- Email: [email protected]
Paramagnetic–ferromagnetic transition in a double-exchange model
ARTICLE IN PRESS
Physica B 378–380 (2006) 286–287 www.elsevier.com/locate/physb
Paramagnetic–ferromagnetic transition in a double-exchange model Eugene Kogana, Mark Auslenderb, a
Jack and Pearl Resnick Institute of Advanced Technology, Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel Department of Electrical and Computer Engineering, Ben-Gurion University of the Negev, P.O. Box 653, Beer-Sheva 84105, Israel
b
Abstract We study paramagnetic–ferromagnetic transition due to exchange interaction between classical localized spins on a lattice and conduction electrons. By solving the equations of dynamical mean field theory, we find explicit formula for Curie temperature at arbitrary Hund coupling, electron band structure and concentration. The results of numerical simulations of the Curie temperature for the semi-circular electron density of states model are presented. r 2006 Elsevier B.V. All rights reserved. PACS: 75.10.Hk; 75.30.Mb; 75.30.Vn Keywords: Strong correlations; Double-exchange model; Magnetism; Ferromagnetic order; Dynamical mean field theory
Double-exchange (DE) model [1] is a basic one in the theories of magnetism and strong electron correlations. Many-electron Hamiltonian of DE model is X X H¼ tll 0 cyla cl 0 a I Sl s^ ab cyla clb , (1) ll 0 a
lab
where c and cy are the fermionic annihilation and creation operators, Sl is the operator of a core spin, tll 0 is the hopping amplitude, I is the Hund exchange coupling between the core spin and electron, r^ is the vector of the Pauli matrices, and a; b are the spin indices. To study paramagnetic (PM) to ferromagnetic (FM) transition vs. electron concentration n and the model parameters we use the classical spin limit Sl ¼ Sml , ml being a unit vector, and a dynamical mean field theory (DMFT) [2]. DMFT universally approximates the averaged local Green’s function G^ loc by ^ (2) G^ loc ¼ hG^ ll ðEÞi ¼ g0 ðE SÞ, P where g0 ðEÞ ¼ N 1 k ðE tk Þ1 , tk is the Fourier trans^ is a local self-energy and the hat indicates form of tll 0 , S Corresponding author. Tel.: +972 8 6461583; fax: +972 8 6472949.
E-mail addresses: [email protected] (E. Kogan), [email protected] (M. Auslender). 0921-4526/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2006.01.009
^ 2 2 matrices in spin space. Specifically in our problem, S satisfies the following matrix equation: 1 ^ þ Jm rÞ ^ 1 i; G^ loc ¼ hðG^ loc þ S
J ¼ IS.
(3)
Here the brackets denote an average over the core spin orientations m at an unspecified site. The probability density PðmÞ for performing the average at a temperature T is calculated within DMFT by Z 1 f ðEÞDDðE; mÞ dE , (4) PðmÞ / exp T where f ðEÞ is the Fermi function and 1 ^ G^ loc DDðE; mÞ ¼ Im ln det½1 þ ðJmr^ þ SÞ (5) p is the electron density of states (DOS) change caused by the interaction with the core spins [3,4]. Eqs. (2)–(5) form the system of self-consistent non-linear ^ To show that FM ordering does equations for G^ loc and S. emerge, we assume finite, but small average spin M, solve Eqs. (2), (3) in the linear approximation with respect to M and substitute the results into Eq. (5). This yields a mean-field like distribution PðmÞ / expðT 1 AM mÞ with an effective exchange constant A expressed via the scalar PM Green’s function g and self-energy S, satisfying
ARTICLE IN PRESS E. Kogan, M. Auslender / Physica B 378–380 (2006) 286–287
the equations
3
1 þ gS ¼ 1. ð1 þ gSÞ2 ðJgÞ2
(6)
Then for ToT C ¼ A=3, the self-consistency condition M ¼ hmi predicts FM state (Ma0), and hence T C is the DMFT Curie temperature. After a lengthy algebra and taking into account the electron degeneracy with a Fermi energy m, we obtain 2 3 Z 6 7 2J 2 m g 7 dE, (7) TC ¼ Im6 4 2 0 3p 1 ðSg gÞð1 þ SgÞ 2J g5 g0 þ g2 3 where g0 ¼ g00 ðE SÞ and g00 ðEÞ ¼ ðd=dEÞg0 ðEÞ. Eqs. (6) and (7) allow one to calculate T C with any realistic, including ab initio, bare band structure. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The use of the semi-circular DOS 2=W 1 E 2 =W 2 , W being half the band width, simplifies Eq. (7) to Z J 2W 2 m TC ¼ Im 6p 1 2 3 6 6 6 4
W 2g E 4
7 g2 7 7 dE, W 2g J 2 W 2 g2 5 E 2 6
ð8Þ
where g, due to Eq. (6), satisfies a quartic equation. The results of computations using Eq. (8) are displayed in Figs. 1 and 2. Given J=W , there exist regions of n where T C o0, which we exclude (see Fig. 1). The curve T C ¼ 0 on the n J=W plane (Fig. 2) is the boundary between the FM states and the states, where the appearance of small spontaneous magnetization is unstable at any T. The FM states destabilize at J=W 41 the easier, the nearer the
0.05
0.04
Tc/W
0.03
2 FM
J/W
g ¼ g0 ðE SÞ;
287
1
NFM 0 0.2
0.4
0.6 n
0.8
1
Fig. 2. The curve T C ¼ 0 on the J=W n plane.
