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.[1]

Background

edit

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,  , 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. Since we are interested in the coefficients   for m in class A, we may eliminate those in class B by an iteration procedure to obtain:

 ,
 

Equivalently, for   ( ):

 

and

 .

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

Schrödinger equation and basis functions

edit

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). Let us denote these states as  , and  ,   and   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:

 ,

where j' is in Class A and   is in Class B. 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.   and   describe the coupling of the  ,   and   states to the other states. The third parameter   relates to the anisotropy of the energy band structure around the   point when  .

Explicit Hamiltonian matrix

edit

The Luttinger-Kohn Hamiltonian   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)

 

Summary

edit

References

edit
  1. ^ S.L. Chuang (1995). Physics of Optoelectronic Devices (First ed.). New York: Wiley. pp. 124–190. ISBN 978-0-471-10939-6. OCLC 31134252.

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