Electronic excitations of bulk LiCl from many-body perturbation theory Yun-Feng Jiang, Neng-Ping Wang, and Michael Rohlfing Citation: The Journal of Chemical Physics 139, 214710 (2013); doi: 10.1063/1.4835695 View online: http://dx.doi.org/10.1063/1.4835695 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/139/21?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Enhanced many-body effects in 2- and 1-dimensional ZnO structures: A Green's function perturbation theory study J. Chem. Phys. 139, 144703 (2013); 10.1063/1.4824078 Communication: Electronic band gaps of semiconducting zig-zag carbon nanotubes from many-body perturbation theory calculations J. Chem. Phys. 136, 181101 (2012); 10.1063/1.4716178 Density functionals from many-body perturbation theory: The band gap for semiconductors and insulators J. Chem. Phys. 124, 154108 (2006); 10.1063/1.2189226 Comparison of low-order multireference many-body perturbation theories J. Chem. Phys. 122, 134105 (2005); 10.1063/1.1863912 Many-body Green’s-function calculations on the electronic excited states of extended systems J. Chem. Phys. 112, 7339 (2000); 10.1063/1.481372
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THE JOURNAL OF CHEMICAL PHYSICS 139, 214710 (2013)
Electronic excitations of bulk LiCl from many-body perturbation theory Yun-Feng Jiang,1 Neng-Ping Wang,1,a) and Michael Rohlfing2 1 2
Science Faculty, Ningbo University, Fenghua Road 818, 315211 Ningbo, People’s Republic of China Institut für Festkörpertheorie, Universität Münster, 48149 Münster, Germany
(Received 4 September 2013; accepted 7 November 2013; published online 6 December 2013) We present the quasiparticle band structure and the optical excitation spectrum of bulk LiCl, using many-body perturbation theory. Density-functional theory is used to calculate the ground-state geometry of the system. The quasiparticle band structure is calculated within the GW approximation. Taking the electron-hole interaction into consideration, electron-hole pair states and optical excitations are obtained by solving the Bethe-Salpeter equation for the electron-hole two-particle Green function. The calculated band gap is 9.5 eV, which is in good agreement with the experimental result of 9.4 eV. And the calculated optical absorption spectrum, which contains an exciton peak at 8.8 eV and a resonant-exciton peak at 9.8 eV, is also in good agreement with experimental data. © 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4835695] I. INTRODUCTION
Excited electronic states and optical spectra play an important role in physics and chemistry. They are used to characterize and manipulate materials and they provide the basis for a wide range of technological applications, such as dye chemicals, photovoltaics, laser technology, and catalysts for chemical reactions. In this context, the detailed understanding and accurate calculation of excited electronic states are of great importance. Li halides can be considered as prototype insulator materials with great technological importance. There are a number of theoretical studies on these materials. The structural and electronic properties of Li halide crystals have been calculated using density-functional theory (DFT) within the local density approximation (LDA)1 or the generalized gradient approximation (GGA).2 However, the band gaps of Li halide crystals are severely underestimated by such calculations. Recently, a self-interaction corrected DFT (SIC-DFT) approach was used to improve the standard LDA band structures,3–7 and this approach partially overcomes the band-gap problem of LDA. However, for LiF (and other insulators) the band widths of the lowest conduction band and the three highest valence bands are still significantly underestimated by SIC-DFT calculations.7 Since the dispersion of these bands affects the formation of excitonic states, both the band gap and the band dispersion are equally important for understanding optical properties. To understand or predict electronic band structures and optical properties, an accurate description of excited electronic states is required. A rigorous approach to excited electronic states is given by many-body perturbation theory (MBPT).8–12 Here, the states are considered as quasiparticle excitations to the electronic ground state, as described by appropriate Green functions. The equation of motion of the single-particle Green function yields the quasiparticle eleca) E-mail: [email protected]
0021-9606/2013/139(21)/214710/5/$30.00
tron and hole states that define the band structure of a system. The equation of motion of the two-particle Green function (the so-called Bethe-Salpeter equation, BSE) yields the charge-neutral electron-hole excitations which are responsible for the optical spectrum. Based on a DFT calculation of the ground state, quasiparticle excitations (electrons and holes) can be described with high accuracy by Hedin’s GW approximation (GWA).13, 14 By introducing GW corrections to electronic exchange and correlation, quasiparticle band structures of LiF and LiCl have been calculated, and the obtained band gaps agree well with available experimental results.7, 15–18 The GWA has become a standard tool for predicting quasiparticle band structures. Optical excitations of semiconductors and insulators are dominated by electron-hole correlation effects9 which are treated within MBPT by solving the BSE for electron-hole pair states.19 By calculating the optical transition matrix elements of the coupled electron-hole excitation states, the entire linear optical spectrum of a material can be evaluated. This procedure has been employed with great success to calculate the optical spectra of a wide range of materials.8, 10–12, 18, 20–26 In Ref. 27, the optical absorption spectra of LiF were calculated using the BSE on top of a SIC-DFT band structure, as well as within MBPT (i.e., GWA + BSE). Since the band widths of the lowest conduction band and the three highest valence bands calculated by SIC-DFT are significantly smaller than those calculated by GWA, using the SIC-DFT electronic energies and wave functions as input of the BSE leads to optical spectra which are considerably shifted towards lower energies in comparison to experimental data, while MBPT yields optical spectra in very good agreement with experimental measurements,7, 27 establishing MPBT as the most accurate approach available today for the calculation of the optical properties of Li halides. In Ref. 28, linear optical responses for Li halides and other materials were studied using the orthogonalized linear-combination-of-atomic-orbitals (OLCAO) method in the LDA. However, the agreement between the calculated
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© 2013 AIP Publishing LLC
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optical absorption spectrum28 for LiCl and the experimental data29 is not satisfying (see below). The calculations in Ref. 28 could not reproduce the first two exciton peaks of the experimental optical spectrum.29 This indicates that some important effects were ignored in the calculations of Ref. 28. To present a theoretical reference, we use many-body perturbation theory to calculate the electronic band structure and optical absorption spectrum of LiCl. In particular, we will investigate the effect of the electron-hole interaction in LiCl and present the details of two-particle wave functions of exciton states in real space. This paper is organized as follows. In Sec. II, we briefly summarize the basic theory of the manybody perturbation theory approach used in this paper. In Sec. III, we present calculation results of the quasiparticle band structure and optical spectrum in comparison with the available experimental results. Finally, conclusions will be given in Sec. IV. II. COMPUTATIONAL METHOD A. Electronic ground state
