k · p method

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

The name comes from the fact that an expression of the form occurs in the energies of the individual energy bands , i.e. the scalar product of the wave vector and the quantum mechanical momentum operator . ${\displaystyle {\vec {k}}\cdot {\vec {p}}}$ ${\displaystyle {\vec {k}}}$ ${\displaystyle {\vec {p}}}$

description

The method is based on a description of the electrons as non -interacting particles in a periodic effective potential . This includes the interaction of the described electron with the electrons and atomic nuclei of the solid.

If the solution of the Schrödinger equation for a wave vector of the electron in reciprocal space is known from other methods (e.g. density functional theory ), then the electron energy for values ​​of in a neighborhood of can be determined as a perturbation of this solution. The band structure of the solid is then determined from the change in energy ( eigenvalues of the Hamilton operator appearing in the Schrödinger equation ) with the wave vector. ${\displaystyle {\vec {k}}_{0}}$ ${\displaystyle {\vec {k}}}$ ${\displaystyle {\vec {k}}_{0}}$

approach

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

${\displaystyle \left({\frac {p^{2}}{2m}}+V({\vec {r}})\right)\psi ({\vec {r}})=E\,\psi ({\vec {r}})}$

With

• the quantum mechanical momentum operator ${\displaystyle {\vec {p}}}$
• the mass of the electron ${\displaystyle m}$
• the effective electrostatic potential , which in a crystalline material is a periodic function with the same periodicity as the crystal itself. ${\displaystyle V}$

Bloch's theorem now states that the solution of such a periodic differential equation can be written as follows:

${\displaystyle \psi _{n,{\vec {k}}}({\vec {r}})=e^{\mathrm {i} {\vec {k}}\cdot {\vec {r}}}\,u_{n,{\vec {k}}}({\vec {r}})}$

is there

• ${\displaystyle n}$ a discrete band index
• ${\displaystyle {\vec {k}}}$ the wave vector
• ${\displaystyle \mathrm {i} }$the imaginary unit
• ${\displaystyle u_{n,{\vec {k}}}}$ a function with the same periodicity as the crystal.

Inserting the one-particle Schrödinger equation, one obtains the following differential equation for : ${\displaystyle \psi _{n,{\vec {k}}}}$ ${\displaystyle u_{n,{\vec {k}}}}$

${\displaystyle \left({\frac {p^{2}}{2m}}+{\frac {\hbar {\vec {k}}\cdot {\vec {p}}}{m}}+{\frac {\hbar ^{2}k^{2}}{2m}}+V({\vec {r}})\right)u_{n,{\vec {k}}}=E_{n,{\vec {k}}}u_{n,{\vec {k}}}}$

with Planck's reduced quantum of action . ${\displaystyle \hbar }$

For a wave vector for which the solutions are known (often at the Γ point ), the k · p method now deals with the term ${\displaystyle {\vec {k}}_{0}}$ ${\displaystyle {\vec {k}}_{0}=0}$

${\displaystyle {\frac {\hbar ({\vec {k}}-{\vec {k}}_{0})\cdot {\vec {p}}}{m}}}$

in the above equation as a disturbance (hence the name). The aim of the perturbation calculation is to find approximate expressions for the energy eigenvalues ​​and the associated eigenstates.

The energies and eigenstates become more precise with increasing order, but the equations become more and more complex. The expressions sought are therefore approximated with perturbations of the second order. For all observed states one obtains equations in which interaction terms occur in the form of transition matrix elements between the observed states and all other states . So one obtains equations with respective interaction terms. ${\displaystyle u_{n,{\vec {k}}}}$ ${\displaystyle u_{n',{\vec {k}}}}$ ${\displaystyle n}$ ${\displaystyle n'}$

For direct applications one only considers states in the vicinity of the band gap , which reduces the number of equations. Furthermore, in crystalline layers, the symmetry properties of the different crystal systems are used in the form of group theory in order to combine many of the interaction terms into effective terms and thus further reduce the number of interaction terms. Finally, there are relatively few equations, which are represented compactly as a matrix in order to then calculate the energy eigenvalues and the associated eigenstates . ${\displaystyle E_{n,{\vec {k}}}}$ ${\displaystyle \psi _{n,{\vec {k}}}}$

From the eigenvalues, expressions for the dispersion , the effective mass of the electrons and selection rules for the interaction with light can be determined with less effort than with a complete calculation. ${\displaystyle {\tfrac {\mathrm {d} }{\mathrm {d} k}}E_{n,{\vec {k}}}}$

It is particularly important in the case of degenerate bands, since the term couples the bands with one another, partially eliminates the degeneracy and determines new selection rules for optical transitions between the bands. ${\displaystyle {\vec {k}}\cdot {\vec {p}}}$

literature

• MS Dresselhaus: Group Theory - Application to the Physics of Condensated Matter. Springer Verlag, Heidelberg 2008.

(Note: a sophisticated textbook in which the KP theory is systematically derived)

• Dorothy G. Bell: Group Theory and Crystall Lattices. Review of Modern Physics, Volume 26, Number 3, p.311, 1954.
• C. Kittel: Quantum theory of the solid. R. Oldenbourg Verlag, Munich 1970, pp. 201ff.
• IJ Robertson, MC Payne: k-point sampling and the kp method in pseudopotential total energy calculations. In: J. Phys .: Condens. Matter. 2, 1990, pp. 9837-9852, doi : 10.1088 / 0953-8984 / 2/49/010 .
• W. Schäfer, M. Wegener: Semiconductor Optics and Transport Phenomena. Springer, 2002, ISBN 3-540-61614-4 , pp. 72ff.
• Christian Köpf: The k · p method. In: Modeling the electron transport in compound semiconductor alloys. Vienna 1997 (dissertation, Vienna University of Technology, online , accessed on January 22, 2010).