SPIN WAVES OF A LAYERED FERROMAGNETIC ELECTRON GAS AND OF A PARAMAGNETIC ELECTRON GAS IN A STRONG MAGNETIC FIELD

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SPIN WAVES OF A LAYERED FERROMAGNETIC

ELECTRON GAS AND OF A PARAMAGNETIC

ELECTRON GAS IN A STRONG MAGNETIC FIELD

W. Gasser, C. Täuber

To cite this version:

JOURNAL DE PHYSIQUE

Colloque C8, Supplkment au no 12, Tome 49, d6cembre 1988

SPIN WAVES OF A LAYERED FERROMAGNETIC ELECTRON GAS AND

OF

A

PARAMAGNETIC ELECTRON GAS IN A STRONG MAGNETIC FIELD W. Gasser and U. C. Tauber

Physik-Department T30, Technische UniversitBt Miinchen, D-8046 Garching, F.R. G.

Abstract.

-

The spin wave spectrum of a layered ferromagnetic electron system is determined in the long-wave length

limit. It is shown that the spin wave stiffness is a highly anisotropic quantity. A discussion of the Stoner condition shows that both electron tunneling processes and a more sophisticated expression of the Coulomb interaction matrix element than the usual Hubbard form are important for the existence of ferromagnetic ordering. The results are compared with those for excitations of a paramagnetic layered electron gas in a strong external magnetic field.

Spin waves may be found in a multiple layered elec- tron gas (LEG) if either the system is ferromagnetic, or if it is even non-magnetic but placed in a strong external magnetic field. In both cases it is important that the Coulomb interaction between the electrons is strong enough to allow the spin wave propagation. To understand these excitations we assume that the elec- tron gas layers are parallel to the xy-plane and that the magnetization or the external magnetic field has the direction of the %axis. Then all operator expres- sions may be expanded with respect to the following complete sets of orthonormal single-particle wave func- tions,

1

qK

(R)

=

-

meikT$'kzv (2) X U _{0 )} in the case of a ferromagnetic LEG, and

for the non-magnetic LEG in an external mag- netic field, where R = (r, z) r (x, y, z) and K = (k,

4 )

=

(L,

k) - = ( h r r ,by,

L).

L.(~) is the normalization leigth i i x (y) -direction, $k,, ( z )

is a Bloch function solving the one-dimensional Schrodinger equation with a periodic potential V (z) and X u is a normalized spin function. The function

Enky (x) is the n-th normalized eigenfunction of a har-

monic oscillator, centered on the point xo = -kya2, where a2 = Ac/eB

=

filmw, is the square of the mag- netic length. Assuming that only one tight-binding band will be important, the band index v will be sup- pressed such that the one-particle energies correspond- ing to (1) are

fi2k2 A E ( K ) =

-

- -

cos k,d

2m 2

and corresponding t o (2) the degenerate energies are ~ ( n , k,, u) =Rw,

A

- g p ~ a B - -COS &d (4)

2

where a = f 1. In the case of the wave functions (1)

the Hamlltonim for the system is given by

with the Coulomb interaction matrix element intro- duced by Fetter [I]

2ne2 sinh qd

V ( Q ) = ~ c o s h q d - cos q,da (6)

In the case of the wave functions (2) we get the Hamil- tonian

nko

-

where V (Q) is again given by the Fetter model (6) and

(8) The spin wave spectra may now be found as poles of the transverse susceptibility

where the spin-density operators S* (fQ) are ex- panded either with respect t o the system (1) or with

C8 - 1612 JOURNAL D E PHYSIQUE

respect t o ( 2 ) . The explicit cdculation of ( 9 ) is repre- sented in [2] and [3] using methods which correspond t o the local-field theory of the electron gas as intro- duced by Devreese et al. [4, 51. In the case of the ferromagnetic LEG the spin wave dispersion relation up t o order Q2 is given by

with

@ is the angle between the three-dimensional vec- tor Q = (q, q z ) and the z-axis which is the axis of the stack. If (18) is written in the usual form, w ( Q ) = D Q ~ , one realizes that the spin wave stiff- ness D for the layered ferromagnetic electron gas is a highly anisotropic quantity, being dependent on the external angle Q in a similar way as the plasmon fre- quency of the LEG [6, 71. The obtained anisotropy of D is first of all a consequence of the anisotropic elec- tron dispersion relation. The details of the Coulomb interaction matrix element are not too important in the long-wave length limit, because only integrals over it appear in the equations in this limit. This, how- ever, is a general experience from the theory of the electron gas [4, 51 - that local field corrections are less important in the small wave number limit. The Stoner condition for the instability of the paramagnetic state is substantially modified for the ferromagnetic LEG

1 _{CV(K-PI--c-=o.}

_a

_{(np )}

_a

_{( n ~ ~ )}

'+

a&

( P ) ~ E ( K )

CKf

-zgEi

. .

(12) The second term of (12) is negative in sign. Further- more one may show that this term diverges logarithmi- cally as log ( E F / A ) with vanishing band width A ( E F is the Fermi energy). This means that equation (12)

cannot be satisfied in the zero band width limit. In other words, an instability of the paramagnetic state is to be expected only for a finite band width, i.e. the ferromagnetic ordering may occur only in the presence of tunneling processes. Equation (12) also shows that it is important to describe the layered system by (5) instead of the simpler Hubbard model, because in the latter case the condition (12) would be reduced imme- diately to the usual statement UD ( E F ) = 1.

For the case of a non-magnetic LEG in a strong ex- ternal magnetic field B, the lowest lying mode in the limits T + 0 , 3 -4 oo and a/d -+ 0 takes the form

Ti" (Q) = fiwcL?/2+

A sin TU ( A 2 / 2 ) sin 21rv

+{

*" - e 2 m u / a ( l - F ) }

where v = N , " ~ T ~ ~ / L , L , is the filling factor (N," is the

number of electrons per layer). The first term in (13) is the Zeeman energy, g,u~=fL~,g/2. The second one cor- responds to spin wave propagation in zdirection (for Q = 0') similar t o (10) and the third term is charac- teristic of electrons in a very strong field B interacting via the Fetter model ( 6 ) .

We note that the long-wavelength spin wave (13) lies outside the Stoner continuum (for small a) and is therefore undamped in that region. The mode (13) should be found in experiments with multiple GaAs- (A1Ga)As heterostructures of inter-layer spacing d of about 50-1000

A

in an applied magnetic field B

=

10 5...io6 G .

[ I ] Fetter, A. L., Ann. Phys. 88 (1974) 1. [2] Gasser, W., 2. Phys. B 63 (1986) 199.

[3] Gasser, W . , Tauber, U. C., 2. Phys. B 69 (1987) 87.

[4] Brosens, F., Lemmens, L. F., Devreese, J. T., Phys. Status Solidi B 74 (1976) 45.

[5] Devreese, J . T . , Brosens, F., Lemmens, L. F., Phys. Rev. B 2 1 (1980) 1349.

[6] Grecu, D., Phys. Rev. B 8 (1973) 1958.