- Email: [email protected]
Orbital polarons in LaMnO3
Physica B 312–313 (2002) 740–742
Orbital polarons in LaMnO3 Jan Ba"aa, Andrzej M. Oles!a,b,*, Peter Horschb b
a ! Poland Institute of Physics, Jagellonian University, Reymonta 4, PL-30059 Krakow, Max-Planck-Institut fur Heisenbergstrasse 1, D-70569 Stuttgart, Germany . Festkorperforschung, .
Abstract We investigate the spectral functions and orbital correlation functions relative to the position of an eg hole moving in the ferromagnetic plane of orbitally ordered LaMnO3 : Polarization of orbitals around the hole leads to a very narrow quasiparticle band and to a confinement of the hole in an orbital polaron with the shape strongly depending on the type of the orbital order. r 2002 Elsevier Science B.V. All rights reserved. PACS: 75.30.Vn; 75.30.Et; 79.60.i Keywords: Orbital polaron; Quasiparticle; Manganites
The orbital degrees of freedom are known to play a central role in the colossal magnetoresistance manganites. The undoped LaMnO3 is a Mott–Hubbard insulator, with an A-type antiferromagnetic (AF) order which consists of ferromagnetic (FM) ða; bÞ planes staggered along the c-axis, and coexists with orbital order with alternating occupied eg orbitals within the ða; bÞ planes, jimS ¼ cosðp=4 fÞjizS7sinðp=4 fÞjixS; where jx2 y2 S jmS and j3z2 r2 S jkS are the components of T ¼ 12 pseudospin. This ground state is well explained and follows from the cooperative Jahn–Teller effect and from the superexchange between the Mn3þ –Mn3þ pairs promoted by the excitations of high-spin configurations pJ ¼ t2 =ðU 3JH Þ [1]. With the on-site Coulomb and exchange elements: UC5:9 eV and JH C0:69 eV; and tC0:41 eV [2], one finds JC44 meV; i.e., J=tC0:1: We investigate the propagation of a single hole in LaMnO3 by considering the orbital t–J model [3], H ¼ Ht þ HJ þ Hz þ HD ; that includes: the kinetic energy of a hole ðHt Þ; the superexchange interaction ðHJ Þ; the orbital splitting in the crystal field ðHz Þ; and the polarization of orbitals around a hole ðHD Þ: The *Corresponding author. Institute of Physics, Jagellonian ! Poland. Tel.: +48University, Reymonta 4, PL-30059 Krakow, 12-632-4888; fax: +48-12-633-4079. E-mail address: [email protected] (A.M. Ole!s).
excitations in the orbital ordered state within the FM planes follow from HJ and form orbital waves (orbitons) [4], with dispersion oq ðfÞ ¼ 3J½1 þ ð2 cos 4f 1Þgq 1=2 ; where gq ¼ 12ðcos qx þ cos qy Þ: The dispersion depends on the orbital angle f; given by sinð2fÞ ¼ E z =4J; where P Ez is the crystal field splitting, Hz ¼ 12Ez i ðnix niz Þ: Due to the lack of SU(2) symmetry, the superexchange model predicts a gap of the order 3JC133 meV in the orbital excitations (orbitons) [4]. Recently, this theoretical prediction was confirmed in Raman scattering by an observation of the orbitons with a gap of about 140 meV in LaMnO3 [5]. A static hole changes locally the orbital order, leading to an orbital polarization P by P its coupling to the g neighboring sites, HD ¼ D g /ijS ni tj ; where ni is the hole number operator, and the pseudospin aðbÞ g operators pffiffitffi j are given by the Pauli matrices: tj ¼ 1 z 1 z x c 4sj 7ð 3=4Þsj and tj ¼ 2sj : The parameter DC0:4 eV stands for energy due to the breathing mode and to the Coulomb interaction between the charged hole and the neighboring eg electrons [6]. In contrast to the t–J model [7], the hole can propagate freely in the orbital ordered background, leading to the hole dispersion, Ek ðfÞ ¼ t½1 2 sinð2fÞ gk : In addition, the hole scatters on the polaronic potential pD and couples to orbitons, creating/absorbing them along its path. In the linear orbital-wave (LOW) order [4] the total effective Hamiltonian represents a coupled
0921-4526/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 1 ) 0 1 2 0 5 - 4
J. Ba!a et al. / Physica B 312–313 (2002) 740–742
741
(a)
(b)
A(k,ω)
(π,π)
(0,0) −4
−2
0 ω/t
4 −4
2
−2
0 ω/t
2
4
Fig. 1. The hole spectral functions found from Eq. (1) for ðjzS þ jxSÞ=ðjzS jxSÞ orbital order ðf ¼ 0Þ for J=t ¼ 0:1; and: (a) D ¼ 0; (b) D ¼ t; along the ð0; 0Þ ðp; pÞ direction.
