Luttinger–Kohn model

The Luttinger–Kohn model is 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.

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 $\phi^_$:

:$\phi = \sum^_ a_ \phi^_$

Assuming the unperturbed eigenstates are orthonormalized, the eigenequations are:

:$(E-H_)a_m = \sum^_H_a_ + \sum^_H_a_$,

where

:$H_ = \int \phi^_ H \phi^_d^3 \mathbf = E^_\delta_+H^_$.

From this expression, we can write:

:$a_ = \sum^_ \frac a_ + \sum^_ \frac a_$,

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 $a_$ for ''m'' in class A, we may eliminate those in class B by an iteration procedure to obtain:

:$a_ = \sum^_ \frac a_$,

:$U^_ = H_ + \sum^_ \frac + \sum_ \frac + \ldots$

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

:$a_ = \sum^_ (U^_ - E\delta_)a_ = 0, m \in A$

and

:$a_ = \sum^_ \frac a_ = 0, \gamma \in B$.

When the coefficients $a_$ belonging to Class A are determined, so are $a_$.

Schrödinger equation and basis functions

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

:$H = H_0 + \frac\bar\cdot\nabla V \times \mathbf$,

where $\bar$ is the Pauli spin matrix vector. Substituting into the Schrödinger equation in Bloch approximation we obtain

:$H u_(\mathbf) = \left( H_0 + \frac\mathbf\cdot\mathbf + \frac \nabla V \times \mathbf \cdot \bar \right) u_(\mathbf) = E_(\mathbf) u_(\mathbf)$,

where

:$\mathbf = \mathbf + \frac\bar \times \nabla V$

and the perturbation Hamiltonian can be defined as

:$H' = \frac\mathbf\cdot\mathbf.$

The unperturbed Hamiltonian refers to the band-edge spin-orbit system (for ''k''=0). At the band edge, the conduction band Bloch waves exhibits s-like symmetry, while the 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:

:$u_ (\mathbf) = \sum^_ a_(\mathbf) u_(\mathbf) + \sum^_ a_(\mathbf) u_(\mathbf)$,

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

:$u_(\mathbf) = u_(\mathbf) = \left , S\frac,\frac \right \rangle = \left, S\uparrow\right\rangle$
:$u_(\mathbf) = u_(\mathbf) = \left , \frac,\frac \right \rangle = \frac , (X+iY)\downarrow\rangle + \frac , Z\uparrow\rangle$
:$u_(\mathbf) = u_(\mathbf) = \left , \frac,\frac \right \rangle = -\frac , (X+iY)\downarrow\rangle + \sqrt , Z\uparrow\rangle$
:$u_(\mathbf) = u_(\mathbf) = \left , \frac,\frac \right \rangle = -\frac, (X+iY)\uparrow\rangle$
:$u_(\mathbf) = \bar_(\mathbf) = \left , S\frac,-\frac \right \rangle = -, S\downarrow\rangle$
:$u_(\mathbf) = \bar_(\mathbf) = \left , \frac,-\frac \right \rangle = \frac , (X-iY)\uparrow\rangle - \frac , Z\downarrow\rangle$
:$u_(\mathbf) = \bar_(\mathbf) = \left , \frac,-\frac \right \rangle = \frac , (X-iY)\uparrow\rangle + \sqrt , Z\downarrow\rangle$
:$u_(\mathbf) = \bar_(\mathbf) = \left , \frac,-\frac \right \rangle = -\frac, (X-iY)\downarrow\rangle$.

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

:$\sum^_ (U^_-E\delta_)a_(\mathbf) = 0,$

where

:$U^_ = H_ + \sum^_ \frac = H_ + \sum^_ \frac$,

:$H^_ = \left \langle u_ \right , \frac \mathbf \cdot \left ( \mathbf + \frac \bar \times \nabla V \right ) \left , u_ \right \rangle \approx \sum_ \fracp^_.$

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 $U^_$

:$D_ \equiv U^_ = E_(0)\delta_ + \sum_ D^_k_k_,$

: D^_ = \frac \left \delta_\delta_ + \sum^_ \frac \right

We now define the following parameters

:$A_0 = \frac + \frac \sum^_ \frac,$

:$B_0 = \frac + \frac \sum^_ \frac,$

:$C_0 = \frac \sum^_ \frac,$

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

:$\gamma_1 = - \frac \frac (A_0 + 2B_0),$

:$\gamma_2 = - \frac \frac (A_0 - B_0),$

:$\gamma_3 = - \frac \frac C_0,$

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 $\mathbf$ 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)

:$\mathbf = \left( \begin E_ & P_z & \sqrtP_z & -\sqrtP_ & 0 & \sqrtP_ & P_ & 0 \\ P_z^ & P+\Delta & \sqrtQ^ & -S^/\sqrt & -\sqrtP_^ & 0 & -\sqrtS & -\sqrtR \\ E_ & P_z & \sqrtP_z & -\sqrtP_ & 0 & \sqrtP_ & P_ & 0 \\ E_ & P_z & \sqrtP_z & -\sqrtP_ & 0 & \sqrtP_ & P_ & 0 \\ E_ & P_z & \sqrtP_z & -\sqrtP_ & 0 & \sqrtP_ & P_ & 0 \\ E_ & P_z & \sqrtP_z & -\sqrtP_ & 0 & \sqrtP_ & P_ & 0 \\ E_ & P_z & \sqrtP_z & -\sqrtP_ & 0 & \sqrtP_ & P_ & 0 \\ E_ & P_z & \sqrtP_z & -\sqrtP_ & 0 & \sqrtP_ & P_ & 0 \\ \end \right)$

Summary

References

2. Luttinger, J. M. Kohn, W., "Motion of Electrons and Holes in Perturbed Periodic Fields", Phys. Rev. 97,4. pp. 869-883, (1955). https://journals.aps.org/pr/abstract/10.1103/PhysRev.97.869

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Condensed matter physics