- Email: [email protected]
Electron correlations in double exchange model with orbital degeneracy
Physica B 281&282 (2000) 536}537
Electron correlations in double exchange model with orbital degeneracy Yoshiki Imai, Seiya Kumada, Norio Kawakami* Department of Applied Physics, Osaka University, Suita, Osaka 565-0871, Japan
Abstract We study the e!ects of the Hubbard interaction among conduction electrons on the double exchange model with the orbital degeneracy. Employing a slave-boson approach to treat the correlation for conduction electrons and a coherentpotential approximation for the Hund coupling, we calculate the density of states and the optical conductivity. The Hund coupling splits the density of states into two parts, which are modi"ed by the band narrowing factor arising from the Hubbard interaction. Identifying the decrease of the magnetization for the localized spin with the increase of the temperature, we discuss the temperature dependence of the optical conductivity. ( 2000 Elsevier Science B.V. All rights reserved. Keywords: Double exchange model; Electron correlations; Orbital degeneracy
There has been a resurgence of interest in doped Perovskite manganites motivated by the recent discovery of the colossal magnetoresistance phenomena. These compounds show various phases depending sensitively on the concentration of doped holes. The ferromagnetic metallic phase is usually studied by employing a double exchange (DE) model [1,2]. The simplest version of the model takes into account the Hund coupling but neglects the Coulomb interaction among conduction electrons as well as the orbital degeneracy. Since the manganese oxide is a strongly correlated electron system with the orbital degeneracy, it is important to discuss the e!ects of the correlation for conduction electrons and the orbital degeneracy [3]. In this paper, we study the e!ects of the Hubbard interaction and the orbital degeneracy on the DE model. We consider the orbitally degenerate DE model with the correlation among conduction electrons, which is
* Corresponding author. Tel.: #81-6-6879-7863; fax: #816-6879-7863. E-mail address: [email protected] (N. Kawakami)
described by the Hamiltonian H" + tmm{ cs c !J + p ) S ij imp jm{p H im i Wi,jXmm{p im # ;+ + n n , imp im{p{ i (m,p)E(m{,p{) where cs (c ) is the creation (annihilation) operator of imp imp a conduction electron with the spin p("C, B) in the orbital m("1, 2) at the site i. The "rst term is the usual tight-binding model with the hopping matrix tmm{"t d , which is assumed to be diagonal in the ij ij m,m{ orbital index for simplicity. The second one describes the Hund coupling between the S"3 localized spin S and 2 i the spin of the conduction electron, p . The last one im represents the Hubbard interaction for conduction electrons. Let us "rst focus our attention on the correlation for conduction electrons by ignoring the Hund coupling for a while. We extend the slave-boson representation proposed by Kotliar and Ruckenstein [4,5], to include the orbital degeneracy. For simplicity, we assume that the density of states for conduction electrons has the semielliptic form N (e, =)"(2/p=2)J=2!e2 (DeD(=), 0 where = is the bandwidth. By exploiting a coherent-state
0921-4526/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 1 0 7 1 - 6
Y. Imai et al. / Physica B 281&282 (2000) 536}537
path integral formalism and then making a static approximation for Bose "elds, we obtain the e!ective density of states N (e, U =),N (e). Here U is the band narrow0 p p p ing factor due to electron correlations, which should be determined after incorporating the e!ect due to the Hund coupling. We further use a coherent-potential approximation (CPA) [6] to take account of the correlation e!ects arising from the localized spin S . i We have performed the self-consistent calculation numerically for the density of states and the optical conductivity [7]. In the above approximations, it is seen that the exchange interaction J splits the density of H states N (e) into two parts, each of which is modi"ed by p the band narrowing factor U . In the case of the nonp doped system (quarter "lling), only the lower-energy band is fully occupied, so that the doped holes are accommodated in this band. In Fig. 1, we show the conductivity p(u) calculated for di!erent Hubbard interactions ;. Two peaks around u/=&0 and u/=&3 originate from the intraband and interband excitations. We "nd that the width of these peaks becomes narrower with increasing ;, which is caused by the strong electron correlation due to the Hubbard interaction. In Fig. 2, we show the conductivity p(u) calculated for several choices of the magnetization M. If we consider the reduction of the magnetization M to be caused by the increase of the temperature ¹ [6], we can deduce the ¹-dependence of the conductivity p(u). For instance, the conductivity for M"3 and M"0 corresponds to that at ¹"0 and 2 ¹ (Curie temperature), respectively. With decreasing # ¹ (increasing M), the width of the Drude-like part (u/=&0) becomes narrower and its height becomes higher, re#ecting the fact that the decrease of the temperature makes the life time of the quasiparticle longer [2]. Here it should be noticed that the width of the Drude-like part in Figs. 1 and 2 may be overestimated because the imaginary part of the self-energy near the Fermi surface becomes larger due to the classical treatment for the local spin [6]. We can also see that the decrease of ¹ reduces the spectral weight of the interband excitation
537
Fig. 2. Optical conductivity p(u) calculated for various M with J "= and ;"2=. The concentration of holes is 17.5%. H
(u/=&3). It is found that the e!ect of the orbital degeneracy only brings about the uniform pre-factor &2 to the physical quantities within the present treatment. Therefore, the weight of the conductivity in the system with the orbital degeneracy is about twice as large as that in the non-degenerate system. This simple result partially comes from that we have used a simpli"ed model for conduction electrons for which the mutual interaction is independent of the orbital index. Concerning the e!ect of the orbital degeneracy, more detailed analysis is now in progress. In summary, we have studied how the correlation e!ect among conduction electrons a!ects the spectral properties of the double exchange model. In this paper, we have neglected the o!-diagonal components for the hopping matrix tmm{(mOm@), which may also play an ij important role in the manganese oxides [6,8]. We leave this problem for future investigation.
