Maximal symmetrization and reduction of fields: Application to wave functions in solid-state nanostructures
A general formalism for the maximal symmetrization and reduction of fields (MSRFs) is proposed and applied to wave functions in solid-state nanostructures. Its primary target is to provide an essential tool for the study and analysis of the electronic and optical properties of semiconductor quantum heterostructures with relatively high point-group symmetry and studied with the k center dot p formalism. Nevertheless the approach is valid in a much larger framework than k center dot p theory; it is applicable to arbitrary systems of coupled partial differential equations (e.g., strain equations or Maxwell equations). This general MSRF formalism makes extensive use of group theory at all levels of analysis. For spinless problems (scalar equations), one can use a systematic spatial domain reduction (SDR) technique which allows, for every irreducible representation, to reduce the set of equations on a minimal domain with automatic incorporation of the boundary conditions at the border, which are shown to be nontrivial in general. For a vectorial or spinorial set of functions, the SDR technique must be completed by the use of an optimal basis in vectorial or spinorial space (in a crystal we call it the optimal Bloch function basis). The full MSR formalism thus consists of three steps: (1) explicitly separate spatial (or Fourier space) and vectorial (spinorial) part of the operators and eigenstates, (2) choose, according to the symmetry and well defined prescriptions (e.g., specific transformation properties), optimal fully symmetrized basis for both spatial and vector (or spin) space, and (3) finally apply the SDR to every individual scalar ultimate component function. We show that with such a formalism the coupling between different vectorial (spinorial) components by symmetry operations becomes minimized and every ultimately reduced envelope function acquires a well-defined specific symmetry. The advantages are numerous: sharper insights on the symmetry properties of every eigenstate, minimal coupling schemes (analytically and computationally exploitable at the component function level), and minimal computing domains. The formalism can be applied also as a postprocessing operation, offering all subsequent analytical and computational advantages of symmetrization. The specific case of a quantum wire with C-3v point group symmetry is used as a concrete illustration of the application of MSRF.
Record created on 2011-12-16, modified on 2016-08-09