Luttinger parameter
Encyclopedia
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.
, 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
The spin-orbit interaction from the relativistic quantum mechanics, lowers the energy of j=1/2 states down.
Phenomenological Luttinger parameters
are defined as
and
If we take
as 
, the Hamiltonian is diagonalized for j=3/2 states.
Two degeneated 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.
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 interactionSpin-orbit interaction
In quantum physics, the spin-orbit interaction is any interaction of a particle's spin with its motion. The first and best known example of this is that spin-orbit interaction causes shifts in an electron's atomic energy levels due to electromagnetic interaction between the electron's spin and...
, 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
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 asPhenomenological Luttinger parameters
are defined asand
If we take
as , the Hamiltonian is diagonalized for j=3/2 states.Two degeneated 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 massEffective mass
In solid state physics, a particle's effective mass is the mass it seems to carry in the semiclassical model of transport in a crystal. It can be shown that electrons and holes in a crystal respond to electric and magnetic fields almost as if they were particles with a mass dependence in their...
of electron in each bands.
Measurement of Luttinger parameters
Luttinger parameter can be measured by Hot-electron luminescence experiment.Example: GaAs