Infoscience

Thesis

A novel formalism for the study of the electronic and optical properties of high symmetry heterostructures

The theories used up to now to model theoretically and numerically nanostructures, and more specifically semiconductor heterostructures, do not allow to include efficiently at the envelope function level, in a k · p approach, the effects imposed by a possible symmetry of the problem. The most elaborated techniques available only allow to take into account the existence of a single symmetry plane, or deal with global symmetry, but not with envelope functions. The most important part of this thesis deals with the development of a novel formalism, very general, which allows to study the electronic and optical properties of high symmetry nanostructures. This new Maximal Symmetrization and Reduction (MSR) formalism allows to maximally symmetrize the eigenstates and significantly reduce the size of the spatial domain of solution. The formalism was explicitly developed with the aim to study a C3v quantum wire, with three symmetry planes at 120°, and allowed to analytically justify some numerical result obtained without an adapted theoretical formalism. In addition, some other physical results related to the effects of symmetry were highlighted and understood. To cite a simple example, it is possible to demonstrate a perfect isotropy polarization with respect to two directions in the cross-section of a wire. In addition, for some transitions new analytical expressions were found for the polarization anisotropy between a direction in the plane and a direction along the wire. The new formalism allows also to understand, in much more details, a number of effects in a qualitative and quantitative way, e.g. symmetry breaking effects. For the example of the C3v wire, the conduction and valence bands are treated separately in the k · p approximation, with respectively one (spinless) band and four bands (describing the valence band mixing). For the scalar wavefunction problem of the conduction band, we propose a new systematic Spatial Domain Reduction (SDR) method. For every different symmetry of the problem (irreducible representation), the independent sub-domains can be identified and a reduced Hamiltonian on the minimal domain can be obtained (numerical optimization). For a spinorial problem, the spatial and spinorial (Bloch functions) parts, have to be considered separately (both for operators and the eigenstates) although treated simultaneously. Whatever the number of bands considered, completely symmetrized bases can be chosen according to the symmetry properties of the heterostructure. This approach allows to classify with respect to the symmetry not only any spinorial states, but every one of their spinorial components (envelope functions) separately. The physical understanding as well as subsequent analytical treatment are considerably simplified. Finally we propose to apply the spatial domain reduction technique to the spinorial components, which ensure to solve the spinorial problem on a minimal domain with numerical optimization. The proposed approach is valid in a much larger framework than k · p theory, and is applicable to arbitrary systems of coupled partial differential equations, e.g. strain equations in heterostructures or the Maxwell equations describing an impurity state in a photonic band-gap.

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