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GW method for 3d and 4f systems
ELSEVIER
Physica B 237-238 (1997) 321-323
GW method for 3d and 4fsystems F. Aryasetiawan ",b,* a Department of Theoretical Physics, University of Lurid, S6lveoatan 14A, S-223 62 Lund, Sweden b Max-Planck-lnstitut fiir Festk6rperforschun9, Heisenberostrasse 1, 70569 Stuttgart, German);
Abstract
Conventional methods for calculating the electronic self-energy of a crystal within the GW approximation are mostly based on a plane-wave approach. While these methods have been successful in treating s-p systems, applications to systems with localized states such as 3d and 4f systems have been hampered by the large size of the computation. We present a method for calculating the self-energy of 3d and 4f systems within the GW approximation. Unlike, conventional plane-wave methods, the method uses products o f L M T O ' s . Due to the small number of basis functions, the method allows full calculations of the dielectric matrix and makes it feasible to treat systems which would otherwise be beyond the present computational capability. Applications have been made to transition-metal systems (Ni, NiO) and more recently to an f-system (Gd) and to the 3d semicore states in ZnSe, GaAs and Ge with encouraging results. The band structure of Ni is greatly improved in particular the overestimated LDA bandwidth is narrowed by ,- 1 eV bringing it much closer to the experimental value. In NiO, the LDA gap, which is much too small (0.2 eV in LDA and 4.0 eV experimentally), is widened to ~5 eV. Since the initial LDA band gap is quite different from the final one, we find that it is important to perform the calculation self-consistently. Application to the semicore states in ZnSe, GaAs has also yielded good agreement with experiment. We derived a simple formula to describe the increasing error from ZnSe to Ge in the LDA eigenvalues. Finally, application to an f-system Gd has revealed some new features: a very small quasiparticle weight (0.3) and the presence of a low-energy satellite structure in the spectral function. Keywords: Self-energy; GW approximation; Quasiparticle; Satellite
I. Introduction
The most widely used method for calculating ground-state properties and electronic structures o f solids is the Kohn-Sham (KS) density functional theory (DFT) [1,2] within the local density approximation ( L D A ) [2]. It is customary to interpret the KS eigenvalues as quasiparticle energies measured in photoemission experiments, although there is no clear theoretical justification, except for the highest occupied states. For s-p systems, it is found that the agreement is often rather good. However, there are * Correspondingaddress: F. Aryasetiawan,Department of Theoretical Physics, University of Lund, S61vegatan 14A, S-233 62 Lund, Sweden. 0921-4526/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved PH S0921-4526(97)001 96-8
serious discrepancies: the band gaps in semiconductors (Si, GaAs, Ge, etc.) are systematically underestimated, as much as 100% in Ge. The bandwidth in Na is about 15% too large. The discrepancies become worse in strongly correlated 3d and 4 f systems. The bandwidth in Ni, for instance, is 30% too large in the LDA. In the so-called Mott-Hubbard insulators o f transition metal oxides, the L D A band gap is much too small compared with experiment. In some cases, the L D A gives qualitatively wrong results. For example, the Mott-Hubbard insulator CoO and the undoped parent compound of the high-Tc material La2CuO4 are predicted to be metals whereas experimentally they are insulators. A proper way o f calculating quasiparticle energies is provided by the Green function theory where the
F Aryasetiawan/ Physica B 237-238 (1997) 321-323
322
many-body effects are contained in the self-energy operator which is non-local and energy dependent. A working approximation to the self-energy is the socalled GW approximation (GWA) [3, 4] which may be thought of as a generalization of the Hartree-Fock approximation (HFA) but with a dynamically screened interaction. The GWA was applied to the electron gas by Hedin [3, 4] and more extensively by Lundqvist [5] many years ago. Calculations for real systems were hampered by the size of the computation and it was not until the mid-1980s that such calculations were possible. Since then, the GWA has been applied with success to many systems ranging from alkali metals [68], semiconductors [9-12], transition metals [13, 14], metal surfaces [15] to clusters [16]. Conventional methods for calculating the selfenergy use plane waves as basis functions which are suitable for s-p systems. However, applications to 3d and 4f systems using plane-wave basis are not feasible up to now due to a large number of plane waves needed to describe the localized 3d or 4f states. We have developed a method for treating systems with localized states. The method is based on the linear muffin-tin orbital (LMTO) approach [17] and basis functions needed to describe the screened Coulomb interaction and the self-energy are products of the LMTO's.