We carry out the standard DFT-LDA calculations with nonlocal norm-conserving ab initio pseudopotentials30, 31 and solve the Kohn-Sham equations32 to obtain the ground state properties of the considered system. The pseudopotential for Li is taken from Ref. 18, and the Li 1s state is included among the valence orbitals in addition to the Li 2s state. For Cl, it is sufficient to treat only the 3s and 3p states as valence states. Gaussian orbitals are used to construct the LDA basis sets. Throughout this work, the Kohn-Sham wave functions are represented in a basis set of s, p, d, and s∗ Gaussian orbitals, as defined in Ref. 33. The decay constants ( in atomic units) of Gaussian orbitals are 0.15, 0.5, 1.8, and 6.0 for Li and 0.2, 0.6, 1.2, and 3.6 for Cl. The same Gaussian orbitals are also used in Sec. II B as basis functions for the representation of all quantities occurring in the GW self-energy operator and the electron-hole interaction. Solving the Kohn-Sham equations self-consistently yields the LDA band structure, wave functions, and total energy. B. Quasiparticle band structure
The exchange-correlation potential Vxc used in the DFT cannot describe correctly the dynamical electron correlation in the solid and is not appropriate for calculating single particle excitation energies. Quasiparticle excitations can be described by MBPT.13, 14 Within MBPT, Vxc is replaced by a nonlocal, energy-dependent self-energy operator (r, r , E), which enters the following quasiparticle equation for periodic systems:34 ¯2 2 QP ∇ + Vps (r) + VH (r) ψnk (r) − 2m QP QP 3 QP QP ψnk (r )d r = Enk ψnk (r), (1) + r, r , Enk QP QP where Enk and ψnk (r) denote the quasiparticle energy and quasiparticle wavefunction, respectively. The self-energy op-
J. Chem. Phys. 139, 214710 (2013)
erator can be calculated within the GWA13, 14, 35 i + (r, r , E) = e−iω0 G1 (r, r , E − ω)W (r, r , ω)dω, 2π (2) where G1 and W are the one-body Green function and the screened Coulomb interaction of the system, respectively. The difference [(E) − Vxc ] constitutes a quasiparticle correction to the LDA band structure. In the present work, the one-body Green function G1 is calculated approximately in terms of the results of the DFT-LDA calculation, and a generalized plasmon-pole approximation (GPPA)15, 34 is used to describe the frequency dependence of W. The static part W(ω = 0) of the interaction is calculated using the random-phase approximation (RPA). The GPPA is known to work well for delocalized valence states. However, for a system with semicore d states which are highly localized, the GPPA leads to a significant discrepancy between the calculated band gap and the experimental result.36–38 For a system without semicore d states, the difference between quasiparticle band structures calculated using the full frequency integration and the GPPA is small, and the band gap obtained from the GPPA calculation is in good agreement with the experimental value.37 Since the present system does not have semicore d states, the GPPA may be used in GWA calculations for the present system. Recently, the GWA calculation based on full potential all-electron method was performed and a comparison with the GWA calculation based on the pseudopotential method was given.39, 40 It has been found that the semicore d states strongly interact with the s and p states of the same shell. If the s and p states are treated as part of the frozen core of the pseudopotential (namely, the s and p states are prevented from interacting with the d states), the bind energies of semicore d states are apparently underestimated.39, 40 When the d states as well as the s and p states of the same shell are included as valence states in the calculation, the agreement of the GWA band gap and the experimental value is clearly improved.36, 37 However, there should be no problem using the pseudopotential method in the present study since the pseudopotential for Cl constructed in Sec. II A allows the Cl 3d states among conduction bands to interact with the Cl 3s and 3p states of the same shell. C. Electron-hole excitations
Due to the interaction between the excited electrons and holes, optical electron-hole excitations cannot be described by an effective one-particle picture. Instead, it is necessary to consider two-particle states, which can be described within the Tamm-Dancoff approximation41 as χS (rh , re ) =
hole elec k
v
∗ ASvck ψvk (rh )ψc,k+Q (re ) ,
(3)
c
where S denotes the correlated electron-hole excitation of the system, rh and re refer to the coordinates of the hole and the electron, respectively. And Q is the total momentum of the electron-hole state which corresponds to the momentum of the absorbed photon. In Eq. (3), ψvk (rh ) and ψ c, k + Q (re ) are
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Jiang, Wang, and Rohlfing
J. Chem. Phys. 139, 214710 (2013)
the single-particle wave functions of hole and electron states, respectively. Using the equation of motion of the two-particle Green function G2 , the electron-hole amplitudes ASvck in Eq. (3) and the corresponding excitation energies S can be determined by the Bethe-Salpeter equation19, 42 hole elec QP QP S Ec,k+Q − Ev,k Avck + vck|K eh |v c k ASv c k v
k
= S ASvck ,
c
(4)
where Keh is the electron-hole interaction kernel19, 42 which contains a screened, direct term Keh,d and an exchange term Keh,x . Generally, the direct interaction term is frequency dependent and the frequency dependence is controlled by the elementary electronic excitations of the system, starting at the band gap. The dynamical effects for Keh,d become considerable if the electron-hole binding energy is close to the characteristic frequency of the screening, i.e., the fundamental gap.19, 42 For the system under consideration, the electronhole binding energy is no more than 0.7 eV, which is much smaller than the fundamental gap of 9.5 eV. Therefore, we use the static screening in the calculation for the screened direct interaction Keh,d . After solving the BSE, the imaginary part of the macroscopic transverse dielectric function of the system can be calculated using the expression 2 hole elec 16π 2 e2 S · v|ck δ(ω−S ), Avck vk|λ 2 (ω) = 2 ω S
k
v
c
(5) is the polarization vector of the light and where λ v = i[H sp , r]/¯ is the single-particle velocity operator (which corresponds to the current operator). Here, Hsp denotes the single-particle Hamiltonian. Without the electronhole interaction, the excitations would be given by vertical interband transition between independent hole and electron states, and Eq. (5) would be changed into hole elec 16π 2 e2 · v|ck|2 δ(ω−(Ec −Ev )) . |vk|λ ω2 v c k (6) The difference of 2 (ω) − 20 (ω) represents the effect of the electron-hole interaction in the optical absorption spectrum.
