Comparing the Eight-Band Luttinger-Kohn-Pikus-Bir-Hamiltonian with the Four-Band Empiric k·p Hamiltonian

Abstract

The purpose of this chapter is to compare on one side the Luttinger Kohn Hamiltonian with the Pikus Bir modification to account for the strained lattices of nanostructured semiconductors, commonly used by the solid state physicists, with, on the other side, the Empiric k·p Hamiltonian proposed and described in Chap. 3, and more oriented to the research and development of optoelectronic devices. The concept of spin is explained, mainly to fix the nomenclature. It is pointed out that the $\left| {\text{S}} \right\rangle ,\left| {\text{X}} \right\rangle ,\left| {\text{Y}} \right\rangle ,\left| {\text{Z}} \right\rangle$ Γ-Bloch functions used in Chap. 3 may have up and down spins becoming $\left| {{\text{S}}{\uparrow}} \right\rangle ,\left| {{\text{S}}{\downarrow}} \right\rangle \ldots$ etc. However these are not the Γ-Bloch functions in this chapter. These are instead $\left| {{\text{cb}} + } \right\rangle ,\left| {{\text{hh}} + } \right\rangle ,\left| {{\text{lh}} + } \right\rangle ,\left| {{\text{so}} + } \right\rangle$, and their four counterparts with minus signs, which are linear combinations of $\left| {{\text{S}}{\uparrow}} \right\rangle ,\left| {{\text{S}}{\downarrow}} \right\rangle \ldots$ etc. The eight-band Luttinger Kohn Pikus Bir Hamiltonian is described, making it possible for the reader to use it with a strain distribution. An approximate four-band Luttinger Kohn Pikus Bir Hamiltonian is also presented. Fittings are made to match the calculated values with the experimental values of the effective masses and bandgaps. The eigenvalues and eigenfunctions of the new Hamiltonians are calculated, and subsequently the sub-bandgap absorption coefficients and the quantum efficiency for the exemplary InAs/GaAs cell modeled along this book. The latter is compared with the experimental data and with the calculations based on the Empiric k·p Hamiltonian. An assessment of the calculation time with the different methods is presented.