ncn inac negf method: capabilities and challenges molecular electronics cnt electronics molecular...
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NCN
INAC
NEGF Method:Capabilities and Challenges
Molecular Electronics
CNT ElectronicsMolecular Sensor M
VG VD
CHANNEL
INSULATOR
DRAIN
SOURCE
I
Supriyo DattaSchool of Electrical &
Computer EngineeringPurdue University
Gate
NCN
INAC
Nanodevices: A Unified View
Molecular Electronics
CNT ElectronicsMolecular Sensor M
VG VD
CHANNEL
INSULATOR
DRAIN
SOURCE
I
H + U
‘s’
€
Σ1
€
Σ2
€
μ1
€
μ2
€
Σs
Unified Model
Gate
NCN
INAC
Hamiltonian, [H]
VG VD
CHANNEL
INSULATOR
DRAIN
SOURCE
I
Effective Mass Equation
2t2t 2t-t -t -t -t
Finite Difference / Finite Element
Damle, Ren, Venugopal, Lundstrom ---> nanoMOS
H + U
‘s’
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Σ1
€
Σ2
€
μ1
€
μ2
€
Σs
€
t ≡ h2 /2ma2
NCN
INAC
Hamiltonian, [H]
VG VD
CHANNEL
INSULATOR
DRAIN
SOURCE
I
Atomistic sp3d basis
€
α
€
α
€
α
€
β
€
β
€
β
€
β, are matrices
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β
€
α
Nanowire Electronics
Rahman, Wang, Ghosh, Klimeck, Lundstrom
H + U
‘s’
€
Σ1
€
Σ2
€
μ1
€
μ2
€
Σs
NCN
INAC
Hamiltonian, [H]
Atomistic pz basis
€
α
€
α
€
α
€
β
€
β
€
β
€
β, are (2x2) matrices
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β
€
α
Gate
CNT Electronics
Guo, Lundstrom
H + U
‘s’
€
Σ1
€
Σ2
€
μ1
€
μ2
€
Σs
NCN
INAC
Hamiltonian, [H]
Atomistic
non-orthogonal basis
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α
€
α
€
α
€
β
€
β
€
β
€
β
Nanowire/CNT Electronics
Siddiqui, Kienle, Ghosh, Klimeck
Gate
EHT
H + U
‘s’
€
Σ1
€
Σ2
€
μ1
€
μ2
€
Σs
NCN
INAC
Hamiltonian, [H]
Atomistic basis:
Molecular Electronics
Ghosh, Rakshit,Liang, Zahid,Siddiqui, Golizadeh, Bevan, Kazmi
Huckel / EHT / Gaussian
H + U
‘s’
€
Σ1
€
Σ2
€
μ1
€
μ2
€
Σs
NCN
INAC
“Self-energy”,
€
Σ
€
Ε−ε1 − t
− t E −ε2
⎡
⎣ ⎢
⎤
⎦ ⎥
gate
€
ε2
€
t
€
ε1 H + U
‘s’
€
Σ1
€
Σ2
€
μ1
€
μ2
€
Σs
NCN
INAC
“Self-energy”,
€
Σ
€
Ε−ε1 − t
− t E −ε2
⎡
⎣ ⎢
⎤
⎦ ⎥
€
E −ε1 −t 2
E −ε2
⎡
⎣ ⎢
⎤
⎦ ⎥
gate
€
ε2
€
t
€
ε1 H + U
‘s’
€
Σ1
€
Σ2
€
μ1
€
μ2
€
Σs
NCN
INAC
“Self-energy”,
€
Σ
€
Ε−ε1 − t
− t E −ε2
⎡
⎣ ⎢
⎤
⎦ ⎥
€
E −ε1 −t 2
E −ε2
⎡
⎣ ⎢
⎤
⎦ ⎥
gate
€
ε2
€
t
€
ε1
€
Σ
€
ε1
€
Σ
H + U
‘s’
€
Σ1
€
Σ2
€
μ1
€
μ2
€
Σs
NCN
INAC
“Self-energy”,
€
Σ
[H]
€
′ Η
€
τ
[H]
€
Σ
€
Σ = τ ΕS − ′ Η [ ]−1
τ +
H + U
‘s’
€
Σ1
€
Σ2
€
μ1
€
μ2
€
Σs
NCN
INAC
From molecule to QPC
€
Σ = τ ΕS − ′ Η [ ]−1
τ +
H + U
‘s’
€
Σ1
€
Σ2
€
μ1
€
μ2
€
Σs
Damle, Ghosh PRB (2001)
€
Au6“molecule”
NCN
INAC
€
H[ ]
€
Σ1[ ]
€
Σ2[ ]
Quantum
ChemistrySurface
Physics
Surface
Physics
Bridging Disciplines
Basis mixing:Ghosh, Liang, Kienle, Polizzi
NCN
INAC
C60 on Silicon
T(E
)
V (V)
dI/d
VdI
/dV
III III
IV
III
III
IV
STS measurements: (a) Dekker, et al., surface science 2002. (b)& (c) Yao, et al, surface science 1996 Theory: Liang, Ghosh
NCN
INAC Molecule on silicon
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
-6.0 -4.0 -2.0 0.0 2.0 4.0 6.0
V (Volts)
I (nA)
Room temperature
Expt:Mark HersamNanoletters, 01/04
Cover story
(a) V = 0 (b) V < 0 (c) V > 0
Quantumchemistry
SurfacePhysics
NCN
INAC
NEGF equations
H + U
‘s’
€
Σ1
€
Σ2
€
μ1
€
μ2
€
Σs
€
G = (ES − H − Σ) −1
€
A = i [G − G+]
€
Γ=i [Σ − Σ+]
€
[Gn ] = [GΓ1G+] f1 + [GΓ2G+] f2
€
˜ I =q
hTrace Γ1 f1 [A(E)] −[Gn (E)][ ]
€
=q
hTrace Γ2 f2 [A(E)] −[Gn (E)][ ]
NCN
INAC
Matrices <--> Numbers
µ1µ2
€
γ1
€
γ2 €
ε ↔ H[ ]
€
γ ↔ Γ[ ] , Σ[ ]
H + U
‘s’
€
Σ1
€
Σ2
€
μ1
€
μ2
€
Σs
€
N ↔ ρ[ ]€
n(E) ↔ [Gn (E)] = − i [G<(E)]
€
D(E) ↔ [A(E)]
NCN
INAC Minimal Model
U --> I
N --> U
U --> N
Self-ConsistentSolution
“Poisson”
€
I =qh
dE∫ D(E −U)γ1γ2
γ1+γ2f1−f2[ ]
€
N = dE∫ D(E −U)γ1f1+γ2f2
γ1+γ2
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
€
U = UL + U0(N − N0)
Nanowires / Nanotubes / Molecules
VG VD
CHANNEL
INSULATOR
DRAIN
SOURCE
I
NCN
INAC FET: Why current “saturates” ?