20 (ω) =
III. RESULTS
LiCl is an ionic insulator material in rocksalt structure. The calculations for band structures as well as optical spectra are performed at the experimental lattice constant of 5.13 Å. Solving self-consistently the Kohn-Sham equations, we obtain the LDA band structure which is shown as dotted curves in Fig. 1 along high-symmetry lines in the Brillouin zone. Including the GW correction, we calculate the quasiparticle band structure and the obtained result is shown as solid curves in Fig. 1. The GW valence-band structure consists of five bands, which may be classified as a Li 1s band at −51.2 eV,
30 20
Energy (eV)
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10
Li 2s
0
Cl 3p
-10 Cl 3s
-20 -30
LDA GWA
LiCl
-40 Li 1s
-50 -60 L
Γ
X
W
K
Γ
FIG. 1. Bulk band structure of LiCl, as calculated within LDA (dotted curves) and within GWA (solid curves).
a Cl 3s band between −14.0 and −13.1 eV, and three Cl 3p bands between −3.4 and 0 eV. The lowest conduction band between 9.5 and 13.6 eV (having its minimum at the point) mainly consists of the Li 2s states, with non-negligible contributions from the Cl orbitals. The width of Cl 3p bands and that of Li 2s band are 3.0 and 3.3 eV in LDA, respectively. They are increased to 3.4 and 4.1 eV by quasiparticle corrections, respectively. The fundamental band gap is increased from 5.9 eV in LDA to 9.5 eV in GWA. The measured band gap of bulk LiCl from experiments is 9.4 eV.43 Our calculated quasiparticle band gap using GWA agrees very well with the experiment result.43 The result of this work is also close to previous GW calculation results for LiCl (9.1 eV and 10.2 eV, respectively) given in Refs. 15 and 17. The Li 1s core level observes a strong QP correction of −9 eV, which is typical for strongly localized electronic states. The Li 1s core level has been observed experimentally by Johansson and Hagström44 at the energy of −53.2 eV. Our GWA energy of −51.2 eV for the Li 1s core level is close to the experimental measurement. Clearly, GWA corrections to the LDA improve the agreement with experiment. Using the approach presented in Sec. II C, the optical absorption spectrum can be calculated. We have included the Cl 3s valence band, three Cl 3p valence bands, and the nine lowest conduction bands in the evaluation of Eq. (4). The lowlying Li 1s band does not need to be taken into account since it cannot contribute to the excitations below 13 eV, which are in the focus of our interest. We use 500 k points inside the Brillouin zone for the fcc structure. After solving Eq. (4), the imaginary part of the dielectric function can be calculated using Eq. (5). The obtained optical absorption spectrum is shown by the solid curve in Fig. 2. Calculations with increasing sets of k points or more conduction bands indicate that the optical spectrum is converged at 500 k points and the four valence and nine conduction bands are sufficient for the description of the optical spectrum of LiCl within the range of excitation energies S ≤ 13 eV. The dashed-dotted curve in Fig. 2 indicates the spectrum without electron-hole interaction effects calculated using Eq. (6). Apparently, significant changes occur due to the electron-hole interaction, in
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J. Chem. Phys. 139, 214710 (2013)
25
Electron
20
ε2
15
Li
10
Cl
5
0
4
5
6
7
8
9
10
11 12
13
Energy (eV) FIG. 2. The optical absorption spectrum of bulk LiCl calculated with the electron-hole interaction included (solid curve) is compared with experimental data from Ref. 29 (dotted curve). The dashed-dotted curve and the long dashed curve indicate the independent-particle spectrum (i.e., without electron-hole interaction) obtained by this work and the one taken from Ref. 28, respectively. Lorentzian broadening of 0.14 eV has been applied to the optical spectra calculated with and without electron-hole interaction in this work to facilitate comparison with experiment.
particular in the low energy range which is now dominated by a characteristic exciton peak at 8.8 eV, i.e., 0.7 eV below the quasiparticle gap. For comparison sake, experimental data29 are also included in Fig. 2. Clearly, a good agreement between experiment and theory is obtained for both the energy and the oscillator strength (amplitude × width) for the main peak of the optical spectrum. The calculated energy for the second peak (9.8 eV) and overall shape of the optical spectrum are also in good agreement with the experimental results. In Fig. 2, we also show the calculation result from Ref. 28, in which the linear optical spectrum was calculated using OLCAO method in the LDA and the electron-hole interaction effect was ignored in this calculation. Our calculated optical spectrum including the electron-hole interaction is in a better agreement with experimental data than the calculation result of Ref. 28. The overall shape of the spectrum from Ref. 28 is similar to that of our independent-particle spectrum, but the low-energy absorption threshold of Ref. 28 is about 3.5 eV lower than that of this work. The reason is that the LDA calculations for LiCl underestimated the band gap and the band widths of the lowest conduction band and the three highest valence bands. And this causes the excited states to shift towards lower energies, as pointed out in Refs. 7 and 27. Using the coefficients ASvck obtained from the solution of the BSE, the wave function of the electron-hole excitation χ S (rh , re ) can be evaluated from Eq. (3). |χ S (rh , re )|2 gives the real-space correlation between the hole (at rh ) and the electron (at re ). We calculate two-dimensional projections of the probability density |χ S (rh , re )|2 for the lowest exciton state (at 8.8 eV). The top panel of Fig. 3 shows the obtained real-space distribution |χ S (rh , re )|2 of the electron relative to the hole, which is fixed at the central Cl atom in (001) plane. The bottom panel of Fig. 3 shows the distribution of the hole relative to the electron, which is also fixed at the central Cl
Hole
Li Cl
FIG. 3. Real-space distribution of the probability density |χ S (rh , re )|2 for the lowest-energy exciton of LiCl (at 8.8 eV), in the (001) plane. The top panel shows the distribution of the electron (re ) with respect to the hole (rh ), which is fixed at the central Cl atom. The bottom panel shows the distribution of the hole (rh ) with respect to the electron (re ), which is also fixed at the central Cl atom.
atom. The hole of the exciton state shown in the bottom panel of Fig. 3 is mainly formed by the Cl 3px or 3py orbitals at the very same central Cl atom where the electron is fixed, with some contributions of the Cl 3px or 3py states at the neighboring Cl atoms. The electron of the exciton state shown in the top panel of Fig. 3, on the other hand, is mainly composed of Cl 4s-like orbitals at the central Cl atom where the hole is localized, accompanied by some contributions of Li 2s orbitals and Cl 4s orbitals at the neighboring Cl atoms. From Fig. 3, one can find some interesting features in the electron (or hole) spatial distribution. The probability density of the electron (relative to the hole fixed at the central Cl atom) has maximal values on the central Cl atom and considerable values on neighboring Cl atoms. However, the probability density is very low at the Li atoms. Since the probability density at Li atoms is very low, the exciton state in the present system does not have a simple Frenkel-exciton feature, which would assume the electron hopping from the Cl atom to the nearest-neighbor Li atoms. Furthermore, the electron distribution on the neighboring Cl atoms is not isotropic. Instead, it is polarized towards the central Cl atom. The distribution of the hole on the neighboring Cl atoms, which has a shape of 3px or
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3py -like orbital, is also polarized towards the central Cl atom where the electron is fixed. A similar polarization effect is also found in the hole (or electron) distribution of the lowest exciton state of LiF,12, 18 and this effect in LiF is even more pronounced than in LiCl. In fact, such polarization effect of the electron (or hole) distribution results from the correlation of the quasielectron (or quasihole) to the effective positive (or negative) charge of the quasihole (or quasielectron) on the central Cl atom, which is affected by the electron-hole interaction as can be seen from Eqs. (3) and (4). The binding energy of the lowest exciton state in LiCl is about 0.7 eV, which is smaller than the binding energy of nearly 1.6 eV in the lowest exciton state of LiF.12, 18 The electron-hole interaction effect in the lowest exciton state for LiF is stronger than the corresponding effect for LiCl. IV. CONCLUSIONS
Using many-body perturbation theory within the GW approximation and the Bethe-Salpeter equation, we have calculated the quasiparticle band structure and optical absorption spectra of bulk LiCl. The calculated band gap and energy of the Li 1s core level within the GWA are in good agreement with the experimental observations.43, 44 From the solution of the BSE, the lowest exciton state is found at 8.8 eV. In this exciton state, the hole is mainly formed by the 3p orbitals, while the electron is mainly composed of the Cl 4s orbitals, accompanied by some contributions from Li 2s orbitals. The real-space distribution of the hole (or electron) for the lowest exciton shows an obvious polarization feature which is attributed to the electron-hole correlations. At low excitation < energies (S ∼ 12 eV), the optical absorption spectrum calculated with the electron-hole interaction included reproduces well the corresponding optical spectrum from experiments.29 A comparison between the optical spectra calculated with and without electron-hole interaction shows that the effects of the electron-hole interaction in LiCl are important and should be taken into account. ACKNOWLEDGMENTS
We are grateful to P. Krüger for useful discussions. N.-P.W. acknowledges financial support from the National Natural Science Foundation of China (NNSFC) under Grant Nos. 11074136 and 61176081, The Natural Science Foundation of Zhejiang Province under Grant No. Y6100467, and K. C. Wong Magna Fund in Ningbo University. 1 P.
