SPIN WAVE INTERACTIONS IN SOLIDS

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SPIN WAVE INTERACTIONS IN SOLIDS

D. Edwards, B. Fisher

To cite this version:

D. Edwards, B. Fisher. SPIN WAVE INTERACTIONS IN SOLIDS. Journal de Physique Colloques, 1971, 32 (C1), pp.C1-697-C1-698. �10.1051/jphyscol:19711245�. �jpa-00214074�

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JOURNAL DE PHYSIQUE Colloque C I, supplkment au no 2-3, Tome 32, Fkvrier-Mars 1971, page C 1

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⁶⁹⁷

SPIN WAVE INTERACTIONS IN SOLIDS

D. M. EDWARDS and B. FISHER

Department of Mathematics, Imperial College, London S. W. 7, England

R6sum6. - On a obtenu une formulation exacte pour l'knergie Dq2 d'une onde de spin dans un solide ferromagnk- tique. Aucun modkle particulier n'est suppose initialement, mais la dependance en tempkrature de D due aux interactions magnon-magnon est calculCe par les moddes de Heisenberg et de l'electron itinerant. Le rksultat de Dyson pour le premier modkle est obtenu simplement, et l'approche de Izuyama et Kawasaki pour le mod6le itinkrant correspond B l'approximation de Born de Dyson. Une estimation de D z , le coefficient de T s l z dans D est faite pour le nickel. La duree de vie des ondes de spin est discutke.

Abstract. - An exact formula is obtained for the energy Dq2 of a long wavelength spin wave in a ferromagnetic solid. No particular model is assumed initially but the temperature dependence of D, due to magnon-magnon interactions, is investigated for the Heisenberg and itinerant electron models. Dyson's result for the former model is obtained simply and the approach of Izuyama and Kawasaki to the itinerant model is seen to correspond to Dyson's Born approximation.

An estimate of D2, the coefficient of the ^T-512term in D, is made for nickel. Spin wave life-times are discussed.

I. Introduction. - Any system which has rotational symmetry with regard t o spin has the property that its Hamiltonian X commutes with S l , the total spin step-down operator. The generalised transverse susceptibility ~ ( q , o) is defined by

From the equations of motion of the Green's function we find [l]

(2) where

X J ( ~ , a ) =

j

^dt^<I^~;(t),^JT,⁺^eFiwi ⁽³⁾

with AqJi defined as [ S i , XI. Thus J i is a spin current operator. For. small q and o the susceptibility is dominated by the spin wave pole at hw = ~ qso ~ , that

By comparing (2) and (4) in the limit o ^-t0, q / o ^-t0, one obtains, to order q2

x

(

^hq< [J;, s'J > - A2 q2 lim lim X, . (5)

a-0 q+O

1

This is an exact formula for D valid for any metallic or non-metallic ferromagnet, or for a non-ferromagnetic material in a static magnetic field. In the latter case o is replaced by o - o,, where o, is the Larmor frequency. Clearly, from (2), X, contains the spin wave pole but the limit in (5) is finite because the residue tends to zero as q ^-,0. For a gas with short-range interactions ~ ~ ( 0 , o ) is equal to x:(O, w), the << irredu- cible ^)> part, and (5) reduces to the result of Ma et al. [I]. An equivalent formula was derived earlier [2]

for a strong itinerant ferromagnet at T = 0. Our

formalism for x ensures that the corresponding neutron scattering intensity is always proportional to < SZ >

in the long wavelength limit, a result stressed by Marshall and Murray [3] for the Heisenberg model and observed in metals by Stringfellow 141.

For a general system, insulating or metallic, of interacting electrons in a periodic potential, the first term in curly brackets in (5) becomes ti2 q2 n/(2 m), where n is the total number of electrons, and the spin current J t in X, is the component of m-I pi a; in the direction of q. To discuss the temperature depen- dence of D, in particular the T5I2 term which arises from magnon-magnon interactions at low temperatures, more specific models are considered in the following sections. In the cases considered D is real, so that the magnon lifetime appears in a higher power of q and cannot be discussed rigorously by the present method. However it is plausible that the inverse lifetime is proportional to q2 ImXJ(O, D ~ ~ ) , and this leads to correct results for the Heisenberg model.

11. Heisenberg ferromagnet. - For the Heisenberg hamiltonian

where ho,(q) = [J(O) - J(q)] S is the non-interacting spin wave energy and for low temperatures

This comes from expanding Si in terms of S - and S' and is exact for S = $ [3] [5]. On substituting (6) in ( 5 ) one finds that (5), with J: replaced everywhere by K:, gives the low temperature correction to hw,(q) due to magnon-magnon interactions. The thermal averages are evaluated simply using the boson representation for the spin operators 161. The equation of motion for the Green's functions appearing in 4 KqF(t), ~ f , s is a soluble integral equation, since the higher order Green's functions which appear are negligible at low temperatures. The procedure is much simpler than that in references [3] and 151, no decou- pling being required because we start with an exact expression for D. The first term in brackets in (5)

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19711245

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C 1

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⁶⁹⁸ ^{D. M.}EDWARDS AND B. FISHER gives Dyson's Born approximation and inclusion of

the second term yields the full T5I2 correction to Aoo(q) for arbitrary K Approximate evaluation of ZmxJ(O, Dq2) leads to known results [7] [8] for magnon lifetimes in the limits Aoo(q) > k T and fioo(q) -4 kT.

