Short Notes K163

phys. stat. sol. (b) 154, K163 (1989)

Subject classification: 75.30; S2

Institute of Physics, Technical University , Wrodaw' Spin Waves with 1=1 in a Degenerate Fermi Liquid of Alkali Metals in DC Magnetic Fields

BY M. AHMAD2)

Spin waves in an interacting electron system are collective oscillations of the spin magnetization of electrons. They are well-known in magnetically ordered materials such as ferro- and antiferromagnets and in non-ferromagnetic metals were first predicted by Silin 111 in his extension of the Landau-Fermi liquid theory 121 to the case of charged systems, the best examples of which are those formed by the conduction electrons in the alkali metals.

Physical reasons for the existence of paramagnetic spin waves in non- ferromagnetic metals are similar to those usually considered in the theory of ferromagnetism. Namely the interaction between particles leads to the dependence of the quasi-particle energy on their distribution and in effect quasi-particles interact via molecular fields coupling spin density variations on different parts of the Fermi surface, described by momenta p' and . These molecular (Landau) fields are analogous to the exchange molecular field which is responsible for the appearance of ferromagnetic spin waves. In this sense paramagnetic spin waves in non- ferromagnetic metals arise from the spin-dependent part B ( 5 , $I) of the Ferrni liquid interaction function, which may be expanded in the series of Legendre polynomials P1 for the spherical Fermi surface:

A A

B ( 5 , 5') = C (21 + l)BIPl(c 5 ' ) , 1

where the numbers B1 denote the spin-antisymmetric Landau parameters. Spin waves excitations were subsequently observed experimentally 13, 41, in

sodium and potassium, and it was shown in 151 that this type of experiment provides .a means to measure the Fermi liquid parameters B1 entering (1). However, the formula given in 151 takes into account only the .(O, 0) spin wave mode and neglects the others, labelled by a pair of subscripts (1, m), Im( 21, correspond to a particular spherical harmonic with larger 1. This is equivalent to the assumption that only two Landau parameters Bo and B1 are nonzero, see the discussion in / 6 / .

Recently it has been explicitly shown 17, 81 for the neutral Fermi liquid, that as long as one keeps B1 = 0 for 1 h 2 in addition to the 1 = 0 mode, there appears the

) Wybrzeie Wyspiafiskiego 2 1 , PL-50-370 Wrochw, Poland. 2 , On leave of absence from Institute of Physics, University of Lattakia, Syria.

K164 physica status solid (b) 154

1 = 1 one, with the the oscillator strength of the order of (kR)2. Here k is the wave vector of the excitation and R = VF(wC)-' denotes the Larmor radius with VF and w being the Fermi velocity and the cyclotron frequency, respectively. In view of that and of the fact that t.he possibility of observing spin waves with 1 = 1 is not

the question of the far future 18, 91 , it is desirable to have a more general theory of spin waves in the charged Fermi liquid.

-P

In this note the dispersion relation of the (1, m) modes will be derived on the assumption that B1 0 for 1 L 2 , extending thus the method elaborated in /8/ and results obtained therein to the case of the charged Fermi liquid.

The theory of paramagnetic spin waves is based on the kinetic equation describing the motion of the transverse phase space dependent density of magnetization M ( 3 ) . The quantity M ( 3 ) is understood to be evaluated on the Fermi surface at sufficiently low temperatures. Let us take, after / 8 / , the reference frame for M from the component slavishingly subordinated to the external ac magnetic field H($,w), i.e. BH, where B = p o ( l t Bo)-' is the Fermi-liquid renormalized magnetic moment of the quasi-particle and P o the magnetic moment of electron. After / 8 / we will reproduce the kinetic equation in the form given in I10 / (cf. also / 6 I ) , namely

BwH + wM - zw% - 2@Ho% + iw _a = if[%] , c A , .

where z = kVF, W = g.z, w denotes the frequency and the circumflex indicates a unit vector, such that p' is the direction of quasi-particle momentum on the Fermi sphere. The z-axis of the system of coordinates used in writing down ( 2 ) is parallel to the dc magnetic field Ho and the spherical coordinates 8 , d, of 5 are referreg to it. Without loss of generality we choose the direction of the wave propagation to lie in the x-z plane at an angle A to the z-axis. In such a way we can write

