M. AHMAD and S. J. GLADYSZ: Spin Collective Oscillations in 2D Fermi Liquids 285

phys. stat. sol. (b) 164, 285 (1991)

Subject classification: 67.50 and 76.30

Institute of Physics, Technical University, Wroclaw ')

Spin Collective Oscillations in Two-Dimensional Charged Fermi Liquids

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

Starting from the transport equation for spin magnetization the transverse dynamic spin susceptibility is calculated in terms of the Landau two-dimensional Fermi liquid parameters B,. The formula allows to estimate the dispersion relation of spin waves with arbitrary I and the oscillator strength of the modes with 1 = 0, 1. The possibility of experimental determination of the parameters B, with higher I is discussed.

Ausgehend von der Transportgleichung fur die Spinmagnetisierung wird die transversale dynamische Spinsuszeptibilitat mit den Landauparametern B, der zweidimensionalen Fermifliissigkeit berechnet. Die Formel erlaubt, die Dispersionsbeziehung von Spinwellen mit beliebigem I und die Oszillatorstarke der Moden mit 1 = 0, 1 zu berechnen. Die Moglichkeit der experimentellen Bestimmung der B,- Parameter mit hoherem I wird diskutiert.

1. Introduction

The existence of paramagnetic spin waves has been predicted by Silin in [l], where he extended Landau's theory of Fermi liquids [2] to the case of conduction electrons of a metal to which an external magnetic H,, field is applied. About ten years later Platzman and Wolff [3] made the next step; they derived a theory for the main propagating spin wave mode with accuracy up to k2 , where k is the wave vector of an external disturbance. With the help of their theory, Schultz and Dunifer [4] extracted from experiment values for the first two Landau parameters. Their experiment together with the Platzman-Wolff theory extended Silin's consideration to the case of propagating spin waves in a metal and became the basis of a new spectroscopic method in metal physics.

For other Fermi liquids, for instance, 3He and ' H H ~ - ~ H ~ mixture Corruccini et al. [5] in 1972 had demonstrated an experiment in which they gave evidence for the existence of spin waves in such materials. They suggested that the spin wave phenomena are a universal feature of any Fermi liquid, whereas this would not seem to be in the case of zero sound, where the degree of freedom provided by the magnetic field is missing.

As to the case of a two-dimensional (2D) Fermi liquid, Beal-Monod et al., in a series of papers [6, 71 have tried to show that the Landau theory of Fermi liquids cannot be simply transcribed from three dimensions to two ones. There appear singularities at very low temperatures and using standard perturbation methods for calculating, for example, the uniform static susceptibility fails in 2D system. This however, does not necessarily imply a violation of the 2D Fermi liquid theory, but can lead to corrections of the Fermi-liquid-like

') Wybrzeze Wyspanskiego 27, PL-50-370 Wroclaw, Poland. ') On leave of absence from Institute of Physics, University of Lattakia, Syria.

286 M. AHMAD and S. J. GLADYSZ

behaviour (but not to a qualitative change). Assuming the full validity of the Landau theory, Freedman [8] calculated some properties of 2D Fermi liquids, in particular the energy of the main spin wave mode. He assumed that only the first Landau parameter B, is nonvanishing. Moreover, he has shown that there are no new effects associated with the reduced dimensionality and obtained results similar to the three-dimensional case.

Later more than once it has been stated that at this stage of investigation, to assume the validity of the Landau theory in the 2D case would not be premature. We mention here papers [9 to 131, the main of which just use the fact that the Landau theory of Fermi liquids is applicable in the 2D case.

Our motivation for the present work comes from the recent interest in the 2D Fermi liquid which is realized in the normal state of the high-temperature superconductors. In connection with this field of investigation it has been proved very recently [14] that there is no breakdown of Fermi liquid theory in the 2D case, at least, in the low-density limit. It thus seemed important to us to study the propagation of spin waves in the 2D charged Fermi liquid, as they provide a strong spectroscopic tool for the investigation of its dynamics.

2. Transport Equation - Method of Solution

The Landau-Silin equation of motion for the transverse dynamic magnetization M b ; k, w) of the 2D Fermi liquid has been intensively investigated recently [12, 13, 151. The appropriate discussion in this case is analogous to the standard treatment of the three-dimensional case [16 to 191, but modified to take into account the two-dimensional character of the system. As discussed exactly by Miyake and Mullin [12, 131 the geometry of collisions in two dimensions restricts the scattering angle to 0 or n (forward of backward scattering). This, however, does not apply to electrons in metals, we are interested in in this paper, in view of scattering on impurities.