Table 1 Width of the QP band W =t; spectral weights ZðkÞ; and the binding energy Eb at the bottom of the QP band calculated for the orbital ordering given by f ¼ 0 and p=12 for J=t ¼ 0:1 f
D=t
W =t
Zð0; 0Þ
Zðp; pÞ
Zðp; 0Þ
Eb =t
0
0.0 0.5 1.0
0.225 0.090 0.045
0.0241 0.0760 0.0961
0.3592 0.2821 0.1924
0.1353 0.1531 0.1416
1.184 1.333 2.669
p=12
0.0 0.5 1.0
0.295 0.295 0.265
0.0010 0.0048 0.0186
0.5003 0.4489 0.3765
0.0077 0.0294 0.0546
0.824 1.014 1.379
hole–orbiton problem, HLOW ¼
X k
Ek ðfÞhwk hk þ
X
oq ðfÞawq aq
q
1 X w t þ pffiffiffiffiffi hkq hk f½Mk;q ðfÞ þ MqD ðfÞ awq N k;q t þ ½Nk;q ðfÞ þ NqD ðfÞ awqþQ þ h:c:g;
ð1Þ
where hwk and awq are hole and orbiton creation operators, respectively, and Q ¼ ðp; pÞ: The hole–orbiton coupling is given both by the hole hopping t t ðptÞ: pffiffiffi Mk;q ðfÞ ¼ 2t cosð2fÞðuq gkq þ vq gk Þ and Nk;q ðfÞ ¼ 3tðuqþQ Zkq þ vqþQ Zk Þ; and by the orbital polarizaD tion ðpDÞ around a hole by: Mp q ffiffiðfÞ ffi ¼ D cosð2fÞðgq 1Þðuq þ vq Þ; and NqD ðfÞ ¼ ð 3ÞD sinð2fÞZq ðuqþQ þ vqþQ Þ; here Zq ¼ 12ðcos qx cos qy Þ and fuq ; vq g are the
coefficients of the respective Bogoliubov transformation [4]. We evaluated the Green function, Gðk; oÞ ¼ ½o Ek ðfÞ Sðk; oÞ 1 ; and the corresponding spectral functions, Aðk; oÞ ¼ ð1=pÞ Im Gðk; o þ ieÞ; using the selfconsistent Born approximation (SCBA) [8]. The ladder type structure of the spectral function, due to gapped orbital excitations, shows a quasiparticle (QP) peak with dispersion pJ; i.e., large effective mass [Fig. 1(a)]. The orbital polarization has a particularly strong effect when the occupied orbitals are far from directional, and leads then to a rapid decrease of the QP bandwidth W with increasing D: The hole localization is confirmed by a large binding energy Eb (Table 1), and by the ladder spectrum at D ¼ 2t [Fig. 1(b)], which resembles that of the t–J z model [7]. On the contrary, for the alternating 3x2 r2 =3y2 r2 orbital order ðf ¼ p=12Þ; the orbital
J. Ba!a et al. / Physica B 312–313 (2002) 740–742
742
.15 NR
.10 .05 0.0
Ry
Rx Fig. 2. Hole–orbiton correlations NR for J=t ¼ 0:1; D ¼ 0 and m ¼ 12 (bottom) for 3x2 r2 =3y2 r2 orbital order ðf ¼ p=12; top).