A

A

w = sin8cosd,kx t cosekz , kx = sinA, kZ = cosA . ( 3 )

The quantity $(p') entering the kinetic equation ( 2 ) is defined as follows:

and denotes the deviation of transverse spin magnetization M from its local equilibrium value. At the end Ja denotes the collision integral, which later will be chosen in the relaxation time approximation.

x(g,w) To obtain the frequency and wave vector dependent spin susceptibility the poles of which determine the energy spectrum of spin waves, the density of spin magnetization M ( 5 ) involved in ( 2 ) should be integrated over the Fermi sphere. Remembering that the special reference frame has been chosen for M we get the

relation (cf. / 8 / )

Short Notes K165

in which N(O) , as usually, denotes the density of states on the Fermi surface. We will search the solution of the kinetic equation (2) expressed in terms of the spherical harmonics Yfn

Now from (4 ) and (6 ) we easily find

Putting the decomposition (6) into the relation (5) and performing the spherical average one can prove with no effort, that the spin susceptibility X(;,W)

is proportional to MZ, so we can find the dispersion relations of the spin wave excitations occurring at the poles of M:(<,W). Following the treatment in / 6 / the formula (7) allows us to write the collision t e r m in the kinetic equation as the sum

Making use of the formulae of the type

we simplify the terms arising from (3) and truncate the series (1) of the Landau parameters on 1 = 1.

In such a way we get the system of coupled equations on Mp(k,w), 1 = 0, 1, ImlS1. After solving we obtain the following equation determining the poles of

+

MZ(kt,63):

1 2 [ ( 9 ) AZ(W)A;'(W)A~(W)A,(W) 1 - z2E A ; l ( ~ ) A 1 ( ~ ) ~ ~ ~ A - (Ay(w))2sin2A] = 0

with 1 E = ~ ( 1 + Bo)(l + B1) , A f n ( ~ ) = w - ~ f n , 1 = 0, 1, Im(<l ,

l+B1 ufn (2@Ho + m w c ) ( l + B1) + i - .

T1 To find the dispersion relation of the spin wave mode (1, m) (9 ) will be solved with the accuracy up to z , namely we will search the energy um(<)of the spin wave as 2

m 2 1

(10) urn(<) W f n + Crl z . 1 Now it is easy to find, after putting expression (10) into (S), the general formula

K166 physica status solid (b) 154

in which zero above the letter should be understood in such a way that if we calculate a: for the particular pair of subscripts (1, m) -then w = w: in all

A:(w). From (11) we now generate

2 sin h

(u: - U ; ) ~ - [uc(l t B1)I2

(w:-w:)sin 2 A

a-1 = -E (11d)

The expression for c iz in ( l l a ) is exactly the same as in / 5 1 . On the other hand, for A = 0 when the wave vector is directed along the external magnetic field and the orbital torque term becomes ineffective, the coefficient a; given by ( l l c ) agrees with that obtained recently 181.

The author wishes to thank Prof. J. Czerwonko for suggesting the problem and

1 (w;-w:)(w;-+

useful discussions.

References /1/ V.P. SILIN, Zh. eksper. teor. Fiz. 35, 1243 (1958). / 2 / L.D. LANDAU, Zh. eksper. teor. Fiz. 30, 1058 (1956); 3'2, 49 (1957). /3/ S . SCHULTZ and G. DUNIFER, Phys. Rev. Letters 18, 283 (1967). / 4 / G . DUNIFER, S. SCHULTZ, and P.H. SCHMIDT, J. appl. Phys. 39, 397

/5/ P.M. PLATZMAN and P.A. WOLFF, Phys. Rev. Letters g, 280 (1967).

/ 6 / P.M. PLATZMAN and P.A. WOLFF, Waves and Interactions in Solid State

(1968).

Plasma, Academic Press, New York 1973 (Chap. X.58). / 7 / J. CZERWONKO, Physica m, 414 (1987).

/ 8 / J. CZERWONKO, Japan. J. appl. Phys., Suppl. 26, 223 (1987).

/91 D. CANDELA, N. MASUHARA, D.S. SHERRIL, and D.O. EDWARDS, J. low -Temp. Phys. 63, 331 (1986).

/ l o / A.A. ABRIKOSOV, Introduction to the Theory of Normal Metals, Academic Press, New York 1972 (Chap. 13).

(Received May 2 , 1989)