Quantitative study of magnetotransport through a (Ga,Mn)As single ferromagnetic domain

We have performed a systematic investigation of the longitudinal and transverse magnetoresistance of a single ferromagnetic domain in ${\mathrm{Ga}}_{1-x}{\mathrm{Mn}}_x\mathrm{As}$. We find that, by taking into account the intrinsic dependence of the resistivity on the magnetic induction, an excellent agreement between experimental results and theoretical expectations is obtained. Our findings provide a detailed and fully quantitative validation of the theoretical description of magnetotransport through a single ferromagnetic domain. Our analysis furthermore indicates the relevance of magnetoimpurity scattering as a mechanism for magnetoresistance in ${\mathrm{Ga}}_{1-x}{\mathrm{Mn}}_x\mathrm{As}$.

Figure 1

(Color online.) Panels (a)-(c) show the longitudinal and transverse magnetoresistance of a single ${\mathrm{Ga}}_{1-x}{\mathrm{Mn}}_x\mathrm{As}$ ferromagnetic domain, measured for three different orientations of the in-plane magnetic field (${\varphi }_H=-20°,+15°$, and $+39°$, respectively). In all panels, full symbols represent the data measured for an increasing magnetic field $H$ (up-sweep) and the open symbols those measured with a decreasing field (down-sweep). The full lines represent the resistances calculated from Eqs. (1, 2), as discussed in the text. Panel (d) defines the angles ${\varphi }_H$ and ${\varphi }_M$ between the field $\mathbf{H}$, the magnetization $\mathbf{M}$, and the current density $\mathbf{j}$. Panel (e) illustrates the orientation of the two easy axes in ${\mathrm{Ga}}_{1-x}{\mathrm{Mn}}_x\mathrm{As}$ with respect to the direction of the current flow. When the field $H$ is swept, magnetization reversal occurs by subsequent, $\approx 90°$ switches, e.g., $\beta \to \overline{\alpha }\to \overline{\beta }$.

Figure 2

Panels (a) and (b) illustrate the behavior of the different contributions to the longitudinal magnetoresistance (thin lines) as well as their sum (i.e., the total longitudinal resistance; thick line), as calculated from Eq. (1) for two different orientations of the in-plane field (${\varphi }_H=-20°$ and $+15°$). All the traces are offset for clarity. The first term, ${\rho }_{\perp }$, is proportional to the magnitude of the magnetic induction vector. The second term $({\rho }_{\Vert }-{\rho }_{\perp }){\cos }^2\left({\varphi }_M\right)=\Delta \rho {\cos }^2\left({\varphi }_M\right)$, reflects the abrupt switches of ${\varphi }_M$. A comparison between calculations and experimental data is shown in Figs. 1a, 1b and in Fig. 3.

Figure 3

(Color online) Comparison between the longitudinal magnetoresistance measured for different orientations ${\varphi }_H$ of the magnetic field (symbols) and the predictions by Eq. (1) of the SDM model (full lines) in which all parameters have been fixed by our analysis. The magnetoresistance traces have been offset for clarity. Full symbols represent the experimental data measured for magnetic field up sweeps, open symbols correspond to the data measured for magnetic field down sweeps.

Figure 4

(Color online) (a) Longitudinal magnetoresistance measured ramping up (full circles) and down (open circles) the magnetic field, reverting the sweep direction at 15 mT, i.e., in between the two switches of the magnetization orientation. The full line is calculated theoretically using Eq. (1) of the SDM model, without any adjustable parameters (all parameters are fixed by the previous analysis). The dash-dotted line illustrates the magnetoresistance obtained if the field sweep direction is reversed after the second switch of the magnetization orientation [see also Fig. 1a]. (b) The magnetic field dependence of ${\rho }_{\perp }$ measured experimentally (open symbols) is reproduced by the magneto-impurity scattering model proposed by Nagaev (full line) over a broad range of magnetic fields.