Short Notes K119

phys. stat. sol. (b) 159, K119 (1990) Subject classification: 75.30; 67.50; S2

Institute of Physics, Technical University of Wroclaw )

The Dynamic Spin Susceptibility and 1 = 0 - 1 Spin Waves in a Charged Fermi Liquid of Alkali Metals in a DC Magnetic Field

M. AHMAD') and S.J. GLADYSZ BY

The experimental observation of spin waves in non-ferromagnetic metals 11 to 41, as a series of side bands in conduction electron spin resonance, indicated unambiguously that to accurately describe the properties of real metals, one must include an analysis of electron-electron interactions, the so-called many-body effects or Fermi liquid effects. The nature of spin-wave excitations is strongly determined by these many-body effects and in fact they exist as a primary effect in themselves. More exactly, they vanish in the limit of zero interactions, and thus the experiments on spin collective oscillations in the system of electrons provided the first really convincing evidence for the validity of Landau's theory of Fermi liquids 151 .

Spin waves are well-known in magnetically ordered materials and in paramagetnic metals were predicted by Silin 161, in his extension of the Landau Fermi liquid theory 1 5 1 to the case of charged systems in a dc magnetic field. Physical reasons for the existence of spin waves in non-ferromagnetic metals are similar to those usually considered in the theory of ferromagnetism. Namely, the interaction between particles leads to a 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($,$' ) of the Fermi liquid interaction function, which may be expanded in the series of Legendre polynomials PI for these spherical Fermi surface:

B( f i , f i ' ) =c(zl + UB1P1(6.$') , (1) 1

where the numbers B1 denote the spin-antisymmetric Landau parameters.

Wybrzeze Wyspiafiskiego 27, PL-50-370 Wrodaw, Poland. ') On leave of absence from: Institute of Physics, University of Lattakia,

Syria.

K120 physica status solidi (b) 159

The experimental data of the papers /1 to 41 were successfully interpreted by means of a phenomenological theory of transport for the tranpverse spin magnetization in a charged Fermi liquid 1 7 , 41. It was shown in / 7 1 that a study of the spin-wave modes provides a means to measure the Fermi liquid parameters Bl entering into (1). However, in / 7 / only the main (0, 0) spin-wave mode has been taken into account and the others, labelled by a pair of subscripts (1, m)

I m l , S 1, corresponding to a particular spherical harmonic with larger 1 have been neglected. The discussion presented in 181 shows that this is equivalent to the assumption that only two Landau parameters Bo and,B1 are nonzero, when the theory is constructed with accuracy up to k , being the wave vector of the excitation. A more refined approach, which includes excitations with higher 1, improving the agreement between theory and experiment and testing the dependence of the experimental results on B2 has been developed in 19, l o / . Both the theories include the interaction, at long wavelengths, of the collective modes

(1, m) with one another and with single-particle excitations, therefore the solution of the transport equation and resulting dispersion relations became extremely complicated. To overcome difficulties the authors of /9 / and / l o / made some approximations and the final results have been obtained by means of some kind of perturbation theory. On the contrary, recently it has been explicitly shown /11, 121, for a neutral Fermi liquid and with accuracy up to k2 how the modes with larger 1 enter into the dynamic spin susceptibility of the system. The oscillator strength of the peak with 1 = 1, which is of the order of (kR)2, in comparison with that of the main peak with 1 = 0, indicates /12, 131 that the detection of the

-1 spin waves with 1 = 1 is not a question of the far future. Here R vF(wc) denotes the Larmor radius with vF and wc being the Fermi velocity and the cyclotron frequency, respectively. In view of that it is desirable to have calculated the dynamic spin susceptibility, for a charged Fermi liquid, more accurately than in / 9 , l o / , to clarify the role of Bq on spin-wave modes.

In this note the transverse dynamic spin susceptibility, containing spin Cave poles (1, m) with 1 = 0, 1, l m l S 1, is found, on the basis of the kinetic equation which is solved in a straightforward manner. The dispersion relations of the modes and their oscillator strengths are exact up to k . Thus we extend the method elaborated in /12/ and results obtained therein to the case of a charged Fermi liquid.

2

2

According to Landau’s theory /5/, the behaviour of the system of interacting electrons to which a strong dc magnetic field Ho is applied, is described by the set of transport equations / 8 / . To investigate spin collective oscillations, one has to analyse the kinetic equation describing the motion of the transverse phase space

dependent density of spin magnetisation M ( C ) . The quantity M ( 3 ) is understood to be evaluated on the Fermi surface at sufficiently low temperatures. After /12/r?e

Short Notes K121

take the reference frame for M from the component slavishingly subordinated to the external ac magnetic field H(k, w), i.e. BH, where f3 = B0(1 + Bo)-' is the Fermi liquid renormalized magnetic moment of the quasi-particle while @ is that of the electron. Following / l 2 / we reproduce the kinetic equation in the form given in /14/ (cf. also / 8 / ) , namely

0

BwH + wM - zwfl - 2@Ho% + iw c a a @ $ = iJa[$] , (2)

where z = kvF, w = 6, k, w denotes the frequency, and the circumflex indicates a unit vector, such that fi 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 , @ of are referred to it. Without loss of generality we choose the direction of the wave propagation k to lie in the x-z plane at an angle A to the z-axis. In such a way we can write

't *

-+

+

w = sinecos@k X + cos8kZ, k = sinA , kZ = COSA . (3) ,.l

The quantity M entering into the kinetic equation (2) denotes the deviation of the transverse spin magnetization M from its local equilibrium value and is defined as follows :

, The collision integral Ja on the right hand side of (2) will be later chosen in the relaxation time approximation, in which the relaxation time depends on the angular behaviour of the deviation of the magnetization from its local equilibrium. The deviation from local not from global equilibrium automatically ensures that this

approximate form of the collision integral is consistent with conservation laws /15/. +

To obtain the nonlocal dynamic spin susceptibility X(k, w), the poles of which determine the energy spectrum of spin waves, the density of the spin magnetization M( p') involved the special reference frame 101 1

in (2) should be integrated over the Fermi sphere. In we have chosen for M we get the relation (cf. /12,

which serves as a definition of x($, w). In (5) , as usually, N ( 0 ) denotes the density of states on the Fermi surface.

