Luttinger parameter explained

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

p _{3/2} and p _{1/2} 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

\left| j, m _{j} \right\rangle as \left| (3)/(2) , \pm (3)/(2) \right\rangle, \left| (3)/(2) , \pm (1)/(2) \right\rangle, \left| (1)/(2) , \pm (1)/(2) \right\rangle

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^[1]

H = {{\hbar ^{2}} \over { 2 m _{0}}} [(\gamma _{1+{{5}} \over { 2 }} \gamma _{2)} k ^{2} -2 \gamma _{2} ( k ⋅ J ) ^{2]}

Phenomenological Luttinger parameters

\gamma _{i} are defined as\alpha = \gamma _{1} + { 5 \over 2 } \gamma _{2}

and

\beta = \gamma _{2}

If we take

k as k = k \hat{e} _{z} , the Hamiltonian is diagonalized for j =3 / 2 states.E = { {\hbar ^{2} k ^{2}} \over { 2 m _{0}} }( \gamma _{1} + {{ 5 } \over { 2 }} \gamma _{2} - 2 \gamma _{2} m ^{2 )} _{j}

Two degenerated resulting eigenenergies are

E _{hh} = { {\hbar ^{2} k ^{2}} \over { 2 m _{0}} }( \gamma _{1} - 2 \gamma _{2)} for m _{j} = \pm { 3 \over 2 } E _{lh} = { {\hbar ^{2} k ^{2}} \over { 2 m _{0}} }( \gamma _{1} + 2 \gamma _{2)} for m _{j} = \pm { 1 \over 2 } E _{hh} (E _{lh} ) 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,

\epsilon _{h,l} = - {{ 1 } \over { 2 }} \gamma _{1} k ^{2} \pm [ {\gamma _{2} }^ k^ + 3 (^ - ^) \times (^ ^ + ^ ^ + ^^)]^

Further reading

• Book: Luttinger Model: The First 50 Years and Some New Directions . World Scientific . 2013 . Mastropietro . Vieri . Mattis . Daniel C. . 978-981-4520-71-3. 10.1142/8875 .
• Luttinger . J. M. . Joaquin Mazdak Luttinger. Quantum Theory of Cyclotron Resonance in Semiconductors: General Theory . Physical Review. 102 . 4 . 1956-05-15 . 0031-899X . 10.1103/physrev.102.1030 . 1956PhRv..102.1030L . 1030–1041.
• Baldereschi . A. . Lipari . Nunzio O. . Spherical Model of Shallow Acceptor States in Semiconductors . Physical Review B. 8 . 6 . 1973-09-15 . 0556-2805 . 10.1103/physrevb.8.2697 . 1973PhRvB...8.2697B . 2697–2709.
• Baldereschi . A. . Lipari . Nunzio O. . Cubic contributions to the spherical model of shallow acceptor states . Physical Review B. 9 . 4 . 1974-02-15 . 0556-2805 . 10.1103/physrevb.9.1525 . 1974PhRvB...9.1525B . 1525–1539.

Notes and References

1. Book: Haug, Hartmut . Koch . Stephan W . Quantum Theory of the Optical and Electronic Properties of Semiconductors . World Scientific. 2004 . 978-981-238-609-0 . 10.1142/5394 . 46. 4th.