The Non-Equilibrium Green Function (NEGF) method

The Non-Equilibrium Green Function (NEGF) method was established in the 1960's through the classic work of Schwinger, Kadanoff, Baym, Keldysh and others.

1. Structure of OPENMX extension: NEGF
Yang XIAO
Nanjing University of Aeronautics and
Astronautics, China

2. The configuration under consideration
(1)L,C,R regions
(2)C is extended.

3. Kohn-Sham Hamiltonian
Scatter’s Green’s function
Non-equilibrium density matrix
Transmission coefficient
Lead’s self-energies
surface Green function
(
)
(
)
L CL CL L LC LC
ES H g ES H Σ = − −
L L L
i
+
 
Γ = Σ −Σ
 
L
g
=
[
]
ˆˆ
i eff i i i
H n T V
ψ ψ εψ
  = + =
 
[ ]
1
R
L R
G ES H n
−
 
= − −Σ −Σ
 
neq eq
ρ ρ ρ
S gd
( )
( )
0
2
k k
i
dEG E i f E
σ σ k r
π
+∞
+
±
−∞
= ± −
∫
( ) ( ) ( ) ( )
,
1
2
k k k k
s
dEG E i E G E i f E
σ σ σ σ
ρ ε ε
π
+∞
−∞
d S g f H d
∫
(
)
(
)
(
)
(
)
, ,
k k k k k
L R
T Tr E G E i E G E i
σ σ σ σ σ
ε ε
 
= Γ + Γ −
 
( )
33
1 1
nn
ik R ik R
k k k
BZBZ
cc
dk e dk e
V V
σ σ σ
ρ ρ ρ
− ⋅ − ⋅
+ −
S H g d∫ ∫
The transport
algorithm in
OPENMX

4. Kohn-Sham Hamiltonian
Scattering’s Green’s function
Non-equilibrium density matrix
Transmission coefficient
Lead’s self-energies
surface Green function
with H0 and H1 of lead (
)
(
)
L CL CL L LC LC
ES H g ES H Σ = − −
L L L
i
+
 
Γ = Σ −Σ
 
L
g
=
[
]
ˆˆ
i eff i i i
H n T V
ψ ψ εψ
  = + =
 
[ ]
1
L R
G ES H n
−
 
= − −Σ −Σ
 
neq eq
ρ ρ ρ
S gd
( )
( )
0
2
k k
i
dEG E i f E
σ σ k r
π
+∞
+
±
−∞
= ± −
∫
( ) ( ) ( ) ( )
,
1
2
k k k k
s
dEG E i E G E i f E
σ σ σ σ
ρ ε ε
π
+∞
−∞
d S g f H d
∫
(
)
(
)
(
)
(
)
, ,
k k k k k
L R
T Tr E G E i E G E i
σ σ σ σ σ
ε ε
 
= Γ + Γ −
 
( )
33
1 1
nn
ik R ik R
k k k
BZBZ
cc
dk e dk e
V V
σ σ σ
ρ ρ ρ
− ⋅ − ⋅
+ −
S H g d∫ ∫
Step I Step IIIStep II DFT+NEGF
L
i+1
L
i
L
i-1

5. Main structure of transport module
double DFT(……)
{
…
do {
Set_Aden_Grid (…);
Set_Hamiltonian (…,
H, …);
TRAN_DFT (…, H,
S, …, DM, …);
Checking self-
consistency;
} while (self-consistency
)
…
}
DFT.c
TRAN_DFT.c
double TRAN_DFT
(……)
{
…
for
(k=0;k<=n_k;k++) {
TRAN_DFT_Kdepende
nt(…k, … CDM,…);
}
Summing DM for all k-
points;
…
}
double
DFT_Kdependent (…)
{
…
for(i=0;i<n_i;i++) {
E(i)=tran_omega_scf(i);
G
R
, G
<
and ρ;
}
…}
3
k
BZ
dk
ρ ρ
=
∫
( )
k k
dEG E
σ
ρ
+∞
−∞
=
∫

