1. KKR Method
Ins$tute
for
Solid
State
Physics,
The
University
of
Tokyo
Hisazumi
Akai
KKR Hands-On 2014

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. 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. 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. 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. Muffin-tin potential model
Muffin-tin potential
Muffin Tin
Spherical potential
v(r)
r
Interstitial region

7. Electron scattering
electrons
attractive potential

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. Scattering t-matrix (transition matrix)
v vgv vgvgv
+ +
vgvgvgv t
+ + ... =
total scattering amplitude ＝ sum of each scattering amplitude ＝ t-matrix

10. Expression of t-matrix
Formal expression
t = v + vgv + vgvgv +
= v 1+ (gv)+ (gv)2
{ (gv)3
+ +}
= v 1
1− gv

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. Total scattering amplitude T
Formal expression
T = t + tgt + tgtgt +
{ +}
= t 1+ (gt) + (gt)2
+ (gt)3
= t 1
1− gt

13. Scattering by a crystal
T
T describes scattering by a crystal.

14. Stationary state of scattering
Electrons stay forever. No incident electron is needed.

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. 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. Energy dispersion relation
Energy eigenvalues for a given k
E(k)
k

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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Spectrum for complex energy
Set of δ functions Cannot be integrated
Can be integrated
Δ
E
E+iΔ
Set of Lorenz curves

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