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 |j,m_j> as | 3 2 , ± 3 2 ⟩ , | 3 2 , ± 1 2 ⟩ , | 1 2 , ± 1 2 ⟩

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

H = ℏ 2 2 m 0 [ ( γ 1 + 5 2 γ 2 ) k 2 − 2 γ 2 ( k ⋅ J ) 2 ]

Phenomenological Luttinger parameters γ i are defined as

α = γ 1 + 5 2 γ 2

and

β = γ 2

If we take k as k = k e ^ z , the Hamiltonian is diagonalized for j=3/2 states.

E = ℏ 2 k 2 2 m 0 ( γ 1 + 5 2 γ 2 − 2 γ 2 m j 2 )

Two degenerated resulting eigenenergies are

E h h = ℏ 2 k 2 2 m 0 ( γ 1 − 2 γ 2 ) for m j = ± 3 2

E l h = ℏ 2 k 2 2 m 0 ( γ 1 + 2 γ 2 ) for m j = ± 1 2

E h h ( E l h ) 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.

Example: GaAs

ϵ h , l = − 1 2 γ 1 k 2 ± [ γ 2 2 k 4 + 3 ( γ 3 2 − γ 2 2 ) × ( k x 2 k z 2 + k x 2 k y 2 + k y 2 k z 2 ) ] 1 / 2