FINITE-SIZE EFFECTS IN ULTRATHIN MAGNETIC LAYERS

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FINITE-SIZE EFFECTS IN ULTRATHIN MAGNETIC

LAYERS

U. Gummich, G. Cabrera, C. E. T. da Silva

To cite this version:

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JOURNAL DE PHYSIQUE

Colloque C8, Suppl6ment au no 12, Tome 49, d6cembre 1988

FINITE-SIZE EFFECTS IN ULTRATHIN MAGNETIC LAYERS

U. Gummich, G. G. Cabrera and C. E. T. Gon~alves da Silva

Instituto de Fisica "Gleb Wataghin" , Universidade Estadual de Campinas, Caixa Postal 61 65, 13.081 Campinas, S6o Paulo, Brazil

Abstract.

-

We study thin ferro and antiferromagnetic films for the Heisenberg model using a modified version of the Onsager reaction field approximation adapted for finite-size layered systems. No phase transition is obtained for the finite system, but spin-spin correlations are enhanced when the number of layers increases.

Introduction

Molecular beam epitaxy techniques have become a sophisticated tool to produce ultrathin magnetic films approaching the monolayer limit. In particular the MnSeIZnSe superlattice in the zinc-blende structure has been recently studied with special interest on the magnetic properties of MnSe ultrathin layers [I]. Al- though bulk MnSe is known to be antiferromagnetic, the ultrathin layer behaves as paramagnetic. It has been suggested that this result can be explained in terms of a dimensionality crossover when the layer thickness is reduced to a few monolayers [I], with no phase transition for the two dimensional system. This fact agrees with some general theorems which preclude the presence of long-range orde in two dimensions [2].

Stimulated by the above experiments, we engaged ourselves in the theoretical study of the size effects in a layered Heisenberg system as a function of the layer thickness, using a modified version of the Onsager reaction field approximation (RFA) [3] adapted to treat a multilayer thin film. Preliminary results for the ferromagnetic system have been published elsewhere [4]. In

Full two-dimensional periodicity in the yz directions is maintained and we use two dimensional lattice vectors {Rj) for sites and an index n to lab el the different layers. In the antiferromagnetic case Rj is a lattice vector in a Bravais latticee with a basis of two spins to describe possible antiferromagnetic ordering.

Partial Fourier transform in two dimensions is done using Q vectors in a two dimensional Brillouin zone,

Q

= ( Q Y , Q Z )

where the subscript v labels the spin components, v =

x,

y, z ; No is the number of spins per layer (and per sublattice in the case of antiferromagnetism), and the spin has been chosen as S = 5 / 2 which is the case of Mn.

The Heisenberg Hamiltonian with nearest neighbors interactions

the present contribution we want to report additional

is treated through a mean field approach where the results which include both, the ferro and antiferromag-

local effective field is given by netic cases. Our analytical results point to the absence

of a phase transition in a film of finite thickness, in

contrast to approaches using different approximations H% (Q, n) = J (Q

I

nm)

(s'

(Q, m ) ) - ~

(9,

n)

-

The model

We model our layered system through layer depen- dent quantities: magnetization parameters for each layer (two sublattices for antiferromagnetic coupling) which measure the long-range order if present; RF parameters which play the role of local short-range order parameters and are normalized t o the number of nearest neighbors for a given layer.

The presence of an external magnetic field paral- lel to the film surface is allowed, in order to avoid demagnetizing effects. The direction of spin quanti- zation (ziaxis) is then chosen along the direction of the field (and corresponding to one of the cube axis).

H ~ E (Q, n) = J (Q

I

nm) (SP (Q, m ) ) - H (Q, n) -

where a and

p

stand for the two different sublattices in the case of antiferromagnetism (for the ferromagnetic system both expressions in (3) are iden- tical), J (Q

I

nm) is the partial Fourier transform of the exchange interaction, and the angle brackets stand for statistical averages. The standard mean field approximation is obtained without the last term in the right hand side of formulae (3), being X u (n) the RF parameter for the wth layer and the cr sublattice (cr = a,

p)

.

