Luttinger parameter

In semiconductors, valence bands are well characterized by 3 Luttinger parameters. At the Г-point in the band structure, ${\displaystyle p_{3/2}}$ and ${\displaystyle p_{1/2}}$ 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 band, l=1 and s=1/2 state generate six state of ${\displaystyle \left|j,m_{j}\right\rangle }$ as ${\displaystyle \left|{\frac {3}{2}},\pm {\frac {3}{2}}\right\rangle ,\left|{\frac {3}{2}},\pm {\frac {1}{2}}\right\rangle ,\left|{\frac {1}{2}},\pm {\frac {1}{2}}\right\rangle }$

The spin–orbit interaction from the relativistic quantum mechanics, lowers the energy of ${\displaystyle j={\frac {1}{2}}}$ states down.

Phenomenological Hamiltonian for the j=3/2 states

Phenomenological Hamiltonian in spherical approximation is written as^[1]

${\displaystyle H={{\hbar ^{2}} \over {2m_{0}}}[(\gamma _{1}+{{5} \over {2}}\gamma _{2})\mathbf {k} ^{2}-2\gamma _{2}(\mathbf {k} \cdot \mathbf {J} )^{2}]}$

Phenomenological Luttinger parameters ${\displaystyle \gamma _{i}}$ are defined as

${\displaystyle \alpha =\gamma _{1}+{5 \over 2}\gamma _{2}}$

and

${\displaystyle \beta =\gamma _{2}}$

If we take ${\displaystyle \mathbf {k} }$ as ${\displaystyle \mathbf {k} =k{\hat {e}}_{z}}$, the Hamiltonian is diagonalized for ${\displaystyle j=3/2}$ states.

${\displaystyle E={{\hbar ^{2}k^{2}} \over {2m_{0}}}(\gamma _{1}+{{5} \over {2}}\gamma _{2}-2\gamma _{2}m_{j}^{2})}$

Two degenerated resulting eigenenergies are

${\displaystyle E_{hh}={{\hbar ^{2}k^{2}} \over {2m_{0}}}(\gamma _{1}-2\gamma _{2})}$ for ${\displaystyle m_{j}=\pm {3 \over 2}}$

${\displaystyle E_{lh}={{\hbar ^{2}k^{2}} \over {2m_{0}}}(\gamma _{1}+2\gamma _{2})}$ for ${\displaystyle m_{j}=\pm {1 \over 2}}$

${\displaystyle E_{hh}}$ (${\displaystyle E_{lh}}$) 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.

Example: GaAs

In gallium arsenide,

${\displaystyle \epsilon _{h,l}=-{{1} \over {2}}\gamma _{1}k^{2}\pm [{\gamma _{2}}^{2}k^{4}+3({\gamma _{3}}^{2}-{\gamma _{2}}^{2})\times ({k_{x}}^{2}{k_{z}}^{2}+{k_{x}}^{2}{k_{y}}^{2}+{k_{y}}^{2}{k_{z}}^{2})]^{1/2}}$