] and by Reining and Del Sole [45], and confirmed by ab-initio calculations done by Rohlfing and Louie [40] of the electron-hole interaction by solving the Bethe Salpeter equation for the two-particles Green s function [46] Ge(100)-(2 1) and Ge(111)-(2 1) The reconstructed Ge(100) surface structure is very similar to Si(100), which was described in a previous subsection. At room temperatures, the surface shows buckled dimers in a 2 1 pattern, that undergoes a phase transition to a c(4 2) pattern at low temperatures. In the 2 1 phase, the buckling angles in Ge and Si are very similar, as well as the relation between the dimer bonds and the bulk bonds. Therefore, the surface band structures of the Ge(100)-(2 1) and of the Si(100)-(2 1) are nearly identical. This has been confirmed by Rohlfing et al. [47] using a Gaussian-basis-set GW calculation. Whereas in Si(100) the surface band gap is 0.70 ev, in Ge(100) it is equal to 0.80 ev (the LDA value is 0.40 ev). The only remarkable difference is that in Ge the surface states are lower in energy than in the Si surface, leading to a resonant coupling of the Ge D up state with bulk states at the point Γ of the surface Brillouin zone. The case of the Ge(111)-(2 1) surface is more interesting, owing to qualitative differences from Si(111)-(2 1). Whereas in Si the Pandey chain reconstruction with positive buckling is the most stable one, in Ge the negative buckling (see Fig. 3) has a total energy almost equal to those exhibiting positive buckling [48], so suggesting the coexistence of both isomers in a real reconstructed surface. Recent total-energy calculations carried out by Rohlfing et al. [49] favors the negative buckling by about 13 mev/surface-(1 1) cell, in contrast to the conventionally assumed positive buckling. Moreover, the same authors show that the surface QP band structure of both structures are very similar, but that corresponding to the negative buckling is marginally closer to experimental direct and inverse photoemission experiments [50]. Note that these results seem to supersede those obtained by Zhu and Louie [51] in which they found a close agreement between experimental QP data and the surface band structure of the positive buckling reconstruction. These discrepancies arises from the different geometry constants used in each calculation. A final support in favor of the ground-state geometry of Ge(111)-(2 1) in a negative buckling reconstruction comes from the excellent agreement between its optical excitation spectra (including excitonic effects) and the low temperature experimental differential reflectivity data. This agreement cannot be reached considering the conventional Pandey chain with positive buckling structure.

8 P. García-González, R.W. Godby / Computer Physics Communications 137 (2001) Other semiconductor surfaces The GaAs(110) surface has been investigated by Zhu et al. [52] and more recently by Pulci et al. [53]. Despite the small size of the supercell used by Zhu et al., the GW band structure obtained by these authors showed a good agreement with the experiment. The calculation by Pulci et al. uses a larger supercell and a better approximation to the dynamical dielectric function, which allows a converged calculation of the reflectance anisotropy spectrum (RAS) of the surface. Although the LDA result is clearly improved, the neglect of local-field effects and, above all, excitonic effects prevents a full reproduction of the experimental RAS data. Pulci et al. [54] have also calculated the QP states at GaAs(110) after a full solution of the quasiparticle Eq. (1), i.e. going beyond first order perturbation theory. Although small, the differences between the true QP wavefunctions and the LDA ones implies that the QP energies can be underestimated up to 0.1 ev if perturbation theory is used. Furthermore, the calculate RAS changes (although not qualitatively) if the true QP wavefunctions are used instead. These results show that the identification of LDA and QP wavefunctions may lead to incorrect results especially at surfaces. InP(110) has been studied, but using simplified GW approaches, by Bechstedt and Del Sole [55] and by Jenkins et al. [56]. Comparison with experiments is difficult due to the resonant character of many of the surface states. However, these simple many-body corrections are enough to improve the LDA energies as well as to show a dramatic change in the shape of the surface bands. Different reconstructions of the β-sic(001) surface have been analyzed by Sabisch et al. [57]. In this case the interest is twofold since the surface reconstruction itself affects the surface electronic structure. For C-terminated surfaces, the number of possible reconstructions hinders a direct comparison with experiments. This has to be taken into account, together with the uncertain interpretation of the experimental data, in the assessment of recent calculations by Gundel et al. for CdTe(001) [58]. For studies on other different surfaces we refer the reader to the review article by Louie [17]. 