polarization has only little influence and the spectra are dominated by the dispersive feature pt F it only reduces somewhat the QP weight at band minimum KCðp; pÞ; while W remains almost unchanged when D increases from 0 to t: The shape of the polaron is described by the spatial distribution of local orbital excitations at distance R from the hole at site Rj ; X ðmÞ nj ð1 þ eiQRj ÞawRj þR aRj þR jCK S; ð2Þ NR ¼ /CðmÞ K j j
jCðmÞ K S
where is the wave function containing up to m orbital excitations in analogy to the SCBA wave function for the t–J model [9], and awRj þR is the Fourier transform of awq : In the localized limit ðt ¼ 0Þ the polaron is well localized on the nearest neighbors around a hole. For realistic values of D and t; the polaron is still confined to the atoms close to the hole, with the largest amplitudes on the hole nearest neighbors. Decreasing superexchange J increases the number of orbital excitations around the hole, with the polaron either increasing its size (at ta0Þ; or constrained to neighboring Mn3þ ions (at t ¼ 0 and Da0). The correlation function (2) depends strongly on the type of orbital order; for instance, when the directional orbitals shown in Fig. 2 are considered, it is large at the sites with an
occupied orbital directed towards the hole, which favors the hole transport along this direction. Summarizing, the results indicate that orbital polarizations cause a strong localization of doped holes and lead to a very narrow QP band, particularly if the orbital order is close to ðjzS þ jxSÞ=ðjzS jxSÞ; as induced by the superexchange pJ: Since the orbitons are gapped, the hole–orbiton correlations decay exponentially and are confined to only first and second neighbors around the moving hole. This work was supported by the Committee of Scientific Research (KBN) of Poland, Project No. 5 P03B 055 20.
References [1] [2] [3] [4] [5] [6] [7] [8] [9]
L.F. Feiner, A.M. Ole!s, Phys. Rev. B 59 (1999) 3295. A.E. Bocquet, et al., Phys. Rev. B 46 (1992) 3771. J. van den Brink, et al., Phys. Rev. Lett. 85 (2000) 5174. J. van den Brink, et al., Phys. Rev. B 59 (1999) 6795. E. Saitoh, et al., Nature 410 (2001) 180. R. Kilian, G. Khaliullin, Phys. Rev. B 60 (1999) 13458. G. Mart!ınez, P. Horsch, Phys. Rev. B 44 (1991) 317. S. Schmitt-Rink, et al., Phys. Rev. Lett. 60 (1989) 2793. A. Ram$sak, P. Horsch, Phys. Rev. B 57 (1998) 4308.
Orbital polarons in LaMnO3 Jan Ba"aa, Andrzej M. Oles!a,b,*, Peter Horschb b
a ! Poland Institute of Physics, Jagellonian University, Reymonta 4, PL-30059 Krakow, Max-Planck-Institut fur Heisenbergstrasse 1, D-70569 Stuttgart, Germany . Festkorperforschung, .
Abstract We investigate the spectral functions and orbital correlation functions relative to the position of an eg hole moving in the ferromagnetic plane of orbitally ordered LaMnO3 : Polarization of orbitals around the hole leads to a very narrow quasiparticle band and to a confinement of the hole in an orbital polaron with the shape strongly depending on the type of the orbital order. r 2002 Elsevier Science B.V. All rights reserved. PACS: 75.30.Vn; 75.30.Et; 79.60.i Keywords: Orbital polaron; Quasiparticle; Manganites
The orbital degrees of freedom are known to play a central role in the colossal magnetoresistance manganites. The undoped LaMnO3 is a Mott–Hubbard insulator, with an A-type antiferromagnetic (AF) order which consists of ferromagnetic (FM) ða; bÞ planes staggered along the c-axis, and coexists with orbital order with alternating occupied eg orbitals within the ða; bÞ planes, jimS ¼ cosðp=4 fÞjizS7sinðp=4 fÞjixS; where jx2 y2 S jmS and j3z2 r2 S jkS are the components of T ¼ 12 pseudospin. This ground state is well explained and follows from the cooperative Jahn–Teller effect and from the superexchange between the Mn3þ –Mn3þ pairs promoted by the excitations of high-spin configurations pJ ¼ t2 =ðU 3JH Þ [1]. With the on-site Coulomb and exchange elements: UC5:9 eV and JH C0:69 eV; and tC0:41 eV [2], one finds JC44 meV; i.e., J=tC0:1: We investigate the propagation of a single hole in LaMnO3 by considering the orbital t–J model [3], H ¼ Ht þ HJ þ Hz þ HD ; that includes: the kinetic energy of a hole ðHt Þ; the superexchange interaction ðHJ Þ; the orbital splitting in the crystal field ðHz Þ; and the polarization of orbitals around a hole ðHD Þ: The *Corresponding author. Institute of Physics, Jagellonian ! Poland. Tel.: +48University, Reymonta 4, PL-30059 Krakow, 12-632-4888; fax: +48-12-633-4079. E-mail address: [email protected] (A.M. Ole!s).
