- Email: [email protected]
Temperature dependence of the optical spectral weights in LaMnO3
ARTICLE IN PRESS
Physica B 359–361 (2005) 1288–1290 www.elsevier.com/locate/physb
Temperature dependence of the optical spectral weights in LaMnO3 $ Andrzej M. Oles´ a,b,, Peter Horschb, Giniyat Khaliullinb a
Marian Smoluchowski Institute of Physics, Jagellonian University, Reymonta 4, PL-30059 Krako´w, Poland b Max-Planck-Institut fu¨r Festko¨rperforschung, Heisenbergstrasse 1, D-70569 Stuttgart, Germany
Abstract We outline a unified approach to magnetic and optical properties of strongly correlated transition metal oxides, and analyze the optical spectral weights for the Mott insulator LaMnO3 : We demonstrate that spin and orbital correlations are responsible for the observed temperature variation and large anisotropy of the optical intensities. r 2005 Elsevier B.V. All rights reserved. PACS: 75.47.Lx; 75.30.Et; 78.20.e Keywords: Superexchange; Spin–orbital model; Optical spectral weight
Recently, pronounced anisotropy of the lowenergy optical absorption was reported for LaMnO3 [1,2], both for the A-type antiferromagnetic (AF) phase below the Ne´el temperature T N ; with ferromagnetic (FM) ða; bÞ planes and AF order along the c-axis, and for the orbital ordered phase at T4T N : Here we show that partial sum rules for high-spin and low-spin excitations [3] explain well these experimental observations.
In a Mott insulator LaMnO3 charge dynamics is quenched due to large intraorbital Coulomb interaction U at Mn3þ ions, and virtual charge excitations d 4i d 4j Ð d 5i d 3j along each bond hiji are responsible for superexchange interactions between S ¼ 2 spins [4]. Due to the multiplet splittings of Mn2þ (d5 ) ions given by Hund’s exchange J H ; the superexchange involves several excited states (upper Hubbard bands) labeled by n, and has a generic form XX H ðgÞ (1) HJ ¼ J n ðijÞ, n
$
This work was supported by the Polish State Committee of Scientific Research (KBN) under Project no. 1 P03B 068 26. Corresponding author. Tel.: +48 12 632 4888; fax: +48 12 633 4079. E-mail address: [email protected] (A.M. Oles´ ).
hijikg
where the terms H ðgÞ n ðijÞ contribute for each bond hiji along a cubic axis g ¼ a; b; c: The superexchange J ¼ 4t2 =U is determined by an effective
0921-4526/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2005.01.359
ARTICLE IN PRESS A.M. Oles´ et al. / Physica B 359– 361 (2005) 1288–1290
Mn3þ –Mn3þ hopping on a s-bond, t ¼ t2pd =D: Here tpd is the manganese–oxygen hopping in the charge transfer model with a gap D [5]. In a Mott insulator H ðgÞ n ðijÞ determine the kinetic energy per bond K ðgÞ n via the Hellman– Feynman theorem [6]: ðgÞ K ðgÞ n ¼ 2hH n ðijÞi: Further, via the optical sum rule, the kineticRenergy K ðgÞ n determines the optical 1 spectral weight 0 sðgÞ n ðoÞ do for each transition n, and the following relation between the so-called effective carrier number N ðgÞ eff;n [1] and the superexchange energy holds [3], m0 a20 ðgÞ 2m0 a20 K ¼ hH ðgÞ (2) n n ðijÞi, _2 _2 with a coefficient ðm0 a20 =_2 Þ ’ 2 eV1 : Here, a0 is the nearest-neighbor Mn–Mn distance (the tightbinding model is implied), and m0 is the electron mass. For LaMnO3 H J includes the orbital degrees of freedom of eg electrons, and consists of the FM (n ¼ 1) and AF (n41) terms [4]: N ðgÞ eff;n ¼
ðgÞ ðgÞ 1 1 ~ ~ H ðgÞ 1 ¼ 20 r1 ðS i S j þ 6Þð4 ti tj Þ,
H ðgÞ 2 H ðgÞ 3 H ðgÞ 4 H ðgÞ 5
¼ ¼ ¼
3 ~ 160 r2 ðS i
