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8436 A. J. MILLIS 53

TABLE I. Values of the parameters deduced by fitting mean- v 5A 2K /b \
2 2
field theory to structural data. ox 1 . ~13!
M ox
2u (0) E 0 /K 1 310 3 k/K 1 310 4 A/k
K 2 /K 1 1 l/K 1 The factor of 2 arises because there are two Mn-O bonds in
0 80.9 0.044 2.9 0 0.70 Eq. ~2!. Estimating 100 meV*\v *30 meV and using
ox
0.1 79.2 0.045 2.8 0.85 0.87 b54 Å gives
0.3 75.7 0.045 2.5 1.9 1.31
300 eV*K *30 eV. ~14!
0.5 72.3 0.046 2.4 2.6 1.92 1
0.75 68.2 0.047 2.4 3.4 3.31 I am unaware of measurements of the phonon spectrum in
1 64.4 0.049 2.4 4.0 6.93 LaMnO . If, however, it is assumed that the phonon spec-
3
trum has a rather weak doping dependence one may use op-
12
tical data from La 1.85 Sr 0.15 MnO . The highest-lying phonon
3
~A7!, u ,u and elastic constants are determined. By requir- modes were observed at v ;70 meV. It is reasonable to
2
1
ph
ing that the deduced u ,u minimize Eq. ~7! the anisotropy assume that these are the bond-stretching oxygen modes of
2
1
energy A is found. The experimental data for the structure interest and that these modes are only weakly dispersive;
are given in Appendix B and the mean field equations are thus, one may identify v ph with v oxy and estimate K '200
1
solved in Appendix C. eV.
It is convenient to express the lattice distortions in terms An alternative estimate may be obtained from the mean-
W
of a staggered oxygen displacement u and a uniform strain field approximation to the structural transition temperature
s
W T '750 K. This is shown in Appendix E to be T MF '3k,
e. By rewriting Eqs. ~A2!, ~A7! we obtain
s
s
and mean-field theory overestimates T ;so
s
l
a
a
a
u 5 @cos2~u 1c !2cos2~u 1c !#,
1
2
s
2K 1 k.20 meV. ~15!
2l This bound on k yields K /K -dependent bounds for
1
2
a
a
a
e 5 $cos2~u 1c !1cos2~u 1c !%. ~10! K ranging from K .220 eV (K /K50.1) to K .50 eV
2
1
2
1
1
1
~K 12K !
2
1
(K /K 51). Values of K /K *0.5 are most consistent with
2 1 2 1
a
1
2
In Appendix B the values e 520.028(21/2,21/2,1) estimates of v oxy &50 meV, those of K /K ,0.5 with
a
1
and u 50.038(1,21,0) are derived from the data of Ref 2. v oxy *50 meV. Combining this with the estimate K '200
s eV suggests A;k. This estimate is consistent with estimates
z
That u 50 implies u 52u 1p; substituting this into Eqs. given in a standard review that typical anharmonicity ener-
13
1
s
2
~10! leads to equations for u and l/K which may be solved gies are of order a few hundred kelvin.
1
1
if K /K is given. Results are listed in Table I. The estimates of K imply Jahn-Teller energies E rang-
2
1
We now turn to the value of A. The assumption of a 1 0
ing from '100 meV at the low end (K ;30 eV! to1eV at
1
two-sublattice distortion and the condition h 50 implies that the high end (K ;300 eV!. The estimate v 570 meV
i
Eq. ~7! becomes 1 oxy
implies E '0.6 eV, slightly larger than the largest Jahn-
0
Teller energy listed in a standard review. 13 In any event, be-
1
E5 A@cos~6u !1cos~6u !#13kcos~2u 22u !. ~11! cause the energy splitting between the two d levels is 4E ,it
0
2
1
2
1
2 is safe to assume that at any reasonable temperature the split-
ting is frozen in. Unfortunately the splitting is difficult to
By minimizing Eq. ~11! and using u 5u 1p we find
1
2
measure directly because most methods for coupling to the
A 22sin~4u ! d level involve changing the valence of the Mn, which would
1
52 . ~12! bring other physics in to play. The transition should be Ra-
k sin6u 1 man active, though.
Values for A/k are also listed in Table I. To summarize, it has been shown in this section that the
The most important information contained in Table I is Jahn-Teller energy of LaMnO may be written
3
that the basic Jahn-Teller energy E is much greater than the
0
stiffness k which orients the distortions from site to site. E x50 ( cos~2u 12c !cos~2u 12c !
5k
Indeed, from Eq. ~6! the ratio may be seen to be 1 3 ia i a i1a a
K /(K 1K ); as it is unlikely that the Mn-Mn force con-
2
2
1
stant K . the Mn-O force constant K , the ratio is less than 1A( cos6u . ~16!
2
1
i
1/6. The structural transition occurring at T '800Kin i
s
LaMnO is therefore of the order-disorder type, and we may
3
expect local distortions to persist for T.T . If A*T , then it is reasonable to assume that at each site
s
s
From Table I it is also clear that the anisotropy energy is u is near one of the three angles favored by the anharmo-
i
not small, although the precise value depends sensitively on nicity term, so that the system may be mapped on to a three-
8
K /K . state Potts model as previously noted. Details are given in
2 1
Now consider magnitudes of energy scales. The basic Appendix D. The result is conveniently written in a notation
scale is K ; this is related to the frequency of an oxygen in which the state of site i is represented by a vector Q with
i
1
bond stretching phonon v ox by a 1 in one place and 0 in the other two places; Q 5(1,0,0)
i