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 |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 as
Phenomenological Luttinger parameters are defined as
If we take as , the Hamiltonian is diagonalized for j=3/2 states.
Two degenerated resulting eigenenergies are
() 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.
- Hartmut Haug, Stephan W. Koch (2004). Quantum Theory of the Optical and Electronic Properties of Semiconductors (4th ed.). World Scientific. p. 46.
- Mastropietro, Vieri; Mattis, Daniel C. (2013). Luttinger Model: The First 50 Years and Some New Directions. World Scientific. ISBN 978-981-4520-71-3.
- J. M. Luttinger, Physical Review, Vol. 102, 1030 (1956). APS
- A. Baldereschi and N.O. Lipari, Physical Review B., Vol. 8, pp. 2675 (1973). APS
- A. Baldereschi and N.O. Lipari, Physical Review B., Vol. 9, pp. 1525 (1974). APS