4. Surface barrier in metals An electron outside a metal surface sees a potential that behaves as V(z) = 1/4(z z 0 ),wherez is the coordinate perpendicular to the surface and z 0 is the effective position of the edge of the metal. This image-like behaviour can be explained classically by the polarization induced on the surface region by an external electron, and it has its analogous counterpart in the quantum XC potential. Thus, although the microscopic details of the surface play a role in V XC ( r), its behaviour far outside the metal surface should approach the above limit [59]. It is likely that this behaviour has very little influence in total-energy calculations, but it is crucial in understanding the series of spectroscopic image states bounded by the image potential [60] as well as for a proper interpretation of scanning tunneling microscopy experiments [61]. In the previous section, we have analyzed the success of the many-body GW approximation to correct some of the limitations of the DFT-LDA when determining the one-electron spectra. DFT, being a ground-state theory, should not be appropriate to describe properties of excited states, but V XC ( r) is a truly ground-state property, which is related to the pair correlation function of the electron system. Hence, the well-known failure of the LDA XC potential to reproduce the image-like behaviour of V XC ( r) [11] is due to the local approximation itself, and cannot be attributed to an intrinsic limitation of DFT. The reason for this failure is easy to understand because in the LDA the XC hole associated with an electron is always centred at the electron s position. However, if the electron is far outside a metal surface, its XC hole should be located in the surface. Moreover, the spreading of the XC over the metal surface is finally responsible for the image-like behavior of V XC ( r), and neither the LDA nor other XC gradient or semi-local approximations can give the desired limit of V XC ( r). Note that the situation is slightly different from that in a localized system, in which V XC ( r) decays as 1/r [10]. The latter can be fulfilled at the exchange-only (Pauli) level, because an exact treatment of the exchange effects gives a localized XC hole with the proper normalization. Correlation (electrostatic effects) provides lower order corrections to V XC (r 0).

9 116 P. García-González, R.W. Godby / Computer Physics Communications 137 (2001) In a metal surface, when considering a slab geometry, the exact V X ( r) behaves as a/z 2 anditisonlyafterthe inclusion of correlation when the exact image-like behaviour is reproduced [62]. Therefore, correlation effects are the main source of the image-like behaviour and the exchange potential is important in the determination of the constant z 0. However, in a semi-infinite geometry the exact V X ( r) behaves as 1/4z [63], but this is due to the extreme delocalization of the exchange hole over the bulk. In this case, correlation is responsible for the final concentration of the XC hole in the surface region and, again, of the asymptotic shape of V XC. All these features reflects the importance of electrostatic correlations even in the classical description of the problem. It is, then, evident that the correct description of the image-potential in DFT is not an easy task. There have been many efforts to formulate prescriptions for the XC functional able to give the right asymptotic behaviour in a surface [5]. Some of them, such as the non-local functional introduced by Gunnarson and Jones [64], seemed to give the image-like behaviour [65], but at the expense of giving unphysical results for the position and shape of the XC hole [66,67] and surface energies [68]. Only recently has a self-consistent prescription been proposed able to solve this problem [69], in which some essential features of the XC hole (normalization and overall shape) are modeled by means of a highly non-local functional. Interpolation procedures [66,70] may be useful, but the lack of self-consistency avoids fully reliable total-energy calculations as well as the use of the time-dependent extension of DFT which has been proved to be a good approximation in the study of electron excitations in surfaces [71]. A different approach to the problem is the use of the GW approximation, because it contains all the physical ingredients that lead to main behaviour of the surface barrier. Moreover, the so-called Sham Schlüter equation [16, 72,73] gives a direct link between the many-body functions and the exact XC potential: + dω d r 1 G 0 ( r, r 1 ; ω)v XC ( r 1 )G( r 1, r; ω) = + dω d r 1 d r 2 G 0 ( r, r 1 ; ω)σ( r 1, r 2 ; ω)g( r 2, r; ω). (9) In the GW framework, the exact expression (9) can be approximated by calculating Σ according to Eq. (3), and by setting G = G 0 in (3) and (9). In addition, since V XC ( r) determines the KS fictitious system, and hence G 0,the final determination of V XC ( r) is a well-defined problem that has to be solved self-consistently. The above calculation has been carried out by Eguiluz et al. for a simple jellium surface [62], and the results for r s = 3.93 (the average density of Na) are displayed in Fig. 5. As can be seen, the V XC (z) obtained from (9) Fig. 5. Exchange-correlation potential for a jellium surface with r s = 3.93 (λ F = 12.9 a.u.). The solid line is the result within the GW approximation. The dotted line is the LDA potential. The dashed curve is the pure image-like potential. From Eguiluz et al. [62].

