# 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 |j,mj> as ${\displaystyle |{3 \over 2},\pm {3 \over 2}\rangle ,|{3 \over 2},\pm {1 \over 2}\rangle ,|{1 \over 2},\pm {1 \over 2}\rangle }$

The spin-orbit interaction from the relativistic quantum mechanics, lowers the energy of j=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 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.

## Measurement of Luttinger parameters

Luttinger parameter can be measured by Hot-electron luminescence experiment.

## Example: GaAs

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

## References

1. ^ Hartmut Haug, Stephan W. Koch (2004). Quantum Theory of the Optical and Electronic Properties of Semiconductors (4th ed.). World Scientific. p. 46.