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| In semiconductors, [[valence bands]] are well characterized by 3 Luttinger parameters. At the ''Г''-point in the [[band structure]], <math>p_{3/2} </math> and <math>p_{1/2} </math> 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]].
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| ==Three valence band state==
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| 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<sub>j</sub>> as <math> |{3 \over 2}, \pm {3 \over 2} \rangle, |{3 \over 2}, \pm {1 \over 2}\rangle, |{1 \over 2}, \pm {1 \over 2}\rangle </math>
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| The spin-orbit interaction from the relativistic quantum mechanics, lowers the energy of j=1/2 states down.
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| ==Phenomenological Hamiltonian for the j=3/2 states==
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| Phenomenological Hamiltonian in spherical approximation is written as<ref name="Hartmut Haug, Stephan W. Koch 0">{{cite book |title=Quantum Theory of the Optical and Electronic Properties of Semiconductors |author=Hartmut Haug, Stephan W. Koch |page=46 |year=2004 |edition=4th |publisher=World Scientific}}</ref>
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| <math> 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]</math>
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| Phenomenological Luttinger parameters <math> \gamma _i </math> are defined as
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| <math> \alpha = \gamma _1 + {5 \over 2} \gamma _2 </math>
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| and
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| <math> \beta = \gamma _2 </math>
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| If we take <math> \mathbf{k} </math> as <math> \mathbf{k}=k \hat{e}_z </math>, the Hamiltonian is diagonalized for j=3/2 states.
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| <math> E = { {\hbar^2 k^2} \over {2m_0} }( \gamma _1 + {{5} \over {2}} \gamma _2 - 2 \gamma _2 m_j^2)</math> | |
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| Two degenerated resulting eigenenergies are
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| <math> E _{hh} = { {\hbar^2 k^2} \over {2m_0} }( \gamma _1 - 2 \gamma _2)</math> for <math> m_j = \pm {3 \over 2} </math>
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| <math> E _{lh} = { {\hbar^2 k^2} \over {2m_0} }( \gamma _1 + 2 \gamma _2)</math> for <math> m_j = \pm {1 \over 2} </math>
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| <math> E_{hh} </math> (<math> E_{lh} </math>) indicates heav-(light-) hole band energy. If we regard the electrons as nearly free electrons, the Luttinger parameters describe [[Effective mass (solid-state physics)|effective mass]] of electron in each bands.
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| ==Measurement of Luttinger parameters==
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| Luttinger parameter can be measured by Hot-electron luminescence experiment.
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| ==Example: GaAs==
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| <math> \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}</math>
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| ==References==
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| {{reflist}}
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| ==See also==
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| * J. M. Luttinger, Physical Review, Vol. '''102''', 1030 (1956). [http://prola.aps.org/abstract/PR/v102/i4/p1030_1 APS]
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| * A. Baldereschi and N.O. Lipari, Physical Review B., Vol. '''8''', pp. 2675 (1973). [http://prb.aps.org/abstract/PRB/v8/i6/p2697_1 APS]
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| * A. Baldereschi and N.O. Lipari, Physical Review B., Vol. '''9''', pp. 1525 (1974). [http://prb.aps.org/abstract/PRB/v9/i4/p1525_1 APS]
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| [[Category:Semiconductors]]
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58 year old Beef Cows Farmer Branden from Ontario, usually spends time with hobbies which include running, brave frontier and compose music. Will soon undertake a contiki journey which will contain taking a trip to the Kilimanjaro National Park.
Feel free to surf to my web site; brave frontier hack