Optical properties of cylindrical nanowires

Chapter 3.Electronic propertiesrelatively unimportant. Also the interband matrix elements in the numeratorreduces the number of bands that contribute to the effective mass. Aswill be explicitly shown in subsection 4.4.1, the matrix elements are subjectto selection rules which are determined by the symmetry properties of thebands in question. Most of the matrix elements become zero by this kind ofsymmetry arguments.Secondly, including the spin-orbit interaction in case of a non-degenerateband makes little practical difference, since its effect is absorbed in the interbandmatrix elements which are determined by experiment. For a degenerateband this is different, because the spin-orbit interaction in general lifts thedegeneracy and will cause small splitting between the bands.A last important remark has to be made concerning the limitations of thek·p method as depicted here, up to second order in k. The above results relyon the assumption that ɛ n (k)−ɛ n remains much smaller then the band edgedifferences ɛ n − ɛ n ′ (and a similar assumption in case of degenerate bands),which is not necessarily satisfied, e.g. in semiconductor compounds with anarrow band gap. Instead of expanding beyond second order in the sameframework, a commonly used approach [14] [15] [17] initiated by Kane [19]solves this problem by diagonalizing the group of neighboring bands exactlyand afterwards treating the coupling with the well separated other bands ina second order perturbation. However, in the remaining part of this paperit is assumed that the bands under investigation are well separated from theothers, i.e. splitting terms as the band gap E g and spin-orbit splitting ∆ 0are assumed to be sufficiently large.3.1.1 Top valence bands in III-V semiconductorsIn principle the above theoretical statements now can be applied to anyband, or group of bands, once the energy and the interband matrix elementsare known. Here the band structure of the six fold degenerate (includingspin) top valence band at the Γ point (k = 0) in III-V semiconductor compoundswill be summarized. However, the explicit diagonalization of thematrix (3.8) will be performed further on in the envelope function frameworksince this is the most convenient way when the nanowire structure isanticipated.Starting with symmetry considerations, it is well known [14][15] that thetop valence bands in III-V materials have Γ 4 like symmetry, apart from spin.The corresponding spatial parts of the valence band wavefunctions at k = 0are p-like, which means that they are triply degenerate and transform underrotations like the three components of a vector. Including spin this leads tosix band edge Bloch functions, which are denoted by |X〉|σ〉, |Y 〉|σ〉, |Z〉|σ〉,with σ =↑,↓.The one-electron Hamiltonian H 0 is diagonalized by linear combinationsof these band edge Bloch functions. Rewriting the spin orbit term in (3.2)44