L O C A L M A G N O N M O D E S IN O N E - D I M E N S I O N A L F E R R O - A N D A N T I F E R R O M A G N E T S *

B. K O I L L E R Departamento de Fisica, Pontificia Universidade Catolica, 20.000 Rio de Janeiro, Brazil

and S. M. R E Z E N D E Departmento de Fisica Universidade Federal de Pernambuco, 50. 000 Recife, Brazil

A Green's function formulation, solved exactly with the use of transfer matrix techniques, is used to calculate the frequencies of the local p- and s-modes associated with a single impurity spin for the typical 1D ferromagnet CsNiF 3 and the antiferromagnet TMMC at zero temperature. Study of the dynamic response of the so-mode shows that its susceptibility is enhanced by the proximity of the k = 0 magnon energy.

In this work we treat the p rob l em of one impur- ity in a magnet ic chain within a formula t ion which does not require knowledge of the pure lattice Green ' s function, and turns out to be much simpler than the usual t rea tments of this p r o b l e m [ l , 2].

We consider initially a chain of spins S~, coupled by isotropic neares t ne ighbor fer romagnet ic ex- change. The host crystal magnet ic ions have spin S, exchange constant J and an iso t ropy field H A. The single impuri ty at site i = 0 has pa ramete r s S ' , J ' and H~. We denote by G~,i the Fourier t rans form of the double- t ime Zuba rev Green ' s function for the deject lattice.

The equat ion of mot ion for G~j is readily de- rived within the o n e - m a g n o n approximat ion . Fo r zero tempera ture we obtain [3]

where

= - E (1) J

= X 2h-lJij, Sy + (2)

o:~, = 2 h - l j i y V ( S i S j , ) , (3)

in which 7i = git~B/h. The infinite set of coupled equat ions (1) can be solved exact ly by the in- t roduct ion of the t ransfer function:

Gn,j _ T(~o) = G,,,-----~; + v ( e 2 - 1) - e, (4)

*Work partially supported by Financiadora de Estudos e Pro- jetos (FINEP) and Conselho Nacional de Desenvolvimento Cienttfico e Tecnolrgico (CNPq).

for In' - j[ = 1 + I n - J[ and n, n' 4= 0. Here e = (~0 - ~01)/2~o T, ~o I = ~O2~o = 4 J S / h + ")'H A and ¢o T = ~o~n±l~ o = 2 J S / h . The sign of the square root in (4) can be shown to be the same as the sign of e. The diagonal e lements of the Green ' s funct ion at the impur i ty site and at its nearest neighbors are obta ined f rom (1) and (4) t a k i n g j = 0 a n d j = 1:

(5) Goo = C(o: ) '

GI 1 (60 -- ~00I)B(~0) - (t .oT1) 2

= B ( o : ) C ( o : ) ' (6)

where B(w) = ~0 - ~0~ + WTT(~0) and C(~o) = (w - ~0I)B(~o) - 2(~0~1) 2. Note that the t ransfer funct ion (5) is complex for c0 in the interval (~o I - 2~o v, ~o I + 2~0v), i.e. - 1 < ~ < 1, which is the f requency range of the propaga t ing spin waves. The poles of Goo cor respond to symmetr ic impur i ty modes. There is one such mode below the m a g n o n band (So) if 8a < 0 and one above the m a g n o n band (sl), if a[ f l + 4 / ( 4 - 8)] > 2, where a = J ' / J , fl = S ' / S and 8 = ( g ' g a H ~ - g g a H A ) / 2 J S is the rela- tive anisotropy parameter . The ant i symmetr ic or p - m o d e frequency cor responds to the zero of B(w). These local modes frequencies are identical to the results of White and H o g a n [1] and Tonegawa [2]. In fig. 1 we show the frequencies of the s- and p - m o d e s as a funct ion of a for several values of 8, keeping fl = 2, which applies to Fe 2+ impurit ies in the 1D anisotropic fe r romagnet ic CsNiF 3.

The spacial localization of the local modes m a y be investigated f rom the weights of the poles corre- sponding to these modes in the local density of states at the impuri ty and at its nearest neighbors.

Journal of Magnetism and Magnetic Materials 15-18 (1980) 336-338 ©North Holland 336

B. Koiller and S. M. Rezende/ Local magnons in I D magnets 337

I ] I I I 6 : 2

-I

4 s' Z $~ -2

T : 3

2

LI.I O t t n u

- I

- 2 S o

- 2 ~ ; J t J _ _ 0 I 2

od-- d ' / J

Fig. 1. Frequencies • = (o~ - ~ i ) / 2 o ~ of the s- and p-modes for a ferromagnet ic linear chain as functions of the exchange con-

stants for various values of the anisotropy parameter 8 =

( g' p, aH'A - g#BH A) / 2 J S .

0 9 .

- - 6 = - 2 / __ 6=_1 j~ 0.6- - - 6 = - 0 2 / -

/" / / :

v , [ i )

~ s , - mode wavefunction ( o ~ , i )

- 3 - 2 -I O I 2 3 i

Fig. 2. W a v e f u n c t i o n s ~ko(i) of the s o loca l m o d e a r o u n d the

i m p u r i t y site i = 0 in a f e r r o m a g n e t i c c h a i n fo r several va lues o f

the a n i s o t r o p y p a r a m e t e r 8. a = J ' / J = 1, fl = S ' / S = 2.

