Quantum point contacts as a subject of the theory of quantum size effects

Detailed calculations are carried out demonstrating conductance quantization with a 2D quantum point contact (QPC). The QPC is defined as a gap of width a in a diaphragm located perpendicularly in a narrow stripe of width L-x. The stripe is considered as a container of an ideal gas of electrons at zero temperature. Emphasis is put on treating the stripe as a quantum-mechanical waveguide of the electrons. For simplicity, collisions of the electrons inside the stripe are neglected. The author focuses attention on the dependence of the conductance of the QPC on the variable u = ak(F). ((h) over bark(F) is the Fermi momentum.) The plot of this dependence exhibits steps manifesting two kinds of singular points. The frontal edge of each conductance step represents a singularity of the type similar to 1/root u - u(mu). This corresponds to the singularity of the type similar to 1/(n(e) - n(e mu)) when the density ne of the electrons is used rather than the variable u. ( If collisions of the electrons inside the stripe are taken into account, this singularity is transformed into a sharp finite-value maximum.) The second kind of singularity is of the type similar to root u - u(v). and is due to the waveguide character of the stripe. The author exemplifies his theory with a configuration defined with the gap ratio a/L-x = 1/2.