Spin splitting of the conduction band by exchange interaction in the valence band through a $k\ifmmode\cdot\else\textperiodcentered\fi{}p$ interband process in ferromagnetic semiconductors

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

Kenji Hayashida and Hiroshi Akera

Phys. Rev. B 105, 235203 – Published 21 June 2022

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 $k \cdot p$ theory. As an example, we consider an $n$ -type ferromagnetic semiconductor (nFMS) of $T_{d}$ 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 $Γ_{6}$ irreducible representation from the eight-band $k \cdot p$ 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 $p − d$ 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 $k \cdot p$ mechanism of orbital-spin coupling is expected to help find remarkable phenomena and useful applications in a wide variety of materials and structures.