principles of green’s function technique including o(n...

Principles of Green’s function technique including O(N) methods

Igor A. Abrikosov (igor.abrikosov@ifm.liu.se ) Theoretical Physics, Department of Physics, Chemistry, and Biology (IFM), Linköping University, Sweden

Contents :

• Green’s function or multiple scattering formalism.

• Coherent potential approximation (CPA). • Beyond the single-site CPA: O(N) LSGF

method

2

2me

2 VKS (x1,R1,R2,...,RN I)

i(x1,R1,R2,...,RN I

) ii(x1,R1,R2,...,RN I

LDA, GGA, etc.

VASP, Wien2k,CASTEP,ABINIT, KKR, etc.

Supercell, CPA, etc.

For the list of codes see Appendix in P. E. A. Turchi, I. A. Abrikosov, B. Burton, S. G. Fries, G. Grimvall, L. Kaufman, P. A. Korzhavyi, V. Rao Manga, M. Ohno, A. Pisch, A. Scott, and W. Zhang, CALPHAD 31, 4 (2007).

in

sR’

out

Atomic-like wave function

Free-electron-like wave function

Green’s functions

Green’s functions

KKR

Rl

RL[2+2] RL (2,rR )=0

VMTZ =0

V(r)=V(|r|)

),()()1()],([22

2

RjRlRjRRRR

RjRlR rrrvrll

rrr

[m(z)-B(k,z)]

2

[ ]-1

LMTO

Rl

RL2 RL (rR )=0

),()()1()],([22

2

RjRlRjRRRR

RjRlR rrrvrll

rrr

EMTO

RL[2+2] RL (2,rR )=0

Rl

Rl

),()()1()],([22

2

RjRlRjRRRR

RjRlR rrrvrll

rrr

EMTO

0)]()([

)(

,''2

'''

,''

a

jRLjaRLLLRRj

RL

aRLLRR

ajRLj

RL

aRLLR

vDSa

vK

LLRRa

RLLRLR

aLRLR kzgkzK ''''''

'''''''''' ),(),(

dzzGi

NF

F )(21)(

RLDRl

aRl

aRl

aLRLR

RLLR BZ

aRLLR

DRl

zzDzD

kdkzKkzgzG

1

)()(

),(),()( ''''

''

EMTO-SCA

)()( rnrn )()( rvrv

EMTO-FCD

)(rn )(rv

[m(z)-B(k,z)]

2

[ ]-1

)~ );g~ (Em(E

)(Ecg A + )()1( Egc B)(~ Eg =

cc BA 1

A B

A A B B B

B B A B A

A B A B B

A B A A

A B B A A

m~ m~ m~ m~ m~

m~ m~ m~ m~ m~

m~ m~ m~ m~ m~

m~ m~ m~ m~ m~

m~ m~ m~ m~ m~

Coherent Potential Approximation (CPA)

(E)g(EmEm(E)g

(E)g BABA ~

)]~)([~11

)()(

kdE)kB((E)mV

(E)gULBZ

ULBZ

31

,~1~

)~ );~2211

~~~~ (g(g UUUU

A C

1~U2

~U

A D D C

B A B C B

C B A D A

D A B C D

C D C B A

C

C

)(1

~1

~1 AA UUgx + )(

1~

1~

1 CC UU

gx)~11

~~ (g UU =

11 )()( 2211DBCA xxxx

DBCA

)~ );g~ (Em(E

)(Ecg A + )()1( Egc B)(~ Eg =

cc BA 1

A B

A A B B B

B B A B A

A B A B B

A B A A

A B B A A

m~ m~ m~ m~ m~

m~ m~ m~ m~ m~

m~ m~ m~ m~ m~

m~ m~ m~ m~ m~

m~ m~ m~ m~ m~

Coherent Potential Approximation (CPA)

(E)g(EmEm(E)g

(E)g BABA ~

)]~)([~11

)()(

kdE)kB((E)mV

(E)gULBZ

ULBZ

31

,~1~

B. Alling, A. V. Ruban, A. Karimi, O.Peil, L. Hultman, and I. A. Abrikosov, Phys. Rev. B 75, 045123 (2007)

Beyond CPA: requirements

• To account for fluctuations in the local environment in a self-consistent way

• To become exact in the limit of large cluster size• To recover the CPA for a single-site cluster• To be relatively easy to implement numerically• To preserve the translational and point group

symmetries of the underlying crystal lattice• To preserve analytical properties of the single-

particle Green’s function.

Locally Self-consistent Green’s Functon method (LSGF)

(E)jigEjm(Ejm(E)ijgLIZj

(E)iig(E)iig )()~~~

(E)jlgEjm(Ejm(E)ljgLIZj

(E)llg(E)llg )()~~~

Locally Self-consistent Green’s Function method (LSGF)

kd)kS((E)mV

(E)gULBZ

ULBZ

31~1~

SCi(E)igEim(Em(E)g(E)g(E)ig , )()~~~

)(1)~ EigSCiN

(EgSC

SCi(E)jigEjm(Ejm(E)ijgLIZj

(E)iig(E)iig

, )()~~~

fcc Pd75 V25