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)2041-172310.1038/s41467-019-12229-y] 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 k·p theory. As an example, we consider an n-type ferromagnetic semiconductor (nFMS) of Td point group symmetry with the p-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 Γ6 irreducible representation from the eight-band k·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 kx=ky=0 in a quantum well with a nonzero quantized momentum kz. An estimation shows that the p-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-d exchange, which provides both the coupling of orbital motion to spin and that of spin to nFMS magnetization. We find that the p-d exchange-induced Hall conductivity exhibits an accelerated increase with Fe density, in contrast to that produced by the s-d exchange and the Dresselhaus SOI. This finding suggests that the extended k·p mechanism of orbital-spin coupling is expected to help find remarkable phenomena and useful applications in a wide variety of materials and structures.