# Korringa–Kohn–Rostoker method

# Korringa–Kohn–Rostoker method

The **Korringa–Kohn–Rostoker method** or **KKR method**
is used to calculate the electronic band structure of periodic solids. In the derivation of the method using multiple scattering theory by Jan Korringa^{[1]} and the derivation based on the Kohn variational method,^{[2]} the muffin-tin approximation was used.^{[3]} Later calculations are done with full potentials having no shape restrictions.^{[4]}^{[5]}

Introduction

`All solids in their ideal state are single crystals with the atoms arranged on a periodic lattice. In condensed matter physics, the properties of such solids are explained on the basis of theirelectronic structure. This requires the solution of a complicated many-electron problem, but thedensity functional theoryofWalter Kohnmakes it possible to reduce it to the solution of a Schroedinger equation with a one-electron periodic potential. The problem is further simplified with the use ofgroup theoryand in particularBloch's theorem, which leads to the result that the energy eigenvalues depend on the crystal momentumand are divided into bands.Band theoryis used to calculate the eigenvalues and wave functions.`

Many band theory methods have been proposed over the years. Some of the most widely used, such as the electronic structure programs VASP and WIEN2k, make use of approximations so that acceptable accuracy can be achieved with a minimum of computer resources. The KKR method is chosen when the primary goal is high accuracy.

The parameters obtained from reliable band-theory calculations are useful in theoretical studies of problems, such as superconductivity, for which the density functional theory does not apply.

Mathematical formulation

The KKR band theory equations for space-filling non-spherical potentials are derived in books^{[4]}^{[5]} and in the article on multiple scattering theory.

`The wave function near siteis determined by the coefficients. According to Bloch's theorem, these coefficients differ only through a phase factor. Thesatisfy the homogeneous equations`

- ,

`whereand.`

`Theis the inverse of the scattering matrixcalculated with the non-spherical potential for the site. As pointed out by Korringa,`

^{[1]}Ewaldderived a summation process that makes it possible to calculate the structure constants,. The energy eigenvalues of the periodic solid for a particular,, are the roots of the equation. The eigenfunctions are found by solving for thewith. By ignoring all contributions that correspond to an angular momentumgreater than, they have dimension.`In the original derivations of the KKR method, spherically symmetric muffin-tin potentials were used. Such potentials have the advantage that the inverse of the scattering matrix is diagonal in`

- ,

`whereis the scattering phase shift that appears in the partial wave analysis in scattering theory. The muffin-tin approximation is good for closely packed metals, but it does not work well for ionic solids like semiconductors. It also leads to errors in calculations of interatomic forces.`

## References

*Physica*.

**XIII**(6–7): 392–400. Bibcode:1947Phy....13..392K. doi:10.1016/0031-8914(47)90013-x.

*Phys. Rev*.

**94**(5): 1111–1120. Bibcode:1954PhRv...94.1111K. doi:10.1103/physrev.94.1111.

*Theoretical Solid State Physics*. Wiley and Sons - Dover Publications. ISBN 0-486-65015-4.

*Electron Scattering in Solid Matter: A Theoretical and Computational Treatise*(Softcover reprint of hardcover 1st ed. 2005 ed.). Springer. ISBN 978-3642061387.

*Multiple Scattering beta edition*(Kindle Interactive ed.). Amazon. ASIN B015NFAN6M.