Analytical formalism for determining quantum-wire and quantum-dot band structure in the multiband envelope-function approximation

We describe a new formalism for determining energy eigenstates of spherical quantum dots and cylindrical quantum wires in the multiple-band envelope-function approximation. The technique is based upon a reformulation of the K⋅P theory in a basis of eigenstates of total angular momentum. Stationary states are formed by mixing bulk energy eigenvectors and imposing matching conditions across the heterostructure interface, yielding dispersion relations for eigenenergies in quantum wires and quantum dots. The bound states are studied for the conduction band and the coupled light and heavy holes as a function of radius for the GaAs/${\mathrm{Al}}_{\mathit{x}}$ ${\mathrm{Ga}}_{1\mathrm{-}\mathit{x}}$As quantum dot. Conduction-band–valence-band coupling is shown to be critical in a ‘‘type-II’’ InAs/GaSb quantum dot, which is studied here for the first time. Quantum-wire valence-subband dispersion and effective masses are determined for GaAs/${\mathrm{Al}}_{\mathit{x}}$ ${\mathrm{Ga}}_{1\mathrm{-}\mathit{x}}$As wires of several radii. The masses are found to be independent of wire radius in an infinite-well model, but strongly dependent on wire radius for a finite well, in which the effective mass of the highest-energy valence subband is as low as 0.16${\mathit{m}}_0$. Implications of the band-coupling effects on optical matrix elements in quantum wires and dots are discussed.