K. de Boer and R. A. de Groot, Eur. Phys. J. B 4, 25 (1998). R. Gopikrishnan, D. Jose, and A. Datta, AIP Adv. 2, 012131 (2012). 3 D. Vogel, P. Krüger, and J. Pollmann, Phys. Rev. B 54, 5495 (1996). 2 C.
J. Chem. Phys. 139, 214710 (2013) 4 A.
Filippetti and V. Fiorentini, Eur. Phys. J. B 71, 139 (2009). C. Erwin and C. C. Lin, J. Phys. C 21, 4285 (1988). 6 B. Baumeier, P. Krüger, J. Pollmann, and G. V. Vajenine, Phys. Rev. B 78, 125111 (2008). 7 C. Sommer, P. Krüger, and J. Pollmann, Phys. Rev. B 85, 165119 (2012). 8 M. S. Hybertsen and S. G. Louie, Phys. Rev. Lett. 55, 1418 (1985). 9 W. Hanke and L. J. Sham, Phys. Rev. B 21, 4656 (1980). 10 G. Onida, L. Reining, R. W. Godby, R. Del Sole, and W. Andreoni, Phys. Rev. Lett. 75, 818 (1995); S. Albrecht, L. Reining, R. Del Sole, and G. Onida, ibid. 80, 4510 (1998). 11 L. X. Benedict, E. L. Shirley, and R. B. Bohn, Phys. Rev. Lett. 80, 4514 (1998); Phys. Rev. B 57, R9385 (1998). 12 M. Rohlfing and S. G. Louie, Phys. Rev. Lett. 80, 3320 (1998); 81, 2312 (1998); 82, 1959 (1999). 13 L. Hedin, Phys. Rev. 139, A796 (1965). 14 L. Hedin and S. Lundqvist, in Solid State Physics: Advances in Research and Application, edited by F. Seitz, D. Turnbull, and H. Ehrenreich (Academic, New York, 1969), Vol. 23, p. 1. 15 M. S. Hybertsen and S. G. Louie, Phys. Rev. B 34, 5390 (1986). 16 E. L. Shirley, L. J. Terminello, J. E. Klepeis, and F. J. Himpsel, Phys. Rev. B 53, 10296 (1996). 17 S. Iwata, S. Ishii, and K. Ohno, Mater. Trans. 46, 1100 (2005). 18 N.-P. Wang, M. Rohlfing, P. Krüger, and J. Pollmann, Phys. Rev. B 67, 115111 (2003). 19 G. Strinati, Phys. Rev. Lett. 49, 1519 (1982); Phys. Rev. B 29, 5718 (1984). 20 J.-W. van der Horst, P. A. Bobbert, M. A. J. Michels, G. Brocks, and P. J. Kelly, Phys. Rev. Lett. 83, 4413 (1999). 21 M. Rohlfing and S. G. Louie, Phys. Rev. Lett. 83, 856 (1999); M. Rohlfing, M. Palummo, G. Onida, and R. Del Sole, ibid. 85, 5440 (2000). 22 J. C. Grossman, M. Rohlfing, L. Mitas, S. G. Louie, and M. L. Cohen, Phys. Rev. Lett. 86, 472 (2001). 23 P. H. Hahn, W. G. Schmidt, and F. Bechstedt, Phys. Rev. Lett. 88, 016402 (2001). 24 N.-P. Wang, M. Rohlfing, P. Krüger, and J. Pollmann, Appl. Phys. A 78, 213 (2004); Phys. Rev. Lett. 92, 216805 (2004); Phys. Rev. B 71, 045407 (2005); 74, 155405 (2006). 25 Y. Ma, M. Rohlfing, and A. Gali, Phys. Rev. B 81, 041204(R) (2010). 26 M. Rohlfing, Phys. Rev. Lett. 108, 087402 (2012). 27 C. Sommer, P. Krüger, and J. Pollmann, Phys. Rev. B 86, 155212 (2012). 28 W. Y. Ching, F. Gan, and M.-Z. Huang, Phys. Rev. B 52, 1596 (1995). 29 J. E. Eby, K. J. Teegarden, and D. B. Dutton, Phys. Rev. 116, 1099 (1959). 30 G. B. Bachelet, D. R. Hamann, and M. Schlüter, Phys. Rev. B 26, 4199 (1982). 31 D. R. Hamann, Phys. Rev. B 40, 2980 (1989). 32 W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965). 33 M. Rohlfing, P. Krüger, and J. Pollmann, Phys. Rev. B 52, 1905 (1995). 34 M. Rohlfing, P. Krüger, and J. Pollmann, Phys. Rev. B 48, 17791 (1993). 35 M. S. Hybertsen and S. G. Louie, Phys. Rev. B 37, 2733 (1988). 36 W. Luo, S. Ismail-Beigi, M. L. Cohen, and S. G. Louie, Phys. Rev. B 66, 195215 (2002). 37 Y. Ma and M. Rohlfing, Phys. Rev. B 75, 205114 (2007). 38 P. Larson, M. Dvorak, and Z. Wu, Phys. Rev. B 88, 125205 (2013). 39 R. Gómez-Abal, X. Li, M. Scheffler, and C. Ambrosch-Draxl, Phys. Rev. Lett. 101, 106404 (2008). 40 X.-Z. Li, R. Gómez-Abal, H. Jiang, C. Ambrosch-Draxl, and M. Scheffler, New J. Phys. 14, 023006 (2012). 41 A. Fetter and J. D. Walecka, Quantum Theory of Many Particle Systems (McGraw-Hill, San Francisco, 1971), p. 538. 42 M. Rohlfing and S. G. Louie, Phys. Rev. B 62, 4927 (2000). 43 A. B. Kunz, Phys. Rev. B 26, 2056 (1982). 44 L. I. Johansson and S. B. M. Hagström, Phys. Scr. 14, 55 (1976). 5 S.