In fig. 2 the wavefunction for the s o mode is plotted for different values of 6, for a = l and fl -- 2. The modes nearest to the band edges are less localized, and in this situation the Ising approximation gives very poor description of the local mode frequency. Modes with frequencies far from the continuum become Ising-like and are well localized at the impurity.

The response of the spin system to k "~ 0 elec- tromagnetic radiation such as used in Raman scattering or infrared absorption experiments can be described by the susceptibility, written in terms

of Green's function [4]

X(60) = ~ 2 (Sisj)l/2gigjl~2Gij(60). (7) l , J

Owing to the fact that the Green's functions matrix elements are related to each other by powers of the transfer function (except Goo), the summation in (7) can be grouped into geometric series, and therefore can be performed exactly. For simplicity we a s s u m e S ' g ' 2 = Sg 2, and for frequencies near a pole of G, i.e. near a local mode with frequency 60x, the susceptibility is

x ~ ( 6 0 ) - - - A 1 + T(60)[1 - 2R x ] + 4RoXl

60 -- 60x 1 - T(60)

+ 2RL, l, [ l - r(60)]2J (8)

where A = 2Sg2l.t2/V, R~ is the residue of Gij(60 ) at 60x. Notice that Xi~,p = A/(oJ - 60x) is the sus- ceptibility of one isolated impurity spin; therefore the lat t ice-impurity spin interaction may change drastically the response of the local mode to the external field. Since T(60k=0)= 1 at the uniform mode (k = 0 magnon frequency) the response of the local mode may be largely enhanced by its proximity to the bot tom of the magnon band. Local mode enhancements of the order of l0 2 have been observed in the 3D ant i ferromagnets CoF2:Mn [5, 6] and FeFE:Mn [7], where the Mn mode frequency is just below the magnon band. This effect makes the impurity mode as intense as the uniform resonance mode. The ratio ~/ = X/Ximp for the mode s o calculated from (8) is presented in table 1. Notice that the s o mode is enhanced by the impurity-latt ice interaction. On the contrary, the s 1 mode is greatly weakened by this interaction. This effect can be understood as resulting from the action of the transverse exchange field of the neighbors at the impurity spin [7]. While the en- hancement of the s o mode increases as its f r equency a p p r o a c h e s the k = 0 m a g n o n frequency, it is not as large as it would be for a 3D spin system with the same local mode frequency relative to the magnon band and the same ex- change constants. This is in part due to the smaller number of neighbors and consequently smaller ex- change field of the 1D system.

338 B. Koiller and S. M. Rezende/ Local magnons in ID magnets

TABLE 1

Energy c a n d e n h a n c e m e n t factor r/of the s o local mode for = 1 a n d f l = 2

- 2 - 1 -0.2 -0.1 c - 1.54 - 1.20 - 1.01 - 1.004 7/ 3.65 5.84 21.86 41.54

The calculat ions can readily be extended to an ant i ferromagnet ic l inear chain with one impur i ty coupled either ferro- or ant i fer romagnet ica l ly to

the host. Depend ing on the values of a, fl and 8 the p and

s 1 modes may lie above the m a g n o n band. Since the anisotropy of T M M C is very small the m a g n o n gap is small and no localized s o mode is expected to exist. The results for 1D ant i fer romagnets will be reported elsewhere.

The observat ion of local m a g n o n modes in one- d imens ional ferro- and ant i fer romagnet ic materials

should not be much more difficult than in the 3D materials extensively explored. All the techniques

used to study magnet ic defects, R a m a n and Neu- t ron scattering, far infrared laser spectroscopy and N M R are appropriate for these observations. The

fact that the ordering temperature of 1D materials is small presents some complicat ions, but these are not major ones. In fact, these measurements should be feasible even at temperatures above the order ing temperature, as long as the s p i n - s p i n correlat ion

length is larger than the spread of the local mode. Since this spread accounts for only a few lattice spacings, depending on the proximity to the mag-

n o n band , local magnons may exist at temperatures as high as 10 K in T M M C .

R e f e r e n c e s

[1] R. M. White and C. M. Hogan, Phys. Rev. 167 (1968) 480. [2] T. Tonegawa, J. Phys. Soc. Jap. 33 (1972) 348. [3] J. B. Salzberg, C. E. T. Gon~alves da Silva and L. M.

Falicov, Phys. Rev. B14 (1976) 1314. [4] R. A. Cowley and W. J. L. Buyers, Rev. Mod. Phys. 44

(1972) 406. [5] B. Enders, P. L. Richards, W. E. Termant and E. Catalano,

AIP Conf. Proc. 10 (1972) 179. [6] A. S. Prokhorov and E. G. Rudashevskii, JETP Lett. 22

(1975) 99. [7] R. W. Sanders, V. Jaccarino and S. M. Rezende, Solid State

Commun. 28 (1978) 907.