Effect of inversion asymmetry on the intrinsic anomalous Hall effect in ferromagnetic (Ga,Mn)As

The relativistic nature of the electron motion underlies the intrinsic part of the anomalous Hall effect, believed to dominate in ferromagnetic (Ga,Mn)As. In this paper, we concentrate on the crystal band structure as an important facet to the description of this phenomenon. Using different $k\cdot p$ and tight-binding computational schemes, we capture the strong effect of the bulk inversion asymmetry on the Berry curvature and the anomalous Hall conductivity. At the same time, we find that it does not affect other important characteristics of (Ga,Mn)As, namely, the Curie temperature and uniaxial anisotropy fields. Our results extend the established theories of the anomalous Hall effect in ferromagnetic semiconductors.

Figure 1

(Color online) The top of the GaAs valence band with spin splitting $\Delta =-0.1\text{eV}$ calculated by different methods. Horizontal lines denote the positions of the Fermi level $E_F$ for the hole densities 0.3 and $1.2{\text{nm}}^{-3}$ in the $spd{s}^{\star }$ tight-binding model.

Figure 2

(Color online) Computed magnitudes of Curie temperatures as a function of the hole concentration according to various band-structure models for ${\text{Ga}}_{1-x}{\text{Mn}}_x\text{As}$ with the Mn content $x=2\mathrm{\%}$, 5%, and 8%. Vertical lines indicate the maximal experimentally realizable hole densities for given $x$. On this and the following graphs, the numerical data were generated more densely than the markers, which are visual aids only.

Figure 3

(Color online) The amplitude of the uniaxial anisotropy field $H_{\text{un}}$ (divided by saturation magnetization, see Ref. 27) for compressive $\left({\epsilon }_{xx}=-1\mathrm{\%}\right)$ and tensile $\left({\epsilon }_{xx}=1\mathrm{\%}\right)$ biaxial strain in (Ga,Mn)As with spin splitting $\Delta =-0.15\text{eV}$. The effect of shape anisotropy is neglected. In the central range of hole concentrations (II), the easy axis is in-plane and perpendicular to the plane for compressive and tensile strain, respectively, while the reorientation transition is expected in either low or high-concentration regimes (I or III).

Figure 4

(Color online) Berry curvature of majority heavy and light hole bands in (Ga,Mn)As calculated using (a) the six-band $k\cdot p$, (b) eight-band $k\cdot p$ and (c) $spd{s}^{\star }$ tight-binding model, with and without the spin splitting $\Delta$. Inset: divergences in the Berry curvature.

Figure 5

(Color online) Anomalous Hall conductivity ${\sigma }_{xy}$ vs spin splitting $\Delta$ for different hole concentrations $p$ and biaxial strain ${\epsilon }_{xx}$ in the six-band $k\cdot p$ model. The hallmark of the diabolic point for unstrained $p=0.5{\text{nm}}^{-3}$ curve is marked with an arrow. Numerical errors are ca 0.5%.

Figure 6

(Color online) Anomalous Hall conductivity ${\sigma }_{xy}$ vs spin splitting $\Delta$ for different hole concentrations $p$ and biaxial strain ${\epsilon }_{xx}$ in the eight-band $k\cdot p$ model.

Figure 7

(Color online) Anomalous Hall conductivity ${\sigma }_{xy}$ vs spin splitting $\Delta$ for different hole concentrations $p$ and biaxial strain ${\epsilon }_{xx}$ in the $spd{s}^{\star }$ tight-binding model.

Figure 8

(Color online) Anomalous Hall conductivity ${\sigma }_{xy}$ vs spin splitting $\Delta$ for different hole concentrations $p$ in the $spd{s}^{\star }$ tight-binding model. The scattering-induced energy-level broadening $\hslash \Gamma$ equal to 50 meV and $\left|E_F\right|$ is included.

Figure 9

(Color online) Anomalous Hall conductivity vs spin splitting for different hole concentrations $p$ and biaxial strain ${\epsilon }_{xx}$ in the $sp{s}^{\star }$ tight-binding model.

Figure 10

(Color online) Reconstruction of the experimental (Ref. 41) anomalous Hall conductivities for nominal Mn concentration $x$, hole concentration $p$ and strain ${\epsilon }_{xx}$, using different theoretical models.