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THE JOURNAL OF CHEMICAL PHYSICS 139, 214710 (2013)
Electronic excitations of bulk LiCl from many-body perturbation theory Yun-Feng Jiang,1 Neng-Ping Wang,1,a) and Michael Rohlfing2 1 2
Science Faculty, Ningbo University, Fenghua Road 818, 315211 Ningbo, People’s Republic of China Institut für Festkörpertheorie, Universität Münster, 48149 Münster, Germany
(Received 4 September 2013; accepted 7 November 2013; published online 6 December 2013) We present the quasiparticle band structure and the optical excitation spectrum of bulk LiCl, using many-body perturbation theory. Density-functional theory is used to calculate the ground-state geometry of the system. The quasiparticle band structure is calculated within the GW approximation. Taking the electron-hole interaction into consideration, electron-hole pair states and optical excitations are obtained by solving the Bethe-Salpeter equation for the electron-hole two-particle Green function. The calculated band gap is 9.5 eV, which is in good agreement with the experimental result of 9.4 eV. And the calculated optical absorption spectrum, which contains an exciton peak at 8.8 eV and a resonant-exciton peak at 9.8 eV, is also in good agreement with experimental data. © 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4835695] I. INTRODUCTION
Excited electronic states and optical spectra play an important role in physics and chemistry. They are used to characterize and manipulate materials and they provide the basis for a wide range of technological applications, such as dye chemicals, photovoltaics, laser technology, and catalysts for chemical reactions. In this context, the detailed understanding and accurate calculation of excited electronic states are of great importance. Li halides can be considered as prototype insulator materials with great technological importance. There are a number of theoretical studies on these materials. The structural and electronic properties of Li halide crystals have been calculated using density-functional theory (DFT) within the local density approximation (LDA)1 or the generalized gradient approximation (GGA).2 However, the band gaps of Li halide crystals are severely underestimated by such calculations. Recently, a self-interaction corrected DFT (SIC-DFT) approach was used to improve the standard LDA band structures,3–7 and this approach partially overcomes the band-gap problem of LDA. However, for LiF (and other insulators) the band widths of the lowest conduction band and the three highest valence bands are still significantly underestimated by SIC-DFT calculations.7 Since the dispersion of these bands affects the formation of excitonic states, both the band gap and the band dispersion are equally important for understanding optical properties. To understand or predict electronic band structures and optical properties, an accurate description of excited electronic states is required. A rigorous approach to excited electronic states is given by many-body perturbation theory (MBPT).8–12 Here, the states are considered as quasiparticle excitations to the electronic ground state, as described by appropriate Green functions. The equation of motion of the single-particle Green function yields the quasiparticle eleca) E-mail: [email protected]
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tron and hole states that define the band structure of a system. The equation of motion of the two-particle Green function (the so-called Bethe-Salpeter equation, BSE) yields the charge-neutral electron-hole excitations which are responsible for the optical spectrum. Based on a DFT calculation of the ground state, quasiparticle excitations (electrons and holes) can be described with high accuracy by Hedin’s GW approximation (GWA).13, 14 By introducing GW corrections to electronic exchange and correlation, quasiparticle band structures of LiF and LiCl have been calculated, and the obtained band gaps agree well with available experimental results.7, 15–18 The GWA has become a standard tool for predicting quasiparticle band structures. Optical excitations of semiconductors and insulators are dominated by electron-hole correlation effects9 which are treated within MBPT by solving the BSE for electron-hole pair states.19 By calculating the optical transition matrix elements of the coupled electron-hole excitation states, the entire linear optical spectrum of a material can be evaluated. This procedure has been employed with great success to calculate the optical spectra of a wide range of materials.8, 10–12, 18, 20–26 In Ref. 27, the optical absorption spectra of LiF were calculated using the BSE on top of a SIC-DFT band structure, as well as within MBPT (i.e., GWA + BSE). Since the band widths of the lowest conduction band and the three highest valence bands calculated by SIC-DFT are significantly smaller than those calculated by GWA, using the SIC-DFT electronic energies and wave functions as input of the BSE leads to optical spectra which are considerably shifted towards lower energies in comparison to experimental data, while MBPT yields optical spectra in very good agreement with experimental measurements,7, 27 establishing MPBT as the most accurate approach available today for the calculation of the optical properties of Li halides. In Ref. 28, linear optical responses for Li halides and other materials were studied using the orthogonalized linear-combination-of-atomic-orbitals (OLCAO) method in the LDA. However, the agreement between the calculated
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optical absorption spectrum28 for LiCl and the experimental data29 is not satisfying (see below). The calculations in Ref. 28 could not reproduce the first two exciton peaks of the experimental optical spectrum.29 This indicates that some important effects were ignored in the calculations of Ref. 28. To present a theoretical reference, we use many-body perturbation theory to calculate the electronic band structure and optical absorption spectrum of LiCl. In particular, we will investigate the effect of the electron-hole interaction in LiCl and present the details of two-particle wave functions of exciton states in real space. This paper is organized as follows. In Sec. II, we briefly summarize the basic theory of the manybody perturbation theory approach used in this paper. In Sec. III, we present calculation results of the quasiparticle band structure and optical spectrum in comparison with the available experimental results. Finally, conclusions will be given in Sec. IV. II. COMPUTATIONAL METHOD A. Electronic ground state
