Luttinger-Kohn Hamiltonian

In the analyses of conventional ZB semiconductors, we frequently assume a symmetric parabolic band for the conduction band state, and the Luttinger-Kohn Hamiltonian is used to describe the valence band states. In general, the effective Hamiltonian is derived from a k.p perturbation theory or from the theory of invariants developed by Pikus and Bir. In the latter theory, the operator form of the effective Hamiltonian can easily be constructed from symmetry consideration alone. Within this framework, the lowest two conduction bands and the upper six valence bands are described to the second order of k. The invariant forms of the Hamiltonians are written as follows [26,27] ... 

For WZ compounds, we must consider hexagonal symmetry in the effective Hamiltonian. The Luttinger-Kohn Hamiltonian is constructed under the condition of cubic symmetry and the form reflects cubic crystal symmetry. Thus, in the analysis of WZ nitrides, we must use a k-dependent parabolic band for the conduction band state and Bir-Pikus Hamiltonians for the valence band states. The Hamiltonians for the upper six valence bands and the lowest two conduction bands are given by [28]... 

In the analyses of zincblende (ZB) compounds, the Luttinger-Kohn Hamiltonian is used to describe the valence band states. The Hamiltonian is written as [1,2]... 

In the analyses of conventional zincblende (ZB) semiconductors, we frequently assume a parabolic band for the conduction bands, and the 6 x 6 Luttinger-Kohn Hamiltonians are used to describe the upper valence bands [1,2], In treating the valence bands together with the conduction bands on an equal footing, as when estimating the momentum matrix elements, we often make use of the 8 x 8 Kane Hamiltonian [3], However, the form of the Hamiltonians reflects the crystal symmetry, and Kane Hamiltonians are constructed under the condition of cubic symmetry. For wurtzite (WZ) materials, therefore, we must consider hexagonal symmetry in the effective Hamiltonian. Let us consider the 8 x 8 k.p Hamiltonian for WZ structure [4,5],... 

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