system is to the half filling. This result looks counterintuitive, but has clear physical explanation. In fact, at J ¼ 1 full spin disordering narrows the lower electron subband without shifting its center [5]. Hence, the disordering increases the electron energy for the partial sub-band filling, but does not make any difference when the subband is completely filled. At large but finite J, virtual transitions between the lower and upper sub-bands emerge, resulting in an indirect antiferromagnetic (AFM) exchange, maximal for n ¼ 1 [4]. The smaller the J, the more the disordering shifts the center of the lower sub-band. All this explains why, in the right part of the phase diagram, the FM states loose competition to the PM ones that means, in particular, nonferromagnetic (NFM) ground states [3]. With the decrease of temperature the PM states may become unstable in other channels, like transition to the AFM, spin density wave or phase separated states, but treatment of such instabilities is beyond the limits of our approximations. To conclude, using DMFT we derived the closed formula for T C in the DE model at arbitrary J, electron band structure and n. With the semi-circular electron DOS, we obtained n J=W magnetic phase diagram. References
0.02
0.01
0 0
0.2
0.4
0.6
0.8
1
n Fig. 1. T C as a function of electron concentration n: J=W ¼ 0:25 (dotted line), J=W ¼ 1 (dash-dotted line), J=W ¼ 2 (dashed line), and J=W ¼ 20 (solid line).
[1] C. Zener, Phys. Rev. 82 (1951) 403; P.W. Anderson, H. Hasegawa, Phys. Rev. 100 (1955) 675. [2] A. Georges, G. Kotliar, W. Krauth, M.J. Rozenberg, Rev. Mod. Phys. 68 (1996) 13. [3] A. Chattopadhyay, A.J. Millis, S. Das Sarma, Phys. Rev. B 64 (2001) 012416; A. Chattopadhyay, A.J. Millis, Phys. Rev. B 64 (2001) 024424. [4] E. Kogan, M. Auslender, Phys. Rev. B 67 (2003) 132410; M. Auslender, E. Kogan, Europhys. Lett. 59 (2002) 277. [5] M. Auslender, E. Kogan, Phys. Rev. B 65 (2002) 012408; M. Auslender, E. Kogan, Physica A 302 (2001) 345.
Physica B 378–380 (2006) 286–287 www.elsevier.com/locate/physb
Paramagnetic–ferromagnetic transition in a double-exchange model Eugene Kogana, Mark Auslenderb, a
Jack and Pearl Resnick Institute of Advanced Technology, Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel Department of Electrical and Computer Engineering, Ben-Gurion University of the Negev, P.O. Box 653, Beer-Sheva 84105, Israel
b
Abstract We study paramagnetic–ferromagnetic transition due to exchange interaction between classical localized spins on a lattice and conduction electrons. By solving the equations of dynamical mean field theory, we find explicit formula for Curie temperature at arbitrary Hund coupling, electron band structure and concentration. The results of numerical simulations of the Curie temperature for the semi-circular electron density of states model are presented. r 2006 Elsevier B.V. All rights reserved. PACS: 75.10.Hk; 75.30.Mb; 75.30.Vn Keywords: Strong correlations; Double-exchange model; Magnetism; Ferromagnetic order; Dynamical mean field theory
Double-exchange (DE) model [1] is a basic one in the theories of magnetism and strong electron correlations. Many-electron Hamiltonian of DE model is X X H¼ tll 0 cyla cl 0 a I Sl s^ ab cyla clb , (1) ll 0 a
lab
where c and cy are the fermionic annihilation and creation operators, Sl is the operator of a core spin, tll 0 is the hopping amplitude, I is the Hund exchange coupling between the core spin and electron, r^ is the vector of the Pauli matrices, and a; b are the spin indices. To study paramagnetic (PM) to ferromagnetic (FM) transition vs. electron concentration n and the model parameters we use the classical spin limit Sl ¼ Sml , ml being a unit vector, and a dynamical mean field theory (DMFT) [2]. DMFT universally approximates the averaged local Green’s function G^ loc by ^ (2) G^ loc ¼ hG^ ll ðEÞi ¼ g0 ðE SÞ, P where g0 ðEÞ ¼ N 1 k ðE tk Þ1 , tk is the Fourier trans^ is a local self-energy and the hat indicates form of tll 0 , S Corresponding author. Tel.: +972 8 6461583; fax: +972 8 6472949.
E-mail addresses: [email protected] (E. Kogan), [email protected] (M. Auslender). 0921-4526/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2006.01.009
^ 2 2 matrices in spin space. Specifically in our problem, S satisfies the following matrix equation: 1 ^ þ Jm rÞ ^ 1 i; G^ loc ¼ hðG^ loc þ S
J ¼ IS.
