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8438 A. J. MILLIS 53

classical holes has been shown to be consistent with the re- a a
sistivity at all x and T.400 K and for x,x cond ;0.3 and all u 5( e 2ik•R i u k ~A1!
i
k
T. 5
a a
Because the outer Mn d-orbital is twofold degenerate, a and similarly d . Now d may be decoupled from E latt , Eq.
k
k
Jahn-Teller distortion of the surrounding oxygen octahedron ~2!, by defining
which lowers the local cubic symmetry to tetragonal may
occur about each occupied Mn site, while a breathing mode d 5d 2 1 K ~11e ik a ! u . ~A2!
1
a
a
¯ a
distortion may occur around each unoccupied site. Each oxy- k k 2 2K 1K 2K cosk a k
2
1
2
gen is shared by two Mn ions, and so distortions on adjacent
¯ a
d 50 gives the equilibrium positions about which the Mn
sites are coupled. The coupling was determined from a clas- k
ions fluctuate. After decoupling, the relevant part of the lat-
sical harmonic approximation to the lattice dynamics. The
tice energy may be written
parameters of the model were determined by fitting to the
structural data obtained for LaMnO . The principal results
3
a a
are the following. E 5K 1( F~k !u u , ~A3!
k 2k
a
latt
~1! The basic energy gained in a local Jahn-Teller distor- ka
tion, E *0.1 eV. The estimate E '0.6 eV was obtained with
0
0
using a phonon frequency estimated from an optical mea-
surement on La 1.85 Sr .15 MnO . A direct measurement of the 1 ~K 12K !~12cosk !
2
1
3
a
splitting 4E between the two d levels would be desirable. F~k !5 . ~A4!
a
0 2 K 1K 2K cosk a
1
2
2
The distortions are in any event well formed at any relevant
temperature and the structural transition is to be regarded as The interaction energies are most conveniently written in
a
an order-disorder transition, at which local Jahn-Teller dis- terms of the variables c defined via
k
tortions become spatially decorrelated, but do not disappear.
~2! The model describing the transition is given in Eq. ~7! c 5( e ik•R i 12h !cos2~u 1c !, ~A5!
a
a
~
and may be approximated either by an antiferromagnetic x- k i i i
y model with a modest threefold anisotropy or by a three- z x y
where c 50, c 52p/3, and c 5p/3 were introduced in
state Potts model with an antiferromagnetic first-neighbor Eq. ~7!. Combining Eqs. ~3!, ~4!, ~A5!, gives
interaction and a weak second-neighbor interaction. Which
model is more nearly correct depends on whether the anhar-
a a
monicity parameter A is larger or smaller than the stiffness E 5l( ~12e 2ik a !u u ,
JT
k 2k
ka
k which orients the distortions. By combining an optical
measurement of the highest phonon frequency in
La 1.85 Sr 0.15 MnO 3 with a calculation of T s the estimate E hole 5bl( h ~12e 2ik a !u a 2k . ~A6!
k
A;k was obtained. ka
~3! Added holes disrupt the long-range order by producing
The displacement u may be eliminated by writing E in terms
an effectively random field, which misorients nearby Jahn-
of
Teller distortions. It would be very interesting if it were pos-
sible to observe directly this local misorientation. This ran- a a l~12e 2ik a ! a
dom field effect was shown by various mean-field u ¯ 5u 2 2K F~k ! ~c 1bh !. ~A7!
k
k
k
k
calculations to lead to a rapidly decreasing T (x), in quali- 1 a
s
tative accord with data. A Monte Carlo investigation of the Again u ¯ 50 defines the average state about which the oxygen
problem would be useful. atoms fluctuate. The electronic part of the energy may then
The results in this paper substantiate to some degree the be written
4,5
proposal that electron-lattice interaction is so strong that 2
the high-T cubic ~or pseudocubic! 0.2&x&0.4 phase of E52 1 l ( 12cosk a ~c 1bh !~c a 1bh 2k !. ~A8!
a
La 12x A MnO should be modeled as a disordered array of 2 K 1 ka F~k ! k k 2k
x
3
a
polarons. The results presented here provide a basis for cal-
Fourier transformation yields Eq. ~7! except for the term pro-
culating polaron binding energies and mobilities, both for
0.2&x&0.4 and high T and for low x at all T. portional to A. This term arises from a lattice anharmonicity
3
of the form ( v . Use of Eq. ~A7! yields several terms, of
i i
which the largest is A cos(6u).
ACKNOWLEDGMENTS
APPENDIX B: ANALYSIS OF STRUCTURE
I thank B. I. Shraiman for many helpful conversations, D.
A. Huse for discussions of random field problems, and C. M.
In this appendix the LaMnO structural data obtained by
Varma for discussions of electronic correlation effects. 2 3
Ellemans et al.. are analyzed. The magnitudes of the atomic
displacements observed in LaMnO in Ref. 2 are somewhat
3
greater than those reported in previous work. 15 Indeed,
APPENDIX A: DERIVATION OF ENERGY
the displacements reported for LaMnO by Ref. 15 are
3
This appendix outlines the derivation of Eq. ~7! from Eqs. very similar to those reported by Ellemans et al., for
2
~2!, ~3!, and ~5! and discusses anharmonic terms. Define La 1.95 Ca 0.05 MnO . It will be assumed here that the larger
3