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Korringa–Kohn–Rostoker method
Technique in solid-state physics

The Korringa–Kohn–Rostoker (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 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.

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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 k {\displaystyle {\bf {k}}} 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 (KKR) 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., (1) 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 (2) 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 (BZ) 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,6 AkaiKKR,7 sprKKR,8 FEFF,9 GNXAS10 and JuKKR.11

Mathematical formulation

The KKR band theory equations for space-filling non-spherical potentials are derived in books1213 and in the article on multiple scattering theory.

The wave function near site j {\displaystyle j} is determined by the coefficients c ℓ ′ m ′ j {\displaystyle c_{\ell 'm'}^{j}} . According to Bloch's theorem, these coefficients differ only through a phase factor c ℓ ′ m ′ j = e − i k ⋅ R j c ℓ ′ m ′ ( E , k ) {\displaystyle c_{\ell 'm'}^{j}={e^{-i{\bf {k}}\cdot {\bf {R}}_{j}}}c_{\ell 'm'}(E,{\bf {k}})} . The c ℓ ′ m ′ ( E , k ) {\displaystyle c_{\ell 'm'}(E,{\bf {k}})} satisfy the homogeneous equations

∑ ℓ ′ m ′ M ℓ m , ℓ ′ m ′ ( E , k ) c ℓ ′ m ′ ( E , k ) = 0 , {\displaystyle \sum _{\ell 'm'}M_{\ell m,\ell 'm'}(E,{\bf {k}})c_{\ell 'm'}(E,{\bf {k}})=0,}

where M ℓ m , ℓ ′ m ′ ( E , k ) = m ℓ m , ℓ ′ m ′ ( E ) − A ℓ m , ℓ ′ m ′ ( E , k ) {\displaystyle {M_{\ell m,\ell 'm'}}(E,{\bf {k}})=m_{\ell m,\ell 'm'}(E)-A_{\ell m,\ell 'm'}(E,{\bf {k}})} and A ℓ m , ℓ ′ m ′ ( E , k ) = ∑ j e i k ⋅ R i j g l m , l ′ m ′ ( E , R i j ) {\displaystyle A_{\ell m,\ell 'm'}(E,{\bf {k}})=\sum \limits _{j}{e^{i{\bf {{k}\cdot {\bf {{R}_{ij}}}}}}}g_{lm,l'm'}(E,{\bf {R}}_{ij})} .

The m ℓ m , ℓ ′ m ′ ( E ) {\displaystyle m_{\ell m,\ell 'm'}(E)} is the inverse of the scattering matrix t ℓ m , ℓ ′ m ′ ( E ) {\displaystyle t_{\ell m,\ell 'm'}(E)} calculated with the non-spherical potential for the site. As pointed out by Korringa,14 Ewald derived a summation process that makes it possible to calculate the structure constants, A ℓ m , ℓ ′ m ′ ( E , k ) {\displaystyle A_{\ell m,\ell 'm'}(E,{\bf {k}})} . The energy eigenvalues of the periodic solid for a particular k {\displaystyle {\bf {k}}} , E b ( k ) {\displaystyle E_{b}({\bf {{k})}}} , are the roots of the equation det M ( E , k ) = 0 {\displaystyle \det {\bf {M}}(E,{\bf {k}})=0} . The eigenfunctions are found by solving for the c ℓ , m ( E , k ) {\displaystyle c_{\ell ,m}(E,{\bf {k}})} with E = E b ( k ) {\displaystyle E=E_{b}({\bf {k}})} . By ignoring all contributions that correspond to an angular momentum l {\displaystyle l} greater than ℓ max {\displaystyle \ell _{\max }} , they have dimension ( ℓ max + 1 ) 2 {\displaystyle (\ell _{\max }+1)^{2}} .

