Casio FX-603P: Difference between revisions

Revision as of 21:10, 28 October 2013

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 |j,m_j> as $| \frac{3}{2}, \pm \frac{3}{2} ⟩, | \frac{3}{2}, \pm \frac{1}{2} ⟩, | \frac{1}{2}, \pm \frac{1}{2} ⟩$

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^[1]

$H = \frac{ℏ^{2}}{2 m_{0}} [(γ_{1} + \frac{5}{2} γ_{2}) k^{2} - 2 γ_{2} (k \cdot J)^{2}]$

Phenomenological Luttinger parameters $γ_{i}$ are defined as

$α = γ_{1} + \frac{5}{2} γ_{2}$

and

$β = γ_{2}$

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

$E = \frac{ℏ^{2} k^{2}}{2 m_{0}} (γ_{1} + \frac{5}{2} γ_{2} - 2 γ_{2} m_{j}^{2})$

Two degenerated resulting eigenenergies are

$E_{h h} = \frac{ℏ^{2} k^{2}}{2 m_{0}} (γ_{1} - 2 γ_{2})$ for $m_{j} = \pm \frac{3}{2}$

$E_{l h} = \frac{ℏ^{2} k^{2}}{2 m_{0}} (γ_{1} + 2 γ_{2})$ for $m_{j} = \pm \frac{1}{2}$

$E_{h h}$ ( $E_{l h}$ ) 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

$ϵ_{h, l} = - \frac{1}{2} γ_{1} k^{2} \pm [{γ_{2}}^{2} k^{4} + 3 ({γ_{3}}^{2} - {γ_{2}}^{2}) \times ({k_{x}}^{2} {k_{z}}^{2} + {k_{x}}^{2} {k_{y}}^{2} + {k_{y}}^{2} {k_{z}}^{2})]^{1 / 2}$

References

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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]].
+==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<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.
+==Phenomenological Hamiltonian for the j=3/2 states==
+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>
+<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>
+Phenomenological Luttinger parameters <math> \gamma _i </math> are defined as
+<math> \alpha = \gamma _1 + {5 \over 2} \gamma _2 </math>
+and
+<math> \beta = \gamma _2   </math>
+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.
+<math> E = { {\hbar^2 k^2} \over {2m_0} }( \gamma _1 + {{5} \over {2}} \gamma _2 - 2 \gamma _2 m_j^2)</math>
+Two degenerated resulting eigenenergies are
+<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>
+<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>
+<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.
+==Measurement of Luttinger parameters==
+Luttinger parameter can be measured by Hot-electron luminescence experiment.
+==Example: GaAs==
+<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>
+==References==
+{{reflist}}
+==See also==
+* J. M. Luttinger, Physical Review, Vol. '''102''', 1030 (1956). [http://prola.aps.org/abstract/PR/v102/i4/p1030_1 APS]
+* A. Baldereschi and N.O. Lipari, Physical Review B., Vol. '''8''', pp.&nbsp;2675 (1973). [http://prb.aps.org/abstract/PRB/v8/i6/p2697_1 APS]
+* A. Baldereschi and N.O. Lipari, Physical Review B., Vol. '''9''', pp.&nbsp;1525 (1974). [http://prb.aps.org/abstract/PRB/v9/i4/p1525_1 APS]
+[[Category:Semiconductors]]

Casio FX-603P: Difference between revisions

Revision as of 21:10, 28 October 2013

Contents

Three valence band state

Phenomenological Hamiltonian for the j=3/2 states

Measurement of Luttinger parameters

Example: GaAs

References

See also

Navigation menu

Casio FX-603P: Difference between revisions

Revision as of 21:10, 28 October 2013

Three valence band state

Phenomenological Hamiltonian for the j=3/2 states

Measurement of Luttinger parameters

Example: GaAs

References

See also

Navigation menu

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