THE MICROWAVE SUSCEPTIBILITY OF A MAGNETIC SUPERLATTICE

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THE MICROWAVE SUSCEPTIBILITY OF A

MAGNETIC SUPERLATTICE

W. Schmidt

To cite this version:

JOURNAL DE PHYSIQUE

Colloque C8, Suppliment au no 12, Tome 49, dhcembre 1988

THE R/ffCROWAVE SUSCEPTIBILITY

OF

A MAGNETIC SUPERLATTICE

W. Schmidt

Institute of Molecular Physics, Polish Academy of Sciences, 60-179 Poznan, Poland

Abstract. - In this paper, the recursion method is used to calculate the HF susceptibility of multilayers. Resonance fields and local amplitudes of eigenmodes are given by the spectral density. The Hamiltonian of the system comprises exchange, dipolar, Zeeman and interface anisotropy contributions. The DC field and the magnetization are perpendicular to the surface.

In microwave absorption spectra, amplitudes of the resonance peaks and the resonance fields give in- dependent pieces of information about eigenmodes of magnon excitations. To study a resonance absorption spectrum, we need to calculate the HF susceptibility. Calculations are made for a system of two- component multilayers, each one of the nA ferromag- netic atomic A layers is separated by layers of the fer- romagnetic B with nB atomic layers. Both materials are sc Heisenberg ferromagnets having exchange con- stant JA and JB and the exchange constant across the interface JAB. The anisotropy coefficient of the inter- face anisotropy K, (I)

#

0 for 1 = n ~ , n*

+

1 , and so on. The local Lorentz and demagnetizing fields are taken into account by the dipolar interaction.

In the Hamiltonian, we make two transformations. The first transforms the spin operators to Bose oper- ators a+ (j, I) and a- (j, I) of the creation and anni- hilation operators of the spin deviations localized at the lattice site rjl by means of the Holstein-Primakoff

transformations. The location of the spins is described by the index j within the atomic layer, and 1 labels an atomic layer parallel to the surface. The second trans- formation is a Fourier transformation in the plane of the multilayer

where N is the number of the spins in the plane, and k the wave vector lying in the plane. When the transfor- mates are inserted into the Hamiltonian, it is brought to the form

H = & + ~ H , ,

where is the energy of the ground state 10). The part Hk of the Hamiltonian with k

#

0 has not an influence on the magnon excitations in resonance, and for k = 0

Here, J (j, 1, j', 1') is the exchange interaction between lattice sites rjl and rjrll, H denotes the DC field and

S (1) the local spin quantum number. The dipole- lattice sum (5) does not contain the element with

rjl = rj~lf.

In this paper, we calculate the susceptibility

x

(I .E') = -7 (S (I) S (1')) ' I 2

G1lr

at absolute temper- ature tending to zero (T -+ 0). The retarded Green function is restricted to the form

Gilt = 211.

<<

bo (I) ; b$ (1') >>"+i~g

<

0

1

b t (1) (w

+

i E-(H-Eo)

/

h ) - I b$ (I)

I

0

>,

₍₆₎

where w is the frequency of the HI? field and y denotes the magnetomechanical ratio, E represents the changes in the spectral density of the rnagnon excitations due to all processes not explicitly considered in the theory. The element

GIIr

of the Green function is calculated by the recursion method [I, 21 which involves setting up a new orthonormal basis set Iu,)

,

n = 1,2,

...,

NL.

Here,

NL

is the number of atomic layers. In the new basis, the Hamiltonian takes a threediagonal form. The one-particle state lun) may be expanded in terms of b$ (1) 10)

.

The kets, b$ (1) 10)

,

are the basis vectors of an

NL

dimensional Hilbert space. An arbitrary one- particle state Hb$ (1) 10) may be expanded in terms of these kets. The recursion method enables us to calculate the set Iu,) and elements of the Hamilto- nian. Secondly, a diagonal element the Green function is expressible by the elements of the Hamiltonian (see [I, 21).

C8 - 1690 JOURNAL DE PHYSIQUE The amplitude of the eigenmode (u) on the t t h

atomic layer may be calculated as the limit, P

,

(I) =

lim (i E G z I ) . The calculations of the limit are per- E -0

formed with the condition Iw - 0,l <E .O, and

Iu) are the eigenfrequency and eigenfunctions of the Schrodinger equation: (H

-

&)

Iu) = fiR, lu)

.

The amplitude P

,

(I) obeys the normalization condition.

The resonance fields and local amplitudes of eigen- modes are calculated from the spectral density. The calculations are performed for parameters: the spin quantum numbers of the sublattices A and B: SA = 0.5, and SB = 1; the exchange integrals: JA

/

g p g = 2.7 x

lo6

G, J B

/

g p ~ = 1.7 x

lo6

G, JAB

/

9 p . f . ~ = 0.6 x

lo6

G; the nearest neighbors distance a = 2.2

x

lo-'

cm; the frequency of the HF field w

/

y = 1.2 x

lo4

G; the magnon damping parameter E

/

y = 150 G. The imaginary part of the susceptibility

x

(I, I) vs. the DC field is illustrated in figure 1. Our calculations are restricted to the case of ferromagnetic resonance with DC field and magnetization perpendicular to the plane'of the sample. The condition for the minimum of the free energy restrictes the region of the DC field. Local amplitudes of eigen-modes C, D and E of fig-

ure 1 are illustrated in figure 2. Effect of the interface anisotropy on the resonance fields is given in figure 3.

We present the dependence of the resonance fields of the four-fold multilayer vs.

NL.

The parts A and B are for K.

/

g p g =

-lo4

G and K.

/

g p ~ =

+lo4

G ,

Fig. 1.

-

Imaginary part of local susceptibility vs. the DC field for K, = 0, N L = 60. Peaks A and B are for resonance fields of ferromagnets A and B, respectively. The peak C

is for the double sample. The four-fold sample has two resonance peaks D and E.

Fig. 2. - Local amplitudes of spin-wave eigenmodes of double and four-fold samples ( K , = 0, NL = 60). The curve C, D, and E correspond t o modes of peaks C, D and E of figure 1.

Fig. 3.

-

Dependences of the resonance fields of the four- fold magnetic layers on NL (the sample thickness) with the

spin quantum number SA = SB = 112. The part A is for

K,

/

g p g =

-lo4

G and the part B for K,

/

g p g =

+lo4

G .

respectively. For simplicity, we put spin quantum num- bers of the sublattices A and B equal 112.

Acknowledgments

The author wishes to thank Professor J. Morkowski for many helpful comments regarding this work. The paper was supported by the Polish Academy of Sci- ences under Project CPBP 01.12.