ðgÞ ~j S tðgÞ i tj Þ, ðgÞ ðgÞ 1 ~ ~ 64 r3 ðS i S j 4Þð1 ti tj Þ, ðgÞ 1 ðgÞ 1 1 ~ ~ 32 r4 ðS i S j 4Þð2 ti Þð2 tj Þ,
4Þð14
~i S ~j 4Þ, ¼ 18 bQðS
ð3Þ ð4Þ ð5Þ ð6Þ ð7Þ
where the coefficients: r1 ¼ 1=ð1 3ZÞ; r2 ¼ 1=ð1 þ 2ZÞ; r3 ¼ 1=ð1 þ 8Z=3Þ and r4 ¼ 1=ð1 þ 16Z=3Þ; with Z ¼ J H =U; follow from the multiplet structure of d5 ions. Two spin operators are involved: 1 ~ ~ (i) the projection 10 ðS i S j þ 6Þ which selects the high-spin part of the low-energy excitation to 6 A1 state at energy U 3J H ; and (ii) the projection 1 ~ ~ 8 ðS i S j 4Þ which suppresses all AF terms when ~i S ~j i ¼ 4: the spin configuration is FM, and hS ðcÞ 1 z The pseudospin p orbital operators, t ¼ 2 s ; and ffiffiffi tða;bÞ ¼ 14 ðsz 3sx Þ; where sz ; sx are Pauli matrices, which refer to the fjx2 y2 i; j3z2 r2 ig orbital basis. The last AF term H ðgÞ 5 (7) follows from t2g charge excitations, and is frequently called core spin superexchange. It also follows from the charge transfer model, and by averaging P over possible excitations one finds Q ¼ 14 i qi ; with: q1 ¼ 1=ð1 þ 8Z=3Þ; q2 ¼ 1=ð1 þ 4ZÞ; q3 ¼ 1=ð1 þ 14Z=3Þ; and q4 ¼ 1=ð1 þ 6ZÞ: The coefficient
1289
b ¼ 19 stands for the squared ratio of the hopping elements for p and s Mn–Mn bonds. Below the structural transition at T s ¼ 780 K; the orbital order with jy i ¼ cosðy=2Þj3z2 r2 i sinðy=2Þjx2 y2 i orbitals, alternating between two sublattices A and B in ða; bÞ planes, is stabilized by a superposition of eg superexchange (3)–(6) and the orbital interactions due to the JT effect [4]. Hence, (unlike in LaVO3 [3]) the orbital dynamics is quenched, and it suffices to consider intersite ~i S ~j ig at increasing temspin correlations sg ¼ hS perature T. We derived them by employing the socalled Oguchi method [7], solving exactly a single bond hiji; while the interactions with neighboring spins were treated within the mean-field approximation, with the order parameter hSz i given by a Brillouin function. Spin correlations in the FM planes (sab ) could be found analytically, while the AF correlations (sc ) were determined by a numerical solution. Both correlation functions change fast close to T N ; and remain finite at TbT N [see Fig. 1(a)]. We used the spectroscopic parameters of the charge transfer model [5]: tpd ¼ 1:5 eV; D ¼ 4:9 eV; J H ¼ 0:69 eV; and fixed the value of U ’ 5:0 eV: We found the exchange constants J ab ¼ 1:66 meV and J c ¼ 1:17 meV; so the values 1:66 meV and 1:16 meV extracted from the neutron experiments [8] were well reproduced. They lead to T N ’ 147 K; reduced from the meanfield result for S ¼ 2 spins by a factor 0:71 [9]. The theory predicts that at T ¼ 0 only lowenergy high-spin optical excitations are allowed for the FM bonds in ða; bÞ planes, and one finds N ðabÞ eff;1 ’ 0:25 [Fig. 1(b)]. In contrast, the optical excitations are mainly low-spin for the AF bonds along c axis, and N ðcÞ eff;1 is very small, resulting in a large anisotropy 10: 1 of the low-energy optical intensities. This anisotropy decreases with increasing T [Fig. 1(b)], but even at T ¼ 300 K, with the calculated effective carrier densities: N ðabÞ eff;1 ’ 0:17 and N ðcÞ ’ 0:07; it exceeds 2:1 due to the eff;1 persisting orbital order. These results reproduce well the behavior of the experimental intensities in the low-energy regime [2]. Note, however, that the total optical intensities, shown also in Fig. 1(b), have a much weaker temperature dependence and anisotropy.
