Angle-dependent magnetotransport in cubic and tetragonal ferromagnets: Application to (001)- and $\left(113\right)A$-oriented $(\mathrm{Ga},\mathrm{Mn})\mathrm{As}$

General expressions for the longitudinal and transverse resistivities of single-crystalline cubic and tetragonal ferromagnets are derived from a series expansion of the resistivity tensor with respect to the magnetization orientation. They are applied to strained $(\mathrm{Ga},\mathrm{Mn})\mathrm{As}$ films, grown on (001)- and $\left(113\right)A$-oriented $\mathrm{GaAs}$ substrates, where the resistivities are theoretically and experimentally studied for magnetic fields rotated within various planes parallel and perpendicular to the sample surface. We are able to model the measured angular dependences of the resistivities within the framework of a single ferromagnetic domain, calculating the field-dependent orientation of the magnetization by numerically minimizing the free-enthalpy density. Angle-dependent magnetotransport measurements are shown to be a powerful tool for probing magnetic anisotropy. The anisotropy parameters of the $(\mathrm{Ga},\mathrm{Mn})\mathrm{As}$ films inferred from the magnetotransport measurements agree with those obtained by ferromagnetic resonance measurements within a factor of 2.

Figure 1

Orientation of the (001) and $\left(113\right)A$ samples with respect to the crystallographic axes.

Figure 2

$G_M$ as a function of $\mathit{m}$, calculated for a given magnetic field ${\mu }_{0}H=0.15\mathrm{T}$ and a set of anisotropy parameters with arbitrarily chosen values $B_{c\Vert }=B_{c\perp }=-0.1\mathrm{T}$, $B_{001}=0.15\mathrm{T}$, and $B_{\overline{1}10}=-0.05\mathrm{T}$. The equilibrium position of $\mathit{M}$ is determined by the minimum of $G_M$.

Figure 3

Simulated polar (left axis) and azimuth (right axis) angles of the magnetic field $\mathit{H}$ (dashed lines) and the magnetization $\mathit{M}$ (solid lines) for $\mathit{H}$ rotated in the (111) plane. The same field strength and anisotropy parameters have been used as in Fig. 2.

Figure 4

The angle-dependent magnetotransport measurements were carried out for $\mathit{H}$ rotated within (a) the layer plane, (b) a plane perpendicular to the current direction $\mathit{j}$, and (c) a plane spanned by $\mathit{j}$ and the normal vector $\mathit{n}$.

Figure 5

Angle-dependent resistivities ${\rho }_{\mathrm{long}}$ (circles) and ${\rho }_{\mathrm{trans}}$ (dots) of the (001) $(\mathrm{Ga},\mathrm{Mn})\mathrm{As}$ sample at $4.2\mathrm{K}$. The measurements were carried out at fixed field strengths of ${\mu }_{0}H=0.1$, 0.25, and $0.7\mathrm{T}$ with $\mathit{H}$ rotated in (a) the (001), (b) the (110), and (c) the $\left(\overline{1}10\right)$ plane. The solid lines represent fits to the experimental data using Eqs. (19) and one set of resistivity and anisotropy parameters. The dashed lines at $0.7\mathrm{T}$ simulate the limiting case where $\mathit{M}$ perfectly aligns with $\mathit{H}$. In (a) the dashed lines completely coincide with the solid lines.

Figure 6

Calculated polar (left axis) and azimuth (right axis) angles of the magnetic field $\mathit{H}$ (dashed lines) and the magnetization $\mathit{M}$ (solid lines) for $\mathit{H}$ rotated in the (110) plane of the (001) $(\mathrm{Ga},\mathrm{Mn})\mathrm{As}$ sample at $0.1\mathrm{T}$. The resistivity and anisotropy parameters used in the calculation were derived from a fit to the angle-dependent resistivities shown in Fig. 5.

Figure 7

Angle-dependent FMR fields of the (001) $(\mathrm{Ga},\mathrm{Mn})\mathrm{As}$ sample at $5\mathrm{K}$ for $\mathit{H}$ rotated in the (001), $\left(\overline{1}10\right)$, and (110) planes. The solid lines represent the result of a least-squares fit; the dashed lines were calculated using $g=2.0$ and the anisotropy parameters estimated from the magnetotransport data.

Figure 8

Angle-dependent FMR fields of the $\left(113\right)A$ $(\mathrm{Ga},\mathrm{Mn})\mathrm{As}$ sample at $5\mathrm{K}$ for $\mathit{H}$ rotated in the $\left(33\overline{2}\right)$, $\left(1\overline{1}0\right)$, and $\left(113\right)A$ planes. The solid and dashed lines represent the results of least-squares fits with and without considering a uniaxial term along [001] in $G_M^{113}$, respectively.

Figure 9

Angle-dependent resistivities ${\rho }_{\mathrm{long}}$ (circles) and ${\rho }_{\mathrm{trans}}$ (dots) of the $\left(113\right)A$ $(\mathrm{Ga},\mathrm{Mn})\mathrm{As}$ sample at $4.2\mathrm{K}$. The measurements were carried out at fixed field strengths of ${\mu }_{0}H=0.1$, 0.25, and $0.7\mathrm{T}$ with $\mathit{H}$ rotated in (a) the (113), (b) the $\left(33\overline{2}\right)$, and (c) the $\left(\overline{1}10\right)$ plane. The solid lines represent fits to the experimental data using Eqs. (20) and one set of resistivity and anisotropy parameters. The dashed lines at $0.7\mathrm{T}$ simulate the limiting case where $\mathit{M}$ perfectly aligns along $\mathit{H}$.