Coherent potential approximation explained

The coherent potential approximation (CPA) is a method, in theoretical physics, of finding the averaged Green's function of an inhomogeneous (or disordered) system. The Green's function obtained via the CPA then describes an effective medium whose scattering properties represent the averaged scattering properties of the disordered system being approximated. It is often described as the 'best' single-site theory for obtaining the averaged Green's function.^[1] It is perhaps most famous for its use in describing the physical properties of alloys and disordered magnetic systems,^{[2] [3]} although it is also a useful concept in understanding how sound waves scatter in a material which displays spatial inhomogeneity. The coherent potential approximation was first described by Paul Soven,^[4] and its application in the context of calculations of the electronic structure of materials was pioneered by Balász Győrffy.^[5]

Electronic Structure (KKR-CPA)

In the context of calculations of the electronic structure of materials, the coherent potential approximation is frequently combined with the Korringa–Kohn–Rostoker (KKR) formulation of density functional theory (DFT) to describe the electronic structure of systems with lattice-based disorder, such as substitutional alloys and magnetic materials at finite temperature.^{[6] [7]} The KKR formulation of DFT is also sometimes referred to as multiple scattering theory (MST). When the KKR formulation of DFT is combined with the CPA, it is sometimes referred to as the KKR-CPA.

The KKR formulation of DFT rephrases the usual eigenvalue-eigenvector problem (i.e. solving some effective Schrödinger equation) into an electronic scattering problem.^[8] It does so by partitioning the one-electron potential of DFT into a collection of spatially-localised potentials around each ionic site, before considering an electron propagating through the system and scattering from these localised potentials. In this manner, the Green's function of the system is obtained. In a system where there is lattice-based disorder (for example, in a substitutional alloy) the CPA provides a means by which to average multiple potentials associated with a single lattice site and obtain an average Green's function (and consequent electron density) in a physically meaningful way. Although the approach was originally formulated for potentials described within either the muffin tin or atomic sphere approximations (where the spatially localised potential is assumed spherically symmetric) it is now commonplace to use so-called full-potential calculations,^[9] where the one-electron potential can have arbitrary spatial dependence.

The KKR-CPA has been used with success to study the physics of a variety of alloy systems,^[10]

