KKR Method

Ins$tute  for  Solid  State  Physics,    The  University  of  Tokyo  

Hisazumi  Akai  

KKR Hands-On 2014

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

Kohn-Sham equations

Equations for　　　 containing N parameters

ϕ i(r)

(−∇2 + veff

)ϕi= ε

iϕ

i

veff

(r) = vext

(r) + d 3 %r2n( %r )r − %r

+ vxc∫

'

()

*)

Where

n(r) = ϕ i (r)

2

i=1

N

∑

（Sum over lowest N states）

（Kohn-Sham equations）

Note:

∇2 =

∂ 2

∂x2+∂ 2

∂ y2+∂ 2

∂ z2

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)

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

Muffin-tin potential model Muffin-tin potential

Muffin Tin

Spherical potential

v(r)

r

Interstitial region

Electron scattering

electrons

attractive potential

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.

Scattering t-matrix (transition matrix)

v vgv vgvgv

vgvgvgv t

+ +

+ + ... =

total scattering amplitude ＝ sum of each scattering amplitude ＝ t-matrix

Expression of t-matrix

Formal expression

t = v + vgv + vgvgv +

= v 1+ gv( ) + gv( )2+ gv( )3 +{ }

= v 11− gv

Multiple scattering due to assembly of potentials

t t

t t

t

g

t

g

t

tgtgt

tgt

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

Total scattering amplitude T

Formal expression

T = t + tgt + tgtgt +

= t 1+ gt( ) + gt( )2+ gt( )3 +{ }

= t 11− gt

Scattering by a crystal

T

T describes scattering by a crystal.

Stationary state of scattering

Electrons stay forever. No incident electron is needed.

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.

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)

determines E for given k

Energy dispersion

Traditional KKR band structure calculation

det 1− gt = 0

E = E(k)

Energy dispersion relation

E(k

)

k

Energy eigenvalues for a given k

A bit different approach

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).

KKR-Green’s function method

Green’s function

Linear partial differential equation

Lf (r) = g(r)

LG(r, !r ) = δ (r − !r ).

f (r) = f0 (r) + d !r G(r, !r )g( !r )∫ ,

To solve

find a Green’s function

where f0 is the solution of Lf0(r)=0.

L: linear operator such as differentiation, Hamiltonian.

The solution is expressed as

(G =

1L

)

(f =

1L

g = Gg

)

Check f (r) = f0 (r) + d !r G(r, !r )g( !r )∫Put

Lf (r) = g(r) :into

Lf (r) = Lf0 (r)+ d !r LG(r, !r )g( !r )∫= d !r δ (r − !r )g( !r ) = g(r)∫

certainly satisfies Lf (r) = g(r).

δDefinition of function

f (r)

An example of Green’s functions

Electro static field

−∇2V =ρ(r)ε0

−∇2G(r, #r ) = 1

ε0

δ (r − #r )

G(r, !r ) = 1

4πε0

1r − !r

Corresponding Green’s function

The solution is expressed as

Poisson equation

Coulomb’s law

KKR Green’s function

Corresponding Green’s function

Kohn-Sham equation

(εi −H)ϕ i = 0

(z − H )G(r, "r ; z) = δ (r − "r )

Green’s function of electrons in a crystal

G =GS +GB

GS, g is calculated, T is calculated using KKR.

GB = g + gtg+ gtgtg+

= g + gTg

GS : Green’s function for a single potential

・・・

Multiple scattering

GS

T

Once Green’s function is known

(z − H )G(r, "r ; z) = δ (r − "r )

G(r, !r ; z) = Ck ( !r )

k∑ ϕ k (r)

Expand G into eigen states of Kohn-Sham equation

Put (2) into (1).

(z − H ) Ck ( "r )k∑ ϕ k (r)

= Ck ( "r )k∑ (z − εk )ϕ k (r) = δ (r − "r )

(1)

(2)

Coefficients Ck(r)

Multiply both side with ϕk*(r) and integrate

using the orthogonality relation, we obtain

C!k ( !r )

!k∑ (z − ε

!k ) d r∫ ϕ k* (r)ϕ

!k (r)

= d r∫ ϕ k* (r)δ (r − !r )

Ck (r) =ϕk*(r)z −εk

Expansion of Green’s function

Using the expansion coefficients

Important relation

G(r, !r ; z) =

ϕ k (r)ϕ k* ( !r )

z − εkk∑

1E + iδ

= P.P. 1E

"

#$%

&'− iπδ (E)

P.P. principal part

P.P. 1E-∞

∞

∫ dE = limε→0

1E-∞

−ε

∫ dE +1Eε

∞

∫ dE' ( )

* + ,

Spectrum representation of G

Using this relation

Setting r=r’ gives the density of states.

G(r, !r ;ε + iδ ) = P.P.ϕ k (r)ϕ k

* ( !r )z − εkk

∑

−iπ ϕ k (r)ϕ k* ( !r )δ (ε − εk )

k∑

ρ(r,ε) = ϕk (r)

2δ (ε − εk )

k∑

Density of states

Electron density is thus expressed as ρ(r,ε) = − 1

πℑG(r,r;ε + iδ )

ρ(r) = − 1πℑ dε

−∞

εF

∫ G(r,r;ε + iδ )

= −1πℑ dz

−∞

εF +iδ

∫ G(r,r; z)

Consider complex energy

Why complex integral? sum of delta function → sum of Lorentzian 　　　　　Numerical integration becomes possible

ρ(r) = − 1

πℑ dz

−∞

εF +iδ

∫ G(r,r; z)

εL εF

ℑz

ℜz

Consider complex contour integral

Spectrum for complex energy

Set of δ functions Cannot be integrated

Can be integrated

Δ

E

E+iΔ

Set of Lorenz curves

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

Not necessary to solve eigenvalue problems

e.g. the energy eigenvalues of Kohn-Sham equations and the density of states are not physical observables.

What can KKR do? Anything  that  normal  band  structure  calcula$on  do.  

In  addi$on  High  speed  High  accuracy  Sca>ering  problem  

Systems  with  defects    Impurity  problem    Disordered  systems    Par$al  disorder  

Problems  that  require  Green’s  func$on.  Transport  proper$es  Many-­‐body  problems  

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.

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                          h>p://ma.cms-­‐ini$a$ve.jp/  

 Query  “akaikkr”  will  hit  the  web-­‐site.                  h>p://kkr.phys.sci.osala-­‐u.ac.jp/  

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  ba>ery  –  Hydrogen  storage  MgH2  

–  Heat  of  forma$on  of  alloys  

Demonstration

•  Run  program  on  a  laptop  •  Calcula$on  of  ferromagne$c  Fe  •  Determina$on  of  the  la`ce  constant