excitations in the orbital ordered state within the FM planes follow from HJ and form orbital waves (orbitons) [4], with dispersion oq ðfÞ ¼ 3J½1 þ ð2 cos 4f 1Þgq 1=2 ; where gq ¼ 12ðcos qx þ cos qy Þ: The dispersion depends on the orbital angle f; given by sinð2fÞ ¼ E z =4J; where P Ez is the crystal field splitting, Hz ¼ 12Ez i ðnix niz Þ: Due to the lack of SU(2) symmetry, the superexchange model predicts a gap of the order 3JC133 meV in the orbital excitations (orbitons) [4]. Recently, this theoretical prediction was confirmed in Raman scattering by an observation of the orbitons with a gap of about 140 meV in LaMnO3 [5]. A static hole changes locally the orbital order, leading to an orbital polarization P by P its coupling to the g neighboring sites, HD ¼ D g /ijS ni tj ; where ni is the hole number operator, and the pseudospin aðbÞ g operators pffiffitffi j are given by the Pauli matrices: tj ¼ 1 z 1 z x c 4sj 7ð 3=4Þsj and tj ¼ 2sj : The parameter DC0:4 eV stands for energy due to the breathing mode and to the Coulomb interaction between the charged hole and the neighboring eg electrons [6]. In contrast to the t–J model [7], the hole can propagate freely in the orbital ordered background, leading to the hole dispersion, Ek ðfÞ ¼ t½1 2 sinð2fÞ gk : In addition, the hole scatters on the polaronic potential pD and couples to orbitons, creating/absorbing them along its path. In the linear orbital-wave (LOW) order [4] the total effective Hamiltonian represents a coupled
0921-4526/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 1 ) 0 1 2 0 5 - 4
J. Ba!a et al. / Physica B 312–313 (2002) 740–742
741
(a)
(b)
A(k,ω)
(π,π)
(0,0) −4
−2
0 ω/t
4 −4
2
−2
0 ω/t
2
4
Fig. 1. The hole spectral functions found from Eq. (1) for ðjzS þ jxSÞ=ðjzS jxSÞ orbital order ðf ¼ 0Þ for J=t ¼ 0:1; and: (a) D ¼ 0; (b) D ¼ t; along the ð0; 0Þ ðp; pÞ direction.
Table 1 Width of the QP band W =t; spectral weights ZðkÞ; and the binding energy Eb at the bottom of the QP band calculated for the orbital ordering given by f ¼ 0 and p=12 for J=t ¼ 0:1 f
D=t
W =t
Zð0; 0Þ
Zðp; pÞ
Zðp; 0Þ
Eb =t
0
0.0 0.5 1.0
0.225 0.090 0.045
0.0241 0.0760 0.0961
0.3592 0.2821 0.1924
0.1353 0.1531 0.1416
1.184 1.333 2.669
p=12
0.0 0.5 1.0
0.295 0.295 0.265
0.0010 0.0048 0.0186
0.5003 0.4489 0.3765
0.0077 0.0294 0.0546
0.824 1.014 1.379
hole–orbiton problem, HLOW ¼
X k
Ek ðfÞhwk hk þ
X
oq ðfÞawq aq
q
1 X w t þ pffiffiffiffiffi hkq hk f½Mk;q ðfÞ þ MqD ðfÞ awq N k;q t þ ½Nk;q ðfÞ þ NqD ðfÞ awqþQ þ h:c:g;
ð1Þ
where hwk and awq are hole and orbiton creation operators, respectively, and Q ¼ ðp; pÞ: The hole–orbiton coupling is given both by the hole hopping t t ðptÞ: pffiffiffi Mk;q ðfÞ ¼ 2t cosð2fÞðuq gkq þ vq gk Þ and Nk;q ðfÞ ¼ 3tðuqþQ Zkq þ vqþQ Zk Þ; and by the orbital polarizaD tion ðpDÞ around a hole by: Mp q ffiffiðfÞ ffi ¼ D cosð2fÞðgq 1Þðuq þ vq Þ; and NqD ðfÞ ¼ ð 3ÞD sinð2fÞZq ðuqþQ þ vqþQ Þ; here Zq ¼ 12ðcos qx cos qy Þ and fuq ; vq g are the