10 P. García-González, R.W. Godby / Computer Physics Communications 137 (2001) Fig. 6. Exchange and correlation contributions to the effective local potential for unoccupied quasiparticle states at the Al(111) surface. From White et al. [26]. exhibits the right asymptotic behaviour 1/4(z z 0 ). It is interesting to note that the position z 0 that determines the effective location of the image-plane is not equal to the value arising in the context of the linear response of the surface to an external field, 3 but closer to the jellium edge. This trend has been confirmed by White et al. from a GW calculation for a real Al surface [26]. Using the GW V XC (z), Eguiluz et al. [62,74] have proposed a new non-local parametrization for V XC ( r) based on the many-body calculation that has been applied to obtain intrinsic and image surface states in Al and Pd. Perhaps the main limitation of this many-body evaluation of V XC ( r) is the underestimation (about 10%) of the absolute value of V XC in the bulk with respect to the exact Monte-Carlo results [75]. A full self-consistent GW calculation (i.e. the self-energyand the Green s functionare not only related through Eq. (3), but also through a Dyson s equation) seems to overcome this problem [76,77]. Effects that selfconsistency and/or inclusion of vertex corrections may have, for instance, in the position of the image-plane remain to be investigated. It is equally interesting to see whether the exact V XC can be used to describe some of the physics contained in Σ in a system as inhomogeneous as a surface. We have already commented that in bulk materials, LDA- KS one electron orbitals agree very well with the QP wavefunctions, but due to the inaccuracy of the LDA this result cannot be extended to a metal surface [26]. However, as shown by Deisz et al. [78], in a jellium surface the first KS image state given by the GW V XC is remarkably similar to the corresponding QP state. 4 Moreover, the KS and QP energies practically overlap. This could lead to the idea of simulating the self-energy effects by means of local potentials, but evidently this is not the way especially when this finding only holds for the real part of Σ but not for the imaginary. In fact, Im Σ is intrinsically non-local and cannot be mimicked by any effective local potential [78]. On the other hand, the frequency-dependence of Im Σ( r 1, r 2 ; ω) is very sensitive to the actual spatial positions r 1, r 2 [79]. Finally, coming back to the real part of Σ, we can see in Fig. 6 that the exchange contribution to the potential felt by an unoccupied state decays exponentially far outside the metal surface [26]. In other words, whereas the exact V XC ( r) describes the potential felt by occupied states, this is no longer true for unoccupied states due to the qualitatively different individual roles played by exchange and correlation for states above the Fermi level. However, as shown by Deisz et al. [78], when exchange and correlation are combined, the exact V XC ( r) and the effective potential felt by unoccupied states are fairly similar. 3 It may be noted that this is the assumption made in Ref. [66] to fix the asymptotic behaviour of V XC. 4 Since there is no gap in jellium, this image state is a resonance.

