Electron transmission through a quantum point contact in a tilted magnetic field and a crossed electric field: Fertig-Halperin transformation and its extension

Highlights: • We found that the Fertig-Halperin (FH) model is only defined under certain conditions. • We extend the FH model to the case when the parameters do not satisfy above condition. • The conductance curves for the extended FH model are studied. We investigate the transmission coefficient of an electron through a quantum point contact (QPC) in a tilted magnetic field and a crossed electric field. It is found that the widely employed model by Fertig and Halperin (FH) (1987) [20] is only defined when the sum of squares of oscillator strengths of parabolic QPC confinement and magnetic field in transverse direction exceeds that of QPC antibounding (repulsing) potential along the transport direction (${\Omega }^2>0$). We therefore extend the FH model to the case when the parameters do not satisfy this condition (${\Omega }^2<0$), and propose a different unitary transformation to obtain the corresponding analytical expression for the transmission coefficient. The electric and magnetic effects on the transmission coefficient are discussed. As an example of application, we also investigate the magnetic effects on the constriction conductance for the extended model in the absence of the electric fields. In contrast to the well-defined quantization in the FH model, it is shown that the constriction conductance for the extended model could have well-defined, poor, or even no quantization.