Electronic structure and effective Hamiltonian in perovskite Mn oxides

PHYSICA ELSEVIER

Physica C 263 (1996) 130-133

Electronic structure and effective Hamiltonian in perovskite Mn oxides S. Ishihara *, J. Inoue, S. Maekawa Department of Applied Physics, Nagoya University,Nagoya 464-01, Japan

Abstract

In order to investigate electronic and magnetic structures in perovskite Mn oxides, we consider the effective Hamiltonian derived from the generalized ferromagnetic Kondo-Hubbard model. The mean-field approximation is adopted to analyze the ordered states of the spin and orbital degrees of freedom in the insulating state. As the excitations in the ordered states, we consider the spin and orbital waves and their dispersion relations are calculated.

1. Introduction

Much recent attention has been attracted to La~_xAxMnO 3 ( A = C a , S r , B a ) and related compounds in which the colossal magnetoresistance and metal-insulator transition are observed [1]. The nature of the electrical transport is changed with the concentration of the divalent ions and it is closely related to the magnetic structures. An insulating state is realized in a region of x - 0.0 where the layer-type antiferromagnetic ordered state (the A-type AF) is observed [2,3]. In this regime, a MnO 6 octahedron is stretched out along the (110) or (110) directions [4] and some kind of orbital ordered state seems to be realized. With the increase in x, a metallic phase accompanied by the ferromagnetic spin polarization appears at low temperature and the colossal magnetoresistance is observed near the critical temperature. In the region of x = 1, the system enters an insulat-

* Corresponding author. Fax: +81 52 789 3724; e-mail: [email protected].

ing state again and the G-type antiferromagnetic ordering, where spins are antiferromagnetically arranged in all directions, emerges. The transport properties correlated with the magnetic structures have been discussed by the double exchange mechanism long ago. However, until now, it has not been fully discussed how the orbital ordering and the lattice distortion influence the magnetic structure in the Mn oxides [5,6]. Also, little attention has been paid to the effects of the electron correlations in the eg orbitals [7], in spite of the metal-insulator transition being observed. Electronic and transport properties in the perovskite Mn oxides should be reconsidered by taking these effects into account. In this paper, electronic structure in the Mn oxides is theoretically studied by using the effective Hamiltonian. The effective Hamiltonian is derived from the ferromagnetic K o n d o - H u b b a r d model where the two eg, orbitals are introduced and the electron correlation in the eg orbitals is included. By excluding the doubly occupied states in the eg orbitals, the effective Hamiltonian is obtained. The mean-field theory is adopted to examine the spin and

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S. lshihara et al./ Physica C 263 (1996) 130-133

131

orbital ordered states in the insulating state and a stability of the A-type AF phase is discussed. To investigate the nature of excitations, dispersion relations of the spin and orbital waves are calculated.

H2- site = Hu + Hu'-J + Hu'+J, where the Coulomb interactions between the ferent eg orbitals, respectively, and change interaction in the eg orbitals. given by,

2. Effective H a m U t o n i a n

Hu=

U and U' are same and difJ is the exEach term is

1

U-2K

E(¼nin,-SiS,) <6)

To describe the electronic and magnetic structures in the Mn oxides, we consider the effective Hamiltonian which is derived by the generalized KondoHubbard model. HKH = H e + H K + Hj. The first term is for the electrons m the eg orbitals and it is written as

X [[4t~Z(±n ij ~k2 i + Tiz)(½nj+

mes'~ ES"dd~gcrdi~,cr + E '~jY'(d:7o-djy'o.+C.C. ) icrT ( ij)~r30/

+2t ,jaa'aMI1 "-' + T/z)~_} ,,j tt-~n; -t--~;..)l;.++(~n,

•

.

+E

+ Atbb2[ 1 atab2l 1

g

,,

.

Tiz)

1 .

1 r,z)(:,T,=)

+ 4tba2Z 1 ,,

+ r,z)

+ 2 taa.ba I 1 + Tjz) + Ti_(Tn 1 j + ij tij (.~ ~,'i+[-~nj

E

io'1°'2 Yl ~ 3'4

t

(½. _

*

X U./.... ~2cr2.~3u:.~4a,di~lcqdiy,.a2 di~3~2di~4~,,

(1) where d~y~ is the electron creation operator and 3' denotes one of the two eg orbitals. The second term in HKH describes the Hund coupling between the eg and t2s spins and the 3rd-term implies the antiferromagnetic superexchange interaction between the t2s spins. The sum of both terms is given by

> (½n,-

+ 2ti~Oti~a{( ½ni - Tiz)Tj+ -I-( ½n i --

Ti:)Tj_ }], (3)

1 Hu,_ J

U'-----~J ~-" (¼ninj + SiSj) (ij) X [ 2 ( t ~ j a2-Ftbb2'llnij )t-4 inj - TizTjz) aa bb - - 2 t i j tij ( T i + T j - - [ - Ti- T j + )

H K + H s = K ~_~ S[ 2, d;~,~, * Orcr]o.2di../o.2 i3"cr10"2

+ J% E S:"Sj~'.

