principles of green’s function technique including o(n...
TRANSCRIPT
Principles of Green’s function technique including O(N) methods
Igor A. Abrikosov ([email protected] ) 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
