The RKKY Coupling of Two Magnetically Doped Monolayers in Thin Films
Karol Szałowski* and Tadeusz Balcerzak
Department of Solid State Physics, University of Łódź,
ul. Pomorska 149/153, 90-236 Łódź, Poland
The studies of interaction between localized magnetic moments and charge carriers
usually assume a contact form of the exchange potential. However, as the localized magnetic
moments originate from the inner electronic orbitals (e.g. of d-type in diluted magnetic
semiconductors), they manifest a finite spatial extension [1]. Consequently, it seems natural to
introduce a diffused exchange potential. This, in turn, modifies the perturbative RKKY-type
interaction between the localized spins. Such an effect for the bulk case has been considered
so far in Refs. [2,3]. In the bulk media, the relevant lengthscale to be compared with the
distribution width is given by an inverse of the Fermi wavevector. However, for the important
case of ultrathin film (quantum well) geometry, the film thickness constitutes another vital
lengthscale and the spatial extension of the potential is not negligible compared with that,
independently on the concentration of the carriers.
In our work we apply the Gauss-type exchange coupling potential to the RKKY
mechanism of localized spins interaction in ultrathin film modeled by an infinite quantum
well [4,5]. We consider a model system based on fcc structure with orientation (001), when
two monolayers inside the film contain magnetic ions. Such choice reflects the geometry of
experimentally investigated structures of magnetically-doped and undoped semiconductors
(such as MnGaAs/GaAs). The model is parametrized by the total thickness of the film (i.e.
thickness of the quantum well), the distance between the magnetic layers as well as the
concentration of free charge carriers inside the film. We especially study the coupling energy
of two magnetic monolayers as a function of their separation, total width of the quantum well
and the charge concentration. Our main aim is to explore the range of parameters for which
the interaction energy favors ferromagnetic or antiferromagnetic orientation of magnetizations
in these layers. In addition we calculate the MFA critical temperatures (Curie/Néel) for the
system. We make comparison of our results with the limiting case when the bulk (3D) RKKY
interaction coupling is taken into account. Such comparison emphasizes the importance of
quantum size effects in the thin films.
The considerations are based on the Ising-type Hamiltonian of interaction between the
localized spins. To calculate the energy, summation is performed over the lattice sites up to
the cut-off distance assuring the satisfactory convergence of the interaction energy per spin, so
that we do not have to use the continuous approximation. We emphasize that the diffusion of
the contact potential has a noticeable effect on the interaction energy. What is crucial, it also
removes the divergence of the RKKY thin-film integral in the direction perpendicular to the
film plane.
[1] A.K. Bhattacharjee and C. Benoit à la Guillaume, Solid State Commun. 113, 17 (1999).
[2] J. Schliemann, J. König, and A. H. MacDonald, Phys. Rev. B 64, 165201 (2001).
[3] L. Brey and G. Gómez-Santos, Phys. Rev. B 68, 115206 (2003).
[4] T. Balcerzak, in “Trends in thin solid films research”, pp. 249-277, ed.: Alyssa R. Jost,
Nova Science Publishers, Inc., New York, 2007.
[5] M. A. Boselli, A. Ghazali, and I. C. da Cunha Lima, Phys. Rev. B 62, 8895 (2000).
*corresponding
author, e-mail address: kszalowski@uni.lodz.pl