ARTICLE IN PRESS A.M. Oles´ et al. / Physica B 359– 361 (2005) 1288–1290
1290
Summarizing, we demonstrated that the spin interactions, the magnetic transition temperature T N itself, and the optical spectra for LaMnO3 are well understood within the correlated superexchange model. This demonstrates an intimate link between the magnetic and optical properties of Mott insulators.
γ ,
4.0 a
2.0 0.0 −2.0
c TN
−4.0 0
(a)
100
200
300
References
300
[1] K. Tobe, et al., Phys. Rev. B 64 (2001) 184421. [2] N.N. Kovaleva, et al., Phys. Rev. Lett. 93 (2004) 147204. [3] G. Khaliullin, P. Horsch, A.M. Oles´ , Phys. Rev. B 70 (2004) 195103. [4] L.F. Feiner, A.M. Oles´ , Phys. Rev. B 59 (1999) 3295. [5] T. Mizokawa, A. Fujimori, Phys. Rev. B 54 (1996) 5368. [6] M. Aichhorn, et al., Phys. Rev. B 65 (2002) 201101 (R). [7] T. Oguchi, Progr. Theor. Phys. 13 (1955) 148. [8] F. Moussa, et al., Phys. Rev. B 54 (1996) 15149; G. Biotteau, et al., Phys. Rev. B 64 (2001) 104421. [9] G.S. Rushbrooke, P.J. Wood, Mol. Phys. 1 (1958) 257; see also: M. Fleck, et al., Eur. Phys. J. B 37 (2004) 439.
0.3 a
Neff(γ)
0.2
0.1 c 0.0 (b)
0
100
200 T (K)
~i Fig. 1. Temperature dependence of: (a) spin correlations hS ~j ig for g ¼ a; c; order parameter hSz i (long-dashed line), and S (b) effective carrier numbers N geff ; standing for optical intensities (2) along two cubic axes: a and c (solid and dashed lines). Bottom (heavy) and top (thin) lines in (b) show lowenergy (n ¼ 1) and total intensities. Parameters: Z ’ 0:12; J ¼ 170 meV; y ¼ p=2 (this orbital order is representative —similar results were also found for larger values of y).
Physica B 359–361 (2005) 1288–1290 www.elsevier.com/locate/physb
Temperature dependence of the optical spectral weights in LaMnO3 $ Andrzej M. Oles´ a,b,, Peter Horschb, Giniyat Khaliullinb a
Marian Smoluchowski Institute of Physics, Jagellonian University, Reymonta 4, PL-30059 Krako´w, Poland b Max-Planck-Institut fu¨r Festko¨rperforschung, Heisenbergstrasse 1, D-70569 Stuttgart, Germany
Abstract We outline a unified approach to magnetic and optical properties of strongly correlated transition metal oxides, and analyze the optical spectral weights for the Mott insulator LaMnO3 : We demonstrate that spin and orbital correlations are responsible for the observed temperature variation and large anisotropy of the optical intensities. r 2005 Elsevier B.V. All rights reserved. PACS: 75.47.Lx; 75.30.Et; 78.20.e Keywords: Superexchange; Spin–orbital model; Optical spectral weight
Recently, pronounced anisotropy of the lowenergy optical absorption was reported for LaMnO3 [1,2], both for the A-type antiferromagnetic (AF) phase below the Ne´el temperature T N ; with ferromagnetic (FM) ða; bÞ planes and AF order along the c-axis, and for the orbital ordered phase at T4T N : Here we show that partial sum rules for high-spin and low-spin excitations [3] explain well these experimental observations.
In a Mott insulator LaMnO3 charge dynamics is quenched due to large intraorbital Coulomb interaction U at Mn3þ ions, and virtual charge excitations d 4i d 4j Ð d 5i d 3j along each bond hiji are responsible for superexchange interactions between S ¼ 2 spins [4]. Due to the multiplet splittings of Mn2þ (d5 ) ions given by Hund’s exchange J H ; the superexchange involves several excited states (upper Hubbard bands) labeled by n, and has a generic form XX H ðgÞ (1) HJ ¼ J n ðijÞ, n
$
This work was supported by the Polish State Committee of Scientific Research (KBN) under Project no. 1 P03B 068 26. Corresponding author. Tel.: +48 12 632 4888; fax: +48 12 633 4079. E-mail address: [email protected] (A.M. Oles´ ).
