NEGF Method: Capabilities and Challenges Supriyo Datta School of Electrical & Computer Engineering Purdue University Molecular Electronics CNT Electronics Molecular Sensor M VG VD CHANNEL INSULATOR DRAIN SOURCE I Gate

Nanodevices: A Unified View Unified Model H + U ‘s’ Molecular Electronics CNT Electronics Molecular Sensor M VG VD CHANNEL INSULATOR DRAIN SOURCE I Gate

Hamiltonian, [H] -t 2t ‘s’ H + U Effective Mass Equation VG VD CHANNEL INSULATOR DRAIN SOURCE I Effective Mass Equation Finite Difference / Finite Element 2t -t Damle, Ren, Venugopal, Lundstrom ---> nanoMOS

Rahman, Wang, Ghosh, Klimeck, Lundstrom Hamiltonian, [H] Nanowire Electronics H + U ‘s’ VG VD CHANNEL INSULATOR DRAIN SOURCE I Atomistic sp3d basis , are matrices Rahman, Wang, Ghosh, Klimeck, Lundstrom

Hamiltonian, [H] ‘s’ Gate H + U Atomistic pz basis Guo, Lundstrom CNT Electronics Gate Atomistic pz basis , are (2x2) matrices Guo, Lundstrom

Siddiqui, Kienle, Ghosh, Klimeck Hamiltonian, [H] Nanowire/CNT Electronics H + U ‘s’ Gate Atomistic non-orthogonal basis EHT Siddiqui, Kienle, Ghosh, Klimeck

Hamiltonian, [H] Molecular Electronics ‘s’ H + U Atomistic basis: Huckel / EHT / Gaussian Ghosh, Rakshit,Liang, Zahid, Siddiqui, Golizadeh, Bevan, Kazmi

“Self-energy”, H + U ‘s’

“Self-energy”, gate H + U ‘s’

“Self-energy”, gate H + U ‘s’

“Self-energy”, gate H + U ‘s’

“Self-energy”, H + U ‘s’ [H] [H]

From molecule to QPC “molecule” H + U ‘s’ Damle, Ghosh PRB (2001)

Basis mixing: Ghosh, Liang, Kienle, Polizzi Bridging Disciplines Quantum Chemistry Surface Physics Basis mixing: Ghosh, Liang, Kienle, Polizzi

C60 on Silicon Theory: Liang, Ghosh dI/dV dI/dV T(E) V (V) II III dI/dV IV I II IV dI/dV III T(E) STS measurements: (a) Dekker, et al., surface science 2002. (b)& (c) Yao, et al, surface science 1996 V (V) Theory: Liang, Ghosh

(a) V = 0 (b) V < 0 (c) V > 0 Molecule on silicon Quantum chemistry Surface Physics Expt:Mark Hersam Nanoletters, 01/04 Cover story Room temperature (a) V = 0 (b) V < 0 (c) V > 0

NEGF equations H + U ‘s’

Matrices <--> Numbers H + U ‘s’ µ1 µ2

Nanowires / Nanotubes / Molecules Minimal Model VG VD CHANNEL INSULATOR DRAIN SOURCE I N --> U U --> N Self- Consistent Solution “Poisson” U --> I Nanowires / Nanotubes / Molecules

FET: Why current “saturates” ? Drain current E D(E) µ1 µ2 INSULATOR z x L W CHANNEL Drain voltage

Self-consistent field, U H + U ‘s’ 3D Poisson solver: Eric Polizzi Method of moments: Jing Guo

Self-consistent field, U H + U ‘s’ 3D Poisson solver: Eric Polizzi Method of moments: Jing Guo Correlations

Self-consistent field, U H + U ‘s’ Quantum Chemistry: Closed System in Equilibrium U H+U, N

Self-consistent field, U H + U ‘s’ Quantum Chemistry: Closed System in Equilibrium U H+U, N

Self-consistent field, U H + U ‘s’ Quantum Chemistry: Closed System in Equilibrium U H+U, N

Which self-consistent field ? H + U ‘s’ µ

Which LDA ? H + U ‘s’

Which LDA ? H + U ‘s’ IP = E(N) - E(N-1) EA = E(N+1) - E(N)

N vs. µ µ N - N0

N vs. µ: SCF Theory µ U0/2 N - N0 Rakshit

Self-interaction Correction µ U0/2 No general method N - N0

One-electron vs. Many-electron N one-electron levels 2^N many electron levels 00 11 01 10

2^N many electron levels Two choices 2^N many electron levels H + U ‘s’ 00 11 01 10 Works for Works for

2^N many electron levels Two choices 2^N many electron levels H + U ‘s’ 00 11 01 10 Works for Works for ? Mott insulator Band theory

What is a contact? H + U ‘s’ Klimeck, Lake et.al. APL (1995)

What is a contact? H + U ‘s’ Klimeck, Lake et.al. APL (1995)

“Hot” contacts ‘s’ H + U Energy has to be removed efficiently from the contacts: otherwise --> “hot” contacts

“Hot” contacts H + U ‘s’ Source Drain Venugopal, Lundstrom

“Hot” contacts H + U ‘s’ Source Drain Venugopal, Lundstrom

Other “contacts” H + U ‘s’

Other “contacts” Hot phonons ? H + U ‘s’

Other “contacts” Hot phonons ? H + U ‘s’ Molecular desorption ?

Hot “contacts” Hot phonons ? ‘s’ H + U Molecular desorption ? Source Drain

Two choices Works for “Contact” State A “Contact” State B H + U ‘s’ 00 11 01 10 00 11 01 10 Supplement NEGF with separate rate equation for “contact” Rate equation for full system Works for

Summary Unified Model Electronics & Sensing H + U ‘s’ Transients? VG VD CHANNEL INSULATOR DRAIN SOURCE I www.nanohub.org Transients? Strong correlations ? “Hot contacts” ? Electrical Resistance: An Atomistic View, Nanotechnology 15 , S433 (2004)

Experiment vs. Theory Zahid, Paulsson, Ghosh EXPT: THEORY: Karlsruhe Purdue Group (cond-mat/0403401) EXPT: Karlsruhe Zahid, Paulsson, Ghosh