Spin splitting of the conduction band by exchange interaction in the valence band through a k·p interband process in ferromagnetic semiconductors

Abstract

The momentum-dependent spin splitting in the conduction band couples orbital motion to spin and enables electrical control of spin. Currently, this control relies on the relativistic spin-orbit interaction (SOI), which limits useful materials to those containing heavy elements. Recently, Naka et al. [Nat. Commun. 10, 4305 (2019)] have found a momentum-dependent spin splitting originating from the exchange interaction, which is expected to extend spintronic materials to those without heavy elements. In this paper, we propose a mechanism of the exchange-induced orbital-spin coupling by extending the $\mathbit{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbit{p}$ theory. As an example, we consider an $n$-type ferromagnetic semiconductor (nFMS) of ${T}_{d}$ point group symmetry with the $p\text{\ensuremath{-}}d$ exchange interaction between an electron in the valence band and the spin of a magnetic ion and evaluate the spin splitting in the conduction band of ${\mathrm{\ensuremath{\Gamma}}}_{6}$ irreducible representation from the eight-band $\mathbit{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbit{p}$ Hamiltonian. We find that the lowest-order spin splitting in bulk is of the second order of momentum, which results in a nonzero splitting at ${k}_{x}={k}_{y}=0$ in a quantum well with a nonzero quantized momentum ${k}_{z}$. An estimation shows that the $p\text{\ensuremath{-}}d$ exchange interaction is the dominant origin of the conduction-band spin splitting in InFeAs nFMS. We also calculate the intrinsic anomalous Hall conductivity of bulk InFeAs generated by the $p\text{\ensuremath{-}}d$ exchange, which provides both the coupling of orbital motion to spin and that of spin to nFMS magnetization. We find that the $p\text{\ensuremath{-}}d$ exchange-induced Hall conductivity exhibits an accelerated increase with Fe density, in contrast to that produced by the $s\text{\ensuremath{-}}d$ exchange and the Dresselhaus SOI. This finding suggests that the extended $\mathbit{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbit{p}$ mechanism of orbital-spin coupling is expected to help find remarkable phenomena and useful applications in a wide variety of materials and structures.