- Email: [email protected]
DMFT band calculation for Ce compounds
ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 310 (2007) 374–376 www.elsevier.com/locate/jmmm
DMFT band calculation for Ce compounds O. Sakaia,, Y. Shimizub a
Department of Physics, Tokyo Metropolitan University, Hachioji, Tokyo 192-0397, Japan b Department of Applied Physics, Tohoku University, Sendai 980-8579, Japan Available online 30 October 2006
Abstract Recent development of DMFT band calculation for Ce compounds is reported. The auxiliary impurity Anderson model is solved by a method named NCAf 2 vc, which can include the correct exchange process of f 1 ! f 0 ; f 2 fluctuation and also the crystalline field (CF) and spin–orbit splitting of self-energy. These are necessary features in the quantitative band theory for Ce compounds. The results of application on CeSb and CeP are presented. The double peaks structure in the photoemission spectra of these compounds is reproduced by the non-empirical calculation. The magnetic excitation spectra are also calculated. It is shown that the CF splitting energy is reversed in CeSb when the temperature decreases. r 2006 Elsevier B.V. All rights reserved. PACS: 71.15.m; 71.27.þa Keywords: DMFT; Band calculation; Ce-pnictides
Non-empirical band calculation for the strongly correlated electron systems has been extensively developed on the basis of the dynamical mean field theory (DMFT) [1]. In 4f electrons of Ce compounds the crystalline field splitting (CFS) and the spin–orbit interaction (SOI) play important roles [2]. A theory, which is named NCAf 2 vc has been recently developed to solve the auxiliary impurity Anderson model [3,4]. It is combined with the LMTO method to carry out the DMFT band calculation in the present study. The calculation can include the CFS and SOI effects and also the correct exchange process of f 1 ! f 0 ; f 2 valence fluctuation, and it can give accurate order of the Kondo temperature (T K ). We report the result of the DMFT band calculation for CeSb and CeP [5]. Ce pnictides have NaCl structure and has semimetal band structure: the p states of pnictogen form the valence band and the 5d states of Ce form the conduction band (see references in Ref. [2]). They overlap and supply a very small number of carriers. The top of the valence band has the G8 character and has very strong mixing matrix with the G8 component of Ce 4f electron, but the density of Corresponding author. Tel.: +81 22 217 6439; fax: +81 22 217 6447.
E-mail address: [email protected] (O. Sakai). 0304-8853/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2006.10.083
states (DOS) of bands at the Fermi energy ðE F Þ is small reflecting the semimetal nature of the band. The mixing between holes and the 4f states plays important roles in the p2f mixing model which explains various anomalous magnetic properties of these compounds [6]. Ce pnictides have characteristic double peaks structure in PES [7]. The structure was qualitatively reproduced on the basis of the impurity model by using the hybridization intensity (HI) calculated from the LDA band calculation [8]. T K was also estimated by the same HI, but it became very low, about 105 K because the DOS at E F is small. Higher T K , about 10 K, had been expected from the transport properties. In Fig. 1, we show the 4f DOS of CeSb at T ¼ 300 K. The solid line gives the total 4f PES. The dashed line is the DOS of the ðj ¼ 52ÞG7 component and the dot-dashed line is the DOS of the ðj ¼ 52ÞG8 component. The two-dots-dashed line is the DOS of the j ¼ 72 component. E F is indicated by the vertical dot-dashed line. The DMFT calculation is carried out in the following way. (I) At first we calculate self-consistent LDA band for CeSb by LMTO method. (II) Next the DMFT self-consistent calculation is done in the manifold of Bloch states obtained in the LDA calculation, i.e. the matrix elements of local self-energy between the Bloch states are calculated and the Greenian equation is
ARTICLE IN PRESS O. Sakai, Y. Shimizu / Journal of Magnetism and Magnetic Materials 310 (2007) 374–376
20
DOS (states/Ry)
DOS (states/Ry)
20
10
0 -0.10
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0.70
0.90
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Fig. 1. DOS of CeSb at T ¼ 300 K. For definition of lines, see the text. E F ¼ 0:3633 Ry, and the occupation number of 4f states is set to be 1.01. Electro static CFS is assumed to be E G8 E G7 ¼ 250 K.
