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MAGNETIC EXCITATIONS IN FERROMAGNETS AND ANTIFERROMAGNETS WITHIN THE LOCAL EXCHANGE APPROXIMATION

D.M. EDWARDS and M.A. R A H M A N Department of Mathematics, Imperial College, London SW7 2BZ, England

An expression for the spin wave stiffness constant D is obtained within Callaway and Wang's [1] scheme for calculating the transverse spin susceptibility x(q, to) of a ferromagnet. The method is related to the random phase approximation and it is shown that the results are greatly simplified by neglecting local field effects. This approximation yields much simpler expressions for x(q, to) in the ferromagnetic, antiferromagnetic or ferrimagnetic case than are obtained in previous multiband schemes.

1. I n t r o d u c t i o n

In the local exchange or local spin density approximation an electron of spin tr moves in a local potential V¢(r) in the ground state of the system. The ground state may be regarded as the Har t r ee -Fock ground state of the hamil- tonian

H = ~ [p~/2m + vo(r3] i

+ ~, ~ f d3rF(r)~(r - r,)~(r - ri), (1)

where p~ and ri are the momentum and position of electron i and

Vo = (p+ V+ -p~ V~)/(p+ -p~),

F = (V~ - Vt)](p~ - p { ) . (2)

Here p~(r) and p~(r) are the spin densities. From this point of view the calculation of the transverse spin susceptibility x(q, to) by Cal- laway and Wang [1] (CW) corresponds to the" t ime-dependent Har t r ee -Fock approximation which is equivalent to the random phase ap- proximation (RPA). CW only considered a fer- romagnet explicitly but in fact their result holds equally well for an antiferromagnet or ferrimag- net. Our work relies on reformulating CW's eq. (34) as an integral equation rather than an infinite determinant. We note that in a transverse mode of wave-vector q the local magnetization Mo(r) precesses about the equilibrium z direction with a small cone angle mq(r). The local exchange field Vf(r) = ½(V ~ - V t) then has a perturbing trans- verse component of amplitude Vf(r)mq(r) exp [i(q • r - tot)] and CW's self-con-

sistency condition may be written

--glib ~ , <lk + q J, l exp ( iq . r)Vf(r)mq(r)lnk ~ )

Ink htonk "f ,lk+q ~, + ~to

x [N~ t (k) - NI~ (k + q)]qJ*~ (k, r)qJt ~ (k + q, r)

x exp ( - i q • r) -- mq(r)Mo(r). (3)

Here ¢J,~(k,r), with the corresponding ket [nktr), is the spatial part of the electron wave function, n being a band index, and the rest of the notation follows CW. The frequencies of the modes, which include spin waves, are those values of to for which eq. (2) has a non-trivial solution m~(r) # O.

2. F e r r o m a g n e t i c case ( c ub ic crystal)

We first consider long wave-length spin waves where hto = D q 2 and mq(r) may be expanded as mq( r )= l + q . m l ( r ) since a q = 0 spin wave corresponds to uniform precession. By mul- tiplying eq. (3) by Vf(r), integrating over a unit cell and expanding in powers of q we find [2]

D _

~2

× Et~ ~ - En~ t J" (4)

Here N~, N~ are the total numbers of 1' and spin electrons, N = N~ + N~ and m~(r) satisfies a certain integral equation. Eq. (4) is of the exact form given by Edwards and Fisher [3] and it may be shown that the second term in

Physica 86-88B (1977) 341-342 © North-Holland

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brackets corresponds to the spin current res- ponse function )O calculated using the Hamil- tonian (1) and the RPA. Calculation of X~ in the Hartree-Fock approximation, thus neglecting the vertex correction, corresponds to putting ml = 0 in eq. (4). This implies that the spin wave amplitude mq(r) is constant within the unit cell so that the local field effect is neglected. This approximation obviously fails for a compound with more than one atom per unit cell as discussed in section 3, but may be good for an elemental ferromagnet such as iron. It may be shown that the important property D ~ 0 as Nr - N ~ ~0 , in the very weakly ferromagnetic limit, is preserved. An explicit formula for x(q, to) may be obtained when local field effects are neglected in this way and the equation for the modes, corresponding to the denominator of g vanishing, is obtained from eq. (3) by putting m~(r) = 1, multiplying by Vf(r) and integrating over a unit cell.

3. Two sublattice case (ferromagnetic compound, antiferromagnet or ferrimagnet)

mq must now be allowed to be different on the two sublattices, although local field effects are neglected within each atom. Thus we put mq(r) = Aq + Bq exp (iQ • r) where Q is a mag- netic reciprocal lattice vector. We substitute this in eq. (3) and obtain two equations by multiply- ing by Vt(r) and Vf(r)exp ( - i Q . r), each equa- tion being integrated over the magnetic unit cell. Elimination of Aq and Bq yields the following equation for the modes:

[h + A(q, q)][h + A(q + Q, q + Q)]

- I A ( o , q + O)t 2 = o, (5)

where

h = tx~ 1 f Vf(r)Mo(r) d3r

and

A(q, q') = ~ (lk + q $ IVf(r)exp (iq'. r)lnk "~ ) In k

× (nk ~ [Vf(r) exp ( - iq • r)[lk + q ~, )

[ N , t ( k ) - N ~ ; ( k +q)] x (6)

(hto~k r,l~q ~ + hto)

The corresponding expression for x(q, to) con- tains form factors correctly and is much simpler than in previous multiband formulations by Gil- lan [4] and Young [5] for the antiferromagnetic case. The full consequences of eq. (5), and the corresponding equation for the Hubbard model, will be discussed in a subsequent paper. Recent work by Liu [6] has some similarities with the CW approach. However, he introduces ad- ditional local spin degrees of freedom which effectively double the moment (the inertia of the system) without increasing the stiffness. Thus, for example, he finds D has one half its correct RPA value.

References

[1] J. Callaway and C.S. Wang, J. Phys. F5 (1975) 2119. [2] CW have recently obtained a similar result independently

(Solid State Commun, 20 (1976) 255). However, they make an approximation equivalent to taking m, = 0, thus ne- glecting local field effects.

[3] D.M. Edwards and B. Fisher, J. de Physique 32C1 (1971) 697.

[4] M.J. Gillan, J. Phys. F3 (1973) 1874. [5] W. Young, J. Phys. F5 (1975) 2343 and to be published. [6] S.H. Liu, Phys. Rev. B13 (1976) 3962.