Elastic electron transmission by barriers in a three-dimensional model quantum wire

Numerical solutions of the time-dependent and time-independent Schrödinger equations for electron tunneling in a three-dimensional model quantum wire have been carried out and used to calculate transmission probabilities over a range of energies. The model simulates a quantum heterostructure corresponding to a cylindrically symetric wire constructed from two different materials such that square barriers to the axial (z) motion occur in the disjoint region B=(${\mathit{z}}_1$≤z≤${\mathit{z}}_2$)∪(${\mathit{z}}_3$≤z≤${\mathit{z}}_4$). In addition, the electron is bound harmonically in the radial distance r transverse to the longitudinal axis z; the electron vibrational frequency changes discontinuously from ${\mathrm{\omega }}_{\mathit{A}}$ in the disjoint region A=(-∞<z<${\mathit{z}}_1$)∪(${\mathit{z}}_2$<z<${\mathit{z}}_3$)∪(${\mathit{z}}_4$<z<∞) to ${\mathrm{\omega }}_{\mathit{B}}$ in the disjoint region B.

A basis-set expansion, using harmonic-oscillator eigenstates appropriate to regions A or B, for the electron’s radial vibration transverse to the cylinder axis is used. The basis-set expansion using the eigenstates of the electrons rotation about z only involved the ground rotational state since, due to cylindrical symmetry, the electron does not experience any torques. This ‘‘close-coupling expansion’’ procedure yields close-coupled-wave-packet (CCWP) equations for the time-dependent expansion coefficients in the time-dependent approach, and a system of linear algebraic equations for the expansion coefficients in the time-independent approach. The change of electronic vibration frequency from region A to region B leads to nonadiabatic transitions, giving rise to transmission amplitudes that depend on the final electron-vibrational state. The model is elastic in that phonon degrees of freedom are not included. A final-state analysis for the time-dependent CCWP method is given which does not require spatial integrals of the wave packets. This enables results over a broad energy range (including resonant energies) to be acquired from a single wave-packet propagation, because the packet is initially taken to be relatively narrow in coordinate space. The method is extremely stable over the whole range of energies. All calculations used an initial state comprised of the ground vibrational and zero angular momentum wave functions and vibrationally resolved transmission probabilities are studied for various frequency ratios ${\mathrm{\omega }}_{\mathit{A}}$/${\mathrm{\omega }}_{\mathit{B}}$ and as a function of total energy. Vibrational nonadiabaticity is found to depend strongly of the frequency ratio, and on collision energy.