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 l {\displaystyle l}

m ℓ m , ℓ ′ m ′ = [ α cot ⁡ δ ℓ ( E ) − i α ] δ ℓ , ℓ ′ δ m , m ′ , {\displaystyle m_{\ell m,\ell 'm'}=\left[\alpha \cot \delta _{\ell }(E)-i\alpha \right]\delta _{\ell ,\ell '}\delta _{m,m'},}

where δ ℓ ( E ) {\displaystyle \delta _{\ell }(E)} 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.

Applications

The KKR method may be combined with density functional theory (DFT) and used to study the electronic structure and consequent physical properties of molecules and materials.15 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 atoms16 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.17

One particularly powerful method which is unique to Green's function-based methods is the coherent potential approximation (CPA), which is an effective medium theory used to average over configurational disorder, such as is encountered in a substitutional alloy.1819 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.20 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 (above its Curie temperature).21 This is referred to as the disordered local moment (DLM) picture.2223

References

  1. J. Korringa (1947). "On the calculation of the energy of a Bloch wave in a metal". Physica. XIII (6–7): 392–400. Bibcode:1947Phy....13..392K. doi:10.1016/0031-8914(47)90013-x. /wiki/Jan_Korringa ↩

  2. W. Kohn, N. Rostoker (1954). "Solution of the Schrödinger Equation in Periodic Lattices with an Application to Metallic Lithium". Phys. Rev. 94 (5): 1111–1120. Bibcode:1954PhRv...94.1111K. doi:10.1103/physrev.94.1111. /wiki/Bibcode_(identifier) ↩

  3. W. Jones, N. H. March (1973). Theoretical Solid State Physics. Wiley and Sons – Dover Publications. ISBN 0-486-65015-4. 0-486-65015-4 ↩

  4. Jan Zabloudil; Robert Hammerling; Laszlo Szunyogh; Peter Weinberger (2010) [2005]. Electron Scattering in Solid Matter: A Theoretical and Computational Treatise (Softcover reprint of hardcover 1st ed. 2005 ed.). Springer. ISBN 978-3642061387. 978-3642061387 ↩

  5. Yang Wang; G. Malcolm Stocks; J. Sam Faulkner (2018). Multiple Scattering Theory: Electronic Structure of Solids (Kindle Interactive ed.). IOP. ISBN 978-0750314886. 978-0750314886 ↩

  6. "MuST:Multiple Scattering Theory". GitHub. 26 October 2022. https://github.com/mstsuite/MuST ↩

  7. "AkaiKKR". Retrieved 2021-02-15. http://kkr.issp.u-tokyo.ac.jp/ ↩

  8. "sprKKR". https://www.ebert.cup.uni-muenchen.de/index.php/en/software-en/13-sprkkr ↩

  9. "FEFF". http://monalisa.phys.washington.edu/index-feffproject.html ↩

  10. "GNXAS". http://gnxas.unicam.it/pag_gnxas.html ↩

  11. "JuKKR". 10 February 2023. https://jukkr.fz-juelich.de/ ↩

  12. Jan Zabloudil; Robert Hammerling; Laszlo Szunyogh; Peter Weinberger (2010) [2005]. Electron Scattering in Solid Matter: A Theoretical and Computational Treatise (Softcover reprint of hardcover 1st ed. 2005 ed.). Springer. ISBN 978-3642061387. 978-3642061387 ↩

  13. Yang Wang; G. Malcolm Stocks; J. Sam Faulkner (2018). Multiple Scattering Theory: Electronic Structure of Solids (Kindle Interactive ed.). IOP. ISBN 978-0750314886. 978-0750314886 ↩

  14. J. Korringa (1947). "On the calculation of the energy of a Bloch wave in a metal". Physica. XIII (6–7): 392–400. Bibcode:1947Phy....13..392K. doi:10.1016/0031-8914(47)90013-x. /wiki/Jan_Korringa ↩