hijikg
where the terms H ðgÞ n ðijÞ contribute for each bond hiji along a cubic axis g ¼ a; b; c: The superexchange J ¼ 4t2 =U is determined by an effective
0921-4526/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2005.01.359
ARTICLE IN PRESS A.M. Oles´ et al. / Physica B 359– 361 (2005) 1288–1290
Mn3þ –Mn3þ hopping on a s-bond, t ¼ t2pd =D: Here tpd is the manganese–oxygen hopping in the charge transfer model with a gap D [5]. In a Mott insulator H ðgÞ n ðijÞ determine the kinetic energy per bond K ðgÞ n via the Hellman– Feynman theorem [6]: ðgÞ K ðgÞ n ¼ 2hH n ðijÞi: Further, via the optical sum rule, the kineticRenergy K ðgÞ n determines the optical 1 spectral weight 0 sðgÞ n ðoÞ do for each transition n, and the following relation between the so-called effective carrier number N ðgÞ eff;n [1] and the superexchange energy holds [3], m0 a20 ðgÞ 2m0 a20 K ¼ hH ðgÞ (2) n n ðijÞi, _2 _2 with a coefficient ðm0 a20 =_2 Þ ’ 2 eV1 : Here, a0 is the nearest-neighbor Mn–Mn distance (the tightbinding model is implied), and m0 is the electron mass. For LaMnO3 H J includes the orbital degrees of freedom of eg electrons, and consists of the FM (n ¼ 1) and AF (n41) terms [4]: N ðgÞ eff;n ¼
ðgÞ ðgÞ 1 1 ~ ~ H ðgÞ 1 ¼ 20 r1 ðS i S j þ 6Þð4 ti tj Þ,
H ðgÞ 2 H ðgÞ 3 H ðgÞ 4 H ðgÞ 5
¼ ¼ ¼
3 ~ 160 r2 ðS i
ðgÞ ~j S tðgÞ i tj Þ, ðgÞ ðgÞ 1 ~ ~ 64 r3 ðS i S j 4Þð1 ti tj Þ, ðgÞ 1 ðgÞ 1 1 ~ ~ 32 r4 ðS i S j 4Þð2 ti Þð2 tj Þ,
4Þð14
~i S ~j 4Þ, ¼ 18 bQðS
ð3Þ ð4Þ ð5Þ ð6Þ ð7Þ
where the coefficients: r1 ¼ 1=ð1 3ZÞ; r2 ¼ 1=ð1 þ 2ZÞ; r3 ¼ 1=ð1 þ 8Z=3Þ and r4 ¼ 1=ð1 þ 16Z=3Þ; with Z ¼ J H =U; follow from the multiplet structure of d5 ions. Two spin operators are involved: 1 ~ ~ (i) the projection 10 ðS i S j þ 6Þ which selects the high-spin part of the low-energy excitation to 6 A1 state at energy U 3J H ; and (ii) the projection 1 ~ ~ 8 ðS i S j 4Þ which suppresses all AF terms when ~i S ~j i ¼ 4: the spin configuration is FM, and hS ðcÞ 1 z The pseudospin p orbital operators, t ¼ 2 s ; and ffiffiffi tða;bÞ ¼ 14 ðsz 3sx Þ; where sz ; sx are Pauli matrices, which refer to the fjx2 y2 i; j3z2 r2 ig orbital basis. The last AF term H ðgÞ 5 (7) follows from t2g charge excitations, and is frequently called core spin superexchange. It also follows from the charge transfer model, and by averaging P over possible excitations one finds Q ¼ 14 i qi ; with: q1 ¼ 1=ð1 þ 8Z=3Þ; q2 ¼ 1=ð1 þ 4ZÞ; q3 ¼ 1=ð1 þ 14Z=3Þ; and q4 ¼ 1=ð1 þ 6ZÞ: The coefficient
1289
b ¼ 19 stands for the squared ratio of the hopping elements for p and s Mn–Mn bonds. Below the structural transition at T s ¼ 780 K; the orbital order with jy i ¼ cosðy=2Þj3z2 r2 i sinðy=2Þjx2 y2 i orbitals, alternating between two sublattices A and B in ða; bÞ planes, is stabilized by a superposition of eg superexchange (3)–(6) and the orbital interactions due to the JT effect [4]. Hence, (unlike in LaVO3 [3]) the orbital dynamics is quenched, and it suffices to consider intersite ~i S ~j ig at increasing temspin correlations sg ¼ hS perature T. We derived them by employing the socalled Oguchi method [7], solving exactly a single bond hiji; while the interactions with neighboring spins were treated within the mean-field