K·p perturbation theory

The k · p method (also known KP method) is a perturbative method in quantum mechanics to calculate the electronic band structure of a solid. It provides an approximation of the solution of the Schrödinger equation for electrons in semiconductors and other crystalline solids.

The name comes from the fact that the energy of the different energy bands an expression of the form k · P, that is, the inner product of the quantum momentum operator ( P) and the wave vector (k) occurs.

Description

The method is based on a description of the electrons as non-interacting particles in an effective periodic potential. This potential includes the interaction of the electron described with the electrons and nuclei of the solid. If the solution of the Schrödinger equation for a wave vector k0 of the electron in reciprocal space known, a solution for obvious k- values ​​can be determined by the method.

The change in energy ( eigenvalues ​​of the Schrödinger equation) with the wave vector the desired band structure of the solid is then determined. The method allows to simulate also the electronic behavior of components of microelectronics.

The solutions at k = k0, however, must be known before. Other methods (such as the density functional theory ) provide such solutions. The electron energies for other k values ​​are determined as a perturbation of this solution.

Approach

The wave function of the electron is sufficient in the one-particle approximation of the Schrödinger equation:

Where p is the quantum mechanical operator pulse, V is the effective electrostatic potential, and m is the electron mass.

In a crystalline material V is a periodic function with the same periodicity as the crystal itself Bloch 's theorem now states that the solution of such a periodic differential equation can be written as follows:

K is the wavevector, n is a discrete index ( band index ) and un, k is a function with the same periodicity as the crystal. Substituting ψn, k in the single-particle Schrödinger equation, we obtain the following differential equation for un, k:

For a wave vector k0 for which the solutions are known (often k0 = 0, the so-called Γ - point ) deals with the k · p method, now the term

In the above equation as a perturbation ( hence the name). The aim of the perturbation theory is to find approximate expressions for the energy eigenvalues ​​and the corresponding eigenstates.

The energies and eigenstates are more precisely with increasing order, but in contrast, the equations become more complex. One therefore approximates the places in interference with the second order. For all the considered states of n equations is obtained in which interaction terms in the form of Überlappintegralen between the considered states and all other states n ' occur. Therefore obtains n equations with n ' interaction terms. For direct applications considering only conditions in the vicinity of the band gap, so that the number of equations is reduced. Furthermore, one in crystalline layers takes advantage of the symmetry properties of the different crystal systems in the form of group theory to combine with the help of many of the interaction terms in effective terms and thus further reduce greatly the number of interaction terms. It does this have relatively few equations which can compactly represent as a matrix and then calculating eigenvalues ​​and eigenstates. These are the desired energy eigenvalues ​​En, k with the corresponding eigenstates un, k

From the eigenvalues ​​can then be determined with less than a full calculation expressions for the dispersion, the effective mass of electrons and selection rules for the interaction with light.

Important it is especially in the case of degenerate bands, since the k · p- Term bands coupled with each other and the degeneracy partially offset determined and new selection rules for optical transitions between the bands.

459183
de