Magnetic anisotropy in (Ga,Mn)As: Influence of epitaxial strain and hole concentration

We present a systematic study on the influence of epitaxial strain and hole concentration on the magnetic anisotropy in (Ga,Mn)As at 4.2 K. The strain was gradually varied over a wide range from tensile to compressive by growing a series of (Ga,Mn)As layers with 5% Mn on relaxed graded (In,Ga)As/GaAs templates with different In concentration. The hole density, the Curie temperature, and the relaxed lattice constant of the as-grown and annealed (Ga,Mn)As layers turned out to be essentially unaffected by the strain. Angle-dependent magnetotransport measurements performed at different magnetic-field strengths were used to probe the magnetic anisotropy. The measurements reveal a pronounced linear dependence of the uniaxial out-of-plane anisotropy on both strain and hole density. Whereas the uniaxial and cubic in-plane anisotropies are nearly constant, the cubic out-of-plane anisotropy changes sign when the magnetic easy axis flips from in-plane to out-of-plane. The experimental results for the magnetic anisotropy are quantitatively compared with calculations of the free energy based on a mean-field Zener model. Almost perfect agreement between experiment and theory is found for the uniaxial out-of-plane and cubic in-plane anisotropy parameters of the as-grown samples. In addition, magnetostriction constants are derived from the anisotropy data.

Figure 1

$\Delta F_{M,\text{int}}$ calculated as a function of the magnetization orientation and the strain within the microscopic model (solid symbols) for $\mathit{M}$ (a) in the (001) plane and (b) in the (010) plane. $\phi$ and $\theta$ denote the azimuth and polar angles of $\mathit{M}$, respectively. The solid lines are least-squares fit curves using (a) Eq. (9) and (b) Eq. (10) with $B_{2\perp }$, $B_{4\parallel }$, and $B_{4\perp }$ as fit parameters. (c) The anisotropy parameter $B_{2\perp }$ obtained from the fit shows a pronounced linear dependence on ${\epsilon }_{zz}$.

Figure 2

Reciprocal space map around the asymmetric (224) reflections of a nearly unstrained (Ga,Mn)As layer $\left({\epsilon }_{zz}=-0.04\%\right)$ grown on relaxed (In,Ga)As/GaAs template. $h$ and $l$ are the coordinates in $k$ space in units of the reciprocal-lattice vectors along [100] and [001] in the GaAs substrate, respectively. $\Delta h$ and $\Delta l$ denote the shift of the peak position relative to that of the substrate. In the plot, $\Delta h$ and $\Delta l$ are only depicted for the (Ga,Mn)As layer.

Figure 3

(a) Lateral lattice parameter $a_{\parallel }$ and (b) tilt angle toward the [110] direction of the (In,Ga)As buffer layer, plotted against the In content. The solid line (as grown) and the dashed line (annealed) are regression lines.

Figure 4

(a) Relaxed lattice parameter $a_{\text{rel}}$ and (b) vertical strain ${\epsilon }_{zz}$ of the (Ga,Mn)As layer plotted against the In content in the (In,Ga)As buffer. Postgrowth annealing only leads to a slight decrease of the values.

Figure 5

Hole density $p$ and Curie temperature $T_C$ of (Ga,Mn)As plotted against the strain ${\epsilon }_{zz}$ for the as-grown (solid symbols) and the annealed samples (open symbols). The fluctuations in the values are mainly attributed to slight variations in the growth temperature. For $p$ we estimate an error margin of about $\pm 10\%$ and for $T_C$ an error margin of up to $\pm 20\%$.

Figure 6

The angular dependence of the resistivities was probed by rotating an external magnetic field $\mathit{H}$ within the three different planes (a) perpendicular to $\mathit{n}$, (b) perpendicular to $\mathit{j}$, and (c) perpendicular to $\mathit{t}$. The corresponding configurations are referred to as I, II, and III.

Figure 7

(Color online) Resistivities ${\rho }_{\text{long}}$ and ${\rho }_{\text{trans}}$ recorded from a nearly unstrained (Ga,Mn)As layer with ${\epsilon }_{zz}=-0.04\%$ at 4.2 K and $\mathit{j}\parallel \left[100\right]$ (red solid circles). The measurements were performed at fixed field strengths of ${\mu }_{0}H=0.11$, 0.26 and 0.65 T with $\mathit{H}$ rotated in the (001) plane corresponding to configuration I. The black solid lines are fits to the experimental data using Eqs. (15, 16), and one single set of resistivity and anisotropy parameters.

Figure 8

(a) Dependence of the uniaxial out-of-plane anisotropy parameter $B_{001}=B_{2\perp }+B_d$ on ${\epsilon }_{zz}$ for the as-grown (solid circles) and annealed samples (open circles). $B_d=60\text{mT}$ is inferred from the intersections between the corresponding regression lines (dotted lines) and the vertical axis. (b) Anisotropy parameter $B_{2\perp }$ obtained by subtracting 60 mT from the measured $B_{001}$ data. The lines are model calculations for $B_{2\perp }$ performed within the microscopic theory described in Sec. 3B using values for the parameter $B_G$ as shown and the averaged $p$ values from Fig. 5. (c) Anisotropy parameter $B_{2\perp }=B_{001}-B_d$ normalized to the hole density $p$. Experimentally, the same linear dependence of $B_{2\perp }/p$ on ${\epsilon }_{zz}$ is obtained for both the as grown and the annealed samples. The dotted line represents a linear regression.

Figure 9

SQUID curves of the as-grown and annealed samples with ${\epsilon }_{zz}\approx 0.2\%$, measured at 5 K for $\mathit{H}$ oriented along [001]. The saturation magnetization of $\sim 40\text{mT}$ is representative for the whole set of (Ga,Mn)As samples. The values of the coercive fields, derived from the hysteresis loops (not shown), were found to be below 10 mT.

Figure 10

Dependence of the cubic anisotropy parameters (a) $B_{4\parallel }$ and (b) $B_{4\perp }$ on ${\epsilon }_{zz}$ for the as grown (solid circles) and annealed samples (open circles). The lines are model calculations within the microscopic theory using the same values for $p$ and $B_G$ as in Fig. 8b.

Figure 11

Dependence of the uniaxial in-plane parameter $B_{\overline{1}10}$ for the as-grown (solid circles) and annealed samples (open circles).