~:)}

+ 2 ( t ; p2+ t i;b a 2)\ l~l n , n ; +

TizTjz )

(2)

Because the magnitudes of the intra- and interorbital Coulomb interactions are larger than other energy parameters in this Hamiltonian, we derive the effective Hamiltonian by excluding the doubly occupied states in the eg orbitals. We make a canonical transformation in He, and obtain the effective Hamiltonian as follows, H e tf = /tI.g e + H K + H j, where /4e, is divided into the following three parts: He~ = H0 + H2- site Jr- H 3_ site" H0 is the unperturbed term in the canonical transformation and H 2_ s~te and H3-site include the electron operators in the nearestneighbor two sites and the next nearest-neighbor three sites, respectively. Further, H 2_ site is classified by the energy in the intermediate states, as follows,

- - 2 t ~ b t b a ( T i + T j + + Ti_Tj_ ) -

-

2(tijab tijaa - tb"tbb~(T ij ij ]\ iz Tj+ + Ti.T ~ i- )

--2(t,Tt,'f --t~btbb)(T~+Tjz + Ti_ Tj:)],

(4) and 1

Hw+ s - v' + J - 2 K

z" W " : ' - s's') <,.,> 'l r,.:,3 u " ' )[-sn~nj-

+ 2 t~°tgb(r,+ r~_ + r,_ rj+) + 2 ( t ~ 2 + tiba2)(¼ninj + TizTjz )

S. lshihara et a l . / P h y s i c a C 263 (1996) 1 3 0 - 1 3 3

132

+ 2t~btba(Ti+Tj+ + T i_ Tj_ ) \ -tj -U

'

+ ' ,ba ',Jbb )(",G

represented by products of the spin and orbital parts, since the second-order processes with respect to the nearest-neighbor transfer change both orbital and spin states at the same time. Therefore, it is expected that this nature of the spin structure is strongly influenced by the orbital one. When we assume that t ia~ = t ieb = t and t~ b = t/~" = 0 in n 2_ ~it~, H2-site is identified with the model proposed by Inagaki [8]. Because magnitudes of the transfer depend on sites and directions in general, the anisotropic character in the orbital space appears in the present model and a variety of the orbital structure will be brought about.

3. R e s u l t s a n d d i s c u s s i o n

Using the effective Hamiltonian derived above, the electronic structure in the insulating state ( x = 0) is investigated in the present paper. First, we adopt the mean-field approximation to study the relation between the magnetic and orbital ordered states. In

I

'

I

J

t

Iorbital(A,G,F)l 0.04

°o.o2

I

0

,

I

i

I

0.002 J'~' (eV)

I

l

'

I

I

(5)

Si is the spin operator for the eg spin defined by 1 ~'f ~ S~ = -~S.rdi~,o%#di~ #. The orbital degree of freedom is described by the pseudospin operator (T i) i 7¢ which is given by Ti=-~S,~div,(rrv, di~, ~. /-],, is

'

'

0.006

)

+{'f'"taf~,-tj-,1 +t~btbb)(Ti+nj + T/_nj)].

0.06

I

,

0.004

Fig. 1. The transition temperaturesof the spin and orbital ordered states. A, G and F represent the plane-type AF, the simple AF and the ferromagneticphases, respectively.

0.004

0.002

,,,

i,,,

1.5

,I

2

....

i,,,

2.5

3

R Fig. 2. The magnetic phase diagram in the (R-J'2~) plane. R represents the effects of the lattice distortion.

