Luttinger parameter

In semiconductors, valence bands are well characterized by 3 Luttinger parameters. At the Г-point in the band structure, and 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 as

The spin–orbit interaction from the relativistic quantum mechanics, lowers the energy of states down.

Phenomenological Hamiltonian for the j=3/2 states

Phenomenological Hamiltonian in spherical approximation is written as

Phenomenological Luttinger parameters are defined as

and

If we take as , the Hamiltonian is diagonalized for states.

Two degenerated resulting eigenenergies are

for

for

() 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,

References

Further reading