coefficients of the respective Bogoliubov transformation [4]. We evaluated the Green function, Gðk; oÞ ¼ ½o Ek ðfÞ Sðk; oÞ 1 ; and the corresponding spectral functions, Aðk; oÞ ¼ ð1=pÞ Im Gðk; o þ ieÞ; using the selfconsistent Born approximation (SCBA) [8]. The ladder type structure of the spectral function, due to gapped orbital excitations, shows a quasiparticle (QP) peak with dispersion pJ; i.e., large effective mass [Fig. 1(a)]. The orbital polarization has a particularly strong effect when the occupied orbitals are far from directional, and leads then to a rapid decrease of the QP bandwidth W with increasing D: The hole localization is confirmed by a large binding energy Eb (Table 1), and by the ladder spectrum at D ¼ 2t [Fig. 1(b)], which resembles that of the t–J z model [7]. On the contrary, for the alternating 3x2 r2 =3y2 r2 orbital order ðf ¼ p=12Þ; the orbital
J. Ba!a et al. / Physica B 312–313 (2002) 740–742
742
.15 NR
.10 .05 0.0
Ry
Rx Fig. 2. Hole–orbiton correlations NR for J=t ¼ 0:1; D ¼ 0 and m ¼ 12 (bottom) for 3x2 r2 =3y2 r2 orbital order ðf ¼ p=12; top).
polarization has only little influence and the spectra are dominated by the dispersive feature pt F it only reduces somewhat the QP weight at band minimum KCðp; pÞ; while W remains almost unchanged when D increases from 0 to t: The shape of the polaron is described by the spatial distribution of local orbital excitations at distance R from the hole at site Rj ; X ðmÞ nj ð1 þ eiQRj ÞawRj þR aRj þR jCK S; ð2Þ NR ¼ /CðmÞ K j j
jCðmÞ K S
where is the wave function containing up to m orbital excitations in analogy to the SCBA wave function for the t–J model [9], and awRj þR is the Fourier transform of awq : In the localized limit ðt ¼ 0Þ the polaron is well localized on the nearest neighbors around a hole. For realistic values of D and t; the polaron is still confined to the atoms close to the hole, with the largest amplitudes on the hole nearest neighbors. Decreasing superexchange J increases the number of orbital excitations around the hole, with the polaron either increasing its size (at ta0Þ; or constrained to neighboring Mn3þ ions (at t ¼ 0 and Da0). The correlation function (2) depends strongly on the type of orbital order; for instance, when the directional orbitals shown in Fig. 2 are considered, it is large at the sites with an
occupied orbital directed towards the hole, which favors the hole transport along this direction. Summarizing, the results indicate that orbital polarizations cause a strong localization of doped holes and lead to a very narrow QP band, particularly if the orbital order is close to ðjzS þ jxSÞ=ðjzS jxSÞ; as induced by the superexchange pJ: Since the orbitons are gapped, the hole–orbiton correlations decay exponentially and are confined to only first and second neighbors around the moving hole. This work was supported by the Committee of Scientific Research (KBN) of Poland, Project No. 5 P03B 055 20.
References [1] [2] [3] [4] [5] [6] [7] [8] [9]
L.F. Feiner, A.M. Ole!s, Phys. Rev. B 59 (1999) 3295. A.E. Bocquet, et al., Phys. Rev. B 46 (1992) 3771. J. van den Brink, et al., Phys. Rev. Lett. 85 (2000) 5174. J. van den Brink, et al., Phys. Rev. B 59 (1999) 6795. E. Saitoh, et al., Nature 410 (2001) 180. R. Kilian, G. Khaliullin, Phys. Rev. B 60 (1999) 13458. G. Mart!ınez, P. Horsch, Phys. Rev. B 44 (1991) 317. S. Schmitt-Rink, et al., Phys. Rev. Lett. 60 (1989) 2793. A. Ram$sak, P. Horsch, Phys. Rev. B 57 (1998) 4308.