We carry out the standard DFT-LDA calculations with nonlocal norm-conserving ab initio pseudopotentials30, 31 and solve the Kohn-Sham equations32 to obtain the ground state properties of the considered system. The pseudopotential for Li is taken from Ref. 18, and the Li 1s state is included among the valence orbitals in addition to the Li 2s state. For Cl, it is sufficient to treat only the 3s and 3p states as valence states. Gaussian orbitals are used to construct the LDA basis sets. Throughout this work, the Kohn-Sham wave functions are represented in a basis set of s, p, d, and s∗ Gaussian orbitals, as defined in Ref. 33. The decay constants ( in atomic units) of Gaussian orbitals are 0.15, 0.5, 1.8, and 6.0 for Li and 0.2, 0.6, 1.2, and 3.6 for Cl. The same Gaussian orbitals are also used in Sec. II B as basis functions for the representation of all quantities occurring in the GW self-energy operator and the electron-hole interaction. Solving the Kohn-Sham equations self-consistently yields the LDA band structure, wave functions, and total energy. B. Quasiparticle band structure
The exchange-correlation potential Vxc used in the DFT cannot describe correctly the dynamical electron correlation in the solid and is not appropriate for calculating single particle excitation energies. Quasiparticle excitations can be described by MBPT.13, 14 Within MBPT, Vxc is replaced by a nonlocal, energy-dependent self-energy operator (r, r , E), which enters the following quasiparticle equation for periodic systems:34 ¯2 2 QP ∇ + Vps (r) + VH (r) ψnk (r) − 2m QP QP 3 QP QP ψnk (r )d r = Enk ψnk (r), (1) + r, r , Enk QP QP where Enk and ψnk (r) denote the quasiparticle energy and quasiparticle wavefunction, respectively. The self-energy op-
J. Chem. Phys. 139, 214710 (2013)
erator can be calculated within the GWA13, 14, 35 i + (r, r , E) = e−iω0 G1 (r, r , E − ω)W (r, r , ω)dω, 2π (2) where G1 and W are the one-body Green function and the screened Coulomb interaction of the system, respectively. The difference [(E) − Vxc ] constitutes a quasiparticle correction to the LDA band structure. In the present work, the one-body Green function G1 is calculated approximately in terms of the results of the DFT-LDA calculation, and a generalized plasmon-pole approximation (GPPA)15, 34 is used to describe the frequency dependence of W. The static part W(ω = 0) of the interaction is calculated using the random-phase approximation (RPA). The GPPA is known to work well for delocalized valence states. However, for a system with semicore d states which are highly localized, the GPPA leads to a significant discrepancy between the calculated band gap and the experimental result.36–38 For a system without semicore d states, the difference between quasiparticle band structures calculated using the full frequency integration and the GPPA is small, and the band gap obtained from the GPPA calculation is in good agreement with the experimental value.37 Since the present system does not have semicore d states, the GPPA may be used in GWA calculations for the present system. Recently, the GWA calculation based on full potential all-electron method was performed and a comparison with the GWA calculation based on the pseudopotential method was given.39, 40 It has been found that the semicore d states strongly interact with the s and p states of the same shell. If the s and p states are treated as part of the frozen core of the pseudopotential (namely, the s and p states are prevented from interacting with the d states), the bind energies of semicore d states are apparently underestimated.39, 40 When the d states as well as the s and p states of the same shell are included as valence states in the calculation, the agreement of the GWA band gap and the experimental value is clearly improved.36, 37 However, there should be no problem using the pseudopotential method in the present study since the pseudopotential for Cl constructed in Sec. II A allows the Cl 3d states among conduction bands to interact with the Cl 3s and 3p states of the same shell. C. Electron-hole excitations
Due to the interaction between the excited electrons and holes, optical electron-hole excitations cannot be described by an effective one-particle picture. Instead, it is necessary to consider two-particle states, which can be described within the Tamm-Dancoff approximation41 as χS (rh , re ) =
hole elec k
v
∗ ASvck ψvk (rh )ψc,k+Q (re ) ,
(3)
c
where S denotes the correlated electron-hole excitation of the system, rh and re refer to the coordinates of the hole and the electron, respectively. And Q is the total momentum of the electron-hole state which corresponds to the momentum of the absorbed photon. In Eq. (3), ψvk (rh ) and ψ c, k + Q (re ) are
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J. Chem. Phys. 139, 214710 (2013)
the single-particle wave functions of hole and electron states, respectively. Using the equation of motion of the two-particle Green function G2 , the electron-hole amplitudes ASvck in Eq. (3) and the corresponding excitation energies S can be determined by the Bethe-Salpeter equation19, 42 hole elec QP QP S Ec,k+Q − Ev,k Avck + vck|K eh |v c k ASv c k v
k
= S ASvck ,
c
(4)
where Keh is the electron-hole interaction kernel19, 42 which contains a screened, direct term Keh,d and an exchange term Keh,x . Generally, the direct interaction term is frequency dependent and the frequency dependence is controlled by the elementary electronic excitations of the system, starting at the band gap. The dynamical effects for Keh,d become considerable if the electron-hole binding energy is close to the characteristic frequency of the screening, i.e., the fundamental gap.19, 42 For the system under consideration, the electronhole binding energy is no more than 0.7 eV, which is much smaller than the fundamental gap of 9.5 eV. Therefore, we use the static screening in the calculation for the screened direct interaction Keh,d . After solving the BSE, the imaginary part of the macroscopic transverse dielectric function of the system can be calculated using the expression 2 hole elec 16π 2 e2 S · v|ck δ(ω−S ), Avck vk|λ 2 (ω) = 2 ω S
k
v
c
(5) is the polarization vector of the light and where λ v = i[H sp , r]/¯ is the single-particle velocity operator (which corresponds to the current operator). Here, Hsp denotes the single-particle Hamiltonian. Without the electronhole interaction, the excitations would be given by vertical interband transition between independent hole and electron states, and Eq. (5) would be changed into hole elec 16π 2 e2 · v|ck|2 δ(ω−(Ec −Ev )) . |vk|λ ω2 v c k (6) The difference of 2 (ω) − 20 (ω) represents the effect of the electron-hole interaction in the optical absorption spectrum.
20 (ω) =
III. RESULTS
LiCl is an ionic insulator material in rocksalt structure. The calculations for band structures as well as optical spectra are performed at the experimental lattice constant of 5.13 Å. Solving self-consistently the Kohn-Sham equations, we obtain the LDA band structure which is shown as dotted curves in Fig. 1 along high-symmetry lines in the Brillouin zone. Including the GW correction, we calculate the quasiparticle band structure and the obtained result is shown as solid curves in Fig. 1. The GW valence-band structure consists of five bands, which may be classified as a Li 1s band at −51.2 eV,
30 20
Energy (eV)
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10
Li 2s
0
Cl 3p
-10 Cl 3s
-20 -30
LDA GWA
LiCl
-40 Li 1s
-50 -60 L
Γ
X
W
K
Γ
FIG. 1. Bulk band structure of LiCl, as calculated within LDA (dotted curves) and within GWA (solid curves).