approximation, with the order parameter hSz i given by a Brillouin function. Spin correlations in the FM planes (sab ) could be found analytically, while the AF correlations (sc ) were determined by a numerical solution. Both correlation functions change fast close to T N ; and remain finite at TbT N [see Fig. 1(a)]. We used the spectroscopic parameters of the charge transfer model [5]: tpd ¼ 1:5 eV; D ¼ 4:9 eV; J H ¼ 0:69 eV; and fixed the value of U ’ 5:0 eV: We found the exchange constants J ab ¼ 1:66 meV and J c ¼ 1:17 meV; so the values 1:66 meV and 1:16 meV extracted from the neutron experiments [8] were well reproduced. They lead to T N ’ 147 K; reduced from the meanfield result for S ¼ 2 spins by a factor 0:71 [9]. The theory predicts that at T ¼ 0 only lowenergy high-spin optical excitations are allowed for the FM bonds in ða; bÞ planes, and one finds N ðabÞ eff;1 ’ 0:25 [Fig. 1(b)]. In contrast, the optical excitations are mainly low-spin for the AF bonds along c axis, and N ðcÞ eff;1 is very small, resulting in a large anisotropy 10: 1 of the low-energy optical intensities. This anisotropy decreases with increasing T [Fig. 1(b)], but even at T ¼ 300 K, with the calculated effective carrier densities: N ðabÞ eff;1 ’ 0:17 and N ðcÞ ’ 0:07; it exceeds 2:1 due to the eff;1 persisting orbital order. These results reproduce well the behavior of the experimental intensities in the low-energy regime [2]. Note, however, that the total optical intensities, shown also in Fig. 1(b), have a much weaker temperature dependence and anisotropy.
ARTICLE IN PRESS A.M. Oles´ et al. / Physica B 359– 361 (2005) 1288–1290
1290
Summarizing, we demonstrated that the spin interactions, the magnetic transition temperature T N itself, and the optical spectra for LaMnO3 are well understood within the correlated superexchange model. This demonstrates an intimate link between the magnetic and optical properties of Mott insulators.
4.0 a
2.0 0.0 −2.0
c TN
−4.0 0
(a)
100
200
300
References
300
[1] K. Tobe, et al., Phys. Rev. B 64 (2001) 184421. [2] N.N. Kovaleva, et al., Phys. Rev. Lett. 93 (2004) 147204. [3] G. Khaliullin, P. Horsch, A.M. Oles´ , Phys. Rev. B 70 (2004) 195103. [4] L.F. Feiner, A.M. Oles´ , Phys. Rev. B 59 (1999) 3295. [5] T. Mizokawa, A. Fujimori, Phys. Rev. B 54 (1996) 5368. [6] M. Aichhorn, et al., Phys. Rev. B 65 (2002) 201101 (R). [7] T. Oguchi, Progr. Theor. Phys. 13 (1955) 148. [8] F. Moussa, et al., Phys. Rev. B 54 (1996) 15149; G. Biotteau, et al., Phys. Rev. B 64 (2001) 104421. [9] G.S. Rushbrooke, P.J. Wood, Mol. Phys. 1 (1958) 257; see also: M. Fleck, et al., Eur. Phys. J. B 37 (2004) 439.
0.3 a
Neff(γ)
0.2
0.1 c 0.0 (b)
0
100
200 T (K)
~i Fig. 1. Temperature dependence of: (a) spin correlations hS ~j ig for g ¼ a; c; order parameter hSz i (long-dashed line), and S (b) effective carrier numbers N geff ; standing for optical intensities (2) along two cubic axes: a and c (solid and dashed lines). Bottom (heavy) and top (thin) lines in (b) show lowenergy (n ¼ 1) and total intensities. Parameters: Z ’ 0:12; J ¼ 170 meV; y ¼ p=2 (this orbital order is representative —similar results were also found for larger values of y).