Notes and References

1. Yonezawa . Fumiko . Morigaki . Kazuo . 1973 . Coherent Potential Approximation: Basic concepts and applications . Progress of Theoretical Physics Supplement . en . 53 . 1–76 . 10.1143/PTPS.53.1 . 0375-9687.
2. Soven . Paul . 1967-04-15 . Coherent-Potential Model of Substitutional Disordered Alloys . Physical Review . 156 . 3 . 809–813 . 10.1103/PhysRev.156.809.
3. Gyorffy . B. L. . 1972-03-15 . Coherent-Potential Approximation for a Nonoverlapping-Muffin-Tin-Potential Model of Random Substitutional Alloys . Physical Review B . 5 . 6 . 2382–2384 . 10.1103/PhysRevB.5.2382.
4. Soven . Paul . 1967-04-15 . Coherent-Potential Model of Substitutional Disordered Alloys . Physical Review . 156 . 3 . 809–813 . 10.1103/PhysRev.156.809.
5. Gyorffy . B. L. . 1972-03-15 . Coherent-Potential Approximation for a Nonoverlapping-Muffin-Tin-Potential Model of Random Substitutional Alloys . Physical Review B . 5 . 6 . 2382–2384 . 10.1103/PhysRevB.5.2382.
6. 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 . 10.1088/0034-4885/74/9/096501 . 0034-4885.
7. Book: Faulkner, Stocks, Wang . Multiple Scattering Theory: Electronic structure of solids . December 2018 . . 978-0-7503-1490-9 . Bristol, UK . 10.1088/2053-2563/aae7d8.
8. Book: Faulkner, J S . Multiple Scattering Theory . Stocks . G Malcolm . Wang . Yang . 2018-12-01 . IOP Publishing . 978-0-7503-1490-9 . 10.1088/2053-2563/aae7d8.
9. Asato . M. . Settels . A. . Hoshino . T. . Asada . T. . Blügel . S. . Zeller . R. . Dederichs . P. H. . 1999-08-15 . Full-potential KKR calculations for metals and semiconductors . Physical Review B . 60 . 8 . 5202–5210 . 10.1103/PhysRevB.60.5202.
10. Johnson . D. D. . Nicholson . D. M. . Pinski . F. J. . Gyorffy . B. L. . Stocks . G. M. . 1986-05-12 . Density-Functional Theory for Random Alloys: Total Energy within the Coherent-Potential Approximation . Physical Review Letters . 56 . 19 . 2088–2091 . 10.1103/PhysRevLett.56.2088.
11. Stocks . G. M. . Butler . W. H. . 1982-01-04 . Mass and Lifetime Enhancement due to Disorder on $__$ Alloys . Physical Review Letters . 48 . 1 . 55–58 . 10.1103/PhysRevLett.48.55.
12. Gyorffy . B. L. . Stocks . G. M. . 1983-01-31 . Concentration Waves and Fermi Surfaces in Random Metallic Alloys . Physical Review Letters . 50 . 5 . 374–377 . 10.1103/PhysRevLett.50.374.
13. Pindor . A J . Temmerman . W M . Gyorffy . B L . March 1983 . KKR CPA for two atoms per unit cell: application to Pd and PdAg hydrides . Journal of Physics F: Metal Physics . 13 . 8 . 1627–1644 . 10.1088/0305-4608/13/8/009 . 0305-4608.
14. Long . N H . Ogura . M . Akai . H . 2009-02-11 . New type of half-metallic antiferromagnet: transition metal pnictides . Journal of Physics: Condensed Matter . 21 . 6 . 064241 . 10.1088/0953-8984/21/6/064241 . 0953-8984.
15. Pindor . A J . Staunton . J . Stocks . G M . Winter . H . May 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 . 10.1088/0305-4608/13/5/012 . 0305-4608.
16. Staunton . J. . Gyorffy . B.L. . Pindor . A.J. . Stocks . G.M. . Winter . H. . November 1984 . The “disordered local moment” picture of itinerant magnetism at finite temperatures . Journal of Magnetism and Magnetic Materials . 45 . 1 . 15–22 . 10.1016/0304-8853(84)90367-6 . 0304-8853.
17. Gyorffy . B L . Pindor . A J . Staunton . J . Stocks . G M . Winter . H . June 1985 . A first-principles theory of ferromagnetic phase transitions in metals . Journal of Physics F: Metal Physics . 15 . 6 . 1337–1386 . 10.1088/0305-4608/15/6/018 . 0305-4608.
18. Gonis . Antonios . Butler . W. H. . Stocks . G. M. . 1983-05-09 . First-Principles Calculations of Cluster Densities of States and Short-Range Order in $__$ Alloys |url=https://link.aps.org/doi/10.1103/PhysRevLett.50.1482 |journal=Physical Review Letters |volume=50 |issue=19 |pages=1482–1485 |doi=10.1103/PhysRevLett.50.1482}}^{[10] [11] [12]} including those where disorder is only present on one sub-lattice^{[13] [14]} (the 'inhomogeneous' CPA). In addition, it has been shown that the CPA can very effectively describe magnetism at finite temperature, by considering (weighted) averages taken over all possible spin orientations. This is referred to as the 'disordered local moment' (DLM) picture^{[15] [16]} and can be used to describe the ferromagnetic phase transition in metals.^[17]

    Further reading

    • Book: Ping Sheng . 1995 . Introduction to Wave Scattering, Localization, and Mesoscopic Phenomena . . 978-0-12-639845-8.
    • Fumiko Yonezawa and Kazuo Morigaki . . Coherent Potential Approximation: Basic Concepts and Applications . 53 . 1–76 . 1973 . 10.1143/PTPS.53.1 . 1973PThPS..53....1Y . free .
    • John R. Klauder . The modification of electron energy levels by impurity atoms . . 14 . 43–76 . 1961 . 10.1016/0003-4916(61)90051-3 . 1961AnPhy..14...43K .

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