11 118 P. García-González, R.W. Godby / Computer Physics Communications 137 (2001) Lifetimes of image states The results discussed in the previous section support the common practice of calculating occupied and unoccupied metal surface states as the KS orbitals obtained with a suitable (i.e. with the correct asymptotic behaviour) XC potential. In this way, the binding energies for surface-induced and image surface states have been calculated for a variety of metal surfaces [74,80 82], showing an overall good agreement with photoemission experiments. Focusing on image states, their location in the vacuum region implies a pronounced free-electron behaviour. Therefore, neglecting corrugation effects along the surface, a simple hydrogenic model yields a Rydberg-like series ε n 1/n 2 for the states at Γ (the vacuum level is at E = 0) with an almost parabolic dispersion on the surface Brillouin zone [60]. In contrast, a first-principles evaluation of the lifetime of imagestates requires the calculation of (at least) the imaginary part of the self-energy operator. Since they are almost decoupled from bulk states, they have much longer lifetimes than bulk QP states. The modern time-resolved twophoton photoemission spectroscopy allows accurate measures of these lifetimes [83]. The GW expression for the self-energy was used to estimate lifetimes of image-states in the seminal work by Echenique et al. [84], who used a simplified description of the one-electron states and of the dielectric function. Very recently, full GW calculations have been performed for model surfaces, and lifetimes of image states have been reported for Cu(100) and Cu(111) [85 87]. The simplified system used assumes translational invariance along the plane of the surface which, as mentioned above, is an acceptable first approximation to the problem. The potential V(z) provides band-structure effects through a periodic bulk part and, of course, builds in the right image-like asymptotic behaviour. The parameters that determines this potential are fixed in such a way that at the point Γ, the width and position of the projected bulk gap, the binding energy of the intrinsic surface state, and the binding energy of the first (n = 1) image state reproduce either experimental or first-principles calculation values [70]. In this way, the most important features of the problem can be taken into account. Once the one-electron wavefunctions have been obtained, the inverse of the lifetime of an image-state φ i is given by minus twice the expectation value of the imaginary part of the self-energy: τ 1 = 2 φ i ImΣ(ε i ) φ i, where ε i is the energy of the state. The translational invariance along the surface plane simplifies the expression for Im Σ(ε i ), and the lifetime can be evaluated in terms of the 2D Fourier transform of Im Σ( r, r ; ε i ). Moreover, the only decay channels for an image state are other unoccupied states with lower energy, and in the final evaluation of Im Σ(ε i ) in terms of the screened Coulomb potential only those states have to be considered (see [85] for details). The value so obtained for the width τ 1 of the first image state at Γ in Cu(100) is 22 mev, which compares favorably with the experimental value of 16.5 mev (see also Table 1). However, when using a jellium model for the metal (i.e. neglecting band-structure effects), the theoretical value is 67 mev. This overestimation of the width of the QP state can be easily understood on the basis of the existence of more decay channels that are absent when a gap is opened [85]. Finally, as expected, the main contribution to the finite width of the lowest energy image state is the decay into intrinsic surface states. However, despite the small penetration into the bulk of the Table 1 Calculated lifetimes (in femtoseconds) of the n 3image states of Cu(100) and the first image state (n = 1) of Cu(111) at Γ [85]. The experimental data are taken from [83] Surface State Theory Experiment Cu(100) n = ± 6 n = ± 10 n = ± 15 Cu(111) n = ± 5

12 P. García-González, R.W. Godby / Computer Physics Communications 137 (2001) image-state, its coupling with bulk states is essential for an accurate determination of the lifetimes [85,86]. In [87] a more sophisticated study is presented, including vertex correction in the evaluation of the polarization as well as corrugation effects by means of an effective mass m. Very recently [88], a similar study has been presented for the intrinsic surface states in Cu(111), Be(0001), and Mg(0001). 6. Metal/semiconductor interfaces In many respects the interface between a metal and a semiconductor (or insulator) closely resembles that between a metal and vacuum. Interface states can exist in analogy to surface states, and electrons in the semiconductor experience a 1/z image potential, albeit screened relative to the surface case by the dielectric constant of the semiconductor. One crucial difference, however, is the existence of Bloch states for holes below the band gap of the semiconductor, in contrast to the surface case where the hole states below the Fermi energy remain evanescent to infinitely low energies. Inkson showed in a model calculation [89] that the effect of the image potential for these holes is to pull the valence band up in energy, towards the band gap, as the interface is approached, whereas the conduction band is lowered in energy in the normal way. Therefore, the band gap narrows. Charlesworth, Godby and Needs [90] performed an ab initio GW calculation for a supercell representation of an Al/GaAs(110) interface which showed that this band-gap narrowing occurred, although modified in detail from the simplified model calculation. The band gap of GaAs narrowed by approximately 0.5 ev, most of which arose from the conduction band (see Fig. 7). The asymmetry of the band-gap narrowing was understood in terms of an extension of Inkson s model. The above calculation contained, as a by-product, information about the alignment between the valence- and conduction-band edges of the GaAs, far from the interface, and the Fermi energy of the metal. These two energy differences are, of course, the p-andn-type Schottky barrier heights for the interface in question, and have a direct experimental consequence for the electronic properties of the interface. A quasiparticle calculation of the quality of GW is crucial if they are to be determined accurately. However, if only these barrier heights are required, GW may be employed in a more straightforward way: an LDA calculation alone gives a good representation of the charge density at the interface and hence the variation in electrostatic potential from one material to the other. This may be combined with two separate GW calculations for bulk Al and GaAs to determine the positions of the quasiparticle Fig. 7. Calculated quasiparticle band edges near the Al/GaAs(100) interface (solid line). For comparison, the dotted line gives the classical band bending. From Charlesworth et al. [90].

