Theory and simulation of spin transport in antiferromagnetic semiconductors: Application to MnTe

In this paper we study the parallel spin current in an antiferromagnetic semiconductor thin film where we take into account the interaction between itinerant spins and lattice spins. The spin model is an anisotropic Heisenberg model. Here we use the Boltzmann equation with numerical data on cluster distribution obtained by Monte Carlo simulations and cluster-construction algorithms. We study the cases of degenerate and nondegenerate semiconductors. The spin resistivity in both cases is shown to depend on the temperature, with a broad maximum at the transition temperature of the lattice spin system. The shape of the maximum depends on the spin anisotropy and on the magnetic field. It shows, however, no sharp peak in contrast to ferromagnetic materials. Our method is applied to MnTe. Comparison to experimental data is given.

Figure 1

Energy map of an itinerant spin in the $xy$ plane with $D_1=2$ in units of the lattice constant $a$ and $A=0.01$, for $T=0.01$, $T=1.0$, $T=2.0$, and $T=2.5$ (from left to right, top to bottom, respectively). The values of energy corresponding to different colors are given on the right.

Figure 2

The successive steps in the application of the algorithm by Wolff to the case of Heisenberg spin (see text for explanation).

Figure 3

Number of clusters versus temperature for anisotropy $A=0.01$ (upper) and $A=1$ (lower). The values of $S_z$ are 1, 0.8, and 0.6 denoted by circles, squares, and triangles, respectively. Lines are guides to the eye.

Figure 4

Average size of clusters versus temperature for anisotropy $A=0.01$ (upper) and $A=1$ (lower). The values of $S_z$ are 1, 0.8, and 0.6 denoted by circles, squares, and triangles, respectively. Lines are guides to the eye.

Figure 5

Ratio $R=\rho \left(\text{Born}2\right)/\rho \left(\text{Born}1\right)$ versus $T$ for $V_0$=0.05 (squares, upper curve) and 0.01 (circles, lower curve) (see text for comments).

Figure 6

Sublattice magnetization versus $T$ for several values of anisotropy $A$. From left to right, $A=0.01$, $A=1$, $A=1.5$, and pure Ising spin.

Figure 7

Spin resistivity versus $T$ for several anisotropy values $A$ in an antiferromagnetic bcc system: $A=0.01$ (circles), 1 (squares), and 1.5 (triangles). Upper (lower) curves: degenerate (nondegenerate) system.

Figure 8

Spin resistivity for pure Heisenberg (circles) and Ising (squares) models in an antiferromagnetic bcc system. Upper (lower) curves: degenerate (nondegenerate) system.

Figure 9

Resistivities of up (circles) and down (squares) spins versus $T$ for two magnetic field’s strengths in the degenerate case. Top (bottom): $B=0.6\left(1.5\right)$.

Figure 10

Resistivities of up (circles) and down (squares) spins versus $T$ for two magnetic field’s strengths in the nondegenerate case. Top (bottom): $B=0.6\left(1.5\right)$.

Figure 11

Upper: number of clusters; lower: average size of clusters, versus $T$ for several values of $S_z$ and for magnetic field $B=1.5$; circles: $S_z=1$; squares: $S_z=0.8$; and triangles: $S_z=0.6$. Lines are guides to the eye.

Figure 12

Structure of the type NiAs is shown with Mn atoms only. This is a stacked hexagonal lattice. Up spins are shown by black circles, down spins by white ones. Nearest-neighbor (NN) bond is marked by 1, next NN bond by 2, and third NN bond by 3.

Figure 13

Number of clusters (upper) and cluster size (lower) versus $T$ for the MnTe structure obtained from Monte Carlo simulations for several values of $S_z$: 1 (circles), 0.8 (squares), and 0.6 (triangles). Lines are guides to the eye.

Figure 14

Normalized spin resistivity versus $T$ in MnTe: theoretical nondegenerate case (circles) and experimental results (stars) from Chandra et al.[42] Experimental data lie on the theoretical line for $T⩾140$ K (see text for comments).