Anisotropic magnetoresistance of spin-orbit coupled carriers scattered from polarized magnetic impurities

Anisotropic magnetoresistance (AMR) is a relativistic magnetotransport phenomenon arising from combined effects of spin-orbit coupling and broken symmetry of a ferromagnetically ordered state of the system. In this work we focus on one realization of the AMR in which spin-orbit coupling enters via specific spin-textures on the carrier Fermi surfaces and ferromagnetism via elastic scattering of carriers from polarized magnetic impurities. We report detailed heuristic examination, using model spin-orbit coupled systems, of the emergence of positive AMR (maximum resistivity for magnetization along current), negative AMR (minimum resistivity for magnetization along current), and of the crystalline AMR (resistivity depends on the absolute orientation of the magnetization and current vectors with respect to the crystal axes) components. We emphasize potential qualitative differences between pure magnetic and combined electromagnetic impurity potentials, between short-range and long-range impurities, and between spin-1/2 and higher spin-state carriers. Conclusions based on our heuristic analysis are supported by exact solutions to the integral form of the Boltzmann transport equation in archetypical two-dimensional electron systems with Rashba and Dresselhaus spin-orbit interactions and in the three-dimensional spherical Kohn-Littinger model. We include comments on the relation of our microscopic calculations to standard phenomenology of the full angular dependence of the AMR, and on the relevance of our study to realistic, two-dimensional conduction-band carrier systems and to anisotropic transport in the valence band of diluted magnetic semiconductors.

Figure 1

(Color online) Rashba model and (a) its spin texture along the Fermi contours. Dominant scattering channels for the states with group velocity pointing to the right when (b) magnetic and (c) electromagnetic impurities (see text) constitute the prevalent source of momentum relaxation. Note the indicated directions of impurity polarization. The current flow is directed from left to right. The reader might consider the limit $k_{-}⪢k_{+}$ for better understanding of this and subsequent figures.

Figure 2

(Color online) Dresselhaus model and (a) its spin texture. In order to determine the current and the AMR along the [100] and [110] crystallographic directions we focus on the states with group velocities pointing in the respective directions, (b) and (c). Dominant momentum relaxation channels for these states and scattering on magnetic impurities are indicated on the bottom panels.

Figure 3

(Color online) (a) Spin texture along the Fermi contours of Rashba-Dresselhaus model with $\alpha =\beta$. The AMR is zero for any type of short-range impurities. However, for long-range magnetic impurities the scattering amplitudes depend on the momentum transfer (illustrated by the length of the arrows) and nonzero AMR arises for current both along (b) [110] and (c) $\left[\overline{1}10\right]$ crystallographic directions.

Figure 4

(Color online) (a) Cross-section (parallel to the $k_x,k_y$ plane) of the 3D radial spin texture belonging to the two lower-energy bands of the Kohn-Luttinger Hamiltonian. The two Fermi surfaces are sketched with different sizes for clarity, although the Hamiltonian (6) implies $k_{-}=k_{+}$ as $h\to 0$. (b) Dominant scattering channels for magnetic impurities, note the difference to Fig. 2b. (c) The same as (b) for electromagnetic scatterers.

Figure 5

(Color online) Pure Rashba system with magnetic impurity, AMR as a function of the Fermi energy $E_F$ in units of $m{\alpha }^2/{\hslash }^2$.

Figure 6

(Color online) AMR for current flowing along [100] crystal axis in a pure (a) Rashba and (b) Dresselhaus systems with electromagnetic impurity $\left(\propto a\mathbb{1}+{\sigma }_{x,y}\right)$, varying $a$, the ratio between the electric and magnetic part of the potential. The Fermi energy $E_F$ is taken much larger than the spin-orbit interaction, so that the Fermi radii of the two bands become almost equal, see details in text.

Figure 7

(Color online) AMR for pure magnetic potential impurity as a function of the ratio $\alpha /\beta$. Pure Rashba (Dresselhaus) interaction corresponds to the left (right) edge. (a) Single band case, $E_F=0$ (very low electron concentration). Insets show the spin textures for several chosen values of $\alpha /\beta$. (b) Two band case with $E_F/{\alpha }^2>0$ fixed. Fermi lines are shown schematically, spin textures of the majority band are qualitatively similar to the single band case. In the limit $E_F\to \infty$, the AMR vanishes for any value of $\alpha /\beta$.