Transconductance and effective Land\'e factors for quantum point contacts: spin-orbit coupling and interaction effects

We analyze the effective $g^*$ factors and their dependence on the orientation of the external magnetic field for a quantum point contact defined in the two-dimensional electron gas. The paper simulates the experimental procedure for evaluation of the effective Land\'e factors from the transconductance of a biased device in external magnetic field. The contributions of the orbital effects of the magnetic field, the electron-electron interaction and spin-orbit (SO) coupling are studied. The anisotropy of the $g^*$ factors for the in-plane magnetic field orientation, which seems counterintuitive from the perspective of the effective SO magnetic field, is explained in an analytical model of the constriction as due to the SO-induced subband mixing. The asymmetry of the transconductance as a function of the gate voltage is obtained in agreement with the experimental data and the results are explained as due to the depletion of the electron gas within the quantum point contact constriction and the related reduction of the screening as described within the DFT approach. The results for transconductance and the $g^*$ factors are in a good agreement with the experimental data [Phys. Rev. B {\bf 81}, 041303, 2010].