Korringa–Kohn–Rostoker method
The Korringa–Kohn–Rostoker 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 and the derivation based on the Kohn and Rostoker variational method, the muffin-tin approximation was used. Later calculations are done with full potentials having no shape restrictions.
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 their electronic structure. This requires the solution of a complicated many-electron problem, but the density functional theory of Walter Kohn makes 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 of group theory and in particular Bloch's theorem, which leads to the result that the energy eigenvalues depend on the crystal momentum and are divided into bands. Band theory is used to calculate the eigenvalues and wave functions.As compared with other band structure methods, the Korringa-Kohn-Rostoker band structure method has the advantage of dealing with small matrices due to the fast convergence of scattering operators in angular momentum space, and disordered systems where it allows to carry out with relative ease the ensemble configuration averages. The KKR method does have a few "bills" to pay, e.g., the calculation of KKR structure constants, the empty lattice propagators, must be carried out by the Ewald's sums for each energy and k-point, and the KKR functions have a pole structure on the real energy axis, which requires a much larger number of k points for the Brillouin Zone integration as compared with other band theory methods. The KKR method has been implemented in several codes for electronic structure and spectroscopy calculations, such as MuST, AkaiKKR, sprKKR, FEFF, GNXAS and JuKKR.
Mathematical formulation
The KKR band theory equations for space-filling non-spherical potentials are derived in books and in the article on multiple scattering theory.The wave function near site is determined by the coefficients. According to Bloch's theorem, these coefficients differ only through a phase factor. The satisfy the homogeneous equations
where and.
The is the inverse of the scattering matrix calculated with the non-spherical potential for the site. As pointed out by Korringa, Ewald derived 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 the with. By ignoring all contributions that correspond to an angular momentum greater 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
where is 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.
Recent Methodological Developments
Full-Potential KKR
Early implementations of KKR employed muffin-tin or atomic sphere approximations, which restrict the shape of the potential and charge density. These approximations limit accuracy for systems with significant interstitial or non-spherical contributions.Full-potential KKR removes shape approximations entirely, treating the crystal potential exactly within each atomic region and the interstitial space. Recent advances implement more efficient treatments of the Dyson Green’s functions and handle non-spherical potential components rigorously, making full-potential KKR practical for total energies, forces, and electric field gradients.
For example, new algorithms ensure exact Wronskian relations for Green functions and avoid unphysical irregular solutions that can otherwise contaminate charge and spin densities. These improvements enable accurate calculations of electric field gradients and other sensitive properties.
Highly scalable implementations extend full-potential KKR to thousands of atoms, which is transformative for studying complex defects, interfaces, and disordered systems at realistic sizes.
Screened KKR
The screened KKR reformulation introduces a screening transformation to the structure constants of multiple scattering theory, yielding exponentially decaying interaction terms. This substantially improves numerical efficiency, especially for systems with layered geometries or open boundary conditions, such as surfaces and multilayers, where naive KKR structure constants converge slowly.Screening accelerates convergence and enables linear scaling with system size along principal directions, a key benefit for transport and surface/interface applications.
Applications
The KKR method may be combined with density functional theory and used to study the electronic structure and consequent physical properties of molecules and materials. As with any DFT calculation, the electronic problem must be solved self-consistently, before quantities such as the total energy of a collection of atoms, the electron density, the band structure, and forces on individual atoms may be calculated.One major advantage of the KKR formalism over other electronic structure methods is that it provides direct access to the Green's function of a given system. This, and other convenient mathematical quantities recovered from the derivation in terms of multiple scattering theory, facilitate access to a range of physically relevant quantities, including transport properties, magnetic properties, and spectroscopic properties.
One particularly powerful method which is unique to Green's function-based methods is the coherent potential approximation, which is an effective medium theory used to average over configurational disorder, such as is encountered in a substitutional alloy. The CPA captures the broken translational symmetry of the disordered alloy in a physically meaningful way, with the end result that the initially 'sharp' band structure is 'smeared-out', which reflects the finite lifetime of electronic states in such a system. The CPA can also be used to average over many possible orientations of magnetic moments, as is necessary to describe the paramagnetic state of a magnetic material. This is referred to as the disordered local moment picture.