Advanced resistivity model for arbitrary magnetization orientation applied to a series of compressive- to tensile-strained (Ga,Mn)As layers

The longitudinal and transverse resistivities of differently strained (Ga,Mn)As layers are theoretically and experimentally studied as a function of the magnetization orientation. The strain in the series of (Ga,Mn)As layers is gradually varied from compressive to tensile using (In,Ga)As templates with different In concentrations. Analytical expressions for the resistivities are derived from a series expansion of the resistivity tensor with respect to the direction cosines of the magnetization. In order to quantitatively model the experimental data, terms up to the fourth order have to be included. The expressions derived are generally valid for any single-crystalline cubic and tetragonal ferromagnet and apply to arbitrary surface orientations and current directions. The model phenomenologically incorporates the longitudinal and transverse anisotropic magnetoresistance as well as the anomalous Hall effect. The resistivity parameters obtained from a comparison between experiment and theory are found to systematically vary with the strain in the layer.

Figure 1

Schematic polar plot in arbitrary units showing the dependence of ${\rho }_{\text{long}}-{\rho }_0$ (solid line) and ${\rho }_{\text{trans}}$ (dashed line) on $\mathit{m}$ for in-plane magnetization perpendicular to $\mathit{n}$. The dotted circle indicates the zero line that separates negative and positive values of ${\rho }_{\text{long}}-{\rho }_0$ and ${\rho }_{\text{trans}}$.

Figure 2

Dependence of ${\rho }_{\text{long}}-{\rho }_0$ (solid line) and ${\rho }_{\text{trans}}$ (dashed line) on $\mathit{m}$ for out-of-plane magnetization perpendicular to $\mathit{j}$. The dash-dotted line illustrates the special case of cubic crystal symmetry and current along [100]. The dotted circle indicates the zero line that separates negative and positive values of ${\rho }_{\text{long}}-{\rho }_0$ and ${\rho }_{\text{trans}}$.

Figure 3

Schematic polar plot in arbitrary units showing the dependence of ${\rho }_{\text{long}}-{\rho }_0$ (solid line) and ${\rho }_{\text{trans}}$ (dashed line) on $\mathit{m}$ for out-of-plane magnetization perpendicular to $\mathit{t}$. The dotted circle indicates the zero line that separates negative and positive values of ${\rho }_{\text{long}}-{\rho }_0$ and ${\rho }_{\text{trans}}$.

Figure 4

Graphical illustration of the different contributions to the normalized density of the free enthalpy $G_M$, plotted as three-dimensional functions of the magnetization orientation. The equilibrium position of $\mathit{M}$ is determined by the minimum of $G_M$.

Figure 5

The angular dependence of the resistivities was probed by rotating an external magnetic field $\mathit{H}$ within 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, respectively.

Figure 6

(Color online) Resistivities ${\rho }_{\text{long}}$ and ${\rho }_{\text{trans}}$ recorded from the 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 carried out at fixed field strengths of ${\mu }_{0}H=0.11$, 0.26, and 0.65 T with $\mathit{H}$ rotated in (a) the (001), (b) the (100), and (c) the (010) plane, corresponding to configurations I, II, and III, respectively. The black solid lines are fits to the experimental data using Eqs. (10, 11) and one single set of resistivity and anisotropy parameters. The dashed line in (b) at 0.65 T represents an attempt to fit the measured curve considering only terms up to the second order.

Figure 7

(Color online) Continuation from Fig. 6. The current direction is now along [110] and the magnetic field $\mathit{H}$ was rotated in (a) the (001), (b) the (110), and (c) the $\left(\overline{1}10\right)$ plane, corresponding to configurations I, II, and III, respectively.

Figure 8

Values of the normalized resistivity parameters ${\rho }_i/{\rho }_0$ $\left(i=1,\dots ,8\right)$ plotted against the vertical strain ${\epsilon }_{zz}$. The solid lines are smoothing splines and are drawn to guide the eye. In (a) the normalized values of ${\rho }_1,\dots ,{\rho }_4$ are shown for $\mathit{j}\parallel \left[100\right]$ and in (b) for $\mathit{j}\parallel \left[110\right]$. The dash-dotted and dashed lines depict the strain dependence of ${\rho }_1+{\rho }_3$ and ${\rho }_2+{\rho }_4$, respectively, and are obtained by adding the individual spline curves. In (c), the normalized values of ${\rho }_5$, ${\rho }_6+{\rho }_8$, and ${\rho }_7$ are shown for both current directions.

Figure 9

Expansion parameters $B+b,\dots ,e_4$ plotted against the vertical strain ${\epsilon }_{zz}$. The values were calculated from ${\rho }_i$ $\left(i=1,\dots ,8\right)$ using Eqs. (B4, B5) and are normalized to that of $A$. The solid lines are smoothing splines and are drawn to guide the eye.

Figure 10

(Color online) Variation of the angle-dependent normalized longitudinal resistivity $\left({\rho }_{\text{long}}-{\rho }_0\right)/{\rho }_0$ with vertical strain ${\epsilon }_{zz}$ at ${\mu }_{0}H=0.65\text{T}$ and $T=4.2\text{K}$. The resistivity was recorded as a function of the magnetic field orientation for (a) $\mathit{j}\parallel \left[110\right]$ and $\mathit{H}$ in the (001) plane, (b) $\mathit{j}\parallel \left[100\right]$ and $\mathit{H}$ in the (100) plane, and (c) $\mathit{j}\parallel \left[110\right]$ and $\mathit{H}$ in the (110) plane. The experimental data are depicted by the red solid circles and the calculated curves by the black solid lines.

Figure 11

(Color online) Variation of the angle-dependent normalized transverse resistivity ${\rho }_{\text{trans}}/{\rho }_0$ with vertical strain ${\epsilon }_{zz}$ at ${\mu }_{0}H=0.65\text{T}$ and $T=4.2\text{K}$. The resistivity was recorded as a function of magnetic field orientation for $\mathit{j}\parallel \left[110\right]$ and $\mathit{H}$ in the (110) plane. The dashed line was calculated setting all magnetic anisotropy parameters to zero and demonstrates the influence of the MA at 0.65 T for ${\epsilon }_{zz}=0.22\%$.