Acknowledgements We wish to thank N. Furukawa for valuable discussions.
References [1] [2] [3] [4] [5] [6] [7] [8] Fig. 1. Optical conductivity p(u) calculated for various ; and J "=. The concentration of holes is 17.5%. H
C. Zener, Phys. Rev. 82 (1951) 403. N. Furukawa, J. Phys. Soc. Japan 64 (1995) 3164. Y. Motome, M. Imada, cond-mat/9903183. G. Kotliar, A.E. Ruckenstein, Phys. Rev. Lett. 57 (1986) 1362. V. Dorin, P. Schlottman, Phys. Rev. B 47 (1993) 5095. P.E. Brito, H. Shiba, Phys. Rev. B 57 (1998) 1539. T. Pruschke, D.L. Cox, M. Jarrel, Phys. Rev. B 47 (1993) 3553. H. Shiba, R. Shiina, A. Takahashi, J. Phys. Soc. Japan 66 (1997) 941.
Electron correlations in double exchange model with orbital degeneracy Yoshiki Imai, Seiya Kumada, Norio Kawakami* Department of Applied Physics, Osaka University, Suita, Osaka 565-0871, Japan
Abstract We study the e!ects of the Hubbard interaction among conduction electrons on the double exchange model with the orbital degeneracy. Employing a slave-boson approach to treat the correlation for conduction electrons and a coherentpotential approximation for the Hund coupling, we calculate the density of states and the optical conductivity. The Hund coupling splits the density of states into two parts, which are modi"ed by the band narrowing factor arising from the Hubbard interaction. Identifying the decrease of the magnetization for the localized spin with the increase of the temperature, we discuss the temperature dependence of the optical conductivity. ( 2000 Elsevier Science B.V. All rights reserved. Keywords: Double exchange model; Electron correlations; Orbital degeneracy
There has been a resurgence of interest in doped Perovskite manganites motivated by the recent discovery of the colossal magnetoresistance phenomena. These compounds show various phases depending sensitively on the concentration of doped holes. The ferromagnetic metallic phase is usually studied by employing a double exchange (DE) model [1,2]. The simplest version of the model takes into account the Hund coupling but neglects the Coulomb interaction among conduction electrons as well as the orbital degeneracy. Since the manganese oxide is a strongly correlated electron system with the orbital degeneracy, it is important to discuss the e!ects of the correlation for conduction electrons and the orbital degeneracy [3]. In this paper, we study the e!ects of the Hubbard interaction and the orbital degeneracy on the DE model. We consider the orbitally degenerate DE model with the correlation among conduction electrons, which is
* Corresponding author. Tel.: #81-6-6879-7863; fax: #816-6879-7863. E-mail address: [email protected] (N. Kawakami)
described by the Hamiltonian H" + tmm{ cs c !J + p ) S ij imp jm{p H im i Wi,jXmm{p im # ;+ + n n , imp im{p{ i (m,p)E(m{,p{) where cs (c ) is the creation (annihilation) operator of imp imp a conduction electron with the spin p("C, B) in the orbital m("1, 2) at the site i. The "rst term is the usual tight-binding model with the hopping matrix tmm{"t d , which is assumed to be diagonal in the ij ij m,m{ orbital index for simplicity. The second one describes the Hund coupling between the S"3 localized spin S and 2 i the spin of the conduction electron, p . The last one im represents the Hubbard interaction for conduction electrons. Let us "rst focus our attention on the correlation for conduction electrons by ignoring the Hund coupling for a while. We extend the slave-boson representation proposed by Kotliar and Ruckenstein [4,5], to include the orbital degeneracy. For simplicity, we assume that the density of states for conduction electrons has the semielliptic form N (e, =)"(2/p=2)J=2!e2 (DeD(=), 0 where = is the bandwidth. By exploiting a coherent-state
0921-4526/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 1 0 7 1 - 6
Y. Imai et al. / Physica B 281&282 (2000) 536}537
