Luttinger parameter

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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[edit]

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[edit]

Phenomenological Hamiltonian in spherical approximation is written as[1]

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[edit]

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

Example: GaAs[edit]


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

See also[edit]