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energy (Ry)
energy (Ry)
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log (E (Ry)) Fig. 2. The magnetic excitation spectra of CeSb at T ¼ 300 K (solid line) and at T ¼ 53 K (dashed line). The horizontal axis is the logarithm of energy in Ry, and the spectral intensities are normalized by the value in the low energy limit.
solved in this manifold of space. The detailed method will be explained in a separate paper [5]. When we do DMFT calculation by putting the energy levels of G7 and G8 to be equal, the occupation on G7 becomes small because the HI of the component is small. We tentatively put additive energy lowering of G7 component, 250 K, in the calculation. This may correspond to the electrostatic CFS of 4f states in CeSb. We have double peaks structure: the deep one is at 0.24 Ry (3.3 eV) and the shallow one is at 0.04 Ry (0.6 eV) below E F . The DOS of 4f at E F is not large, but has a finite value. These results are consistent with experimental facts. The f 1 ! f 2 peak in the BIS region has sharp structure because the multiplet splitting of f 2 state is tentatively neglected.
Fig. 3. DOS of CeP at T ¼ 300 K. For definition of lines, see the text. E F ¼ 0:4000 Ry, and the occupation number of 4f states is set to be 1.01. Electro static CFS is assumed to be E G8 E G7 ¼ 350 K.
As seen from the figure, the relative occupation of G7 to G8 (0.64) is larger than the ratio of the degeneracy factor 24. The effective CF energy is lower for G7 component at T ¼ 300 K. We show the magnetic excitation spectra in Fig. 2. The solid line gives spectrum at T ¼ 300 K. The peak at log E 3:3 has the excitation energy about 80 K, and corresponds to the G7 ! G8 transition. At the same time the quasi-elastic component appears indicating the effective T K of 5 K. When we do the calculation at T ¼ 53 K, the relative occupation of G7 decreases to 0.33. The inversion of CFS occurs. The magnetic transition spectrum has a shape of merged excitation of quasielastic and CFS one. The shoulder at about log E 3:7 (30 K) corresponds to the G8 ! G7 transition. These temperature dependence are caused by the appearance of the correlated 4f bands at low temperatures in DMFT calculation. In actual CeSb, the magnetic ordering of the G8 state takes place at T ¼ 17 K. The present calculation is carried out by assuming the paramagnetic state. DMFT band calculation can give energy scale of comparable magnitude to that of actual material, but a little higher. In Fig. 3, we show DOS of CeP at T ¼ 300 K. The intensity of shallow peak becomes large, but it has finite binding energy. These features well correspond to experimental results. The systematics of pnictogen dependence of PES spectra of Ce pnictides can be semi-quantitatively reproduced by non-empirical DMFT band calculation. The effective CFS changes with temperature. References [1] K. Held, et al., Proc. Winter School on Quantum Simulation of Complex Many-Body Systems: From Theory to Algorithms, February 25–March 1 (2002), Rolduc/Kerkrade (NL); cond-mat/0112079.
ARTICLE IN PRESS 376
O. Sakai, Y. Shimizu / Journal of Magnetism and Magnetic Materials 310 (2007) 374–376
[2] O. Sakai, et al., J. Phys. Soc. Japan 74 (2005) 2517. [3] J. Otsuki, Y. Kuramoto, J. Phys. Soc. Japan 75 (2006) 064707. [4] O. Sakai, et al., in: J. Kanamori, A. Kotani (Eds.), Core-Level Spectroscopy in Condensed Systems Theory, Springer, Berlin, 1988, p. 45. [5] O. Sakai, Y. Shimizu, J. Phys. Soc. Japan, submitted for publication.
[6] H. Takahashi, T. Kasuya, J. Phys. C 18 (1985) 2697, 2721, 2731, 2745, 2755. [7] W. Gudat, et al., in: P. Wachter, H. Boppart (Eds.), Valence Instabilities, North-Holland, Amsterdam, 1982, p. 249. [8] M. Takeshige, et al., J. Magn. Magn. Mater. 52 (1985) 363.
