## Position operators and interband matrix elements of scalar and vector potentials in the 8-band Kane model

I. A. Ado, M. Titov, Rembert A. Duine, Arne Brataas

SciPost Phys. 17, 009 (2024) · published 10 July 2024

- doi: 10.21468/SciPostPhys.17.1.009
- Submissions/Reports

### Abstract

We diagonalize the 8-band Kane Hamiltonian with a proper inclusion of the interband matrix elements of the scalar and vector potentials. This leads, among other results, to a modification of the conventional expression for the spin-orbit coupling (SOC) strength in narrow-gap semiconductors with the zinc blende symmetry. We find that in GaAs, at low temperatures, the correct expression for the SOC strength is actually twice as large as usually considered. In InSb it is $1.76$ times larger. We also provide a proper treatment of the interband matrix elements of the position operator. We show that the velocity operator in a crystal should be defined as a time-derivative of a fictitious position operator rather than the physical one. We compute the expressions for both these position operators projected to the conduction band of the 8-band Kane model. We also derive an expression for the projected velocity operator and demonstrate that the SOC strength in it differs from the SOC strength in the Hamiltonian. The ratio between them is not equal to $1$, as it is often assumed for the Rashba model. It does not equal $2$ either. The correct result for this ratio is given by a rational function of the parameters of the model. This function takes values between $4(23+3\sqrt{2})/73≈ 1.49$ and $2$. Our findings modify a vast number of research results obtained using the Rashba model and provide a path for consistent treatment of the latter in future applications.

### Authors / Affiliations: mappings to Contributors and Organizations

See all Organizations.-
^{1}Ivan A. Ado, -
^{1}M. Titov, -
^{2}^{3}Rembert A. Duine, -
^{4}Arne Brataas

^{1}Radboud Universiteit Nijmegen / Radboud University Nijmegen [RUN]^{2}Universiteit Utrecht / University of Utrecht [UU]^{3}Technische Universiteit Eindhoven / Eindhoven University of Technology [TU/e]^{4}Norges Teknisk-Naturvitenskapelige Universitet / Norwegian University of Science and Technology [NTNU]