In solid-state physics, the k·p perturbation theory is an approximated semi-empirical approach for calculating the band structure (particularly effective mass) and optical properties of crystalline solids.
According to quantum mechanics (in the single-electron approximation), the quasi-free electrons in any solid are characterized by wavefunctions which are eigenstates of the following stationary Schrödinger equation: where p is the quantum-mechanical momentum operator, V is the potential, and m is the vacuum mass of the electron.
Bloch's theorem proves that the solutions to this differential equation can be written as follows: where k is a vector (called the wavevector), n is a discrete index (called the band index), and un,k is a function with the same periodicity as the crystal lattice.
The periodic function un,k satisfies the following Schrödinger-type equation (simply, a direct expansion of the Schrödinger equation with a Bloch-type wave function):[1] where the Hamiltonian is Note that k is a vector consisting of three real numbers with dimensions of inverse length, while p is a vector of operators; to be explicit, In any case, we write this Hamiltonian as the sum of two terms: This expression is the basis for perturbation theory.
The analysis that results is called k·p perturbation theory, due to the term proportional to k·p.
The result of this analysis is an expression for En,k and un,k in terms of the energies and wavefunctions at k = 0.
Therefore, k·p perturbation theory is most accurate for small values of k. However, if enough terms are included in the perturbative expansion, then the theory can in fact be reasonably accurate for any value of k in the entire Brillouin zone.
For a nondegenerate band (i.e., a band which has a different energy at k = 0 from any other band), with an extremum at k = 0, and with no spin–orbit coupling, the result of k·p perturbation theory is (to lowest nontrivial order):[1] Since k is a vector of real numbers (rather than a vector of more complicated linear operators), the matrix element in these expressions can be rewritten as: Therefore, one can calculate the energy at any k using only a few unknown parameters, namely En,0 and
The latter are called "optical matrix elements", closely related to transition dipole moments.
In practice, the sum over n often includes only the nearest one or two bands, since these tend to be the most important (due to the denominator).
However, for improved accuracy, especially at larger k, more bands must be included, as well as more terms in the perturbative expansion than the ones written above.
This denominator is then approximated as the band gap Eg, leading to an energy expression: The effective mass in direction ℓ is then: Ignoring the details of the matrix elements, the key consequences are that the effective mass varies with the smallest bandgap and goes to zero as the gap goes to zero.
One finds there are two types of hole, named heavy and light, with anisotropic masses.
is introduced, and to the first order, its matrix elements can be expressed as After solving it, the wave functions and energy bands are obtained.