In semiconductors, valence bands are well characterized by 3 Luttinger parameters. At the Г-point in the band structure, p 3 / 2 {\displaystyle p_{3/2}} and p 1 / 2 {\displaystyle p_{1/2}} 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.
Three valence band state In the presence of spin–orbit interaction, total angular momentum should take part in. From the three valence bands, l=1 and s=1/2 state generate six states of | j , m j ⟩ {\displaystyle \left|j,m_{j}\right\rangle } as | 3 2 , ± 3 2 ⟩ , | 3 2 , ± 1 2 ⟩ , | 1 2 , ± 1 2 ⟩ {\displaystyle \left|{\frac {3}{2}},\pm {\frac {3}{2}}\right\rangle ,\left|{\frac {3}{2}},\pm {\frac {1}{2}}\right\rangle ,\left|{\frac {1}{2}},\pm {\frac {1}{2}}\right\rangle }
The spin–orbit interaction from the relativistic quantum mechanics, lowers the energy of j = 1 2 {\displaystyle j={\frac {1}{2}}} states down.
Phenomenological Hamiltonian for the j=3/2 states Phenomenological Hamiltonian in spherical approximation is written as
H = ℏ 2 2 m 0 [ ( γ 1 + 5 2 γ 2 ) k 2 − 2 γ 2 ( k ⋅ J ) 2 ] {\displaystyle H={{\hbar ^{2}} \over {2m_{0}}}[(\gamma _{1}+{{5} \over {2}}\gamma _{2})\mathbf {k} ^{2}-2\gamma _{2}(\mathbf {k} \cdot \mathbf {J} )^{2}]}
Phenomenological Luttinger parameters γ i {\displaystyle \gamma _{i}} are defined as
α = γ 1 + 5 2 γ 2 {\displaystyle \alpha =\gamma _{1}+{5 \over 2}\gamma _{2}}
and
β = γ 2 {\displaystyle \beta =\gamma _{2}}
If we take k {\displaystyle \mathbf {k} } as k = k e ^ z {\displaystyle \mathbf {k} =k{\hat {e}}_{z}} , the Hamiltonian is diagonalized for j = 3 / 2 {\displaystyle j=3/2} states.
E = ℏ 2 k 2 2 m 0 ( γ 1 + 5 2 γ 2 − 2 γ 2 m j 2 ) {\displaystyle E={{\hbar ^{2}k^{2}} \over {2m_{0}}}(\gamma _{1}+{{5} \over {2}}\gamma _{2}-2\gamma _{2}m_{j}^{2})}
Two degenerated resulting eigenenergies are
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