2. The GW approximation The GWA is formally obtained as the first term in the expansion of the self-energy in the screened interaction W. This is, however, not very useful from the physical point of view. If we do include higher-order diagrams, we do not necessarily get better results (in the sense of closer agreement with experiment). Straightforward higher-order diagrams can give a selfenergy with wrong analytic properties which in turn results in an unphysically negative spectral function (density of states) for some energies. It is more physical and useful to regard the GWA as a Hartree-Fock approximation with frequency-dependent screening which cures the most serious deficiency of the HFA. The self-energy in the GWA is given by S(r, r'; 09) =
±
f
/ d09' G(r, r'; 09 + J ) W ( r , r'; o9'). 2rt J
(1)
The screened potential W is given by W(r,r';09) =
f d3r" e-~(r,r";09)v(r '' - r'),
(2)
where e-l is an inverse dielectric matrix and v is a bare Coulomb potential. The calculation of E- l within the random-phase approximation (RPA) is described in detail in a previous publication [18]. This is done without making the plasmon-pole or other similar approximations. The band structure is calculated using the LMTO [17] method which can be applied to systems with d and f electrons. The LMTO basis within the atomic sphere approximation [17] has the following form:
ze~ = 4~RL+ ~ q~R,L,hR'L',e~,
(3)
RZLI
where RL denote the site and angular momentum (l,m), respectively, ~b is the solution to the Schrrdinger equation inside the muffin-tin sphere and q~ is its energy derivative taken at some fixed energy ev. The response function within the RPA consists of products of Bloch states so that the Hilbert space spanned by the response function is composed of products of the type ~b~b, q~q~, and ~q~. A large fraction of these products is linearly dependent and we construct an optimized basis for e- 1 by forming linear combinations of these product functions. The number of basis functions per atom is typically 50-100 [18].
3. Applications The method has been applied to a number of systems containing localized states. Application to Ni gives similar results to those from a previous calculation using a different method [13]. The LDA band structure is significantly improved, in particular, a band narrowing of ,-~ 1 eV from the LDA value is obtained, bringing the bandwidth in much closer agreement with experiment. The famous 6 eV satellite is not reproduced and the LDA exchange splitting, which is too large by 0.3 eV, remains essentially unchanged. Although this discrepancy is of the same order as the computational error, we believe that it is inherent in the GWA due to the absence of spin-dependent interaction. We are currently performing first-principles
F Aryasetiawan/Physica B 237-238 (1997) 321-323
T-matrix calculations to account for the discrepancy in the exchange splitting and the 6 eV satellite. We have also applied our method to study the elctronic structure of the highly correlated transition metal oxide NiO [14]. The LDA band gap for this prototype of Mott-Hubbard insulators is far to small (0.2 versus 4.0 eV experimentally). The GWA yields a band gap of ~5 eV, in reasonable agreement with experiment. We found in the case of NiO that it was important to perform the calculation in a self-consistent way. It is crucial to modify not only the quasiparticle energies in the Green function but also the singleparticle wave functions. Regarding the character of the top of the valence band, the GW result still has too much 3d character compared to the currently accepted interpretation of the photoemission spectra. A very recent work on the effect of self-consistency on the spectra shows that although a satellite structure at 10 eV is obtained in the first iteration, the intensity decreases towards self-consistency. Application to the 3d semicore states in ZnSe, GaAs and Ge which are a few eV too high in the LDA yields quasiparticle energies in much better agreement with experiment [19]. It is not a priori clear that the GWA should work in this case. Unlike core states which are inert and therefore contribute little to the polarization, the semicore states contribute considerably to the polarization. To further guage the applicability of the GWA, we recently calculated the self-energy and spectral function of the f-system Gd [20]. Several interesting features have arisen from this calculation. Conventionally, the self-energy correction simply shifts the LDA eigenvalue to the correct quasiparticle energy. In Gd, the following occurs: at the LDA f-state eigenvalue, the imaginary part of the self-energy, Im,r, shows a peak structure which results in an unusually small quasiparticle weight (0.3). Experimentally there is no quasiparticle weight at the LDA eigenvalue. The peak structure in Im L" causes the appearance of a satellite which curiously lies at the experimental peak position. The calculation indicates the necessity to go beyond
