Spin-orbit coupling effect in (Ga,Mn)As films: Anisotropic exchange interactions and magnetocrystalline anisotropy

The magnetocrystalline anisotropy (MCA) of (Ga,Mn)As films has been studied on the basis of ab initio electronic structure theory by performing magnetic torque calculations. An appreciable contribution to the in-plane uniaxial anisotropy can be attributed to an extended region adjacent to the surface. Calculations of the exchange tensor allow to ascribe a significant part to the MCA to the exchange anisotropy, caused either by a tetragonal distortion of the lattice or by the presence of the surface or interface.

Figure 1

(a) Geometry for the torque calculations; $J^{xx}-J^{yy}$ (b) and $J^{zz}-J^{yy}$ (c) difference for bulk (Ga,Mn)As with 5$\%$ Mn, with a tetragonal distortion $c/a=1.01$; ${\sum }_j(J_{ij}^{xx}-J_{ij}^{yy})$ (d) and ${\sum }_j(J_{ij}^{zz}-J_{ij}^{yy})$ (e) over all lattice sites up to $R_{ij}⩽5a$; comparison of $J^{xx}-J^{yy}$ (f) and $J^{zz}-J^{yy}$ (g) for $R_{ij}$ along [100] and [010] directions.

Figure 2

(a) Layer-resolved contributions to the MCA energy in the uncovered 8-ML (Ga,Mn)As film with 5 at. $\%$ Mn (circles) and 11 at.$\%$ Mn (triangles), for two directions of magnetization: $\vec{M}\left|\right|\left[110\right]$ and $\vec{M}\left|\right|\left[1\overline{1}0\right]$; (b) layer-resolved contributions to the MCA energy in the 8-ML (Ga,Mn)As film with 5 at. $\%$ Mn, with three capping layers of Au.

Figure 3

Variation of $(J_{ij}^{xx}-J_{ij}^{yy})$ with distance $R_{ij}$ of pairs $(i,j)$ of Mn atoms taken in the $\left[110\right]$ and $\left[1\overline{1}0\right]$ directions in the (Ga,Mn)As/GaAs film: (a) Bulk vs (Ga,Mn)As/GaAs interface and (b) bulk vs (Surf.-3) layer. (c) Layer-resolved sum $-{\sum }_j(J_{ij}^{yy}-J_{ij}^{xx})$ calculated within the sphere of radius $2.9a$ in comparison with the MCA energy $E^{\left[110\right]}-E^{\left[1\overline{1}0\right]}$ evaluated by magnetic torque calculations for the 8-ML (Ga,Mn)As film with 5 at.$\%$ Mn.