Luttinger parameter

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 \left| j, m_j \right\rangle as \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 = \frac{1}{2} states down.

Phenomenological Hamiltonian for the ''j''=3/2 states

Phenomenological Hamiltonian in spherical approximation is written as

H= [(\gamma _1+ \gamma _2) \mathbf{k}^2 -2\gamma_2 (\mathbf{k} \cdot \mathbf{J})^2]

Phenomenological Luttinger parameters \gamma _i are defined as

\alpha = \gamma _1 + {5 \over 2} \gamma _2

and

\beta = \gamma _2

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

E = { {\hbar^2 k^2} \over {2m_0} }( \gamma _1 + \gamma _2 - 2 \gamma _2 m_j^2)

Two degenerated resulting eigenenergies are

E _{hh} = { {\hbar^2 k^2} \over {2m_0} }( \gamma _1 - 2 \gamma _2) for m_j = \pm {3 \over 2}

E _{lh} = { {\hbar^2 k^2} \over {2m_0} }( \gamma _1 + 2 \gamma _2) for m_j = \pm {1 \over 2}

E_{hh} ( 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,

\epsilon {h,l} = - \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

References

1. Haug, Hartmut. (2004). "Quantum Theory of the Optical and Electronic Properties of Semiconductors". World Scientific.