Embedding method for electronic
structure and transport in large molecules
O.R. Davies &
J.E. Inglesfield
School of Physics and Astronomy,
Cardiff University, Cardiff, Wales
Experimental and theoretical studies on DNA -
Porath, Bezryadin, de Vries, and
Dekker, Nature 403, 635 (2000)
Hjort and Stafström, Phys. Rev.
Letts. 87, 228101 (2001)
Embedding - a way of solving for region I joined on to II
• surface and semi-infinite substrate
• molecule and surface
• part of molecule joined on to rest
Replace region II by embedding potential on S - then solve
problem only in region I!
Region I
∂ψ (rS )
= ∫ drS ' Σ(rS , rS ' )ψ (rS ' )
∂nS
(J.E. Inglesfield, J. Phys. C 14, 3795 (1981), Comp. Phys. Commun. 137, 89 (2001))
Embedding potential Σ is the same as self-energy in -
(
Tlr ( E ) = 4Tr Glr ( E )ℑmΣ r ( E )G ( E )ℑmΣ l ( E )
*
rl
)
Embedding method gives accurate surface electronic
structure of surface of SEMI-INFINITE solid, e.g. Fowler-Nordheim plot for
field emission
30
Pt(001)
Pt(111)
Pd(001)
Pt6×(001)
25
20
15
10
5
ln(J/F2)
Stepped Pt(001) surface in electric
field
0
−5
−10
−15
(Ian Merrick and JEI, using programs of
Ishida)
−20
−25
−30
0
1
2
3
1/F ( A° /V)
4
5
6
Embedding in tight-binding
Hψ = Eψ
Write wave-function in terms of atomic orbitals -
ψ = ∑ aiϕ i
i
∑H
ij
a j = Eai
j
where hopping is given by
H ij = ∫ drϕ i (r ) Hϕ j (r )
Embedding potential in tight-binding
Σ = −δH 12G22δH 21
Growing long molecules by embedding
•Calculate Greens function to get embedding potential
•Add embedding potential to Hamiltonian of next
section
•Calculate new Green function
Density of states of DNA
Contour integration of Green functions for static properties
φ
π
DoS ∝ ∫ dφ (sin φℑmG − cos φℜeG )
0
Transmission and embedding:
⎡
⎤
∂ψ
∂G
χ ( r ) = 2i ∫ d rS ⎢G ( r , rS )
( rS ) −
( r , rS )ψ ( rS ) ⎥
∂n S
∂n S
⎣
⎦
2
= 2 i ∫ d 2 rS
∫
d 2 r ' S G ( r , rS ) ℑ m Σ ( rS , r ' S )ψ ( r ' S )
Total transmission between left and right
(
Tlr ( E ) = 4Tr Glr ( E )ℑmΣ r ( E )G ( E )ℑmΣ l ( E )
*
rl
(formula seems to have been first given by Levi Yeyati and
Büttiker, Phys. Rev. B52, 14 360 (1995)
)
DOS and transmission, C12 chain
DOS of C12 embedded onto contacts
OPE a highly conjugated
molecule
Charge density in transmission peak of OPE
I-V curve for OPE
Transmission from different
forms of DNA
X-ray structure
Energy-minimised poly(G)-poly(C)
Charge distribution for
different states
High transmission state
Low transmission state
Transmission at fixed energy for
various contact combinations
I-V for poly(G)-poly(C) DNA
Stretched DNA
1-hole stretched DNA
2-hole stretched DNA