Luttinger–Kohn model

A flavor of the k·p perturbation theory used for calculating the structure of multiple, degenerate electronic bands in bulk and quantum well semiconductors. The method is a generalization of the single band k.p theory.

In this model the influence of all other bands is taken into account by using Löwdin's perturbation method.^[1]

Background

All bands can be subdivided into two classes:

• Class A: six valence bands (heavy hole, light hole, split off band and their spin counterparts) and two conduction bands.
• Class B: all other bands.

The method concentrates on the bands in Class A, and takes into account Class B bands perturbatively.

We can write the perturbed solution $\phi _{{}}^{{}}$ as a linear combination of the unperturbed eigenstates $\phi _{{i}}^{{(0)}}$:

$\phi =\sum _{{n}}^{{A,B}}a_{{n}}\phi _{{i}}^{{(0)}}$

Assuming the unperturbed eigenstates are orthonormalized, the eigenequation are:

$(E-H_{{mm}})a_{m}=\sum _{{n\neq m}}^{{A}}H_{{mn}}a_{{n}}+\sum _{{\alpha \neq m}}^{{B}}H_{{m\alpha }}a_{{\alpha }}$,

where

$H_{{mn}}=\int \phi _{{m}}^{{(0)\dagger }}H\phi _{{n}}^{{(0)}}d^{3}{\mathbf {r}}=E_{{n}}^{{(0)}}\delta _{{mn}}+H_{{mn}}^{{'}}$.

From this expression we can write:

$a_{{m}}=\sum _{{n\neq m}}^{{A}}{\frac {H_{{mn}}}{E-H_{{mm}}}}a_{{n}}+\sum _{{\alpha \neq m}}^{{B}}{\frac {H_{{m\alpha }}}{E-H_{{mm}}}}a_{{\alpha }}$,

where the first sum on the right-hand side is over the states in class A only, while the second sum is over the states on class B. Since we are interested in the coefficients $a_{{m}}$ for m in class A, we may eliminate those in class B by an iteration procedure to obtain:

$a_{{m}}=\sum _{{n\neq m}}^{{A}}{\frac {U_{{mn}}^{{A}}-H_{{mn}}}{E-H_{{mm}}}}a_{{n}}$,

$U_{{mn}}^{{A}}=H_{{mn}}+\sum _{{\alpha \neq m}}^{{B}}{\frac {H_{{m\alpha }}H_{{\alpha n}}}{E-H_{{\alpha \alpha }}}}+\sum _{{\alpha ,\beta \neq m,n;\alpha \neq \beta }}{\frac {H_{{m\alpha }}H_{{\alpha \beta }}H_{{\beta n}}}{(E-H_{{\alpha \alpha }})(E-H_{{\beta \beta }})}}+\ldots$

Equivalently, for $a_{{n}}$ ($n\in A$):

$a_{{n}}=\sum _{{n}}^{{A}}(U_{{mn}}^{{A}}-E\delta _{{mn}})a_{{n}}=0,m\in A$

and

$a_{{\gamma }}=\sum _{{n}}^{{A}}{\frac {U_{{\gamma n}}^{{A}}-H_{{\gamma n}}\delta _{{\gamma n}}}{E-H_{{\gamma \gamma }}}}a_{{n}}=0,\gamma \in B$.

When the coefficients $a_{{n}}$ belonging to Class A are determined so are $a_{{\gamma }}$.

Schrödinger equation and basis functions

The Hamiltonian including the spin-orbit interaction can be written as:

$H=H_{0}+{\frac {\hbar }{4m_{{0}}^{{2}}c^{{2}}}}{\bar {\sigma }}\cdot \nabla V\times {\mathbf {p}}$,

where ${\bar {\sigma }}$ is the Pauli spin matrix vector. Substituting into the Schrödinger equation we obtain

$Hu_{{n{\mathbf {k}}}}({\mathbf {r}})=\left(H_{0}+{\frac {\hbar }{m_{{0}}}}{\mathbf {k}}\cdot {\mathbf {\Pi }}+{\frac {\hbar ^{2}k^{2}}{4m_{{0}}^{{2}}c^{{2}}}}\nabla V\times {\mathbf {p}}\cdot {\bar {\sigma }}\right)u_{{n{\mathbf {k}}}}({\mathbf {r}})=E_{{n}}({\mathbf {k}})u_{{n{\mathbf {k}}}}({\mathbf {r}})$,

where

${\mathbf {\Pi }}={\mathbf {p}}+{\frac {\hbar }{4m_{{0}}^{{2}}c^{{2}}}}{\bar {\sigma }}\times \nabla V$

and the perturbation Hamiltonian can be defined as

$H'={\frac {\hbar }{m_{0}}}{\mathbf {k}}\cdot {\mathbf {\Pi }}.$

The unperturbed Hamiltonian refers to the band-edge spin-orbit system (for k=0). At the band edge, conduction band Bloch waves exhibit s-like symmetry, while valence band states are p-like (3-fold degenerate without spin). Let us denote these states as $|S\rangle$, and $|X\rangle$, $|Y\rangle$ and $|Z\rangle$ respectively. These Bloch functions can be pictured as periodic repetition of atomic orbitals, repeated at intervals corresponding to the lattice spacing. The Bloch function can be expanded in the following manner:

