We recall the theory of the spin-dependent Hall effect in semiconductors and give an elementary presentation, stressing the physical aspects of the problem. The spin-dependent Hall effect arises from the spin-orbit interaction in the crystal, via the admixture of $p$ states into the conduction-band Bloch functions. A remarkable consequence of this admixture is the existence of the so-called periodic part of $\overrightarrow{\mathrm{r}}$, which can be interpreted as a transverse displacement of the spin-polarized electron. This transverse displacement yields a first contribution to the spin-dependent Hall effect; the displacement contribution corresponds to a side jump of the electron upon scattering and gives a transverse conductivity independent of the scattering process. A second contribution to the spin-dependent Hall effect is the skew scattering, due to a left-right asymmetry of the scattering cross section. Next, we report the experimental study of the spin-dependent Hall effect in $n$-type indium antimonide. The spin-dependent Hall effect is unambiguously separated from the much larger ordinary Hall effect by using a spin-resonance method. The study of the effect on a series of samples of various doping levels evidences the presence of the two contributions. In the weakly doped samples, only the displacement contribution remains, and the measurements agree, without any adjustable parameter, with the theoretical predictions for the transverse mobility ${\mu }_{yx}^D=-160\mathrm{}$ ${\mathrm{cm}}^2$/V sec. For higher concentrations, the skew scattering becomes notable, cancelling the displacement contribution and changing the sign of the over-all effect. The contribution of multiple scattering to this process appears to be dominant; a semiempirical calculation is given, which agrees with experiment within a factor of 2. Finally, we report the study of the spin-dependent Hall effect in highly doped $n$-type germanium. The necessary extensions of the theory are presented. The study of the effect as a function of temperature in a sample with $N_D\approx 3.1\times {10}^{17}$ ${\mathrm{cm}}^{-3\mathrm{}}$ reveals a behavior quite similar to that observed in the case of InSb. At low temperature, the displacement contribution is observed alone and is found to be in good agreement with theory (${\mu }_{yx}^D=-0.20\mathrm{}$ ${\mathrm{cm}}^2$/V sec) without any adjustable parameter. At higher temperature, the skew scattering increases and changes the sign of the observed effect. A model with two kinds of carriers is shown to account for the observed temperature dependence.

• Received 25 November 1974

DOI:https://doi.org/10.1103/PhysRevB.11.3918