The solution of (2) we will search in the form of a series of spherical harmonics

K122

Inserting expression (6) in (4 ) we easily find

physica status solidi (b) 159

(6)

Now putting the decomposition (6) into (5) and performing the spherical average one can find with no effort that

It follows immediately from ( 8 ) that the dispersion relations of spin-wave excitations and their oscillator strength are related only to the poles and residues of MZ - the spherical average of M. Following the treatment in /15/ the formula ( 7 ) allows us to write the collision term in the kinetic equation as the sum

where use was made of the observation / l o / that the collision times appear to be modified by Landau parameters. Terms arising from (3) in the kinetic equation have the form cos8Yy($), sineexp(+i$)Y;($) and we simplify them with the help of the formulae of the Clebsch-Gordan type, see for example /16/. In such a way we get the system of coupled equations on My, I m 1 51, and solve it with an accuracy up to k . In the next step, using ( 8 ) , we find the polar terms corresponding to the spin-wave excitations together with their residues. With the mentioned accuracy

2

only 1 = 0 , 1,

XCZ, u> b a t

where

with

I ml51 modes (1, m) appear and finally we get the formula

1 t B1 up = (2BHo t mwc)(l + B1) t i - '1

and

Short Notes

2 2b + cos A ,

cos 2 A , ( l lc )

We have designated: a = (1 + Bo)(l + B1), b = (1 + B1)(l + B2). The residues of the modes (1, m) designated by Rh are equal to

0

0 0 2 > L j 1 m ' R~~ = a:($) + aOwOz

1=1, Im111 - w1

m m

We see that the susceptibility at the long wavelength limit is strongest in the (0, 0) mode and varies with z in the (1, m) mode as z21 which does not support / 9 / but is in agreement with / l o / and with results obtained recently /11, 121. As

z increases, more and more modes become mixed and the resulting dispersion relations are complicated. Our results ( l l b ) to ( l l d ) suggest that mixing is more complicated than that obtained by perturbation in / l o / . Let us notice that the expression for a. in ( l l a ) is exactly the same as that in / 7 / . Moreover, for A = 0, when the wave vector is parallel to the dc magnetic field Ho and the orbital torque term becomes ineffective, the coefficient al @ven by ( l lc ) reproduces

0

0

K124 physica status solidi (b) 159

that in /12/. Resu l t s f o r Rlm of /lo/ arise from e x p r e s s i o n s (12) of the present note when in a? we neglect t e r m s dependent o n b.

W e wish to thank Prof . J. Czerwonko f o r suggesting the problem and u s e f u l discussions.

Refe rences /1/ S. SCHULTZ and G. DUNIFER, Phys. Rev . L e t t e r s 3, 283 (1967). 121 G. DUNIFER, S. SCHULTZ, and P.H. SCHMIDT,

/3/ P.M. PLATZMAN, W.M. WALSH, and E-NI FOO,

/4/ G . DUNIFER, D. PINKEL, and S. SCHULTZ,

J. appl. Phys. 39, 397 (1968).

Phys. Rev. 172, 680 (1968).

Phys. Rev. B lJ, 3159 (1974). /5/ L.D. LANDAU, Zh. eksper. t eo r . Fiz. 30, 1058 (1956);

/6/ V.P. SILIN, Zh. eksper. teor. Fiz. 35, 1243 (1958). /7/ P.M. PLATZMAN and P.A. WOLFF, Phys. Rev . L e t t e r s 3, 280 (1967). /8/ P.M. PLATZMAN and P.A. WOLFF, Waves and In te rac t ions in Solid

/9/ S.C . YING and J.J. QUINN, Phys. Rev. 180, 218 (1969).

- 32, 49 (1957).

State Plasma, Academic Press, New York 1973 (Chap . X.58).

/lo/ A. WILSON and D.R. FREDKIN, Phys. Rev. B 2, 4656 (1970). /11/ J. CZERWONKO, Physica m, 414 (1987). /12/ J. CZERWONKO, Japan. J. appl. Phys. 26, Suppl. 223 (1987). /13/ D. CANDELA, N. MASUHARA, D.S. SHERRIL, and D.O. EDWARDS,

/14/ A.A. ABRIKOSOV, Introduct ion t o the T h e o r y of Normal Metals, J. low-Temp. Phys. 63, 331 (1986).

Academic Press, New York 1972 (Chap . 13). /15/ G. BAYM and C. J . PETHICK, in: The Physics of Liquid

and Solid Helium, Ed. K.H. BENNEMAN and J . B . KETTERSON, Wiley, New York 1978 ( p . 1).

/IS/ G . ARFKEN, Mathematical Methods f o r Physicists, Academic Press, New York 1970 (Chap . 12).

(Received January 30, 1990; in revised form Apri l 5, 1990)