Anomalous Hall Effect in Ferromagnetic Semiconductors in the Hopping Transport Regime

Abstract

We present a theory of the Anomalous Hall Effect (AHE) in ferromagnetic (Ga,Mn)As in the regime when conduction is due to phonon-assisted hopping of holes between localized states in the impurity band. We show that the microscopic origin of the anomalous Hall conductivity in this system can be attributed to a phase that a hole gains when hopping around closed-loop paths in the presence of spin-orbit interactions and background magnetization of the localized Mn moments. Mapping the problem to a random resistor network, we derive an analytic expression for the macroscopic anomalous Hall conductivity σ AH xy . We show that σ AH xy is proportional to the first derivative of the density of states ̺(ǫ) and thus can be expected to change sign as a function of impurity band filling. We also show that σ AH xy depends on temperature as the longitudinal conductivity σxx within logarithmic accuracy.PACS numbers: 75.50. Pp,73.50.Jt,72.20.Ee Diluted magnetic semiconductors (DMS) are very interesting new materials, still not understood in full detail. Although it is well accepted that the origin of ferromagnetism in Ga 1−x Mn x As, the most thoroughly studied DMS, is a hole-mediated effective interaction between the localized S = 5/2 Mn spins, the appropriate description of this interaction, and a closely related question of the nature of hole carriers, are still controversial issues. Samples with the highest Curie temperatures show metallic behavior [1] and are well described by the valenceband hole-fluid model [2]. On the other hand, for low Mn concentrations, i.e. when x is less than about 0.03, (Ga,Mn)As is insulating and the hole transport likely occurs in the Mn impurity band [3]. The low-temperature conduction in this case is by phonon-assisted hopping.Anomalous Hall Effect (AHE) has been an important characterization tool for itinerant ferromagnets [4,5,6] and played an important role in the experimental study of DMS ferromagnetism as well [7]. Experimentally AHE is manifested as an additional term in the Hall resistivity of the sample, usually assumed to be proportional to the magnetization:Standard theories of AHE in metallic ferromagnets [4,6] attribute it to the modification of impurity scattering in the presence of spin-orbit interactions. Recently, however, it was realized that a purely geometrical mechanism of AHE is possible [8,9,10,11]. In particular, a k-space Berry phase theory of AHE [8] has been proposed, and applied specifically to (Ga,Mn)As in the metallic transport regime. This theory is very successful in describing AHE in metallic (Ga,Mn)As, and we believe that the Berry-phase mechanism of AHE should be relevant in general for all itinerant ferromagnets. In this Letter we propose an analogous geometricalphase theory of AHE in (Ga,Mn)As, but in the case when conduction is by hopping between strongly localized states in the impurity band, rather than through extended valence-band states. Our approach is based on a modified Holstein's theory of the Hall effect in hopping conduction [12] (for related w...

Anomalous Hall Effect and Skyrmion Number in Real and Momentum Spaces

“…This is analogous to the the integer quantum Hall effect (IQHE) with the strong external magnetic field [8,9]. It was also proposed that AHE arises even without the spin-orbit interaction if the spin configuration is non-coplanar with finite spin chirality, i.e., the solid angle subtended by the spins where the electron hops successively [10,11,12,13,14]. Consider the double-exchange model…”

Interplay between Carrier and Impurity Concentrations in AnnealedGa1−xMnxAs: Intrinsic Anomalous Hall Effect

“…The observed quadratic relationship between R S and xx for xx < 10 m cm can be interpreted as a manifestation of either intrinsic or extrinsic origins such as the side-jump mechanism. Approximately linear relationship for xx < 10 m cm can be only attributable to dominance of extrinsic origins, either from skew-scattering mechanism or from phonon-assisted hopping of holes between localized states [26]. It is also worth noting that the length scales of the extrinsic side-jump mechanism (0.01-0.1 nm) [15] is thought to be orders of magnitude smaller than skew scattering with predominance of transition from linear to quadratic scaling behavior for increasing resistivities for magnetic systems dominated by extrinsic origins [1].…”

Scaling relations between anomalous Hall and longitudinal transport coefficients in metallic (Ga,Mn)As films

“…However, both M 0 and σ xy deviate from this power-law form in highly conducting samples (σ xx > 100 Ω −1 cm −1 ), which leads to the unanticipated dependence of M 0 and σ xy on the hole concentration 14 . Second, near the metal-insulator transition, M 0 and σ xy at zero temperature should decrease as p decreases toward p c , and should eventually vanish in the localized regime, as predicted by the theory of AHE in ferromagnetic (Ga,Mn)As in the regime where conduction is due to phonon-assisted hopping of holes between localized states in the impurity band 45 . As a result, we speculate that there exists another phase transition as the temperature approaches zero in an insulating ferromagnetic (Ga,Mn)As sample.…”