a Cl 3s band between −14.0 and −13.1 eV, and three Cl 3p bands between −3.4 and 0 eV. The lowest conduction band between 9.5 and 13.6 eV (having its minimum at the point) mainly consists of the Li 2s states, with non-negligible contributions from the Cl orbitals. The width of Cl 3p bands and that of Li 2s band are 3.0 and 3.3 eV in LDA, respectively. They are increased to 3.4 and 4.1 eV by quasiparticle corrections, respectively. The fundamental band gap is increased from 5.9 eV in LDA to 9.5 eV in GWA. The measured band gap of bulk LiCl from experiments is 9.4 eV.43 Our calculated quasiparticle band gap using GWA agrees very well with the experiment result.43 The result of this work is also close to previous GW calculation results for LiCl (9.1 eV and 10.2 eV, respectively) given in Refs. 15 and 17. The Li 1s core level observes a strong QP correction of −9 eV, which is typical for strongly localized electronic states. The Li 1s core level has been observed experimentally by Johansson and Hagström44 at the energy of −53.2 eV. Our GWA energy of −51.2 eV for the Li 1s core level is close to the experimental measurement. Clearly, GWA corrections to the LDA improve the agreement with experiment. Using the approach presented in Sec. II C, the optical absorption spectrum can be calculated. We have included the Cl 3s valence band, three Cl 3p valence bands, and the nine lowest conduction bands in the evaluation of Eq. (4). The lowlying Li 1s band does not need to be taken into account since it cannot contribute to the excitations below 13 eV, which are in the focus of our interest. We use 500 k points inside the Brillouin zone for the fcc structure. After solving Eq. (4), the imaginary part of the dielectric function can be calculated using Eq. (5). The obtained optical absorption spectrum is shown by the solid curve in Fig. 2. Calculations with increasing sets of k points or more conduction bands indicate that the optical spectrum is converged at 500 k points and the four valence and nine conduction bands are sufficient for the description of the optical spectrum of LiCl within the range of excitation energies S ≤ 13 eV. The dashed-dotted curve in Fig. 2 indicates the spectrum without electron-hole interaction effects calculated using Eq. (6). Apparently, significant changes occur due to the electron-hole interaction, in
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25
Electron
20
ε2
15
Li
10
Cl
5
0
4
5
6
7
8
9
10
11 12
13
Energy (eV) FIG. 2. The optical absorption spectrum of bulk LiCl calculated with the electron-hole interaction included (solid curve) is compared with experimental data from Ref. 29 (dotted curve). The dashed-dotted curve and the long dashed curve indicate the independent-particle spectrum (i.e., without electron-hole interaction) obtained by this work and the one taken from Ref. 28, respectively. Lorentzian broadening of 0.14 eV has been applied to the optical spectra calculated with and without electron-hole interaction in this work to facilitate comparison with experiment.
particular in the low energy range which is now dominated by a characteristic exciton peak at 8.8 eV, i.e., 0.7 eV below the quasiparticle gap. For comparison sake, experimental data29 are also included in Fig. 2. Clearly, a good agreement between experiment and theory is obtained for both the energy and the oscillator strength (amplitude × width) for the main peak of the optical spectrum. The calculated energy for the second peak (9.8 eV) and overall shape of the optical spectrum are also in good agreement with the experimental results. In Fig. 2, we also show the calculation result from Ref. 28, in which the linear optical spectrum was calculated using OLCAO method in the LDA and the electron-hole interaction effect was ignored in this calculation. Our calculated optical spectrum including the electron-hole interaction is in a better agreement with experimental data than the calculation result of Ref. 28. The overall shape of the spectrum from Ref. 28 is similar to that of our independent-particle spectrum, but the low-energy absorption threshold of Ref. 28 is about 3.5 eV lower than that of this work. The reason is that the LDA calculations for LiCl underestimated the band gap and the band widths of the lowest conduction band and the three highest valence bands. And this causes the excited states to shift towards lower energies, as pointed out in Refs. 7 and 27. Using the coefficients ASvck obtained from the solution of the BSE, the wave function of the electron-hole excitation χ S (rh , re ) can be evaluated from Eq. (3). |χ S (rh , re )|2 gives the real-space correlation between the hole (at rh ) and the electron (at re ). We calculate two-dimensional projections of the probability density |χ S (rh , re )|2 for the lowest exciton state (at 8.8 eV). The top panel of Fig. 3 shows the obtained real-space distribution |χ S (rh , re )|2 of the electron relative to the hole, which is fixed at the central Cl atom in (001) plane. The bottom panel of Fig. 3 shows the distribution of the hole relative to the electron, which is also fixed at the central Cl
Hole
Li Cl
FIG. 3. Real-space distribution of the probability density |χ S (rh , re )|2 for the lowest-energy exciton of LiCl (at 8.8 eV), in the (001) plane. The top panel shows the distribution of the electron (re ) with respect to the hole (rh ), which is fixed at the central Cl atom. The bottom panel shows the distribution of the hole (rh ) with respect to the electron (re ), which is also fixed at the central Cl atom.
atom. The hole of the exciton state shown in the bottom panel of Fig. 3 is mainly formed by the Cl 3px or 3py orbitals at the very same central Cl atom where the electron is fixed, with some contributions of the Cl 3px or 3py states at the neighboring Cl atoms. The electron of the exciton state shown in the top panel of Fig. 3, on the other hand, is mainly composed of Cl 4s-like orbitals at the central Cl atom where the hole is localized, accompanied by some contributions of Li 2s orbitals and Cl 4s orbitals at the neighboring Cl atoms. From Fig. 3, one can find some interesting features in the electron (or hole) spatial distribution. The probability density of the electron (relative to the hole fixed at the central Cl atom) has maximal values on the central Cl atom and considerable values on neighboring Cl atoms. However, the probability density is very low at the Li atoms. Since the probability density at Li atoms is very low, the exciton state in the present system does not have a simple Frenkel-exciton feature, which would assume the electron hopping from the Cl atom to the nearest-neighbor Li atoms. Furthermore, the electron distribution on the neighboring Cl atoms is not isotropic. Instead, it is polarized towards the central Cl atom. The distribution of the hole on the neighboring Cl atoms, which has a shape of 3px or
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3py -like orbital, is also polarized towards the central Cl atom where the electron is fixed. A similar polarization effect is also found in the hole (or electron) distribution of the lowest exciton state of LiF,12, 18 and this effect in LiF is even more pronounced than in LiCl. In fact, such polarization effect of the electron (or hole) distribution results from the correlation of the quasielectron (or quasihole) to the effective positive (or negative) charge of the quasihole (or quasielectron) on the central Cl atom, which is affected by the electron-hole interaction as can be seen from Eqs. (3) and (4). The binding energy of the lowest exciton state in LiCl is about 0.7 eV, which is smaller than the binding energy of nearly 1.6 eV in the lowest exciton state of LiF.12, 18 The electron-hole interaction effect in the lowest exciton state for LiF is stronger than the corresponding effect for LiCl. IV. CONCLUSIONS
Using many-body perturbation theory within the GW approximation and the Bethe-Salpeter equation, we have calculated the quasiparticle band structure and optical absorption spectra of bulk LiCl. The calculated band gap and energy of the Li 1s core level within the GWA are in good agreement with the experimental observations.43, 44 From the solution of the BSE, the lowest exciton state is found at 8.8 eV. In this exciton state, the hole is mainly formed by the 3p orbitals, while the electron is mainly composed of the Cl 4s orbitals, accompanied by some contributions from Li 2s orbitals. The real-space distribution of the hole (or electron) for the lowest exciton shows an obvious polarization feature which is attributed to the electron-hole correlations. At low excitation < energies (S ∼ 12 eV), the optical absorption spectrum calculated with the electron-hole interaction included reproduces well the corresponding optical spectrum from experiments.29 A comparison between the optical spectra calculated with and without electron-hole interaction shows that the effects of the electron-hole interaction in LiCl are important and should be taken into account. ACKNOWLEDGMENTS
We are grateful to P. Krüger for useful discussions. N.-P.W. acknowledges financial support from the National Natural Science Foundation of China (NNSFC) under Grant Nos. 11074136 and 61176081, The Natural Science Foundation of Zhejiang Province under Grant No. Y6100467, and K. C. Wong Magna Fund in Ningbo University. 1 P.
