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

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.