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
TRANSCRIPT
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KKR Method
Ins$tute for Solid State Physics, The University of Tokyo
Hisazumi Akai
KKR Hands-On 2014
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Introduction
What does KKR do?
Condensed Matter Physics Computational Materials Design
Materials Science ... Quantum simulation
for many-body systmes
Density Functional Theory Hohenberg-Kohn theorem
Kohn-Sham equation Local Density Approximation (LDA)
KKR method ・・・method ・・・method ・・・method
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Kohn-Sham equations
Equations for containing N parameters
ϕ i(r)
(−∇2 + veff
)ϕi= ε
iϕ
i
veff
(r) = vext
(r) + d 3 %r2n( %r )r − %r
+ vxc∫
'
()
*)
Where
n(r) = ϕ i (r)
2
i=1
N
∑
(Sum over lowest N states)
(Kohn-Sham equations)
Note:
∇2 =
∂ 2
∂x2+∂ 2
∂ y2+∂ 2
∂ z2
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Band structure calculation
(−∇2 + veff )ϕ i (r) = ε iϕ i (r)
How to solve this partial differential equations (boundary value problem) efficiently?
One of the ways
KKR method Korringa-Kohn-Rostoker method (Green’s function method)
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KKR method • Sca>ering method • Sta$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
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Muffin-tin potential model Muffin-tin potential
Muffin Tin
Spherical potential
v(r)
r
Interstitial region
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Electron scattering
electrons
attractive potential
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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.
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Scattering t-matrix (transition matrix)
v vgv vgvgv
vgvgvgv t
+ +
+ + ... =
total scattering amplitude = sum of each scattering amplitude = t-matrix
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Expression of t-matrix
Formal expression
t = v + vgv + vgvgv +
= v 1+ gv( ) + gv( )2+ gv( )3 +{ }
= v 11− gv
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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
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Total scattering amplitude T
Formal expression
T = t + tgt + tgtgt +
= t 1+ gt( ) + gt( )2+ gt( )3 +{ }
= t 11− gt
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Scattering by a crystal
T
T describes scattering by a crystal.
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Stationary state of scattering
Electrons stay forever. No incident electron is needed.
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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.
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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)
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Energy dispersion relation
E(k
)
k
Energy eigenvalues for a given k
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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
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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
)
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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)
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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
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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 )
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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
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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)
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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
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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' ( )
* + ,
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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∑
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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)
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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
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Spectrum for complex energy
Set of δ functions Cannot be integrated
Can be integrated
Δ
E
E+iΔ
Set of Lorenz curves
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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.
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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$es Many-‐body problems
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In short What is the Green’s function method(KKR)?
1.Sum up all the scattering amplitudes =KKR method
2.Divergence of the amplitude gives eigen states. 3. The imaginary part of the probability amplitude
is proportional to the number of state at that energy (density of states).
4.Electron density is obtained from the density of states.
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Summary • Basic idea of KKR method • Green’s func$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$ve.jp/
Query “akaikkr” will hit the web-‐site. h>p://kkr.phys.sci.osala-‐u.ac.jp/
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Hands-on tutorial • KKR and KKR-‐CPA calcula$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
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Demonstration
• Run program on a laptop • Calcula$on of ferromagne$c Fe • Determina$on of the la`ce constant