- Email: [email protected]
Ab initio calculation of the optical and magneto-optical properties of moderately correlated magnetic solids
ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 272–276 (2004) 523–524
Ab initio calculation of the optical and magneto-optical properties of moderately correlated magnetic solids A. Perlova,*, S. Chadova, H. Eberta, L. Chioncelb, A.I. Lichtensteinb, M.I. Katsnelsonc a
Munich University, Butenandtstr. 5-13, Munich D-81377, Germany University of Nijmegen, Nijmegen NL-6525 ED, The Netherlands c Uppsala University, Uppsala SE-751 21, Sweden
b
Abstract An approach for the calculation of the optical and magneto-optical properties of solids based on the one-particle Green function is introduced in the framework of the linear muffin-tin orbital (LMTO) method. The approach keeps all advantages of the more accurate Korringa–Kohn–Rostoker (KKR) scheme as the possibility to account for many-body effects in terms of the non-local energy-dependent self-energy but is numerically much more efficient. In particular an incorporation of the single-site self-energy coming from the dynamical mean-field theory (DMFT) is implemented. An application of the approach to bulk Ni and Fe showed rather good agreement with the experimental data, in contrast with the results of standard local spin density approximation (LSDA) computations. r 2003 Elsevier B.V. All rights reserved. PACS: 78.20.L; 71.27; 71.20.B Keywords: Magneto-optics; Transition metals; Correlations
Much information on the electronic structure of magnetic solids is gained by optical and magneto-optical measurements, being useful tools for analyzing the dispersion of (quasi-particle) bands. However, measured optical and magneto-optical spectra can hardly be interpreted without theoretical calculations. To give such an interpretation one in general has to solve manyelectron problem, which is impossible without more or less severe approximations. For materials where the kinetic energy of the electrons is more important than the Coulomb interactions, the most successful first principles method is the density functional theory (DFT) within the local (spin-) density approximation (L(S)DA). Nevertheless, calculations based on the LSDA often give only qualitative agreement with experimental data. *Corresponding author Tel.: +49-89218077589; fax: +4989218077568. E-mail address: [email protected] (A. Perlov).
For an in principle correct description of the excitation energies the non-local self-energy has to be considered. This, however, constitutes a many-body problem. Therefore, DFT–LDA calculations must be supplemented by many-body methods to arrive at a realistic description of the one-particle excitations in correlated systems. For moderately correlated system, where Coulomb interactions (U) and the kinetic energy (bandwidth W ) are of the same order of magnitude which is the case for a most 3d and 5f elements and their compound to get a reasonable description of the electronic structure one has to take into account nonhermitian energy-dependent self-energy. As concerns to the calculations of the optical spectra we have to face the following problem: one-particle wave functions are not defined any more and the formalism has to be taken in the Green function representation. Such a representation has been already derived [1] and successfully applied for the calculations in the framework of Korringa– Kohn–Rostoker (KKR) Green-function method for LSDA calculations. The only drawback of such an
0304-8853/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2003.11.224
ARTICLE IN PRESS
The dispersive part is connected to the absorptive one via Kramers–Kronig relationship. The one-particle Green function entering the expression for the conductivity in general is the solution of the Dyson equation: # # EÞGðEÞ ¼ I;# ð2Þ ðH# 0 þ SðEÞ where H# 0 stands for the L(S)DA one-particle Hamilto# nian and complex self-energy SðEÞ contains information about correlation effects. For moderately correlated # systems a rather good approximation for SðEÞ can be obtained in the framework of LDA þ DMFT method, where an exact many-body problem for a crystal is split into a one-particle impurity problem for the crystal and a many-body problem for one site in an effective medium (effective impurity method, see Refs. [3,4]). To solve the impurity problem and to find the self-energy S as a functional of the bath Green function G we used the spin-polarized FLEX þ T-matrix approach [5]. Within our approach to deal with Eq. (2) the Green function is represented as a sum over energy-independent basis functions jiS: X 1 # GðEÞ ¼ jiS/ijðE H# 0 SðEÞÞ jjS/jj: ð3Þ ij
This leads to a computationally manageable formula for the conductivity, as the corresponding matrix elements of the current operator /jjJjjS are energy-independent. In Fig. 1, we present results of our calculations of the optical conductivity and polar Kerr rotation spectra for Fe and Ni. As one can see for Fe already LSDA gives reasonable agreement with experimental data though the
100
σ1(ω), 10-14s-1
200
60
Kerr rotation (deg)
Fe 300
0.2
Ni
40 20
0 Kerr rotation (deg)
approach is that it is highly demanding as to both computational resources and computational time. In this paper we propose a simplified way to calculate optical and magneto-optical properties of solids in the Green function representation based on variational methods of band structure calculations and present results of the implementation of proposed technique in the framework of the linear muffin-tin orbital (LMTO) method. Optical properties of solids are conventionally described in terms of the optical conductivity tensor sab ðoÞ: Microscopic calculations of sab ðoÞ are based on the Kubo linear response formalism [2]. Using a quasi-particle description of the excitation spectra in the zero-temperature limit an expression for the absorptive part of sab ðoÞ reads Z EF 1 sð1Þ dE tr½j#a I GðEÞj#b I GðE þ _oÞ: ð1Þ ab ðoÞ ¼ po EF o
ωσ1(ω), 10-29s-2
A. Perlov et al. / Journal of Magnetism and Magnetic Materials 272–276 (2004) 523–524
524
0.0 -0.2 -0.4 -0.6 0 1 2 3 4 5 6 7 Energy (eV)
0.0 -0.2
0 1 2 3 4 5 6 7 Energy (eV)
Fig. 1. Calculated and measured optical and magneto-optical properties of Fe and Ni. Solid line—LDA þ DMFT calculations, dashed line—LDA calculations, symbols—experiments [6].