6. Kohn-Sham Hamiltonian [
]
ˆˆ
i eff i i i
H n T V
ψ ψ εψ
  = + =
 
DFT module
Main difference
(
)
(
)
( )
( )
,
,
r r r r
L L R R
H H
C L bulk
z z
H H
C R bulk
z z
V V
V V
=
=
DFT.c

7. DFT.c
Kohn-Sham Hamiltonian [
]
ˆˆ
i eff i i i
H n T V
ψ ψ εψ
  = + =
 
n
i
V
eff
H
ρ
n
i+1
if (abs[n
ikC
i+1]<δ) {stop}
else {charge mixing; iterate}
DFT module

8. NEGF calculation
Band calculation
The first 3 steps are not NEGF cal.
NEGF module
TRAN_DFT.c
3

9. NEGF module
subroutine
( )
33
1 1
nn
ik R ik R
neq eq k k k
BZBZ
cc
dk e dk e
V V
σ σ σ
ρ ρ ρ ρ ρ ρ
− ⋅ − ⋅
+ −
S gd S H g d
∫ ∫
TRAN_DFT.c

10. NEGF module
subroutine
TRAN_DFT.c
( )
( )
0
2
k k
i
dEG E i f E
σ σ k r
π
+∞
+
±
−∞
= ± −
∫
(1) Equilibrium part
( )
,0
1 1
4
p
N
k k
p p
p
G R
σ σ
r l
β
=±
∑
m
(2) Nonequilibrium part
T.Ozaki, PRB 75, 035123 (2007)
( ) ( ) ( ) ( )
,
1
2
k k k k
s
dEG E i E G E i f E
σ σ σ σ
ρ ε ε
π
+∞
−∞
d S g f H d
∫
(
)
(
)
(
)
1 2
f E f E f E
r r
d S H H H

11. NEGF module
subroutine
Green function and self energy
[
]
1
CC CC L R
G ES H
−
= − −Σ −Σ
(
)
(
)
R CR CR L RC RC
ES H g ES H Σ = − −
Hamiltonian and overlap matrix
TRAN_DFT.c
Central region

12. NEGF module
subroutine
Retarded terms
[
]
1
CC CC L R
G ES H
−
= − −Σ −Σ
TRAN_DFT.c
(
)
(
)
L CR CR L RC RC
ES H g ES H Σ = − −
surface Green functiong
L
M.P.L.Sancho, JPF 15,851 (1985)
L
0
L
1
L
4
L
3
L
2

13. NEGF module
subroutine
Advanced terms
[
]
1
CC CC L R
G ES H
−
= − −Σ −Σ
TRAN_DFT.c
0:
k
=
0:
k
≠
(
)
*
A R
G G=
(
)
A
G G E i
ε
= −

14. NEGF module
subroutine
TRAN_DFT.c
(1) Equilibrium part
( )
,0
1 1
4
p
N
k k k
p p
p
G R
σ σ σ
k r l
β
±
=±
∑
m
(2) Nonequilibrium part
( ) ( ) ( ) ( )
,
1
2
k k k k
s
dEG E i E G E i f E
σ σ σ σ
ρ ε ε
π
+∞
−∞
d S g f H d
∫
( )
33
1 1
nn
ik R ik R
k k k
BZBZ
cc
dk e dk e
V V
σ σ σ
ρ ρ ρ ρ
− ⋅ − ⋅
+ −
S H g d
∫ ∫

15. NEGF module
subroutine
TRAN_DFT.c
( )
33
1 1
nn
ik R ik R
k k k
BZBZ
cc
dk e dk e
V V
σ σ σ
ρ ρ ρ ρ
− ⋅ − ⋅
+ −
S H g d
∫ ∫ ρ
Return the new density matrix to DFT module
ρ

16. Thank you！！！！