25 March 19%

PHYSICS LETTERS A

ELSE.WER Physics Letters A 2 12 ( 1996) 237-240

Theory of spin waves in a ferrimagnet with hopping impurities

Min Kang, Ma&i Tomita Department of Physics, Kobe University, Rokkodai, Kobe 657, Japan

Received 10 November 1995; accepted for publication 12 January 1996 Communicated by J. Flouquet

Abstract

Spin waves in a two-sublattice ferrlmagnet with hopping impurities are studied ~eo~tically. We exactly solve our problem within the one maguon and one imp~ty framework by the use of the Green function method. The conditions for the existence of bound states are discussed.

Theoreticai and experimental investigations of magnets including localized impurities have been widely performed in the cases of ferro- and antiferromagnetism [l-6]. However, as far as we know, a theoretical study of a ferrimagnet with impurities has not yet been made. On the other hand, studies of systems with hopping holes seem to be important due to the discovery of high T, superconductors [7,8]. Wang et al. have given the exact solution of magnon-hole bound states of a ferromagnet with hopping holes [9]. The holes were assumed to be due to the excess electrons located at the magnetic ions. We recently considered ferro- and antiferromag- nets with hopping magnetic impurities instead of non-magnetic holes [lo]. In this paper, we present a theory of ferrimagnets with hopping magnetic impurities. Of course, our theory can be easily applied to the case of localized impurities, One of the main differences between antiferromagnetism and ferrimagnetism is the existence of the gap between two bands. We see how the impurity-magnon bound states appear outside the bands, especially in the band gap, and we also see how the bound states are affected by the existence of the hopping parameter t.

Let us consider a ferrimagnet composed of two sublattices, an up-sublattice and a down-sublattice, called A-sublattice and B-sublattice, respectively. We confine ourselves to the case of one impurity and one magnon. This will be experimentally relevant if the impurity concentration is very small and theoretically our problem can be solved exactly. We assume that the impurity spin hops only on the sites of the A-sublattice. This assumption enables us to solve our problem, as can be seen below. Our Hamiltonian can be written as

H= 2f c si l &(I - t&J - rc CC;+&, (1) (id i P2

where C,j ,+ denotes the summation over the nearest neighbor site pairs and Ci the summation over the sites in the sublattice A. i, j denote the A-sublattice sites and II, m denote the B-sublattice sites ~roughout this paper. p2 is a vector between next-nearest neighbors. Sj (S,) denotes a spin operator at the site i (n). ci and ci are the

0375~%01/%/$12.00 0 1996 Elsevier Science B.V. All rights reserved PII SO375-9601(96)0005 l-5

238 M. Kang, M. Tomiru/Physics Letters A 212 (19961237-240

creation and annihilation operators for the impurity at the lattice site i obeying the Fermi commutation relations. t and E are numerical parameters. The first term in Eq. (1) represents the exchange coupling between nearest neighbor spin pairs. It is given by J(1 - E) = J’, if the impurity is on the site i. In this paper, we assume E < 1. That is, the impurity spin is antiparallel to the neighboring spins in the classical ground state. When E = 1, our system is composed of isolated spins and a ferrimagnetic system with hopping holes. The second term represents the hopping term of impurities. When f = 0, the impurities are localized. The magnitude of the spin operator Si (S,) is denoted by S, (S,). ‘Ihe ma~i~de of the impu~ty spin, s”, is denoted by 5” = ,!&(I - 51, where 5 is a numerical constant with t< 1. Thus, Si is generally written as Sj = &(I - ,$c$TJ When the origin of the hopping impurity is an excess electron or a hole, Si = S, + 4 or S,, - 3. Therefore, - 1 < 5 < 1, On the other hand, if we consider a substituted localized impurity, LJ can take various values. For example, when the host ion is Cu2+ and the impurity ion is Co*+, S, = + and S = f. Hence c= - 2.

Let us consider a one-magnon and one-impurity state which is defined by

~Aija;cf + ~B,,b,c; i,i i.n (2)

Here IO) denotes the spin wave ground state for the ferrimagnet in which there are neither magnons nor impurities. at and b, which are Bose operators, are spin deviation operators belonging to the A- and B-sublattices, respectively [ 1 1 - 131.

