Energy subbands, envelope states, and intersubband optical transitions in one-dimensional quantum wires: The local-envelope-states approach

Optical properties associated with intersubband transitions in one-dimensional quantum-wire structures require an accurate knowledge of the energies and the envelope states of the wire’s subbands. In general, there is no analytical method to calculate the quantum-wire envelope states and the subband energies. However, in many practical cases the wire geometry is composed of a strong-confinement direction for which the confinement potential varies on short length scales, as compared to a weaker-confinement direction that provides the additional confinement. For this class of quantum wires we have derived the local-envelope-states (LENS) expansion that provides a simple and intuitive way to analyze the subband structure of the quantum wire. The LENS approach involves a solution of two sets of one-dimensional Ben Daniel-Duke Hamiltonians along the two axes of the wire and a diagonalization of the LENS Hamiltonian that takes into account deviation from an ideal wire. We show that this approach provides a simple way to introduce the symmetry of the structure into the calculations and to derive the intersubband selection rules for a one-dimensional quantum wire. Finally, our formalism is applied for two examples: a hyperbolic quantum wire for which an analytical solution is derived, and V-groove crescent-shaped quantum wires that are grown on nonplanar substrates. The analysis provides an estimate of the allowed transitions in these structures, and the selection rules for intersubband transitions. We show that the intersubband selection rules are very sensitive to the additional confinement of the wire and provide a powerful experimental tool to study the properties of the wire.