Journal of Magnetism and Magnetic Materials 310 (2007) 374–376 www.elsevier.com/locate/jmmm
DMFT band calculation for Ce compounds O. Sakaia,, Y. Shimizub a
Department of Physics, Tokyo Metropolitan University, Hachioji, Tokyo 192-0397, Japan b Department of Applied Physics, Tohoku University, Sendai 980-8579, Japan Available online 30 October 2006
Abstract Recent development of DMFT band calculation for Ce compounds is reported. The auxiliary impurity Anderson model is solved by a method named NCAf 2 vc, which can include the correct exchange process of f 1 ! f 0 ; f 2 fluctuation and also the crystalline field (CF) and spin–orbit splitting of self-energy. These are necessary features in the quantitative band theory for Ce compounds. The results of application on CeSb and CeP are presented. The double peaks structure in the photoemission spectra of these compounds is reproduced by the non-empirical calculation. The magnetic excitation spectra are also calculated. It is shown that the CF splitting energy is reversed in CeSb when the temperature decreases. r 2006 Elsevier B.V. All rights reserved. PACS: 71.15.m; 71.27.þa Keywords: DMFT; Band calculation; Ce-pnictides
Non-empirical band calculation for the strongly correlated electron systems has been extensively developed on the basis of the dynamical mean field theory (DMFT) [1]. In 4f electrons of Ce compounds the crystalline field splitting (CFS) and the spin–orbit interaction (SOI) play important roles [2]. A theory, which is named NCAf 2 vc has been recently developed to solve the auxiliary impurity Anderson model [3,4]. It is combined with the LMTO method to carry out the DMFT band calculation in the present study. The calculation can include the CFS and SOI effects and also the correct exchange process of f 1 ! f 0 ; f 2 valence fluctuation, and it can give accurate order of the Kondo temperature (T K ). We report the result of the DMFT band calculation for CeSb and CeP [5]. Ce pnictides have NaCl structure and has semimetal band structure: the p states of pnictogen form the valence band and the 5d states of Ce form the conduction band (see references in Ref. [2]). They overlap and supply a very small number of carriers. The top of the valence band has the G8 character and has very strong mixing matrix with the G8 component of Ce 4f electron, but the density of Corresponding author. Tel.: +81 22 217 6439; fax: +81 22 217 6447.
E-mail address: [email protected] (O. Sakai). 0304-8853/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2006.10.083
states (DOS) of bands at the Fermi energy ðE F Þ is small reflecting the semimetal nature of the band. The mixing between holes and the 4f states plays important roles in the p2f mixing model which explains various anomalous magnetic properties of these compounds [6]. Ce pnictides have characteristic double peaks structure in PES [7]. The structure was qualitatively reproduced on the basis of the impurity model by using the hybridization intensity (HI) calculated from the LDA band calculation [8]. T K was also estimated by the same HI, but it became very low, about 105 K because the DOS at E F is small. Higher T K , about 10 K, had been expected from the transport properties. In Fig. 1, we show the 4f DOS of CeSb at T ¼ 300 K. The solid line gives the total 4f PES. The dashed line is the DOS of the ðj ¼ 52ÞG7 component and the dot-dashed line is the DOS of the ðj ¼ 52ÞG8 component. The two-dots-dashed line is the DOS of the j ¼ 72 component. E F is indicated by the vertical dot-dashed line. The DMFT calculation is carried out in the following way. (I) At first we calculate self-consistent LDA band for CeSb by LMTO method. (II) Next the DMFT self-consistent calculation is done in the manifold of Bloch states obtained in the LDA calculation, i.e. the matrix elements of local self-energy between the Bloch states are calculated and the Greenian equation is
ARTICLE IN PRESS O. Sakai, Y. Shimizu / Journal of Magnetism and Magnetic Materials 310 (2007) 374–376
20
DOS (states/Ry)
DOS (states/Ry)
20
10
0 -0.10
0.10
0.30
0.50
0.70
0.90
1.10
10
0 -0.10
Fig. 1. DOS of CeSb at T ¼ 300 K. For definition of lines, see the text. E F ¼ 0:3633 Ry, and the occupation number of 4f states is set to be 1.01. Electro static CFS is assumed to be E G8 E G7 ¼ 250 K.
