# On derivation of Dresselhaus spin-splitting Hamiltonians in one-dimensional electron systems

## TL;DR

This paper develops a general method to derive Dresselhaus spin-splitting Hamiltonians for one-dimensional electron systems, accounting for both linear and cubic wavevector terms, with applications to quantum wires and rings.

## Contribution

It introduces a universal procedure for deriving Dresselhaus Hamiltonians in curvilinear 1D structures, extending previous models to more complex geometries.

## Key findings

- Derived Hamiltonians for quantum wire and ring geometries.
- Showed importance of cubic k-terms in low-dimensional structures.
- Provided explicit examples demonstrating the method's applicability.

## Abstract

Two-dimensional (2D) semiconductor structures of materials without inversion center (e.g. zinc-blende ${\rm A^{III}B^V}$) possess the zero-field conduction band spin-splitting (Dresselhaus term), which is linear and cubic in wavevector $k$, that arises from cubic in $k$ splitting in a bulk material. At low carrier concentration the cubic term is usually negligible. However, if we will be interested in the following dimensional quantization (in 2D plane) and the character width in this direction is comparable with the width of 2D-structure, then we have to take into account $k^3$-terms as well (even at low concentrations), that after quantization leads to comparable contribution that arises from $k$-linear term. We propose the general procedure for derivation of Dresselhaus spin-splitting Hamiltonian applicable for any curvilinear 1D-structures. The simple examples for the cases of a quantum wire (QWr) and a quantum ring (QR) defined in usual [001]-grown 2D-structure are presented.