The function B(p, p’) which describes the Landau Fermi liquid interaction can be defined in a way similar to three dimensions [8, 111. Here p and p‘ are the quasi-particle momenta and they both lie on the 2D Fermi surface. For the circular Fermi surface B(p, p’) can be expanded in a Fourier series as

where the numbers B, denote the spin-antisymmetric 2D Landau parameters and cp - cp’ is the angle between p and p‘.

In order to obtain the dispersion relations of spin waves (and their oscillator strengths) we should solve the equation for M , then calculate frequency and wave-vector dependent transverse susceptibility x(k, of and, at the end, find its poles and residues. To fulfil this programme we start from the equation for M in the form given recently by Czerwonko [19], namely

where Z = kv,, W = 8 . k, w denotes the frequency of the external perturbing field and the circumflex above the letter denotes, as usually, the unit vector, such that is the direction of the quasiparticle momentum on the Fermi circle. Ha, = H,,(k, w ) stands for the space

Spin Collective Oscillations in Two-Dimensional Charged Fermi Liquids 287

and time Fourier transformed perturbing magnetic field, w, is the cyclotron frequency, and b denotes the Fermi liquid renormalized magnetic moment of the quasiparticle, i.e. p = po(l + B0)-' , while Po is that of a free particle. Without loss of generality we choose k in the x-direction. The quantity M = M ( p ; k, w ) stands for the phase space dependent density of transverse magnetization and from general theory we know that it is understood to be evaluated on the Fermi surface at sufficiently low temperatures. A = A(p; k, w ) denotes the deviation of M from its local equilibrium value and, for a 2D system, is defined by

n?(p) = M(p) + 7 !!? B(p,p') M(p ' ) . 2.n

0

(3)

The collision term in the right-hand side of (2) will be later chosen in the partial relaxation time approximation, in which it depends on the angular behaviour of the deviation of magnetization from its local equilibrium. This approximation automatically assures that the theory is consistent with conservation laws [20].

The solution of (2) can be expressed for the 2D case as

m

M ( W = ( M , cos lq + N , sin lq), I = O

where N o = 0 everywhere. Hence and from (3) we have

(4)

Now this formula allows us to express the collision integral in the spirit of the considerations presented in [20], that is

( M I cos lq + N , sin lq) . J"[A] = c ___ (6) 1 + B,/2

1=1 z,

3. Dynamic Spin Susceptibility: Spin Wave Poles and Their Residues

Inserting (4) to (6) into (2) and truncating the decomposition (1) at a certain 1 (but arbitrarily large) we obtain the system of coupled algebraic equations for M l ( k , w) and N l ( k , a),

288 M. AHMAD and S. J. GLADYSZ

in which we have defined

YI TI

y l = 1 + B, , b, = 4 B,(1 + dl0), and further A, = w - 2PHdcy, + i - .

This system consists of 21 + 1 coupled equations which are solved with the help of the standard Cramers method. We remember that in the long-wave limit all components of order higher than Z 2 should be dropped. In such a way, after long algebraic transformations, we get the final result for M o , the only Fourier component of M which survives in the equation defining the susceptibility ~ ( k , w), namely

In this equation we have taken, after Czerwonko [19], the reference frame for M ( p ; k, w ) from the component strictly subordinated to the external ac perturbing magnetic field. This special reference frame is exactly the same as that used in writing down the kinetic equation (2). N(0) in (8) denotes the density of states on the Fermi circle and is equal to m*/nh2 in the 2D case.

With no effort from (8) with the help of (4) and the solution of the system (7) for Mo, we get

where

and

I 1 1 w, w ) = n (A: - 42w,zY:) - 4z2 c II YqY4+1AqAq+1(A,Z - H2W,2Y,Z). q = 1 q = l n = l

n + q , q + l

(lob)

We will search for the energy of spin waves as poles of ~ ( k , o), in the long-wave limit, in the form

w / r w = %(O) + % * k 2 , (1 1)

where

Spin Collective Oscillations in Two-Dimensional Charged Fermi Liquids 289

In such a way we get the general formula

c L I + = f v; (yoYIA:A,+A; ... $ A ; + ylyzAoA:A;A;A; ... A:A; + . . . + y ( l - l)ylAoA;- ,)ApA:A; . . . A;-2)A;4)} x(AfA; ... $A; + A,A;A;A, ... ATA; + ... + AoAfA; ... A;-l)A(l+l)A:}-l, (12)

where we have used designations similar to those involved in (7) and the zero above the letter means that if we calculate ctl* for a certain, particular 1, then w = ol* (0) in all A,(w). Now it is easy to generate the formulas