  15. Faulkner, J. S.; Stocks, G. Malcolm; Wang, Yang (2018-12-01). Multiple Scattering Theory: Electronic structure of solids. IOP Publishing. doi:10.1088/2053-2563/aae7d8. ISBN 978-0-7503-1490-9. 978-0-7503-1490-9 ↩

  16. Papanikolaou, N; Zeller, R; Dederichs, P H (2002-03-25). "Conceptual improvements of the KKR method". Journal of Physics: Condensed Matter. 14 (11): 2799–2823. doi:10.1088/0953-8984/14/11/304. ISSN 0953-8984. https://iopscience.iop.org/article/10.1088/0953-8984/14/11/304 ↩

  17. Ebert, H; Ködderitzsch, D; Minár, J (2011-09-01). "Calculating condensed matter properties using the KKR-Green's function method—recent developments and applications". Reports on Progress in Physics. 74 (9): 096501. doi:10.1088/0034-4885/74/9/096501. ISSN 0034-4885. https://iopscience.iop.org/article/10.1088/0034-4885/74/9/096501 ↩

  18. Stocks, G. M.; Temmerman, W. M.; Gyorffy, B. L. (1978-07-31). "Complete Solution of the Korringa-Kohn-Rostoker Coherent-Potential-Approximation Equations: Cu-Ni Alloys". Physical Review Letters. 41 (5): 339–343. doi:10.1103/PhysRevLett.41.339. https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.41.339 ↩

  19. Stocks, G. M.; Temmerman, W. M.; Györffy, B. L. (1979), Phariseau, P.; Györffy, B. L.; Scheire, L. (eds.), "Aspects of the Numerical Solution of the KKR-CPA Equations", Electrons in Disordered Metals and at Metallic Surfaces, Boston, MA: Springer US, pp. 193–221, doi:10.1007/978-1-4684-3500-9_5, ISBN 978-1-4684-3500-9, retrieved 2024-09-21 978-1-4684-3500-9 ↩

  20. Robarts, Hannah C.; Millichamp, Thomas E.; Lagos, Daniel A.; Laverock, Jude; Billington, David; Duffy, Jonathan A.; O’Neill, Daniel; Giblin, Sean R.; Taylor, Jonathan W.; Kontrym-Sznajd, Grazyna; Samsel-Czekała, Małgorzata; Bei, Hongbin; Mu, Sai; Samolyuk, German D.; Stocks, G. Malcolm (2020-01-30). "Extreme Fermi Surface Smearing in a Maximally Disordered Concentrated Solid Solution". Physical Review Letters. 124 (4): 046402. arXiv:2001.11416. doi:10.1103/PhysRevLett.124.046402. https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.124.046402 ↩

  21. Staunton, J.; Gyorffy, B. L.; Pindor, A. J.; Stocks, G. M.; Winter, H. (1984-11-01). "The "disordered local moment" picture of itinerant magnetism at finite temperatures". Journal of Magnetism and Magnetic Materials. 45 (1): 15–22. doi:10.1016/0304-8853(84)90367-6. ISSN 0304-8853. https://linkinghub.elsevier.com/retrieve/pii/0304885384903676 ↩

  22. Pindor, A J; Staunton, J; Stocks, G M; Winter, H (1983). "Disordered local moment state of magnetic transition metals: a self-consistent KKR CPA calculation". Journal of Physics F: Metal Physics. 13 (5): 979–989. doi:10.1088/0305-4608/13/5/012. ISSN 0305-4608. https://iopscience.iop.org/article/10.1088/0305-4608/13/5/012 ↩

  23. Gyorffy, B L; Pindor, A J; Staunton, J; Stocks, G M; Winter, H (1983). "A first-principles theory of ferromagnetic phase transitions in metals". Journal of Physics F: Metal Physics. 15 (6): 1337–1386. doi:10.1088/0305-4608/15/6/018. ISSN 0305-4608. https://iopscience.iop.org/article/10.1088/0305-4608/15/6/018 ↩