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53 COOPERATIVE JAHN-TELLER EFFECT AND ELECTRON-... 8435

relevant to the considerations of this paper, and so the elec- One expects b@1 because the force exerted on the surround-
tronic correlation effects need not be explicitly considered. ing oxygen ions by a Mn of the wrong charge must be much
For each fixed configuration of electrons, the phonon part greater than the force exerted by rearranging the proper
of the free energy is minimized; the result of this minimiza- charge among different d orbitals.
tion is the energy of that configuration of electrons. The For fixed values of u and h , Eqs. ~2!, ~3!, and ~5! may
i
i
phonons are treated in the harmonic approximation. The ef- be minimized. The details are given in Appendix A. The
fect of the undoubtedly important anharmonic terms in the result is most naturally expressed in terms of the parameters
lattice energy is parametrized. Only some of the lattice de- 2
grees of freedom are considered. These are ~1! the vector E 52 3 l K 1K 2 ,
1
0
W
displacements d of the manganese ~Mn! ion on site i and ~2! 2 K K 12K 2
1
1
i
the scalar displacement u (a) of the oxygen ~O! ion along the 2
i
x
Mn-O-Mn bond direction. Thus u is the displacement, in the 1 l K 2 , ~6!
i k5
x direction, of the O atom which sits between the Mn ion on 2 K ~K 12K !
2
1
1
site i and the Mn ion on site i1x ˆ . With this restricted set of
as
displacements one may discuss the Jahn-Teller distortion and
the uniform strain, but not the buckling of the Mn-O-Mn
2 2
2
0
bond or the associated rotation of the octahedra. These latter E5E ( ~12h ! 1b h 1Acos6u 1k( ~12h !~1
i
i
i
i
i ia
lattice distortions occur but, I believe, are not fundamental.
If an electron is present on site i, it will be in a state
uc (u)& given by a linear combination of the two outer d 2h i1a ˆ !cos2~u 1c !cos2~u i1a 1c !12bk( h ~1
a
i
a
i
i
ia
orbitals. In the classical approximation used here the phase
of the electron is of no significance, and so one may write
2
2h i1a !cos2~u i1a 1c !1b k( h h . ~7!
i i1a
a
uc ~u !&5cosu ud 3z 2r 2&1sinu ud 2 2&, ~1! ia
2
x 2y
i
i
i
i
with 0<u ,p. Here a56x,y,z, c 6z 50, c 6x 52p/3, c 6y 5p/3, and we
i
7
The lattice energy E latt is taken to depend on the Mn-O have followed Kanamori by adding a phenomenological an-
distance and the Mn-Mn distance. The unit cell i is taken to harmonicity term with coefficient A. Cubic anharmonicities
W
W
include the Mn ion at position R 1bd and the three O ions exist in any realistic model of lattice dynamics. The anhar-
i
i
monicity is important for two reasons: it couples a staggered
W
1
at positions R 1( 2 b1bu (a) )a ˆ , where a5x, y,or z and b is distortion to a uniform one, and it breaks the perfect rota-
i
i
the lattice constant. Here d and u are dimensionless displace- tional (u) symmetry found otherwise if h 50. The term
ments defined with reference to the ideal perovskite lattice i
added to Eq. ~7! is the simplest one which accomplishes
with lattice constant b. In the harmonic approximation
these two effects and goes into itself under u→u1p as re-
quired. It is derived in Appendix A.
1 a
a
a 2
a
E 5 K 1( ~d 2u ! 1~d 2u i2a ˆ ! 2 To each configuration of orbital occupancies $u % corre-
latt
i
i
i
i
2 i sponds an average distortion from the ideal cubic peroviskite
structure. This may be written in terms of the oxygen (u)
1 a
a
2
1 K 2( ~d 2d i2a ˆ ! . ~2! and Mn (d) displacements as
i
2 i
Here K and K have the dimension of energy; one expects u 5( f ~R 2R !@~12h !cos2~u 1c !1bh #,
a
a
a
2
1
i
j
j
j
j
i
u
j
K >K . j
2
1
If an electron is present on site i, there is an electron-
lattice energy given by d 5( f ~R 2R !@~12h !cos~2u 1c !1bh #. ~8!
a
a
a
E 5l( ~12h !F cos2u v 2 ~v 1v !D j
iS
d
j
j
i
j
j
i
j
1
z
x
y
JT
i i i 2 i i The elastic kernels are
l e ik•R 2ik a !@K 1K ~12cosk !#
A3 f ~R!5 ( ~12e 1 2 a
a
x
y
1sin2u i @v 2v #G . ~3! u ~K 12K !~12cosk ! ,
2 i i K 1 k 1 2 a
Here l e ik•R K cos~k /2!~12e 2ik a !
f ~k!5 ( 1 a . ~9!
9
a a a d K K 1K ~12cosk !
v 5u 2u i2a ˆ ~4! 1 k 1 2 a
i
i
and h 50 if an electron is present on site i and h 51 if not. III. FIT TO DATA
i
i
Finally, if there is no electron present on site i, all of the
neighboring oxygen ions are equally attracted to it, leading In this section the structural information of Ref. 2 is used
to to estimate model parameters. The analysis is essentially that
7
of Kanamori. A two-sublattice ordering of Jahn-Teller dis-
x y z tortions parametrized by angles u and u is assumed. By
E hole 5bl( h @v 1v 1v #. ~5! 1 2
i
i
i
i
i fitting the observed atomic displacements to Eqs. ~A2! and