Using * and H in the SchrWinger equation, we obtain the secular equations for A,, and B,,. Since our Hamiltonian and also the secular equations for A,, and B,,, are invariant under the translation of the origin of coordinates, one may use the following substitution,

Aij = exp(iK l Rjj) A,( rij), Bi,=exp(iK*Ri,)A,(ri,), (3)

where Rij = i(i +j), Ri, = $(i + n), rii = i -j and ri,, = i - n. By using the above relations, we obtain the secular equations for A,(rij) and BR(rin),

(-U+2JSBZ)A,(rij)-2J~~~e~i”‘P/Z~~(~ij-~)-t~e~iK’~2~2A,(~ijf~2) P P2

+ S( i, j) f

-~JS,ZEA,( rij) + 2fa( E + 82’) ~e-iK*p/2BK P

frij-Pf) =O,

(-U-2JS,Z)B,(~,.,) +2J~~~e-iK’P’ZAK(fin-~) -t~e-iK’P2’28~(fin+~~) P PZ

+s(n, ~+P~)[~Js,(E+~~)B~(T~,) -~J~(E+E~‘)~~~‘~~‘~A~(~~,+P*)] =0, (4b)

where U represents the energy measured from the ground state energy. 5 denotes the number of nearest neighbors. S( i, j) is the usual Kronecker delta function. p and pi denote the vectors directed between nearest neighbors. E’ and 5’ are, respectively, defined by E’ = 1 - E and 6’ = 1 - m.

In order to solve the secular equations, we introduce the following Green functions,

M. Kung, M. Tomita/Physics Letters A 2f2 (1996) 237-240 239

G( n, j) = ; c- * t

q) (-2Jf53&)e’4*‘~J,

G(n, M) = ; c 1

9 W(K* 4) ( -Uf2JSs~---2;ry,)eiq”m~. (5)

Here, W(K, q), y and yz are defined by

W(K, q)=(-u+2Js,s-_?-*ty,)(-u-2Js*z-z,ty,) +(2J~-?+y)z

YE _!.$ ~e-w/2+q)*P,

P

yz = & ~e-‘tk+W,_

Pz

X2 is the number of next-nearest neighbors. We note that the Green functions agree with those of Ref. [6], if we assume that S, =Su, t=Oand K=O.

By use of the Green functions of Eq. (51, we can obtain the bound states outside of the continuous energy bands. The energy bands, ub *, can be obtained from Eq. (4) as f 14,151

‘bt= u, ,/252~ = -z,nt’+i[~-1/\lB*yi(J8-I/J8)i+4(1-r2~]~ (6)

where t’ = t/2 JXm and j3 = S,,‘S,. We describe in this paper the result concerning the impute-Macon bound states. Roughly speaking, when

J’ is larger than J, bound states appear above the ub+ band and below the ub_ band but no bound states appear in the gap. On the other hand, when J’ is smaller than J, bound states appear in the gap but not above the u,,+ band and below the ub _ band. The bound state energy above the z++ band, uA, and thy below the Mb_ band, ug, are defined by

uA = (U- E+)/2J.T@3& uB=(U-E-)/232-,

where E+(K) is the top of the &,f band and E_(K) is the bottom of the u,,_ band. In this paper, we show some calculations for the two-dimensional base-centered lattice. A detailed

calculation for the tree-dimensional my-centered-cubic lattice will be reported elsewhere [lo]. First, we show the t-dependence of the bound state energies, in Fig. 1, at K = (0,O) in the case of

S/S, = 2.0, J’/J = 2.5 and S,/S, = 2.0. One s-like mode appears in the region above the Us+ band and s-, p-

Fig. I. r-de~dence of bound state energies at Ki = (0,O) in the case of S/S,, = 2.0, J’/J = 2.5 and 3, /S,, = 2.0. Solid lines: bound states below the I+,__ band; dashed line: bound state above the r+,+ baud. See text for the definition of f’, uA and I(~.

240 M Kang, M. Tomitu /Physics Letters A 2 I2 (19%) 237-240

Fig. 2. Bound states below the ub_ band and above the q,+ band as J’/ J = 2.5 and Sa /S, = 2.0. Bound states do not appear in the gap.

functions of K in the case of f /2 J.Zm = 0.125, s/s, = 2.0,

Fig. 3. Bound states in the band gap for the case of r,/‘2Jsm= 0.125, S/S, = 2.0, S,, /.S, = 2.0 and several values of J’fJ between 0 and 1.1. Bound states above the ub+ band and below the I+,_ band do not appear.

and d-like modes appear in the region below the ub_ band. Bound states do not appear in the band gap for these parameter values.

In Fig. 2, we show the energy, u = U/2 J.Zm, as a function of K for the case of S/S, = 2.0, J’/J = 2.5, S,/S, = 2.0 and f’ = 0.125. Bound states do not appear in the gap for these parameter values.

In Fig. 3, the bound states in the gap are shown in the case of S/S, = 2.0, S,/S,, = 2.0 and t’ = 0.125 for several values of J’/J between 0 and 1.1. Here E = 1 - J’/J which should be smaller than unity. The case of E > 1 is meaningless.

The authors wish to express their sincere thanks to Professor 0. Nagai for suggesting considering this problem and valuable discussions. One of the authors (MK) wishes to thank Professor H.T. Diep of the Universite de Cergy-Pontoise and Professor E. Ilisca of Universite Paris Vi1 for valuable discussions.

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