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第5回CCMSハンズオン(ソフトウェア講習会): AkaiKKRチュートリアル 1. KKR法

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第5回CCMSハンズオン(ソフトウェア講習会): AkaiKKRチュートリアル 1. KKR法

  1. 1. KKR Method Ins$tute for Solid State Physics, The University of Tokyo Hisazumi Akai KKR Hands-On 2014
  2. 2. Introduction What does KKR do? Condensed Matter Physics Computational Materials Design Materials Science ... Quantum simulation for many-body systmes Density Functional Theory Hohenberg-Kohn theorem Kohn-Sham equation Local Density Approximation (LDA) KKR method ・・・method ・・・method・・ ・method
  3. 3. Kohn-Sham equations ϕ i(r) (−∇2 + veff )ϕ i = εi Equations for    containing N parameters ϕ i veff (r) = vext (r) + d3r% 2n(r%) r − r% + v∫ xc ' ( ) * ) Where NΣ n(r) = ϕ i (r) 2 i=1 (Sum over lowest N states) (Kohn-Sham equations) Note: ∇2 = ∂ 2 ∂ x2 + ∂ 2 ∂ y2 + ∂ 2 ∂ z2
  4. 4. Band structure calculation (−∇2 + veff )ϕ i (r) = ε iϕ i (r) How to solve this partial differential equations (boundary value problem) efficiently? One of the ways KKR method Korringa-Kohn-Rostoker method (Green’s function method)
  5. 5. KKR method • Sca>ering method • Sta$onary state of sca>ering → energy eigen state • Muffin-­‐$n poten$al model: prototype • Calcula$on of sca>ering → impurity sca>ering, sca>ering due to random poten$al are also dealt with. Incident electrons Scattered electrons Incident wave Scattered wave crystal
  6. 6. Muffin-tin potential model Muffin-tin potential Muffin Tin Spherical potential v(r) r Interstitial region
  7. 7. Electron scattering electrons attractive potential
  8. 8. Quantum mechanical scattering Single scattering v Double scattering v v g g: probability amplitude that the electron propagate freely V: probability amplitude that the electron is scattered once The electron is scattered once, twice,・・・・,n times are possible.
  9. 9. Scattering t-matrix (transition matrix) v vgv vgvgv + + vgvgvgv t + + ... = total scattering amplitude = sum of each scattering amplitude = t-matrix
  10. 10. Expression of t-matrix Formal expression t = v + vgv + vgvgv + = v 1+ (gv)+ (gv)2 { (gv)3 + +} = v 1 1− gv
  11. 11. Multiple scattering due to assembly of potentials t t t t t g t g t tgt tgtgt g t-matrix describes scattering due to each potential. Multiple scattering is successive scattering due to many potentials. The total scattering amplitude is the sum of the amplitudes of those processes
  12. 12. Total scattering amplitude T Formal expression T = t + tgt + tgtgt + { +} = t 1+ (gt) + (gt)2 + (gt)3 = t 1 1− gt
  13. 13. Scattering by a crystal T T describes scattering by a crystal.
  14. 14. Stationary state of scattering Electrons stay forever. No incident electron is needed.
  15. 15. Divergence of T (transition) matrix As long as T is finite, any incident electron will escape after all. Therefore it cannot be a stationary state. For a stationary state T is not finite. → diverges T = t 1 1− gt → ∞ det 1− gt = 0 T diverges. → The incident electron states will decay immediately.
  16. 16. Stationary state= energy eigenstate g is a function of E and k in a crystal  g=g(E,k) t is a function of E  t=t(E) det 1− gt = 0 determines E for given k Energy dispersion E = E(k) Traditional KKR band structure calculation
  17. 17. Energy dispersion relation Energy eigenvalues for a given k E(k) k
  18. 18. A bit different approach KKR-Green’s function method Instead of calcula$ng eigenvalue E(k) and the corresponding eigen states... Calcula$ng Green’s func$on of the system directly without knowing E(k).
  19. 19. Green’s function Linear partial differential equation To solve L: linear operator such as differentiation, Hamiltonian. Lf (r) = g(r) LG(r,r!) =δ (r − r!). f (r) = f0 (r) + ∫ dr!G(r,r!)g(r!), find a Green’s function The solution is expressed as G = where f0 is the solution of Lf0(r)=0. 1 L f = 1 L g = Gg