path integral formalism and then making a static approximation for Bose "elds, we obtain the e!ective density of states N (e, U =),N (e). Here U is the band narrow0 p p p ing factor due to electron correlations, which should be determined after incorporating the e!ect due to the Hund coupling. We further use a coherent-potential approximation (CPA) [6] to take account of the correlation e!ects arising from the localized spin S . i We have performed the self-consistent calculation numerically for the density of states and the optical conductivity [7]. In the above approximations, it is seen that the exchange interaction J splits the density of H states N (e) into two parts, each of which is modi"ed by p the band narrowing factor U . In the case of the nonp doped system (quarter "lling), only the lower-energy band is fully occupied, so that the doped holes are accommodated in this band. In Fig. 1, we show the conductivity p(u) calculated for di!erent Hubbard interactions ;. Two peaks around u/=&0 and u/=&3 originate from the intraband and interband excitations. We "nd that the width of these peaks becomes narrower with increasing ;, which is caused by the strong electron correlation due to the Hubbard interaction. In Fig. 2, we show the conductivity p(u) calculated for several choices of the magnetization M. If we consider the reduction of the magnetization M to be caused by the increase of the temperature ¹ [6], we can deduce the ¹-dependence of the conductivity p(u). For instance, the conductivity for M"3 and M"0 corresponds to that at ¹"0 and 2 ¹ (Curie temperature), respectively. With decreasing # ¹ (increasing M), the width of the Drude-like part (u/=&0) becomes narrower and its height becomes higher, re#ecting the fact that the decrease of the temperature makes the life time of the quasiparticle longer [2]. Here it should be noticed that the width of the Drude-like part in Figs. 1 and 2 may be overestimated because the imaginary part of the self-energy near the Fermi surface becomes larger due to the classical treatment for the local spin [6]. We can also see that the decrease of ¹ reduces the spectral weight of the interband excitation
537
Fig. 2. Optical conductivity p(u) calculated for various M with J "= and ;"2=. The concentration of holes is 17.5%. H
(u/=&3). It is found that the e!ect of the orbital degeneracy only brings about the uniform pre-factor &2 to the physical quantities within the present treatment. Therefore, the weight of the conductivity in the system with the orbital degeneracy is about twice as large as that in the non-degenerate system. This simple result partially comes from that we have used a simpli"ed model for conduction electrons for which the mutual interaction is independent of the orbital index. Concerning the e!ect of the orbital degeneracy, more detailed analysis is now in progress. In summary, we have studied how the correlation e!ect among conduction electrons a!ects the spectral properties of the double exchange model. In this paper, we have neglected the o!-diagonal components for the hopping matrix tmm{(mOm@), which may also play an ij important role in the manganese oxides [6,8]. We leave this problem for future investigation.
Acknowledgements We wish to thank N. Furukawa for valuable discussions.
References [1] [2] [3] [4] [5] [6] [7] [8] Fig. 1. Optical conductivity p(u) calculated for various ; and J "=. The concentration of holes is 17.5%. H
C. Zener, Phys. Rev. 82 (1951) 403. N. Furukawa, J. Phys. Soc. Japan 64 (1995) 3164. Y. Motome, M. Imada, cond-mat/9903183. G. Kotliar, A.E. Ruckenstein, Phys. Rev. Lett. 57 (1986) 1362. V. Dorin, P. Schlottman, Phys. Rev. B 47 (1993) 5095. P.E. Brito, H. Shiba, Phys. Rev. B 57 (1998) 1539. T. Pruschke, D.L. Cox, M. Jarrel, Phys. Rev. B 47 (1993) 3553. H. Shiba, R. Shiina, A. Takahashi, J. Phys. Soc. Japan 66 (1997) 941.