323
the GWA in order to completely remove the remaining quasiparticle peak and to transfer the weight to the satellite region. In summary, the GWA gives in many cases good quasiparticle energies, but a theory beyond GW is needed to provide a better description of the satellite structure. References [1] [2] [3] [4]
P. Hohenberg and W. Kohn, Phys. Rev. 136 (1964) B864. W. Kohn and L. J. Sham, Phys. Rev. 140 (1965) A1133. L. Hedin, Phys. Rev. 139 (1965) A796. L. Hedin and S. Lundqvist, in: Solid State Physics Vol. 23, eds. H. Ehrenreich, F. Seitz and D. Tumbull (Academic, New York, 1969). [5] B. I. Lundqvist, Phys. Kondens. Mater. 6 (1967a) 193; ibid 6 (1967b) 206; ibid 7 (1968) 117; B. I. Lundqvist, Phys. Stat. Sol. 32 (1969) 273. [6] M. P. Surh, J. E. Northurp and S. G. Louie, Phys. Rev. B 38 (1988) 5976. [7] K. W. K. Shung, B. E. Semelius and G. D. Mahan, Phys. Rev. B 36 (1987) 4499. [8] J. E. Northurp, M.S. Hybertsen and S. G. Louie, Phys. Rev. Lett. 59 (1987) 819. [9] M. S. Hybertsen and S. G. Louie, Phys. Rev. B 34 (1986) 5390. [10] M. S. Hybertsen and S. G. Louie, Comments Condens. Matter Phys. 13 (1987) 223. [11] R. W. Godby, M. Schliiter and L. J. Sham, Phys. Rev. B 37 (1988) 10 159. [12] W. von der Linden and P. Horsch, Phys. Rev. B 37 (1988) 8351. [13] F. Aryasetiawan, Phys. Rev. B 46 (1992) 13051. [14] F. Aryasetiawan and O. Gunnarsson, Phys. Rev. Lett. 74 (1995) 3221. [i5],.J.J. Diesz, A. G. Eguiluz and W. Hanke, Phys. Rev. Lett. 71 (1993) 2793. [16] G. Onida, L. Reining, R. W. Godby, R. Del Sole and W. Andreoni, Phys. Rev. Lett. 75 (1995) 818. [17] O. K. Andersen, Phys. Rev. B 12 (1975) 3060. [18] F. Aryasetiawan and O. Gunnarsson, Phys. Rev. B 49 (1994) 16 214. [19] F. Aryasetiawan and O. Gunnarsson, Phys. Rev. B 54 (1996) 17 564. [20] F. Aryasetiawan and K. Karlsson, Phys. Rev. B 54 (1996) 5353.
Physica B 237-238 (1997) 321-323
GW method for 3d and 4fsystems F. Aryasetiawan ",b,* a Department of Theoretical Physics, University of Lurid, S6lveoatan 14A, S-223 62 Lund, Sweden b Max-Planck-lnstitut fiir Festk6rperforschun9, Heisenberostrasse 1, 70569 Stuttgart, German);
Abstract
Conventional methods for calculating the electronic self-energy of a crystal within the GW approximation are mostly based on a plane-wave approach. While these methods have been successful in treating s-p systems, applications to systems with localized states such as 3d and 4f systems have been hampered by the large size of the computation. We present a method for calculating the self-energy of 3d and 4f systems within the GW approximation. Unlike, conventional plane-wave methods, the method uses products o f L M T O ' s . Due to the small number of basis functions, the method allows full calculations of the dielectric matrix and makes it feasible to treat systems which would otherwise be beyond the present computational capability. Applications have been made to transition-metal systems (Ni, NiO) and more recently to an f-system (Gd) and to the 3d semicore states in ZnSe, GaAs and Ge with encouraging results. The band structure of Ni is greatly improved in particular the overestimated LDA bandwidth is narrowed by ,- 1 eV bringing it much closer to the experimental value. In NiO, the LDA gap, which is much too small (0.2 eV in LDA and 4.0 eV experimentally), is widened to ~5 eV. Since the initial LDA band gap is quite different from the final one, we find that it is important to perform the calculation self-consistently. Application to the semicore states in ZnSe, GaAs has also yielded good agreement with experiment. We derived a simple formula to describe the increasing error from ZnSe to Ge in the LDA eigenvalues. Finally, application to an f-system Gd has revealed some new features: a very small quasiparticle weight (0.3) and the presence of a low-energy satellite structure in the spectral function. Keywords: Self-energy; GW approximation; Quasiparticle; Satellite
I. Introduction