  20. 20. Check Put f (r) = f0 (r) + ∫ dr!G(r,r!)g(r!) into Lf (r) = g(r) : Lf (r) = Lf0 (r) + ∫ dr!LG(r,r!)g(r!) = ∫ dr!δ (r − r!)g(r!) = g(r) Definition of δ function certainly satisfies Lf (r) = g(r). f (r)
  21. 21. An example of Green’s functions Electro static field −∇2V = ρ (r) ε0 Poisson equation Corresponding Green’s function −∇2G(r,r#) = 1 ε 0 δ (r − r#) The solution is expressed as G(r,r!) = 1 4πε 0 1 r − r! Coulomb’s law
  22. 22. KKR Green’s function Kohn-Sham equation (εi − H)ϕ i = 0 Corresponding Green’s function (z − H)G(r,r; z) =δ (r − r)
  23. 23. Green’s function of electrons in a crystal G = GS + GB GS : Green’s function for a single potential GB = g + gtg + gtgtg + = g + gTg ・・・ Multiple scattering GS T GS, g is calculated, T is calculated using KKR.
  24. 24. Once Green’s function is known (z − H)G(r,r; z) =δ (r − r) Expand G into eigen states of Kohn-Sham equation k Σ G(r,r!; z) = Ck (r!) ϕ k (r) Put (2) into (1). k Σ (z − H) Ck (r) ϕ k (r) k Σ = Ck (r) (z − ε k )ϕ k (r) =δ (r − r) (1) (2)
  25. 25. Coefficients Ck(r) Multiply both side with ϕk *(r) and integrate Ck! (r!) ! k Σ (z − ε k! )∫ drϕk * (r)ϕ k! (r) = ∫ drϕk * (r)δ (r − r!) using the orthogonality relation, we obtain Ck (r) = *(r) z −εk ϕ k
  26. 26. Expansion of Green’s function Using the expansion coefficients G(r,r!; z) = Important relation ϕ k (r)ϕ k * (r!) z − ε k k Σ 1 E + iδ = P.P. 1 E # $ % ' − iπδ (E) P.P. principal part P.P. 1E ∞ ∫ dE ' ∞ ∫ dE = lim -∞ ε →0 1E 1E ∫ −ε dE + -∞ ε ( ) * + ,
  27. 27. Spectrum representation of G Using this relation z − ε k k Σ G(r,r!;ε + iδ ) = P.P. ϕ k (r)ϕ k * (r!) k Σ −iπ ϕ k (r)ϕ k * (r!)δ (ε −ε k ) Setting r=r’ gives the density of states. ρ(r,ε ) = ϕ k (r) 2 δ (ε − ε k ) k Σ
  28. 28. Density of states ρ (r,ε ) = − 1 π ℑG(r,r;ε + iδ ) Electron density is thus expressed as ρ (r) = − 1 π ε F ∫ G(r,r;ε + iδ ) ℑ dε −∞ = − 1 π ε F +iδ ∫ G(r,r; z) ℑ dz −∞
  29. 29. Consider complex energy ρ (r) = − 1 π ε F +iδ ∫ G(r,r; z) ℑ dz −∞ ℑz εL εF Why complex integral? sum of delta function → sum of Lorentzian      Numerical integration becomes possible ℜz Consider complex contour integral
  30. 30. Spectrum for complex energy Set of δ functions Cannot be integrated Can be integrated Δ E E+iΔ Set of Lorenz curves
  31. 31. All we need is Green’s function Density functional method determines density. Other quantities are of secondary importance in this context. Energy integral of Green’s function e.g. the energy eigenvalues of Kohn-Sham equations and the density of states are not physical observables. Not necessary to solve eigenvalue problems
  32. 32. What can KKR do? Anything that normal band structure calcula$on do. In addi$on High speed High accuracy Scaering problem Systems with defects Impurity problem Disordered systems Par$al disorder Problems that require Green’s func$on. Transport proper$es Many-­‐body problems
  33. 33. In short What is the Green’s function method(KKR)? 1.Sum up all the scattering amplitudes =KKR method 2.Divergence of the amplitude gives eigen states. 3. The imaginary part of the probability amplitude is proportional to the number of state at that energy (density of states). 4.Electron density is obtained from the density of states.
  34. 34. Summary • Basic idea of KKR method • Green’s func$on method • Applica$on of KKR method Program package for KKR cpa2002v009c (AkaiKKR) (MACHIKANEYAMA2000) has been developed. Catalogues in MateriApps hp://ma.cms-­‐ini$a$ve.jp/ Query “akaikkr” will hit the web-­‐site. hp://kkr.phys.sci.osala-­‐u.ac.jp/
  35. 35. Hands-on tutorial • KKR and KKR-­‐CPA calcula$on – Pure Fe – Curie temperature of Fe and Co – Fe-­‐Ni random alloys – Impurity systems • Applica$ons – Half-­‐metallic Heusler alloys – Li-­‐ion baery – Hydrogen storage MgH2 – Heat of forma$on of alloys
  36. 36. Demonstration • Run program on a laptop • Calcula$on of ferromagne$c Fe • Determina$on of the la`ce constant

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