$u_{{n{\mathbf {k}}}}({\mathbf {r}})=\sum _{{j'}}^{{A}}a_{{j'}}({\mathbf {k}})u_{{j'0}}({\mathbf {r}})+\sum _{{\gamma }}^{{B}}a_{{\gamma }}({\mathbf {k}})u_{{\gamma 0}}({\mathbf {r}})$,

where j' is in Class A and $\gamma$ is in Class B. The basis functions can be chosen to be

$u_{{10}}({\mathbf {r}})=u_{{el}}({\mathbf {r}})=\left|S{\frac {1}{2}},{\frac {1}{2}}\right\rangle =\left|S\uparrow \right\rangle$

$u_{{20}}({\mathbf {r}})=u_{{SO}}({\mathbf {r}})=\left|{\frac {1}{2}},{\frac {1}{2}}\right\rangle ={\frac {1}{{\sqrt 3}}}|(X+iY)\downarrow \rangle +{\frac {1}{{\sqrt 3}}}|Z\uparrow \rangle$

${\displaystyle u_{30}(\mathbf {r} )=u_{lh}(\mathbf {r} )=\left|{\frac {3}{2}},{\frac {1}{2}}\right\rangle =-{\frac {1}{\sqrt {6}}}|(X+iY)\downarrow \rangle +{\sqrt {\frac {2}{3}}}|Z\uparrow \rangle }$

${\displaystyle u_{40}(\mathbf {r} )=u_{hh}(\mathbf {r} )=\left|{\frac {3}{2}},{\frac {3}{2}}\right\rangle =-{\frac {1}{\sqrt {2}}}|(X+iY)\uparrow \rangle }$

$u_{{50}}({\mathbf {r}})={\bar {u}}_{{el}}({\mathbf {r}})=\left|S{\frac {1}{2}},-{\frac {1}{2}}\right\rangle =-|S\downarrow \rangle$

$u_{{60}}({\mathbf {r}})={\bar {u}}_{{SO}}({\mathbf {r}})=\left|{\frac {1}{2}},-{\frac {1}{2}}\right\rangle ={\frac {1}{{\sqrt 3}}}|(X-iY)\uparrow \rangle -{\frac {1}{{\sqrt 3}}}|Z\downarrow \rangle$

${\displaystyle u_{70}(\mathbf {r} )={\bar {u}}_{lh}(\mathbf {r} )=\left|{\frac {3}{2}},-{\frac {1}{2}}\right\rangle ={\frac {1}{\sqrt {6}}}|(X-iY)\uparrow \rangle +{\sqrt {\frac {2}{3}}}|Z\downarrow \rangle }$

$u_{{80}}({\mathbf {r}})={\bar {u}}_{{hh}}({\mathbf {r}})=\left|{\frac {3}{2}},-{\frac {3}{2}}\right\rangle =-{\frac {1}{{\sqrt 2}}}|(X-iY)\downarrow \rangle$.

Using Löwdin's method, only the following eigenvalue problem needs to be solved

$\sum _{{j'}}^{{A}}(U_{{jj'}}^{{A}}-E\delta _{{jj'}})a_{{j'}}({\mathbf {k}})=0,$

where

$U_{{jj'}}^{{A}}=H_{{jj'}}+\sum _{{\gamma \neq j,j'}}^{{B}}{\frac {H_{{j\gamma }}H_{{\gamma j'}}}{E_{0}-E_{{\gamma }}}}=H_{{jj'}}+\sum _{{\gamma \neq j,j'}}^{{B}}{\frac {H_{{j\gamma }}^{{'}}H_{{\gamma j'}}^{{'}}}{E_{0}-E_{{\gamma }}}}$,

$H_{{j\gamma }}^{{'}}=\left\langle u_{{j0}}\right|{\frac {\hbar }{m_{0}}}{\mathbf {k}}\cdot \left({\mathbf {p}}+{\frac {\hbar }{4m_{0}c^{2}}}{\bar {\sigma }}\times \nabla V\right)\left|u_{{\gamma 0}}\right\rangle \approx \sum _{{\alpha }}{\frac {\hbar k_{{\alpha }}}{m_{0}}}p_{{j\gamma }}^{{\alpha }}.$

The second term of $\Pi$ can be neglected compared to the similar term with p instead of k. Similarly to the single band case, we can write for $U_{{jj'}}^{{A}}$

$D_{{jj'}}\equiv U_{{jj'}}^{{A}}=E_{{j}}(0)\delta _{{jj'}}+\sum _{{\alpha \beta }}D_{{jj'}}^{{\alpha \beta }}k_{{\alpha }}k_{{\beta }},$