The most widely used method for calculating ground-state properties and electronic structures o f solids is the Kohn-Sham (KS) density functional theory (DFT) [1,2] within the local density approximation ( L D A ) [2]. It is customary to interpret the KS eigenvalues as quasiparticle energies measured in photoemission experiments, although there is no clear theoretical justification, except for the highest occupied states. For s-p systems, it is found that the agreement is often rather good. However, there are * Correspondingaddress: F. Aryasetiawan,Department of Theoretical Physics, University of Lund, S61vegatan 14A, S-233 62 Lund, Sweden. 0921-4526/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved PH S0921-4526(97)001 96-8
serious discrepancies: the band gaps in semiconductors (Si, GaAs, Ge, etc.) are systematically underestimated, as much as 100% in Ge. The bandwidth in Na is about 15% too large. The discrepancies become worse in strongly correlated 3d and 4 f systems. The bandwidth in Ni, for instance, is 30% too large in the LDA. In the so-called Mott-Hubbard insulators o f transition metal oxides, the L D A band gap is much too small compared with experiment. In some cases, the L D A gives qualitatively wrong results. For example, the Mott-Hubbard insulator CoO and the undoped parent compound of the high-Tc material La2CuO4 are predicted to be metals whereas experimentally they are insulators. A proper way o f calculating quasiparticle energies is provided by the Green function theory where the
F Aryasetiawan/ Physica B 237-238 (1997) 321-323
322
many-body effects are contained in the self-energy operator which is non-local and energy dependent. A working approximation to the self-energy is the socalled GW approximation (GWA) [3, 4] which may be thought of as a generalization of the Hartree-Fock approximation (HFA) but with a dynamically screened interaction. The GWA was applied to the electron gas by Hedin [3, 4] and more extensively by Lundqvist [5] many years ago. Calculations for real systems were hampered by the size of the computation and it was not until the mid-1980s that such calculations were possible. Since then, the GWA has been applied with success to many systems ranging from alkali metals [68], semiconductors [9-12], transition metals [13, 14], metal surfaces [15] to clusters [16]. Conventional methods for calculating the selfenergy use plane waves as basis functions which are suitable for s-p systems. However, applications to 3d and 4f systems using plane-wave basis are not feasible up to now due to a large number of plane waves needed to describe the localized 3d or 4f states. We have developed a method for treating systems with localized states. The method is based on the linear muffin-tin orbital (LMTO) approach [17] and basis functions needed to describe the screened Coulomb interaction and the self-energy are products of the LMTO's.
2. The GW approximation The GWA is formally obtained as the first term in the expansion of the self-energy in the screened interaction W. This is, however, not very useful from the physical point of view. If we do include higher-order diagrams, we do not necessarily get better results (in the sense of closer agreement with experiment). Straightforward higher-order diagrams can give a selfenergy with wrong analytic properties which in turn results in an unphysically negative spectral function (density of states) for some energies. It is more physical and useful to regard the GWA as a Hartree-Fock approximation with frequency-dependent screening which cures the most serious deficiency of the HFA. The self-energy in the GWA is given by S(r, r'; 09) =
±
f
/ d09' G(r, r'; 09 + J ) W ( r , r'; o9'). 2rt J
(1)
The screened potential W is given by W(r,r';09) =
f d3r" e-~(r,r";09)v(r '' - r'),
(2)
where e-l is an inverse dielectric matrix and v is a bare Coulomb potential. The calculation of E- l within the random-phase approximation (RPA) is described in detail in a previous publication [18]. This is done without making the plasmon-pole or other similar approximations. The band structure is calculated using the LMTO [17] method which can be applied to systems with d and f electrons. The LMTO basis within the atomic sphere approximation [17] has the following form:
ze~ = 4~RL+ ~ q~R,L,hR'L',e~,
(3)
RZLI
where RL denote the site and angular momentum (l,m), respectively, ~b is the solution to the Schrrdinger equation inside the muffin-tin sphere and q~ is its energy derivative taken at some fixed energy ev. The response function within the RPA consists of products of Bloch states so that the Hilbert space spanned by the response function is composed of products of the type ~b~b, q~q~, and ~q~. A large fraction of these products is linearly dependent and we construct an optimized basis for e- 1 by forming linear combinations of these product functions. The number of basis functions per atom is typically 50-100 [18].