111. Itinerant model. - For a one-band model with short-range interactions (5) leads directly to a genera- lisation of Edwards' previous formula for D [2].

However this expression is inconvenient for discussing magnon-magnon interactions. Instead, for a strong ferromagnet, we consider the function x,(q, o) which is defined by (1) with -4 S,(t), s:, % replaced by -4 A;(t), A, % . Here A; is the operator which creates a spin wave on the ground state I 0 > and an exact expression for it is implicit in earlier work [2]. It is sufficient to note that A: = n-'/'So, where n is the number of electrons, and we may write, t o order q2, [A:, HI = - Do q2 A: + ^{Rq Kfg.} ⁽⁸⁾

Here Do q2 is the magnon energy at T = 0 and (8) defines K:. Since ~ , ( q , w) has a spin wave pole we may follow the procedure of section I and obtain an equation corresponding to (5). On using (8) we find l)q2 = Do q2

+

^fiq< [Kt,, A,] > -

- fi2 q 2 lim ~ ~ ( 0 , o ) (9)

o + O

where x,(q, o) is defined similarly to x,.

For a strong ferromagnet the low-lying excitations are either spin waves or single particle excitations without spin flip. At low temperatures these are inde- pendent and the single particle excitations give a correction Dl T 2 to Do [9]. To determine the effect of magnon-magnon interactions the thermal averages in (9) may be evaluated using

Refer [I] MA (S.), B ~ L - N O N O D (M.) and FREDKIN (D.), Phys.

Rev., 1968, 174, 227.

[2] EDWARDS (D. M.), Proc. Roy. Soc. A, 1967, 300, 373.

[3] MARSHALL (W.) and MURRAY (G.), J. Phys. C, 1969,

9 e m A, 2-97.

[4] STRINGFELLOW (M. W.), J. Phys. C., 1968, 1, 950.

[5] COOKE (J.) and GERSCH (H.), Phys. Rev., 1967, 153, 641.

[6] DYSON (F. J.), Phys. Rev., 1956, 102, 1217 and 1230.

171 KASHCHEEV (V. N.), and KRIVOGLAZ (M. A.), Fiz.

Tverd. Tela, 1961, 3 , 1541.

where v p is a boson occupation number. The first correction term to Do q2 in (9) is then equivalent to

which is the expression used by Izuyama [lo] and Kawasaki [Ill. Clearly, from the discussion of section 11, it is analogous to Dyson's Born approximation in the Heisenberg model. Both correction terms in (9) involve the factor C ^p2^{v p}which gives a T5/' dependence [12] 191.

Following Izuyama and Kawasaki, we may evaluate (11) approximately using the r. p. a. expression for A: and assuming a parabolic band. I t appears that Izuyama's result [lo] is the correct one and is inde- pendent of whether the magnetic carriers are electrons or holes. D,, the coefficient of T5/' in D, has been estimated in this way for nickel assuming the band structure t o be six groups of holes [13] with effective mass 5.5 and exchange splitting 0.4 ev. We find D2 is positive with D2/Do = 6 x 10-9(0K)-5'2. NO experi- mental value has been quoted for nickel but for iron, Phillips [14] finds 1 x lo-* and Stringfellow [4] finds 2 x

Thompson's result 1151 on the magnon lifetime z corresponds to the simplest process giving a non-zero contribution to ImxJ(O, Do q2). Since z-' -- q6 the spin wave is well defined at T = 0 for a strong ferromagnet. However Ma, BCal-Monod and Fredkin [I]

find ^{2 - I}

-

q2 for a paramagnet in a field and a similar result is expected for a weak ferromagnet. Thus spin waves are not well defined, and the phenomenological arguments [12] [9] for T2 and T5I2 behaviour of D are not obviously correct in this case.

[8] HARRIS (A. B.), Phys. Rev., 1968, 175, 674.

[9] IZUYAMA (T.) and KUBO (R.), J. Appl. Phys., 1964, 35, 1074.

[lo] IZUYAMA (T.), Phys. Lett., 1964, 9, 293.

[ll] KAWASAKI (K.), Phys. Rev., 1964, 135, A1371.

[12] MARSHALL (W.), Proc. 8th Intern. Conf. Low. Temp.

Phys., 1962, p. 215 (Butterworths, London, 1963).

[13] HERRING (C.), Magnetism, vol. 4 (edited by Rado and Suhl), p. 356 (Academic Press).

[14] PHILLIPS (T. G.), PYOC. Roy. SOC. A, 1966, 292, 224.

1151 THOMPSON (E. D.), J. Appl. Phys., 1965, 36, 1133.