A flavor of the
k·p perturbation theory used for calculating the structure of multiple, degenerate electronic bands in bulk and
quantum well
A quantum well is a potential well with only discrete energy values.
The classic model used to demonstrate a quantum well is to confine particles, which were initially free to move in three dimensions, to two dimensions, by forcing them to occupy ...
semiconductors
A semiconductor is a material which has an electrical resistivity and conductivity, electrical conductivity value falling between that of a electrical conductor, conductor, such as copper, and an insulator (electricity), insulator, such as glas ...
. 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
as a linear combination of the unperturbed eigenstates
:
:
Assuming the unperturbed eigenstates are orthonormalized, the eigenequation 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
The
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
including the spin-orbit interaction can be written as:
:
,
where
is the
Pauli spin matrix
In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () when used in c ...
vector. Substituting into the
Schrödinger equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the ...
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, conduction band
Bloch waves
In condensed matter physics, Bloch's theorem states that solutions to the Schrödinger equation in a periodic potential take the form of a plane wave modulated by a periodic function. The theorem is named after the physicist Felix Bloch, who di ...
exhibit s-like symmetry, while 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 parameter
In semiconductors, valence bands are well characterized by 3 Luttinger parameters. At the ''Г''-point in the band structure, p_ and p_ orbitals form valence bands. But spin–orbit coupling splits sixfold degeneracy into high energy 4-fold and ...
s) 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
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
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
{{DEFAULTSORT:Luttinger-Kohn model
Condensed matter physics