3. Applications The method has been applied to a number of systems containing localized states. Application to Ni gives similar results to those from a previous calculation using a different method [13]. The LDA band structure is significantly improved, in particular, a band narrowing of ,-~ 1 eV from the LDA value is obtained, bringing the bandwidth in much closer agreement with experiment. The famous 6 eV satellite is not reproduced and the LDA exchange splitting, which is too large by 0.3 eV, remains essentially unchanged. Although this discrepancy is of the same order as the computational error, we believe that it is inherent in the GWA due to the absence of spin-dependent interaction. We are currently performing first-principles
F Aryasetiawan/Physica B 237-238 (1997) 321-323
T-matrix calculations to account for the discrepancy in the exchange splitting and the 6 eV satellite. We have also applied our method to study the elctronic structure of the highly correlated transition metal oxide NiO [14]. The LDA band gap for this prototype of Mott-Hubbard insulators is far to small (0.2 versus 4.0 eV experimentally). The GWA yields a band gap of ~5 eV, in reasonable agreement with experiment. We found in the case of NiO that it was important to perform the calculation in a self-consistent way. It is crucial to modify not only the quasiparticle energies in the Green function but also the singleparticle wave functions. Regarding the character of the top of the valence band, the GW result still has too much 3d character compared to the currently accepted interpretation of the photoemission spectra. A very recent work on the effect of self-consistency on the spectra shows that although a satellite structure at 10 eV is obtained in the first iteration, the intensity decreases towards self-consistency. Application to the 3d semicore states in ZnSe, GaAs and Ge which are a few eV too high in the LDA yields quasiparticle energies in much better agreement with experiment [19]. It is not a priori clear that the GWA should work in this case. Unlike core states which are inert and therefore contribute little to the polarization, the semicore states contribute considerably to the polarization. To further guage the applicability of the GWA, we recently calculated the self-energy and spectral function of the f-system Gd [20]. Several interesting features have arisen from this calculation. Conventionally, the self-energy correction simply shifts the LDA eigenvalue to the correct quasiparticle energy. In Gd, the following occurs: at the LDA f-state eigenvalue, the imaginary part of the self-energy, Im,r, shows a peak structure which results in an unusually small quasiparticle weight (0.3). Experimentally there is no quasiparticle weight at the LDA eigenvalue. The peak structure in Im L" causes the appearance of a satellite which curiously lies at the experimental peak position. The calculation indicates the necessity to go beyond
323
the GWA in order to completely remove the remaining quasiparticle peak and to transfer the weight to the satellite region. In summary, the GWA gives in many cases good quasiparticle energies, but a theory beyond GW is needed to provide a better description of the satellite structure. References [1] [2] [3] [4]
P. Hohenberg and W. Kohn, Phys. Rev. 136 (1964) B864. W. Kohn and L. J. Sham, Phys. Rev. 140 (1965) A1133. L. Hedin, Phys. Rev. 139 (1965) A796. L. Hedin and S. Lundqvist, in: Solid State Physics Vol. 23, eds. H. Ehrenreich, F. Seitz and D. Tumbull (Academic, New York, 1969). [5] B. I. Lundqvist, Phys. Kondens. Mater. 6 (1967a) 193; ibid 6 (1967b) 206; ibid 7 (1968) 117; B. I. Lundqvist, Phys. Stat. Sol. 32 (1969) 273. [6] M. P. Surh, J. E. Northurp and S. G. Louie, Phys. Rev. B 38 (1988) 5976. [7] K. W. K. Shung, B. E. Semelius and G. D. Mahan, Phys. Rev. B 36 (1987) 4499. [8] J. E. Northurp, M.S. Hybertsen and S. G. Louie, Phys. Rev. Lett. 59 (1987) 819. [9] M. S. Hybertsen and S. G. Louie, Phys. Rev. B 34 (1986) 5390. [10] M. S. Hybertsen and S. G. Louie, Comments Condens. Matter Phys. 13 (1987) 223. [11] R. W. Godby, M. Schliiter and L. J. Sham, Phys. Rev. B 37 (1988) 10 159. [12] W. von der Linden and P. Horsch, Phys. Rev. B 37 (1988) 8351. [13] F. Aryasetiawan, Phys. Rev. B 46 (1992) 13051. [14] F. Aryasetiawan and O. Gunnarsson, Phys. Rev. Lett. 74 (1995) 3221. [i5],.J.J. Diesz, A. G. Eguiluz and W. Hanke, Phys. Rev. Lett. 71 (1993) 2793. [16] G. Onida, L. Reining, R. W. Godby, R. Del Sole and W. Andreoni, Phys. Rev. Lett. 75 (1995) 818. [17] O. K. Andersen, Phys. Rev. B 12 (1975) 3060. [18] F. Aryasetiawan and O. Gunnarsson, Phys. Rev. B 49 (1994) 16 214. [19] F. Aryasetiawan and O. Gunnarsson, Phys. Rev. B 54 (1996) 17 564. [20] F. Aryasetiawan and K. Karlsson, Phys. Rev. B 54 (1996) 5353.