application of DMFT makes agreement even better. Somewhat another situation takes place for Ni. The high-energy peak in the diagonal part of the optical conductivity calculated within LSDA is shifted approximately 1 eV with respect to experiment. This shift is due to well-known overestimating by LSDA of the dbandwidth for Ni. Applying DMFT cures the situation completely giving practically perfect agreement with measured spectra. At the same time changes in the calculated Kerr rotation spectra are not so pronounced although they are in the required directions. Apparently, self-energy originated from the present realization of DMFT is not enough pronounced. To summarize, we have developed and realized an efficient approach to calculate optical and magnetooptical properties of the moderately correlated systems where correlation effects are taken into account via DMFT formalism. This allowed us to improve the description of the optical and magneto-optical properties of Fe and Ni crystals.
References [1] T. Huhne, H. Ebert, Phys. Rev. B 60 (1999) 12982. [2] R. Kubo, J. Phys. Soc. Jpn. 12 (1957) 570. [3] A. Georges, G. Kotliar, W. Krauth, M. Rozenberg, Rev. Mod. Phys. 68 (1996) 13. [4] A.I. Lichtenstein, M.I. Katsnelson, Phys. Rev. B 57 (1998) 6884. [5] M.I. Katsnelson, A.I. Lichtenstein, Eur. Phys. J. B 30 (2002) 9. [6] H. Ebert, Rep. Prog. Phys. 59 (1996) 1665 references to the experimental data can be found in the review article.
Journal of Magnetism and Magnetic Materials 272–276 (2004) 523–524
Ab initio calculation of the optical and magneto-optical properties of moderately correlated magnetic solids A. Perlova,*, S. Chadova, H. Eberta, L. Chioncelb, A.I. Lichtensteinb, M.I. Katsnelsonc a
Munich University, Butenandtstr. 5-13, Munich D-81377, Germany University of Nijmegen, Nijmegen NL-6525 ED, The Netherlands c Uppsala University, Uppsala SE-751 21, Sweden
b
Abstract An approach for the calculation of the optical and magneto-optical properties of solids based on the one-particle Green function is introduced in the framework of the linear muffin-tin orbital (LMTO) method. The approach keeps all advantages of the more accurate Korringa–Kohn–Rostoker (KKR) scheme as the possibility to account for many-body effects in terms of the non-local energy-dependent self-energy but is numerically much more efficient. In particular an incorporation of the single-site self-energy coming from the dynamical mean-field theory (DMFT) is implemented. An application of the approach to bulk Ni and Fe showed rather good agreement with the experimental data, in contrast with the results of standard local spin density approximation (LSDA) computations. r 2003 Elsevier B.V. All rights reserved. PACS: 78.20.L; 71.27; 71.20.B Keywords: Magneto-optics; Transition metals; Correlations
Much information on the electronic structure of magnetic solids is gained by optical and magneto-optical measurements, being useful tools for analyzing the dispersion of (quasi-particle) bands. However, measured optical and magneto-optical spectra can hardly be interpreted without theoretical calculations. To give such an interpretation one in general has to solve manyelectron problem, which is impossible without more or less severe approximations. For materials where the kinetic energy of the electrons is more important than the Coulomb interactions, the most successful first principles method is the density functional theory (DFT) within the local (spin-) density approximation (L(S)DA). Nevertheless, calculations based on the LSDA often give only qualitative agreement with experimental data. *Corresponding author Tel.: +49-89218077589; fax: +4989218077568. E-mail address: [email protected] (A. Perlov).
