The k・p and envelope-function methods are widely applied to study III�V semiconductor heterostructures. The strong interaction between the N resonant states and the conduction band edge means that the conventional eight-band k・p method cannot be applied to GaInNAs and related hetero-structures. We must include the interaction between the N resonant states and the conduction band edge to describe the variation of the (zonecenter) conduction-band-edge energy with N. This leads to a modified ten-band k・p Hamiltonian for GaInNAs, with the modified Hamiltonian giving a good description of the conduction-band dispersion over an energy range at least on the order of 200 meV [49], sufficient for most analyses.
We illustrate this by comparing the band structure of a Ga32As32 and a Ga32N1As31 supercell in Figure 3.8, where the dotted lines show the sp3s* band structure plotted with the spin-orbit interaction Eso set to zero. The GaAs eight-band k・p Hamiltonian reduces to a two-band Hamiltonian for the conduction and light-hole valence bands along the [0,0,1] direction when Eso=0, as illustrated by the thick solid lines, which show the dispersion of these two bands calculated using ψc0 and the light-hole zone-center wavefunction, ψlh0, as the k・p basis states. The k・p matrix elements were found by explicitly evaluating <ψi0|H(kz)|ψj0> using the tight binding Hamiltonian [49].
We must add the nitrogen resonant state ψN0 to the k・p Hamiltonian for Ga32NAs31. The conduction and light-hole band dispersion are then
found by diagonalizing a 3×3 k・p model. The most general form of this 3×3 Hamiltonian includes k-dependent diagonal and off-diagonal matrix elements linking the ψN0, ψc0.
The thick solid lines in Figure 3.8b show the band structure of Ga32NAs31 calculated using Equation 3.8, where we evaluate the matrix elements directly using the tightbinding Hamiltonian. This Hamiltonian gives an excellent fit to the conduction-band dispersion within about 200 meV of the band edge. However, it is notable that the N impurity band in Figure 3.8b does not correspond to a specific higher-lying conduction b and in the supercell.
ROSSANA HERNANDEZ
Electrónica del estado solido
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