Antiferromagnetic interlayer coupling in diluted magnetic thin films with RKKY interaction

We study a model thin film containing diluted bilayer structure with the Ruderman-Kittel-Kasuya-Yosida long-range interaction. The magnetic subsystem is composed of two magnetically doped layers, separated by an undoped nonmagnetic spacer, and placed inside a wider film modeled by a quantum well of infinite depth. We focus our study on the range of parameters for which the antiferromagnetic coupling between the magnetic layers can be expected. The critical temperatures for such system are found and their dependence on magnetic layer thickness and charge-carrier concentration is discussed. The magnetization distribution within each magnetic layer is calculated as a function of layer thickness. The external field required to switch the mutual orientation of layer magnetizations from antiferromagnetic to ferromagnetic state is also discussed.

Figure 1

(Color online) Schematic cross section of an ultrathin film containing two magnetically doped layers. In this figure the total thickness of the film amounts to $n=11$, whereas the thickness of each magnetic layer is $n_L=3$ and the thickness of nonmagnetic spacer is $n_S=3$.

Figure 2

Density of states at the Fermi level in an ultrathin film, normalized to the bulk value, as a function of number of monolayers. The charge-carrier concentration is assumed to be $\rho =x/n$ for three representative values of $x$. In the inset, a similar plot is shown for the charge-carrier concentration $\rho =x/2$.

Figure 3

Interplanar coupling energy for an ultrathin film containing $n$ monolayers, as a function of separation between the planes, for the charge-carrier concentration $\rho =x/n$ and $x=0.05$.

Figure 4

Interplanar coupling energy in an ultrathin film for $n=20$, $x=0.05$, and the charge-carrier concentration $\rho =x/n$. Full points denote the values obtained from summation of the RKKY thin-film-modified exchange integral. The empty points represent the values obtained from summation of the usual RKKY bulk formula. In the inset, the very exchange integrals $J\left(r=0,z\right)$ are plotted: the solid line is for the RKKY interaction in thin film, while the dashed line corresponds to the bulk 3D RKKY coupling.

Figure 5

Interplanar coupling energy for an ultrathin film characterized by $n=20$ and $x=0.05$ and for three representative charge-carrier concentrations.

Figure 6

Interlayer coupling energy for the system characterized by $x=0.05$, $n=20$, and $n_S=3$ as a function of magnetic layer thickness. The two charge-carrier concentrations: $\rho =x/n$ (circles) and ${\rho }_{L}x/n$ (squares) are assumed.

Figure 7

External magnetic field required to switch the magnetizations of the layers from AF to $F$ state as a function of layer thickness. The results are presented for $T\to 0$ and selected parameters, where AF interlayer coupling is predicted.

Figure 8

Critical temperatures in MFA approximation for an ultrathin film with two magnetic layers. In (a) and (b) the filled symbols denote the Curie temperatures, while the empty ones are for the Néel temperatures. (a) Comparison of the results for charge-carrier concentration ${\rho }_{L}x/n$ (squares) and $\rho =x/n$ (circles). (b) Results for $\rho =x/n$ and various parameter ranges which are advantageous for AF interlayer coupling. (c) Comparison of the results obtained from the approximate [Eq. (15)] (empty symbols) and without this approximation (filled symbols).

Figure 9

Spatial distribution of magnetizations in subsequent monolayers for various thicknesses $n_L$ and for two sets of parameters: (a) $x=0.05$, $n=32$, and $n_S=14$; and (b) $x=0.025$, $n=28$, and $n_S=5$. The temperature is $T=\left(2/3\right)T_{\text{c}}$.