13 120 P. García-González, R.W. Godby / Computer Physics Communications 137 (2001) band edges relative to the LDA band edges. This approach was taken by the same authors in a study [91] of the dependence of the Schottky barrier height of Al/GaAs(110) on the detailed atomic structure of the interface (in particular, the atomically relaxed structure corresponding to various translational alignments of the two bulk crystals). The Schottky barrier height varied by up to 0.7 ev between different translation states, even though the total energies were rather similar. While one, novel, translation state was found to be energetically favorable, it is likely that many translation states, and hence many barrier heights, exist in real, imperfect interfaces. Returning for a moment to the ground-state world of Kohn Sham density-functional theory, we note that the Kohn Sham exchange-correlation functional exhibits some peculiar behaviour at metal-semiconductor interfaces. Godby, Sham and Schlüter [92] found that the exact Kohn Sham exchange-correlation potential exhibits a longrange spatial variation, accompanying the long-range electrostatic potential that pins the semiconductor band edge to the Fermi energy at very large distances. This exchange-correlation potential has an unfortunate ultra-non-local dependence on the electron density, and is therefore absent from all density-based practical approximations for the exchange-correlation energy functional and potential. 7. Summary and conclusions In this article we have reviewed some of the applications of the many-bodygw approximation for the calculation of surface properties in real materials. Regarding semiconductors, the surface state energies calculated by means of the GW method clearly improve the corresponding LDA values, in close resemblance with the bulk case. Extensions to other more complex surfaces may be envisaged, although for some materials, such a noble metals, the dynamical dependence of the dielectric function would require a full treatment beyond the level of the plasmonpole models sometimes used. An a metal surface, the energies of the KS states seem to be remarkably similar to the QP energies as long as the V XC potential used in the KS calculation exhibits the right image-like behaviour. However, the true QP states are the solutions of the quasiparticle equation (1) and, so far, this coincidence has been only fully verified for simple jellium surfaces and, indirectly, by the good agreement between KS calculations and experiments. In any case, the lack of physical meaning of the KS states must always be taken into account, besides the fact that a fully first-principle evaluation of a well-behaved V XC potential for a real material has not yet been performed. On the other hand, the energies of image states clearly depend on the details of the surface barrier potential experienced by unoccupied states. Therefore, the best strategy would be the many-body evaluation of QP properties by means of a standard GW calculation, starting from KS Green s functions evaluated using a V XC potential with the proper asymptotic behaviour. Obviously, since the common GW procedure is not fully self-consistent, there will be always a dependence on the particular V XC potential used. Nevertheless, this is a minor limitation if compared with the crude approximation made by identifying the energies of QP and KS states. Full GW calculations of QP properties at metal surfaces are even more motivated by recent time-resolved twophoton photoemission experiments which provide information on the electron dynamics at the femtosecond scale. Indeed, lifetimes of surface states can be well described using the GW method, although present calculations have not reached the ab initio level achieved for the corresponding bulk materials [93 95]. The actual screening effects may be very different in a real system due, for instance, to the influence of occupied d bands. Acknowledgements The authors thank Prof. Rodolfo Del Sole, Dr. Ansgar Liebsch, and Dr. Giovanni Onida for a through reading of the manuscript and for many valuable suggestions. They also thanks Dr. G. Onida for providing a copy of Ref. [49] prior to its publication. P.G.G. acknowledges financial support by the Spanish Ministerio de Educación.

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