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.
[1] All bands can be subdivided into two classes: The method concentrates on the bands in Class A, and takes into account Class B bands perturbatively.
We can write the perturbed solution,
, as a linear combination of the unperturbed eigenstates
: Assuming the unperturbed eigenstates are orthonormalized, the eigenequations are: where From this expression, we can write: 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.
for m in class A, we may eliminate those in class B by an iteration procedure to obtain: Equivalently, for
belonging to Class A are determined, so are
The Hamiltonian including the spin-orbit interaction can be written as: where
is the Pauli spin matrix vector.
Substituting into the Schrödinger equation in Bloch approximation we obtain where and the perturbation Hamiltonian can be defined as 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).
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: where j' is in Class A and
The basis functions can be chosen to be Using Löwdin's method, only the following eigenvalue problem needs to be solved where The second term of
can be neglected compared to the similar term with p instead of k. Similarly to the single band case, we can write for
We now define the following parameters and the band structure parameters (or the Luttinger parameters) can be defined to be These parameters are very closely related to the effective masses of the holes in various valence bands.
describe the coupling of the
relates to the anisotropy of the energy band structure around the
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) 2.
Luttinger, J. M. Kohn, W., "Motion of Electrons and Holes in Perturbed Periodic Fields", Phys.