1.50
1.00
0.50
0.00 -5
-4
0.10
0.30
0.50
0.70
0.90
1.10
energy (Ry)
energy (Ry)
int. of mag. ex.
375
-3
-2
log (E (Ry)) Fig. 2. The magnetic excitation spectra of CeSb at T ¼ 300 K (solid line) and at T ¼ 53 K (dashed line). The horizontal axis is the logarithm of energy in Ry, and the spectral intensities are normalized by the value in the low energy limit.
solved in this manifold of space. The detailed method will be explained in a separate paper [5]. When we do DMFT calculation by putting the energy levels of G7 and G8 to be equal, the occupation on G7 becomes small because the HI of the component is small. We tentatively put additive energy lowering of G7 component, 250 K, in the calculation. This may correspond to the electrostatic CFS of 4f states in CeSb. We have double peaks structure: the deep one is at 0.24 Ry (3.3 eV) and the shallow one is at 0.04 Ry (0.6 eV) below E F . The DOS of 4f at E F is not large, but has a finite value. These results are consistent with experimental facts. The f 1 ! f 2 peak in the BIS region has sharp structure because the multiplet splitting of f 2 state is tentatively neglected.
Fig. 3. DOS of CeP at T ¼ 300 K. For definition of lines, see the text. E F ¼ 0:4000 Ry, and the occupation number of 4f states is set to be 1.01. Electro static CFS is assumed to be E G8 E G7 ¼ 350 K.
As seen from the figure, the relative occupation of G7 to G8 (0.64) is larger than the ratio of the degeneracy factor 24. The effective CF energy is lower for G7 component at T ¼ 300 K. We show the magnetic excitation spectra in Fig. 2. The solid line gives spectrum at T ¼ 300 K. The peak at log E 3:3 has the excitation energy about 80 K, and corresponds to the G7 ! G8 transition. At the same time the quasi-elastic component appears indicating the effective T K of 5 K. When we do the calculation at T ¼ 53 K, the relative occupation of G7 decreases to 0.33. The inversion of CFS occurs. The magnetic transition spectrum has a shape of merged excitation of quasielastic and CFS one. The shoulder at about log E 3:7 (30 K) corresponds to the G8 ! G7 transition. These temperature dependence are caused by the appearance of the correlated 4f bands at low temperatures in DMFT calculation. In actual CeSb, the magnetic ordering of the G8 state takes place at T ¼ 17 K. The present calculation is carried out by assuming the paramagnetic state. DMFT band calculation can give energy scale of comparable magnitude to that of actual material, but a little higher. In Fig. 3, we show DOS of CeP at T ¼ 300 K. The intensity of shallow peak becomes large, but it has finite binding energy. These features well correspond to experimental results. The systematics of pnictogen dependence of PES spectra of Ce pnictides can be semi-quantitatively reproduced by non-empirical DMFT band calculation. The effective CFS changes with temperature. References [1] K. Held, et al., Proc. Winter School on Quantum Simulation of Complex Many-Body Systems: From Theory to Algorithms, February 25–March 1 (2002), Rolduc/Kerkrade (NL); cond-mat/0112079.
ARTICLE IN PRESS 376
O. Sakai, Y. Shimizu / Journal of Magnetism and Magnetic Materials 310 (2007) 374–376
[2] O. Sakai, et al., J. Phys. Soc. Japan 74 (2005) 2517. [3] J. Otsuki, Y. Kuramoto, J. Phys. Soc. Japan 75 (2006) 064707. [4] O. Sakai, et al., in: J. Kanamori, A. Kotani (Eds.), Core-Level Spectroscopy in Condensed Systems Theory, Springer, Berlin, 1988, p. 45. [5] O. Sakai, Y. Shimizu, J. Phys. Soc. Japan, submitted for publication.
[6] H. Takahashi, T. Kasuya, J. Phys. C 18 (1985) 2697, 2721, 2731, 2745, 2755. [7] W. Gudat, et al., in: P. Wachter, H. Boppart (Eds.), Valence Instabilities, North-Holland, Amsterdam, 1982, p. 249. [8] M. Takeshige, et al., J. Magn. Magn. Mater. 52 (1985) 363.