Eigenfunction-expansion method for solving the quantum-wire problem: Formulation

We present a method of formulating the multiband-envelope-function equations for a quantum structure whose internal interfaces are perpendicular planes. The method can be used for quantum wells, quantum wires, or quantum dots (one-, two-, or three-dimensional confinement of the electronic wave function), as well as for periodic repetitions (superlattices) of these elementary structures. The technique used is expansion of the multiband envelope functions in a Fourier series for each of the coordinates x, y, and z. Special attention is paid to formulating interface-matching conditions that impose Hermiticity on the resulting systems of equations. This demand leads to the usual condition that the normal component of the current must be continuous across each internal interface. The method we have devised is similar to the one used by Altarelli for the quantum-well problem in that it leads to a secular equation that is solved by diagonalizing an energy-independent matrix. It differs in that here, the envelope functions are expanded in smooth continuous functions using the same expansion coefficients in all regions of the structure. Using this method, one can now calculate the optical absorption, its frequency, and polarization dependence, with the same amount of detail that has previously been possible only for confinement in one dimension, namely, in the quantum well and one-dimensional superlattice.