In semiconductors, valence bands are well characterized by 3 Luttinger parameters.
At the Г-point in the band structure,
orbitals form valence bands.
But spin–orbit coupling splits sixfold degeneracy into high energy 4-fold and lower energy 2-fold bands.
Again 4-fold degeneracy is lifted into heavy- and light hole bands by phenomenological Hamiltonian by J. M. Luttinger.
In the presence of spin–orbit interaction, total angular momentum should take part in.
From the three valence band, l=1 and s=1/2 state generate six state of
The spin–orbit interaction from the relativistic quantum mechanics, lowers the energy of
Phenomenological Hamiltonian in spherical approximation is written as[1]
γ
γ
γ
Phenomenological Luttinger parameters
γ
γ
γ
γ
, the Hamiltonian is diagonalized for
γ
γ
γ
Two degenerated resulting eigenenergies are
γ
γ
γ
γ
) indicates heav-(light-) hole band energy.
If we regard the electrons as nearly free electrons, the Luttinger parameters describe effective mass of electron in each bands.
In gallium arsenide,
γ