0 0.2 0.4 0.60
1
2
3
4
5
6x 10-4
Drain voltage
Drain current
INSULATOR
z
xLW
CHANNEL
E
D(E)
µ1
µ2
€
U = UL + U0(N − N0)
NCN
INAC
Self-consistent field, U
Method of moments:Jing Guo
3D Poisson solver:
Eric Polizzi
H + U
‘s’
€
Σ1
€
Σ2
€
μ1
€
μ2
€
Σs
NCN
INAC
Self-consistent field, U
Method of moments:Jing Guo
3D Poisson solver:
Eric Polizzi
H + U
‘s’
€
Σ1
€
Σ2
€
μ1
€
μ2
€
Σs
Correlations
NCN
INAC
Self-consistent field, U
Quantum Chemistry:
Closed System
in Equilibrium
U
H+U, N
H + U
‘s’
€
Σ1
€
Σ2
€
μ1
€
μ2
€
Σs
NCN
INAC
Self-consistent field, U
Quantum Chemistry:
Closed System
in Equilibrium
U
H+U, N
H + U
‘s’
€
Σ1
€
Σ2
€
μ1
€
μ2
€
Σs
NCN
INAC
Self-consistent field, U
Quantum Chemistry:
Closed System
in Equilibrium
U
H+U, N
H + U
‘s’
€
Σ1
€
Σ2
€
μ1
€
μ2
€
Σs
NCN
INAC
Two choices
H + U
‘s’
€
Σ1
€
Σ2
€
μ1
€
μ2
€
Σs
2^N many electron levels
€
Γ <<UWorks forWorks for
€
Γ≥U
00
11
01
10
NCN
INAC
Two choices
H + U
‘s’
€
Σ1
€
Σ2
€
μ1
€
μ2
€
Σs
2^N many electron levels
€
Γ <<UWorks forWorks for
€
Γ≥U
00
11
01
10
Band theory Mott insulator?
NCN
INAC
“Hot” contacts
H + U
‘s’
€
Σ1
€
Σ2
€
μ1
€
μ2
€
Σs
€
γ1
€
γ2
Energy has to be removed efficiently
from the contacts: otherwise
--> “hot” contacts
NCN
INAC
Other “contacts”
H + U
‘s’
€
Σ1
€
Σ2
€
μ1
€
μ2
€
Σs
• Plate Size ~ 10nm X 1μ 3 m X μm
• ~ Wire Size 10nm X 10nm 3 X μm
P type Silicon
oxide
Metal
+N+N
N-/P-
Integrated Bottom Gate
Hot phonons ?
NCN
INAC
Other “contacts”
H + U
‘s’
€
Σ1
€
Σ2
€
μ1
€
μ2
€
Σs
Molecular desorption ?
• Plate Size ~ 10nm X 1μ 3 m X μm
• ~ Wire Size 10nm X 10nm 3 X μm
P type Silicon
oxide
Metal
+N+N
N-/P-
Integrated Bottom Gate
Hot phonons ?
NCN
INAC
Hot “contacts”
H + U
‘s’
€
Σ1
€
Σ2
€
μ1
€
μ2
€
Σs
Molecular desorption ?
• Plate Size ~ 10nm X 1μ 3 m X μm
• ~ Wire Size 10nm X 10nm 3 X μm
P type Silicon
oxide
Metal
+N+N
N-/P-
Integrated Bottom Gate
Hot phonons ?
Source Drain
NCN
INAC
Two choices
H + U
‘s’
€
Σ1
€
Σ2
€
μ1
€
μ2
€
Σs
€
Γ <<UWorks for
Supplement NEGF with
separate rate equation
for “contact”
“Contact”State A
“Contact”State B
00
11
0110
00
11
01
10
Rate equation for full system
NCN
INAC
Summary
Electronics & Sensing
M
VG VD
CHANNEL
INSULATOR
DRAIN
SOURCE
I
Unified Model
Transients?Strong correlations ?
“Hot contacts” ?
Electrical Resistance: An Atomistic View,
Nanotechnology 15 , S433 (2004)
H + U
‘s’
€
Σ1
€
Σ2
€
μ1
€
μ2
€
Σs
www.nanohub.org