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’

€

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

€

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

€

β

€

α

Gate

CNT Electronics

Guo, Lundstrom

H + U

‘s’

€

Σ1

€

Σ2

€

μ1

€

μ2

€

Σs

NCN

INAC

Hamiltonian, [H]

Atomistic

non-orthogonal basis

€

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

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

Which self-consistent field ?

H + U

‘s’

€

Σ1

€

Σ2

€

μ1

€

μ2

€

Σs

µ

NCN

INAC

Which LDA ?

H + U

‘s’

€

Σ1

€

Σ2

€

μ1

€

μ2

€

Σs

NCN

INAC

Which LDA ?

H + U

‘s’

€

Σ1

€

Σ2

€

μ1

€

μ2

€

Σs

IP = E(N) - E(N-1)EA = E(N+1) - E(N)

NCN

INAC

N vs. µ

µ

N - N0

NCN

INAC

N vs. µ: SCF Theory

µ

N - N0 Rakshit

U0/2

€

Ui = U0 (N − N0)

NCN

INAC

Self-interaction Correction

µ

N - N0 No general method

U0/2

€

Ui = U0 (N − N0 + 0.5 − ni )

NCN

INAC

One-electron vs. Many-electron

N one-electron levels 2^N many electron levels

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

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

What is a contact?

H + U

‘s’

€

Σ1

€

Σ2

€

μ1

€

μ2

€

Σs

Klimeck, Lake et.al.APL (1995)

NCN

INAC

What is a contact?

H + U

‘s’

€

Σ1

€

Σ2

€

μ1

€

μ2

€

Σs

Klimeck, Lake et.al.APL (1995)

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

“Hot” contacts

H + U

‘s’

€

Σ1

€

Σ2

€

μ1

€

μ2

€

Σs

Venugopal, Lundstrom

Source Drain

NCN

INAC

“Hot” contacts

H + U

‘s’

€

Σ1

€

Σ2

€

μ1

€

μ2

€

Σs

Venugopal, Lundstrom

Source Drain

NCN

INAC

Other “contacts”

H + U

‘s’

€

Σ1

€

Σ2

€

μ1

€

μ2

€

Σs

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

NCN

INAC

Experiment vs. Theory

Zahid, Paulsson, Ghosh

THEORY:Purdue Group(cond-mat/0403401)

EXPT:Karlsruhe