In our approach the mean field is corrected by the reaction field in order to take into account

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C8

-

1702 JOURNAL DE PHYSIQUE spin-spin correlations. The inclusion of this effect s u p

presses the phase transition in two dimensions. Since the

Xu

(n) parameters are essentially spin-spin correlations, they are directly linked to the susceptibility through the fluctuation-dissipation theorem. A self- consistent equation is obtained if one notes that the susceptibility can also be obtained from the magnetization as a derivative in relation to the external field [41.

Summation over the two dimensional Brillouin zone has been done using the special point scheme devel- oped by Cunningham [6]. The particular cases N = 1

(two dimensions) and N --, co (three dimensions) can be studied analytically. In the limit N --, co one re- covers the three dimensional behavior (71 with a phase transition at a critical temperature given by

for the simple cubic structure, for both ferro (Jij

<

0) and antiferromagnetism (Jij

>

0) .Tc, is the standard

Curie-Weiss (mean field) transition temperature. Numerical results and discussions

For the ferromagnetic case, we have obtained numerical results up to 10 layer thicknesses for different values of temperature and magnetic field. All the cases display the same qualitative behavior with the absence of a phase transition. Surface effects for free end boundary conditions fall off very quickly when go- ing to the interior of the film: for all practical purpose, the N = 3 case illustrates the general situation. When an external magnetic field is applied the magnetization parameters saturate a t low temperature to the value S = 512, with a typical paramagnetic behavior. The

e

Fig. 1. - Magnetization parameters for the three layer system as a function of temperature. The magnetic field is

H = 2 I J I , where J is the exchange constant, and 6 is the reduced temperature (1 J] 0 = k B T )

.

Magnetization for the inner layer is M ( 2 ) , and the superscripts F and AF label the ferro and antiferromagnetic cases. For the surfaces we get M (1) = M (3) a s required by symmetry. The continu- ous line depicts the Brillouin function for S = 512 (spin of Mn), as a reference.

Fig. 2. - RF parameters as functions of temperature for the three layer system. Notation is the same as in figure I, and the external magnetic field is also H = 2

1

JI

.

Displayed quantities for t h e antiferromagnetic case are negative.

same occurs for the RF parameters, where normaliza- tion at T = 0 is given by the coordination number. In figures 1 and 2 we are displaying their variation as a function of temperature for the three layer system, and for a particular value of the magnetic field.

Concerning the antiferromagnetic case (Jij

>

0)

,

the RE' parameters are mainly negative indicating the presence of distinctive antiferromagnetic correlations of short range. Long-range order does not develop in the system, as can be seen in figure 1 for the magnetiza, tion parameters. Since we are always in the paramagnetic region, both sublattice magnetizations are equal. The small values shown in figure 1 can be understood based on physical grounds: the spins are frustrated as a result of the competition between the antiferromagnetic coupling and the tendency of alignment along the applied field.

Is is still not clear, within the approximation de- veloped here, if the inclusion of anisotropic effects will induce a phase transition in finite-width systems. This point is currently under study.

[I] Nurmikko, A. V., Lee, D;, Hefetz, Y., Kolodziejski, L. A. and Gunshor, R. L., Proc. 18th Intern. Conf. on Physics of Semiconductors, Ed. 0. Engstrom (World Scientific, Singapore) 1987, p. 775.

[2] Mermin, N. M. and Wagner, H., Phys. Rev. Lett. 17 (1966) 1133.

[3] Brout, R. and Thomas, H., Physics 3 (1967) 317. [4] Gummich, Ute, Cabrera, G. G. and Gon~alves da

Silva, C. E. T., Surf. Sci. 196 (1988) 643. [5] Diep-The-Hung, Levy, J. C. S. and Nagai, O.,

Phys. Status Solidi B 93 (1979) 351.

[6] Cunningham, S. L., Phys. Rev. B 12 (1974) 4988. [7] Scherer, C. and Aveline, I., Phys. Status Solidi B