For an in principle correct description of the excitation energies the non-local self-energy has to be considered. This, however, constitutes a many-body problem. Therefore, DFT–LDA calculations must be supplemented by many-body methods to arrive at a realistic description of the one-particle excitations in correlated systems. For moderately correlated system, where Coulomb interactions (U) and the kinetic energy (bandwidth W ) are of the same order of magnitude which is the case for a most 3d and 5f elements and their compound to get a reasonable description of the electronic structure one has to take into account nonhermitian energy-dependent self-energy. As concerns to the calculations of the optical spectra we have to face the following problem: one-particle wave functions are not defined any more and the formalism has to be taken in the Green function representation. Such a representation has been already derived [1] and successfully applied for the calculations in the framework of Korringa– Kohn–Rostoker (KKR) Green-function method for LSDA calculations. The only drawback of such an
0304-8853/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2003.11.224
ARTICLE IN PRESS
The dispersive part is connected to the absorptive one via Kramers–Kronig relationship. The one-particle Green function entering the expression for the conductivity in general is the solution of the Dyson equation: # # EÞGðEÞ ¼ I;# ð2Þ ðH# 0 þ SðEÞ where H# 0 stands for the L(S)DA one-particle Hamilto# nian and complex self-energy SðEÞ contains information about correlation effects. For moderately correlated # systems a rather good approximation for SðEÞ can be obtained in the framework of LDA þ DMFT method, where an exact many-body problem for a crystal is split into a one-particle impurity problem for the crystal and a many-body problem for one site in an effective medium (effective impurity method, see Refs. [3,4]). To solve the impurity problem and to find the self-energy S as a functional of the bath Green function G we used the spin-polarized FLEX þ T-matrix approach [5]. Within our approach to deal with Eq. (2) the Green function is represented as a sum over energy-independent basis functions jiS: X 1 # GðEÞ ¼ jiS/ijðE H# 0 SðEÞÞ jjS/jj: ð3Þ ij
This leads to a computationally manageable formula for the conductivity, as the corresponding matrix elements of the current operator /jjJjjS are energy-independent. In Fig. 1, we present results of our calculations of the optical conductivity and polar Kerr rotation spectra for Fe and Ni. As one can see for Fe already LSDA gives reasonable agreement with experimental data though the
100
σ1(ω), 10-14s-1
200
60
Kerr rotation (deg)
Fe 300
0.2
Ni
40 20
0 Kerr rotation (deg)
approach is that it is highly demanding as to both computational resources and computational time. In this paper we propose a simplified way to calculate optical and magneto-optical properties of solids in the Green function representation based on variational methods of band structure calculations and present results of the implementation of proposed technique in the framework of the linear muffin-tin orbital (LMTO) method. Optical properties of solids are conventionally described in terms of the optical conductivity tensor sab ðoÞ: Microscopic calculations of sab ðoÞ are based on the Kubo linear response formalism [2]. Using a quasi-particle description of the excitation spectra in the zero-temperature limit an expression for the absorptive part of sab ðoÞ reads Z EF 1 sð1Þ dE tr½j#a I GðEÞj#b I GðE þ _oÞ: ð1Þ ab ðoÞ ¼ po EF o
ωσ1(ω), 10-29s-2
A. Perlov et al. / Journal of Magnetism and Magnetic Materials 272–276 (2004) 523–524
524
0.0 -0.2 -0.4 -0.6 0 1 2 3 4 5 6 7 Energy (eV)
0.0 -0.2
0 1 2 3 4 5 6 7 Energy (eV)
Fig. 1. Calculated and measured optical and magneto-optical properties of Fe and Ni. Solid line—LDA þ DMFT calculations, dashed line—LDA calculations, symbols—experiments [6].
application of DMFT makes agreement even better. Somewhat another situation takes place for Ni. The high-energy peak in the diagonal part of the optical conductivity calculated within LSDA is shifted approximately 1 eV with respect to experiment. This shift is due to well-known overestimating by LSDA of the dbandwidth for Ni. Applying DMFT cures the situation completely giving practically perfect agreement with measured spectra. At the same time changes in the calculated Kerr rotation spectra are not so pronounced although they are in the required directions. Apparently, self-energy originated from the present realization of DMFT is not enough pronounced. To summarize, we have developed and realized an efficient approach to calculate optical and magnetooptical properties of the moderately correlated systems where correlation effects are taken into account via DMFT formalism. This allowed us to improve the description of the optical and magneto-optical properties of Fe and Ni crystals.
References [1] T. Huhne, H. Ebert, Phys. Rev. B 60 (1999) 12982. [2] R. Kubo, J. Phys. Soc. Jpn. 12 (1957) 570. [3] A. Georges, G. Kotliar, W. Krauth, M. Rozenberg, Rev. Mod. Phys. 68 (1996) 13. [4] A.I. Lichtenstein, M.I. Katsnelson, Phys. Rev. B 57 (1998) 6884. [5] M.I. Katsnelson, A.I. Lichtenstein, Eur. Phys. J. B 30 (2002) 9. [6] H. Ebert, Rep. Prog. Phys. 59 (1996) 1665 references to the experimental data can be found in the review article.