$D_{{jj'}}^{{\alpha \beta }}={\frac {\hbar ^{2}}{2m_{0}}}\left[\delta _{{jj'}}\delta _{{\alpha \beta }}+\sum _{{\gamma }}^{{B}}{\frac {p_{{j\gamma }}^{{\alpha }}p_{{\gamma j'}}^{{\beta }}+p_{{j\gamma }}^{{\beta }}p_{{\gamma j'}}^{{\alpha }}}{m_{0}(E_{0}-E_{{\gamma }})}}\right].$

We now define the following parameters

$A_{0}={\frac {\hbar ^{2}}{2m_{0}}}+{\frac {\hbar ^{2}}{m_{0}^{2}}}\sum _{{\gamma }}^{{B}}{\frac {p_{{x\gamma }}^{{x}}p_{{\gamma x}}^{{x}}}{E_{0}-E_{{\gamma }}}},$

$B_{0}={\frac {\hbar ^{2}}{2m_{0}}}+{\frac {\hbar ^{2}}{m_{0}^{2}}}\sum _{{\gamma }}^{{B}}{\frac {p_{{x\gamma }}^{{y}}p_{{\gamma x}}^{{y}}}{E_{0}-E_{{\gamma }}}},$

$C_{0}={\frac {\hbar ^{2}}{m_{0}^{2}}}\sum _{{\gamma }}^{{B}}{\frac {p_{{x\gamma }}^{{x}}p_{{\gamma y}}^{{y}}+p_{{x\gamma }}^{{y}}p_{{\gamma y}}^{{x}}}{E_{0}-E_{{\gamma }}}},$

and the band structure parameters (or the Luttinger parameters) can be defined to be

$\gamma _{1}=-{\frac {1}{3}}{\frac {2m_{0}}{\hbar ^{2}}}(A_{0}+2B_{0}),$

$\gamma _{2}=-{\frac {1}{6}}{\frac {2m_{0}}{\hbar ^{2}}}(A_{0}-B_{0}),$

$\gamma _{3}=-{\frac {1}{6}}{\frac {2m_{0}}{\hbar ^{2}}}C_{0},$

These parameters are very closely related to the effective masses of the holes in various valence bands. $\gamma_1$ and $\gamma_2$ describe the coupling of the $|X\rangle$, $|Y\rangle$ and $|Z\rangle$ states to the other states. The third parameter $\gamma_3$ relates to the anisotropy of the energy band structure around the $\Gamma$ point when $\gamma _{2}\neq \gamma _{3}$.

Explicit Hamiltonian matrix

The Luttinger-Kohn Hamiltonian ${\mathbf {D_{{jj'}}}}$ can be written explicitly as a 8X8 matrix (taking into account 8 bands - 2 conduction, 2 heavy-holes, 2 light-holes and 2 split-off)

${\mathbf {H}}=\left({\begin{array}{cccccccc}E_{{el}}&P_{z}&{\sqrt {2}}P_{z}&-{\sqrt {3}}P_{{+}}&0&{\sqrt {2}}P_{{-}}&P_{{-}}&0\\P_{z}^{{\dagger }}&P+\Delta &{\sqrt {2}}Q^{{\dagger }}&-S^{{\dagger }}/{\sqrt {2}}&-{\sqrt {2}}P_{{+}}^{{\dagger }}&0&-{\sqrt {3/2}}S&-{\sqrt {2}}R\\E_{{el}}&P_{z}&{\sqrt {2}}P_{z}&-{\sqrt {3}}P_{{+}}&0&{\sqrt {2}}P_{{-}}&P_{{-}}&0\\E_{{el}}&P_{z}&{\sqrt {2}}P_{z}&-{\sqrt {3}}P_{{+}}&0&{\sqrt {2}}P_{{-}}&P_{{-}}&0\\E_{{el}}&P_{z}&{\sqrt {2}}P_{z}&-{\sqrt {3}}P_{{+}}&0&{\sqrt {2}}P_{{-}}&P_{{-}}&0\\E_{{el}}&P_{z}&{\sqrt {2}}P_{z}&-{\sqrt {3}}P_{{+}}&0&{\sqrt {2}}P_{{-}}&P_{{-}}&0\\E_{{el}}&P_{z}&{\sqrt {2}}P_{z}&-{\sqrt {3}}P_{{+}}&0&{\sqrt {2}}P_{{-}}&P_{{-}}&0\\E_{{el}}&P_{z}&{\sqrt {2}}P_{z}&-{\sqrt {3}}P_{{+}}&0&{\sqrt {2}}P_{{-}}&P_{{-}}&0\\\end{array}}\right)$

References

1. S.L. Chuang (1995). Physics of Optoelectronic Devices (First ed.). New York: Wiley. pp. 124–190. ISBN 0-471-10939-8.