Physica B 156 & 157 (1989) 205-207 North-Holland, Amsterdam

SPIN CORRELATION FUNCTIONS IN RANDOM HEISENBERG ANTIFERROMAGNETS

Jun-ichi IGARASHI Department of Physics, Faculty of Science, Osaka University, Toyonaka. Osaka 560, Japan

Localization effects of spin waves on spin correlation functions are studied for Heisenberg antiferromagnets with random uniaxial anisotropy. Transverse spin correlation function is shown to be strongly affected by spin-wave interaction near the mobility edge.

Recently there has been much progress in the understanding of Anderson localization of elec- tron in a random potential [l] as well as light in a random media, while only little works have been done on the localization of spin waves [2,3]. In this paper we study the localization effects of spin waves in random antiferromagnets on the spin correlation functions which are determined by neutron scattering experiments.

We employ the Heisenberg model whose Hamiltonian is described by

H = 25 Zj s;sj - 2 Q(S,‘)’ ) (1) I I

where J is assumed to be a positive constant. Di denotes a uniaxial anisotropy energy which is a random variable with configurational average: (D,)imp = Do, ((D, - Do)*)imp = A/(2S)*, with S being the magnitude of spin. In our units, h=k,=l.

Applying the Holstein-Primakov transforma- tion to spin operators and using the Bogoliubov transformation, we get

H= c &+Q + KPk) k

+ c

k,k,k,k,

(C~P,k,,k,,k,(Yk:(Tk:(Yk,(Yk,

+ C~~k2,k,,kqP~,PK:Pk,Pkq

+ C~f.kz,k),k~(Y~,P~~(Yk,Pk4)

x S(k, + k, - k, - k4)

(2)

where (Ye and Pk represent the annihilation operators of two kinds of spin waves with momentum k in a uniform medium with the anisotropy energy D,. The second and third terms represent the spin-wave interaction and the scattering of spin waves due to the anisot- ropy energy deviating from the average value; neglected are the terms with nonconserving spin- wave number, whose effects can be included by renormalizing A [3].

First we consider the transverse spin correla- tion functions for total spin and staggered spin. Within the harmonic approximation, both are the same except for unimportant prefactors:

P(o))-’

FT(k, w, - I1 + nT(w)l (o _ Ek)2 + [2T(o)1-2

(2+))-’

+ IzT(@) (0 + Eky + [27(w)]-* ’ (3)

where n,(w) is the Bose distribution function, ll(e”‘r - 1). T(W) denotes the elastic lifetime, which ’ estimated l/T(O) - 2np(o)A~~~(4D,), with p(o) ard z being the density of states of spin waves and the number of nearest neighbour sites. The spin-wave peak is broadened by the elastic scattering, while there is no critical behavior with o passing through the mobility edge, .sc, below which spin waves are localized.

Spin-wave interaction may drastically change the situation through a temperature dependent shift of the spin-wave peak, although (3) still holds. We estimate the shift by taking account of

0921-4526/89 I$03 SO 0 Elsevier Science Publishers B .V. (North-Holland Physics Publishing Division)

206 J. Igarashi I In random Heisenberg antiferromagnets

k k k k k k

Fig. 1. Self-energy due to spin-wave interaction (dashed

lines) in the lowest order approximation. (a) For a nonran-

dom system; (b) for a random system. The hatched blocks

represent the diffusion propagator.

the lowest-order terms of interaction, which shown in fig. 1. For a nonrandom system, becomes

‘k - Ek = 7 (C;;yk + c;;k, + C;;kq

+ q&k + c;~k,)n,(Eq) ’

is it

where ck is the renormalized energy of spin wave. Note that the leading terms in (4) are cancelled in the long-wavelength limit that k and q -+ 0, and hence (4) gives only a small contri- bution.

This cancellation breaks down in random sys- tem, since only the interaction between the same kind of modes (a - (Y and p - p) is enhanced by the diffusive motion of spin waves, leading to interaction effects larger than those in ferromag- nets, where the spin-wave interaction is neglig- ible in the long-wavelength limit even in a ran- dom system. We use a perturbational expansion with respect to l/ kl with I being the elastic mean three path; by using the diffusion approximation in the vertices shown in fig. l(b), we get the energy shift

‘k - ‘k - Jzn,(E,)a:

X

+

1 1 2 4’rr DO(Ek)C(Ek)T2(Ek)

1 1 -

I 8~ [DO(~k)~(~k)]3’2 ’ (5)

for TT(F~) 9 1, where u,, and c(sk) denote the lattice constant and the velocity of spin wave, and Dg(ek) = { c(F~)~~(E~). The imaginary part of the self-energy in this approximation is found to be negligible with TT(E,) % 1. Note that by

reversing the direction of the loop in fig. 1 (b). we get the diagrams involving the Cooper prop- agator, whose contribution should be the same as that of the diffusion propagator neglecting the momentum dependence of the interaction. Near the mobility edge, the higher-order corrections due to randomness get important to reduce the diffusion constant, which we estimate as D(k =

0, w = 0, E) = DO(~)(~’ - F~)/(E* - E:) with F,, denoting the gap in the spin-wave dispersion, by taking account of the “maximally crossed” diag-

rams. Replacing D,, by D in (5), we get large shift near the mobility edge in the spin-wave dispersion, as shown in fig. 2 with parameters roughly simulating the mixed antiferromagnet. Fe,Mn ,_*F2. This is a crude approximation which may break down near the mobility edge,

since the interplay between randomness and in- teraction becomes important there. Discussing spin waves in the “critical region” is beyond the

i_i JSz

t

t

129 i +

I EC I %

1.28

Fig. 2. Spin-wave dispersion in the random system with

S =2, z = 8, D,, = l.SJ, Ai(% (1.2J)‘. a, is the lattice

constant. The dashed line represents the critical momentum

corresponding to the mobility edge, in the vicinity of which

our theory is inapplicable.

J. Igarashi I In random Heisenberg antiferromagnets 207

scope of this paper. Note that the velocity of spin wave gets close to zero leading to an enhance- ment of the density of states, and the elastic lifetime may hence become shorter. Such a large linewidth could be observed in the spin-wave peak.

The effects of random exchange interaction

In summary, we found that randomness in uniaxial anisotropy could cause the localization of spin wave in the long-wavelength region, and thereby the lifetime becomes anomalously short due to the enhanced effect of the spin-wave interaction for nearly localized spin waves.

are usually overwhelmed by the effects of ran- dom anisotropy, without which spin waves are not localized in the long-wavelength region but in the high-energy region just as with phonons and photons. The mixed antiferromagnet, FexMnl_xF2, may be a candidate to test our prediction, since the anisotropy energy is quite different between the Fe and Mn atoms.

References

[l] For a review, see P.A. Lee and T.V. Ramakrishnan, Rev. Mod. Phys. 57 (1985) 287.

[2] R. Bruinsma and N.S. Coppersmith, Phys. Rev. B 33 (1986) 6541.

[3] J. Igarashi, Phys. Rev. B 3.5 (1987) 5151.