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53 COOPERATIVE JAHN-TELLER EFFECT AND ELECTRON-... 8437

implies there is a Jahn-Teller distortion with the long axis e 2E~u,$h a %!/T
n
~
along x, Q 5(0,1,0) means y and ~0,0,1! means z. Then P~u!5( x h 12x! 62n h . ~19!
i
$h a % Z~$h %!
a
W
a W
W
W
5k
E Potts ( Q I Q i1c 1J8( Q •Q i12a , ~17! Here $h % is a distribution of holes on sites adjacent to the
a
i
i
ia ia distinguished one, n is the number of holes in that particular
h
configuration, and
a
with I a bond-direction-dependent interaction given in Eq.
~D3! and J8,0 a ferromagnetic interaction between Z~$h %!5E p du
‘‘straight-line’’ second neighbors, which is of order k/A and a 0 p e 2E~u,$h a %!/T . ~20!
was apparently not neglected in previous work. The second-
neighbor interaction is an approximation to the true interac- The energy may be written in terms of the average values
tion, as discussed in Appendix D. The three state Potts model of the cosine and sine on the other sublattice, c5^cos2u&
has a transition in the x-y universality class as, therefore, and s5^sin2u&,as
does Eq. ~16!. The second-neighbor ‘‘ferromagnetic’’ cou-
pling lifts the degeneracies which lead to subtleties in the
behavior of the usual Potts model. The estimates of A sug- E~u,$h %!52k( ~12h !cos~2u12c !
a
a
c
a
gest that the extreme Potts limit will not provide a good
quantitative description of LaMnO . 3@~ccos2c 2ssin2c !12h 1bh #. ~21!
3
a
a
c
c
The quantities c and s satisfy a self-consistency equation; the
IV. HOLES
linearized equation giving T may be written
s
This section discusses the effects of added holes. It is
clear from Eq. ~7! that a hole on site i eliminates the Jahn- c52E p du P~u!cos2u. ~22!
Teller distortion on site i and leads to a potential, 0 p
bkcos(2u i1b 12c ), which acts to orient the distortion on
b
ˆ
site i1b so that its long axis is along b. Thus added holes The derivation and evaluation of this equation are given
in Appendix E. An analytic treatment is not simple except in
lead both to site dilution and to a field which tends to orient
some of the neighbors of the hole in directions not compat- the limits A→0or A→`~arbitrary b) and b→0or b→`
ible with long-range order. ~arbitrary A!. For A50,
If A.0 ~as seems to occur in LaMnO !, the angles fa- 3k 2
3
I
2
1
vored by holes are compatible with the angles favored by 15 F 12xS 11 4I 2 DG . ~23!
1
anharmonicity; if A,0 an interesting competition arises, T s I 0 I 0
which will not be discussed here. Here the I are Bessel functions of imaginary argument
In the A@0 limit the effect of added holes is particularly n
s
transparent. By following the derivation that led to Eq. ~17! ib/T .
x
one finds that a hole on site i produces a term in the energy In the A→` limit, T (b,x) satisfies
3k 623e 23b/2T s 16e 23b/T s 2
W b W
!
E i 5bk( R •Q , ~18! 15 F 12x ~112e 23b/2T s 2 1Ox G . ~24!
T
hole i1b s
b
x
with R 5(21,1/2,1/2), etc. Thus in this limit a hole mani- For b50, the x 2 and higher terms vanish and
14
festly produces a field which tends to orient the spins on T 53k(12x) as expected for simple site dilution. The
s
mean-field theory overestimates the x at which T vanishes
neighboring sites. c s
A Monte Carlo investigation based on Eq. ~7! or on Eqs. because it does not contain the physics of percolation. As
b/T is increased, the coefficient of dT /dx increases; for
~17!, ~18! would be desirable. Here simple arguments are s s
given to estimate T (x). Assume the hole positions are un- b/T →`, T →3k(126x), suggesting x '0.16. Compari-
c
s
s
s
correlated with each other or with the configuration of Jahn- son to the percolation argument given previously suggests
that this is an underestimate. The general result, however, of
Teller orderings. To estimate the critical concentration x ,of
c
holes at which ordering vanishes, note that for site-diluted a T (x) which drops rapidly as x is increased and depends
s
somewhat on model parameters ~and so on materials! is in
systems (b50), T vanishes when the occupied sites do not
s
percolate. 14 For the simple cubic lattice, the percolation reasonable accord with data. Note, however, that at low T
threshold is about p 50.3, 14 and so x (b50)50.7. Of quantum effects involving motion of holes will become im-
c
c
course for such large values of x the model is not valid. For portant.
b→`, each hole eliminates five sites ~itself and four neigh-
bors, two remain approximately correctly oriented!, implying V. CONCLUSION
125x 50.3 or x (b→`)>0.14.
c
c
Alternatively, one may use mean-field theory to estimate A classical model for La 12x A MnO has been analyzed.
3
x
T (x). The fundamental object in mean-field theory is the It is known that doping on the La site changes the valence of
s
probability distribution P(u) of the angle on a distinguished the Mn site in such a way that the mean number of outer
site in an effective field depending on the average values of d-shell electrons on the Mn is 12x. The holes were assumed
the angles on the adjacent sites and on whether or not holes to be classical, so that each Mn site is occupied, with prob-
are present. The assumption of uncorrelated holes implies ability 12x, or empty, with probability x. The hypothesis of