K. de Boer and R. A. de Groot, Eur. Phys. J. B 4, 25 (1998). R. Gopikrishnan, D. Jose, and A. Datta, AIP Adv. 2, 012131 (2012). 3 D. Vogel, P. Krüger, and J. Pollmann, Phys. Rev. B 54, 5495 (1996). 2 C.
J. Chem. Phys. 139, 214710 (2013) 4 A.
Filippetti and V. Fiorentini, Eur. Phys. J. B 71, 139 (2009). C. Erwin and C. C. Lin, J. Phys. C 21, 4285 (1988). 6 B. Baumeier, P. Krüger, J. Pollmann, and G. V. Vajenine, Phys. Rev. B 78, 125111 (2008). 7 C. Sommer, P. Krüger, and J. Pollmann, Phys. Rev. B 85, 165119 (2012). 8 M. S. Hybertsen and S. G. Louie, Phys. Rev. Lett. 55, 1418 (1985). 9 W. Hanke and L. J. Sham, Phys. Rev. B 21, 4656 (1980). 10 G. Onida, L. Reining, R. W. Godby, R. Del Sole, and W. Andreoni, Phys. Rev. Lett. 75, 818 (1995); S. Albrecht, L. Reining, R. Del Sole, and G. Onida, ibid. 80, 4510 (1998). 11 L. X. Benedict, E. L. Shirley, and R. B. Bohn, Phys. Rev. Lett. 80, 4514 (1998); Phys. Rev. B 57, R9385 (1998). 12 M. Rohlfing and S. G. Louie, Phys. Rev. Lett. 80, 3320 (1998); 81, 2312 (1998); 82, 1959 (1999). 13 L. Hedin, Phys. Rev. 139, A796 (1965). 14 L. Hedin and S. Lundqvist, in Solid State Physics: Advances in Research and Application, edited by F. Seitz, D. Turnbull, and H. Ehrenreich (Academic, New York, 1969), Vol. 23, p. 1. 15 M. S. Hybertsen and S. G. Louie, Phys. Rev. B 34, 5390 (1986). 16 E. L. Shirley, L. J. Terminello, J. E. Klepeis, and F. J. Himpsel, Phys. Rev. B 53, 10296 (1996). 17 S. Iwata, S. Ishii, and K. Ohno, Mater. Trans. 46, 1100 (2005). 18 N.-P. Wang, M. Rohlfing, P. Krüger, and J. Pollmann, Phys. Rev. B 67, 115111 (2003). 19 G. Strinati, Phys. Rev. Lett. 49, 1519 (1982); Phys. Rev. B 29, 5718 (1984). 20 J.-W. van der Horst, P. A. Bobbert, M. A. J. Michels, G. Brocks, and P. J. Kelly, Phys. Rev. Lett. 83, 4413 (1999). 21 M. Rohlfing and S. G. Louie, Phys. Rev. Lett. 83, 856 (1999); M. Rohlfing, M. Palummo, G. Onida, and R. Del Sole, ibid. 85, 5440 (2000). 22 J. C. Grossman, M. Rohlfing, L. Mitas, S. G. Louie, and M. L. Cohen, Phys. Rev. Lett. 86, 472 (2001). 23 P. H. Hahn, W. G. Schmidt, and F. Bechstedt, Phys. Rev. Lett. 88, 016402 (2001). 24 N.-P. Wang, M. Rohlfing, P. Krüger, and J. Pollmann, Appl. Phys. A 78, 213 (2004); Phys. Rev. Lett. 92, 216805 (2004); Phys. Rev. B 71, 045407 (2005); 74, 155405 (2006). 25 Y. Ma, M. Rohlfing, and A. Gali, Phys. Rev. B 81, 041204(R) (2010). 26 M. Rohlfing, Phys. Rev. Lett. 108, 087402 (2012). 27 C. Sommer, P. Krüger, and J. Pollmann, Phys. Rev. B 86, 155212 (2012). 28 W. Y. Ching, F. Gan, and M.-Z. Huang, Phys. Rev. B 52, 1596 (1995). 29 J. E. Eby, K. J. Teegarden, and D. B. Dutton, Phys. Rev. 116, 1099 (1959). 30 G. B. Bachelet, D. R. Hamann, and M. Schlüter, Phys. Rev. B 26, 4199 (1982). 31 D. R. Hamann, Phys. Rev. B 40, 2980 (1989). 32 W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965). 33 M. Rohlfing, P. Krüger, and J. Pollmann, Phys. Rev. B 52, 1905 (1995). 34 M. Rohlfing, P. Krüger, and J. Pollmann, Phys. Rev. B 48, 17791 (1993). 35 M. S. Hybertsen and S. G. Louie, Phys. Rev. B 37, 2733 (1988). 36 W. Luo, S. Ismail-Beigi, M. L. Cohen, and S. G. Louie, Phys. Rev. B 66, 195215 (2002). 37 Y. Ma and M. Rohlfing, Phys. Rev. B 75, 205114 (2007). 38 P. Larson, M. Dvorak, and Z. Wu, Phys. Rev. B 88, 125205 (2013). 39 R. Gómez-Abal, X. Li, M. Scheffler, and C. Ambrosch-Draxl, Phys. Rev. Lett. 101, 106404 (2008). 40 X.-Z. Li, R. Gómez-Abal, H. Jiang, C. Ambrosch-Draxl, and M. Scheffler, New J. Phys. 14, 023006 (2012). 41 A. Fetter and J. D. Walecka, Quantum Theory of Many Particle Systems (McGraw-Hill, San Francisco, 1971), p. 538. 42 M. Rohlfing and S. G. Louie, Phys. Rev. B 62, 4927 (2000). 43 A. B. Kunz, Phys. Rev. B 26, 2056 (1982). 44 L. I. Johansson and S. B. M. Hagström, Phys. Scr. 14, 55 (1976). 5 S.
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