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

TL;DR

This paper refines the 8-band Kane model by accurately including interband matrix elements, revealing significant corrections to spin-orbit coupling estimates and clarifying the velocity operator's proper definition in narrow-gap semiconductors.

Contribution

It introduces a proper diagonalization of the Kane Hamiltonian with interband matrix elements, correcting the SOC strength and velocity operator definitions, impacting previous Rashba model applications.

Findings

SOC strength in GaAs is twice the conventional estimate at low temperatures.

In InSb, the SOC strength is 1.76 times larger than previously thought.

The ratio between the velocity operator's SOC strength and the Hamiltonian's is a rational function between 1.49 and 2.

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 $4(23+3\sqrt{2})/73\approx 1.49$ and $2$. Our findings modify a vast number of research results obtained using the Rashba model and provide a path for a consistent treatment of the latter in future applications.