To obtain the oscillator strength of the modes we must calculate the residua R,, of the poles in (9). Having calculated L and M only with an accuracy up to k2 we convince ourselves that R, , (k ) = 0 for 1 2 2, and

A,+ A, - 'I 1 Ro(k) = w0(k) - crok2o0 +7 + ... + 0 + D

It is clear that only the 1 = 0 mode contains the term of order unity in its residuum, while the others fulfil the relation R, - k z z as it happens in the three-dimensional case.

4. Final Remarks

In the preceding sections we derived the long-wavelength dispersion coefficients of spin waves in the 2D charged Fermi liquid and the oscillator strength for the first three modes. No restriction on the number of Landau parameters has been done, thus our results are most general. They provide a baseline with which experimentalists could compare their results. This is especially important in new materials which display 2D-like behaviour.

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290 M. AHMAD and S. J. GLADYSZ: Spin Collective Oscillations in 2D Fermi Liquids

Results for residues (oscillator strengths) give information on the possibility of exciting a given mode. Our results suggest that the possibility of observing higher modes with 1 = 1 increases with increasing k, while it decreases for the main 1 = 0 mode. It would be of much interest if one could modify the experimental set-up so as to observe higher modes, because they could provide an alternative information on Landau parameters. At the end, let us remark that our results are in qualitative agreement with the recent work [21] on the subject, done in the framework of autocorrelation function theory.

Acknowledgement

Suggesting the problem and useful discussions with Prof. J. Czerwonko are gratefully acknowledged.

References

[l] V. P. SILIN, Zh. eksper. teor. Fiz. 33, 1227 (1957); 35, 1243 (1958). [2] L. D. LANDAU, Zh eksper. teor. Fiz. 30, 1058 (1956); 32, 49 (1957). [3] P. M. PLATZMAN and P. A. WOLFF, Phys. Rev. Letters 18, 280 (1967). [4] S. SCHULTZ and G. DUNIFER, Phys. Rev. Letters 18, 283 (1967). [5] L. R. CORRUCCINI, P. D. OSHEROFF, D. M. LEE, and R. C. RICHARDSON, J. low-Temp. Phys. 8,229

[6] A. THEUMANN and M. T. BEAL-MONOD, Phys. Rev. B 29, 2567 (1984). [7] M. GABAY and M. T. BEAL-MONOD, Phys. Rev. B 18, 5033 (1978). [8] R. FREEDMAN, Phys. B 18, 2482 (1978). [9] M. D. MILLER and L. H. NOSANOV, J. low-Temp. Phys. 32, 145 (1978).

[lo] A. SACHRAJDA, 0. F. BREWER, and W. S. TRUSCATT, J. low-Temp. Phys. 56, 617 (1984). [11] S. M. HAVENS-SACCO and A. WIDOM, J. low-Temp. Phys. 40, 357 (1980). [12] W. J. MULLIN and K. MIYAKE, J. low-Temp. Phys. 53, 313 (1983). [13] K. MIYAKE and W. J. MULLIN, J. low-Temp. Phys. 56, 499 (1984). [14] J. R. ENCELBRECHT and M. RANDERIA, Phys. Rev. Letters 65, 1032 (1990). [15] M. AHMAD, phys. stat. sol. (b) 160, K65 (1990). [16] M. AHMAD, phys. stat. sol. (b) 154, K163 (1989). [17] M. AHMAD and S. J. GLADYSZ, phys. stat. sol. (b) 159, K119 (1990). [18] G. DUNIFER, D. PINKEL, and S. SCHULTZ, Phys. Rev. B 10, 3159 (1974). [I91 J. CZERWONKO, J. appl. Phys., Suppl. 26, 223 (1987). [20] G. BAYM and C. J. PETHICK, in: The Physics of Liquid and Solid Helium, Ed. K. H. BENNEMAN

[21] J. CZERWONKO, Proc. Internat. Conf. Current Topics of Field Theory and Statistical Mechanics,

(1972).

and J. B. KETTERSON, Wiley, 1978 (p. 1).

World Scientific Publ. Co., Singapore 1990 (in the press).

(Received November 26, 1990)