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, ${\displaystyle p_{3/2}}$ and ${\displaystyle p_{1/2}}$ 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.

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 ${\displaystyle \left|j,m_{j}\right\rangle }$ as ${\displaystyle \left|{\frac {3}{2}},\pm {\frac {3}{2}}\right\rangle ,\left|{\frac {3}{2}},\pm {\frac {1}{2}}\right\rangle ,\left|{\frac {1}{2}},\pm {\frac {1}{2}}\right\rangle }$

The spin–orbit interaction from the relativistic quantum mechanics, lowers the energy of ${\displaystyle j={\frac {1}{2}}}$ states down.

Phenomenological Hamiltonian in spherical approximation is written as^[1]

${\displaystyle H={{\hbar ^{2}} \over {2m_{0}}}[(\gamma _{1}+{{5} \over {2}}\gamma _{2})\mathbf {k} ^{2}-2\gamma _{2}(\mathbf {k} \cdot \mathbf {J} )^{2}]}$

Phenomenological Luttinger parameters ${\displaystyle \gamma _{i}}$ are defined as

${\displaystyle \alpha =\gamma _{1}+{5 \over 2}\gamma _{2}}$

and

${\displaystyle \beta =\gamma _{2}}$

If we take ${\displaystyle \mathbf {k} }$ as ${\displaystyle \mathbf {k} =k{\hat {e}}_{z}}$, the Hamiltonian is diagonalized for ${\displaystyle j=3/2}$ states.

${\displaystyle E={{\hbar ^{2}k^{2}} \over {2m_{0}}}(\gamma _{1}+{{5} \over {2}}\gamma _{2}-2\gamma _{2}m_{j}^{2})}$

Two degenerated resulting eigenenergies are

${\displaystyle E_{hh}={{\hbar ^{2}k^{2}} \over {2m_{0}}}(\gamma _{1}-2\gamma _{2})}$ for ${\displaystyle m_{j}=\pm {3 \over 2}}$

${\displaystyle E_{lh}={{\hbar ^{2}k^{2}} \over {2m_{0}}}(\gamma _{1}+2\gamma _{2})}$ for ${\displaystyle m_{j}=\pm {1 \over 2}}$

${\displaystyle E_{hh}}$ (${\displaystyle E_{lh}}$) 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.

In gallium arsenide,

${\displaystyle \epsilon _{h,l}=-{{1} \over {2}}\gamma _{1}k^{2}\pm [{\gamma _{2}}^{2}k^{4}+3({\gamma _{3}}^{2}-{\gamma _{2}}^{2})\times ({k_{x}}^{2}{k_{z}}^{2}+{k_{x}}^{2}{k_{y}}^{2}+{k_{y}}^{2}{k_{z}}^{2})]^{1/2}}$

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