Fig. 1, transition temperatures for the spin and orbital ordered phases are shown. Parameter values are chosen to be K = - 5 e V , U = S e V , U ' = 4 e V a n d J = 1 eV and the orbital ordering is assumed to be (3x 2 - r 2 : 3 y 2 - r 2) type. The A-type AF phase turns up in the region between the ferromagnetic and the G-type AF phase owing to the competition between the ferromagnetic interactions (K, J ) and the antiferromagnetic one ( J % ) . The A-type AF phase is supported by the orbital ordering, since the ferromagnetic interaction in the ab-plane becomes larger than that in the c-direction under the (3x 2 - r2: 3y 2 - r 2) type orbital ordering. In Fig. 2, we show how the magnetic structure is influenced by the distortion in the MnO 6 octahedron as is observed in the orthorhombic phase. It is assumed that the lattice distortion changes the transfer intensity and the ratio between the intensities in the short and long M n - O bonds is denoted by R. Also, J,2 is changed with R. With the increase in the lattice ~listortion, the region of the ferromagnetic phase is extended and the A-type phase is pushed up in the region with the higher values of J%. These results originate from the intensification of the ferromagnetic interaction along the c-direction by the lattice distortion. The present results are different from those obtained by the recent Hartree-Fock calculations [5,6] where the ferromagnetic phase becomes unstable with the increase in the lattice distortion. Discrepancies are caused by effects of the electron correlation which produces the energy gap even in the lattice undistorted case. To study excitations in the A-type AF phase accompanied by the (3x 2 - r 2 : 3 y 2 - r 2) type or-

S. lshihara et a l . / Physica C 263 (1996) 130-133

0. 04 0.03

-?(? Jt'~O. C332S

..(a)

• ~ Jt2r=-O.C 1275 ........ Jt~*=O.C '225 ," ",

...... ,,,.%,,,,,

0.02 0.01 0. 00

Z 0. 20 ,~ O. 15

~o. lo

F

X

M

A

F

•l ORBITAL"1

(b)

/

\

°¢Z

F

X

M

F

modify the phonon dispersion relation. Also, the orbital wave affects the nature of the spin wave through the strong correlation between the orbital and spin degrees of freedom. A dispersion relation of the orbital wave is shown in Fig. 3(b). The energy gap for the orbital wave reflects an anisotropy in the orbital space, and the flat dispersion relation along the F - X direction is caused by the (3x 2 - r2: 3y 2 - r 2) type orbital ordering. By experimental observation of the orbital wave, existence of the (3x 2 - r 2 : 3 y 2 - r 2) type orbital ordering in the insulating phase is able to be confirmed.

Acknowledgments

O. 05

0.

133

A

Fig. 3. Dispersion relations of the spin wave (a) and orbital wave (b). Dispersions are plotted in the reduced Brillouin zone.

bital ordering, dispersion relations of the spin and orbital waves are calculated by the Holstein-Primakoff theory. The Hartree approximation is used in the interaction between the spin and orbital waves and details of the calculation will be presented elsewhere. In Fig. 3(a), dispersion relations of the spin wave for several values of J % are presented. When the state becomes close to the phase boundary between the A-type AF and the ferromagnetic phases with the decrease in J %, the slope of the dispersion along the F - X direction increases. On the other hand, the slope along the F - Z direction is softened. The softening of the spin wave implies a weakening of the antiferromagnetic interaction in the c-direction and it reflects the stability of the A-type AF phase as we discussed above. As for the excitations in the orbital degree of freedom, they propagate as the orbital wave through the inter-site interaction. The orbital wave is able to be detected by the light scattering or the electron energy loss spectroscopy and it may

One of the authors (SI) is supported by a Research Fellowship of the Japan Society for the Promotion of Science for Young Scientists. This work was supported by Priority-Areas Grants from the Ministry of Education, Science and Culture of Japan, and the New Energy and Industrial Technology Development Organization (NEDO). Parts of the numerical calculations have been performed at the Supercomputing Center of the Institute for Materials Research, Tohoku University. We would like to thank A.J. Millis for his valuable comments.

References [1] Y. Tokura, A. Urushibara, Y. Moritomo, T. Arima, A. Asamitsu, G. Kido and N. Furukawa, J. Phys. Soc. Jpn. 63 (1994) 3931. [2] E.O. Wollan and W.C. Koehler, Phys. Rev. 100 (1955) 545. [3] J.B. Goodenough, in: Metallic Oxides, Progress in Solid State Chemistry, (Pergamon, London, 1971). [4] G. Matsumoto, J. Phys. Soc. Jpn. 29 (1970) 606. [5] T. Mizokawa and A. Fujimori, Phys. Rev. B 51 (1995) 12880. [6] N. Hamada, H. Sawada and K. Terakura, in: Spectroscopy of Mott insulators and correlated metals, Solid-State Sciences I 19 (Springer, Berlin, 1995). [7] J. lnoue and S. Maekawa, Phys. Rev. Lett. 74 0995) 3407. [8] S. Inagaki, J. Phys. Soc. Jpn. 39 (1975) 596; see also K.1. Kugel and D.I. Khomskii, Sov. Phys. JETP 37 (1973) 725.