(3)
Here the brackets denote an average over the core spin orientations m at an unspecified site. The probability density PðmÞ for performing the average at a temperature T is calculated within DMFT by Z 1 f ðEÞDDðE; mÞ dE , (4) PðmÞ / exp T where f ðEÞ is the Fermi function and 1 ^ G^ loc DDðE; mÞ ¼ Im ln det½1 þ ðJmr^ þ SÞ (5) p is the electron density of states (DOS) change caused by the interaction with the core spins [3,4]. Eqs. (2)–(5) form the system of self-consistent non-linear ^ To show that FM ordering does equations for G^ loc and S. emerge, we assume finite, but small average spin M, solve Eqs. (2), (3) in the linear approximation with respect to M and substitute the results into Eq. (5). This yields a mean-field like distribution PðmÞ / expðT 1 AM mÞ with an effective exchange constant A expressed via the scalar PM Green’s function g and self-energy S, satisfying
ARTICLE IN PRESS E. Kogan, M. Auslender / Physica B 378–380 (2006) 286–287
the equations
3
1 þ gS ¼ 1. ð1 þ gSÞ2 ðJgÞ2
(6)
Then for ToT C ¼ A=3, the self-consistency condition M ¼ hmi predicts FM state (Ma0), and hence T C is the DMFT Curie temperature. After a lengthy algebra and taking into account the electron degeneracy with a Fermi energy m, we obtain 2 3 Z 6 7 2J 2 m g 7 dE, (7) TC ¼ Im6 4 2 0 3p 1 ðSg gÞð1 þ SgÞ 2J g5 g0 þ g2 3 where g0 ¼ g00 ðE SÞ and g00 ðEÞ ¼ ðd=dEÞg0 ðEÞ. Eqs. (6) and (7) allow one to calculate T C with any realistic, including ab initio, bare band structure. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The use of the semi-circular DOS 2=W 1 E 2 =W 2 , W being half the band width, simplifies Eq. (7) to Z J 2W 2 m TC ¼ Im 6p 1 2 3 6 6 6 4
W 2g E 4
7 g2 7 7 dE, W 2g J 2 W 2 g2 5 E 2 6
ð8Þ
where g, due to Eq. (6), satisfies a quartic equation. The results of computations using Eq. (8) are displayed in Figs. 1 and 2. Given J=W , there exist regions of n where T C o0, which we exclude (see Fig. 1). The curve T C ¼ 0 on the n J=W plane (Fig. 2) is the boundary between the FM states and the states, where the appearance of small spontaneous magnetization is unstable at any T. The FM states destabilize at J=W 41 the easier, the nearer the
0.05
0.04
Tc/W
0.03
2 FM
J/W
g ¼ g0 ðE SÞ;
287
1
NFM 0 0.2
0.4
0.6 n
0.8
1
Fig. 2. The curve T C ¼ 0 on the J=W n plane.
system is to the half filling. This result looks counterintuitive, but has clear physical explanation. In fact, at J ¼ 1 full spin disordering narrows the lower electron subband without shifting its center [5]. Hence, the disordering increases the electron energy for the partial sub-band filling, but does not make any difference when the subband is completely filled. At large but finite J, virtual transitions between the lower and upper sub-bands emerge, resulting in an indirect antiferromagnetic (AFM) exchange, maximal for n ¼ 1 [4]. The smaller the J, the more the disordering shifts the center of the lower sub-band. All this explains why, in the right part of the phase diagram, the FM states loose competition to the PM ones that means, in particular, nonferromagnetic (NFM) ground states [3]. With the decrease of temperature the PM states may become unstable in other channels, like transition to the AFM, spin density wave or phase separated states, but treatment of such instabilities is beyond the limits of our approximations. To conclude, using DMFT we derived the closed formula for T C in the DE model at arbitrary J, electron band structure and n. With the semi-circular electron DOS, we obtained n J=W magnetic phase diagram. References
0.02
0.01
0 0
0.2
0.4
0.6
0.8
1
n Fig. 1. T C as a function of electron concentration n: J=W ¼ 0:25 (dotted line), J=W ¼ 1 (dash-dotted line), J=W ¼ 2 (dashed line), and J=W ¼ 20 (solid line).
[1] C. Zener, Phys. Rev. 82 (1951) 403; P.W. Anderson, H. Hasegawa, Phys. Rev. 100 (1955) 675. [2] A. Georges, G. Kotliar, W. Krauth, M.J. Rozenberg, Rev. Mod. Phys. 68 (1996) 13. [3] A. Chattopadhyay, A.J. Millis, S. Das Sarma, Phys. Rev. B 64 (2001) 012416; A. Chattopadhyay, A.J. Millis, Phys. Rev. B 64 (2001) 024424. [4] E. Kogan, M. Auslender, Phys. Rev. B 67 (2003) 132410; M. Auslender, E. Kogan, Europhys. Lett. 59 (2002) 277. [5] M. Auslender, E. Kogan, Phys. Rev. B 65 (2002) 012408; M. Auslender, E. Kogan, Physica A 302 (2001) 345.