Trying to build a model to compute I-Vs Saturation S-D Current ID Turn-on S-D Current ID Rise Gate Voltage Source-Drain Voltage What does a device engineer look for?

Top-down vs. Bottom-up L > 1 mm Semiclassical Transport/ “Meso”scopics Critical Role of Quantum Mechanics L ~ 100s nm Ballistic QM “Nano” Critical role of Contacts Quantum Dynamics in Channel + Stat Mech in source drain L ~ 10 nm Drift-Diffusion (“Macro/Micro”) Classical Mechanics + Statistical Mechanics Mixed up

Small is different ! Ohm’s Law Where is the Transition? R = rL/A R = h/2q2 ~ 13 KW R = r(L + l)/A M: # of modes, l : mean-free-path = h/2q2[(1 + L/l)/M]

What is resistance? R not a simple sum of R1 and R2 !

Starting point: Band-diagram Solids have energy bands (Ch 5)

E Filling up the Bands with Electrons Empty levels Filled levels Metal (Copper) (charges move) Semiconductor (Si) (charge movement can be controlled) Insulator (Silica) (charges can’t move)

Separating filled and empty levels Source Drain conduction band Fermi Energy (electro- chemical potential) EF valence band

Average occupancy (Fermi function) Source Drain Filled Empty 1 f0(E-EF) E f0(E) =1/(eE/kBT + 1) f0(E-EF) kBT EF Boltzmann Constant: kB = 1.38 x 10-23 JK-1 Room Temperature T=300K, Thermal Energy kBT = 4.14 x 10-21 J (To get eV, divide by q=1.6 x 10-19C), so Room temperature ~ 25 meV

Role of Gate  Create levels at Fermi Energy VG = 0 VG > 0 n type operation Vacuum Level Vacuum Level Fermi energy held fixed by source and drain Conduction depends on availability of states at EF

In plain English… Positive gate bias attracts electrons into channel VG > 0 n type operation Vacuum Level More electrons Positive gate bias attracts electrons into channel Channel now becomes more conductive

p type operation Negative gate repels electrons VG < 0 Negative gate repels electrons and draws holes which conduct Vacuum Level Fewer electrons Sufficient to have states at EF (filled or empty!) Two electrons per level (up and down spins)

Current onsets controlled by gate VG VD CHANNEL INSULATOR DRAIN SOURCE I + Positive onset (EF to CB) Negative onset (EF to VB) Energy bands (VD small, fixed)

From solid to molecule V+ V- ID VG Current rises when level encountered Current falls when level crossed Gate scans states of the channel Discrete levels (smaller the molecule, more the separation) HOMO LUMO

How else could we bring levels near EF? Workfunction engineering (choose a contact material whose EF naturally lies near VB or CB) Vacuum level Low work function Cathode (eg. Ca) Molecular Layer High work function Anode (eg. ITO) Organic LEDs or OLEDS (not shown at equilibrium since EFs aren’t aligned)

How else could we bring levels near EF? Workfunction engineering (choose a contact material whose EF naturally lies near VB or CB) hn

What drives current? µ1 µ2 VD splits Fermi levels in source and drain VG VD CHANNEL INSULATOR DRAIN SOURCE I µ1 qVD µ2 VD splits Fermi levels in source and drain  Non-equilibrium Levels in the window sense a ‘difference of opinion’ between two contacts, and conduct as a result

Current flows due to difference in ‘agenda’ µ1 E µ2 f1 f2 Only levels near EF conduct Need to complete loop to restore lost/gained charges

p vs n-type conduction µ1 µ1 µ2 µ2 Go thro’ level that is full at equilibrium (first eject charge then inject) Go thro’ level that is empty at equilibrium (first inject charge then eject) Battery removes the extra electron and hole in the contacts

Escape time E µ1 E µ2 f1 f2 Large Small g: strength of bonding between Electrons spread out over molecule Electron sitting away from ends f2 g: strength of bonding between contact and channel electrons g/ћ: inverse escape time into leads ћ = h/2p = 1.06 x 10-34 Js g1 = 1 meV  g1/ћ ≈ 1012/s = 1 THz

Toy Model : Current E µ1 E µ2 f1 f2 (weighted average of f1,2) (Only levels near EF conduct) (Net escape time 1/g = 1/g1 + 1/g2)

Finite temperature E µ1 µ2 f1 f2 (-df0/dE)qVD (-df0/dE)(m2-m1) Width kBT Height 1/4kBT µ1 µ2 E f1 f2 (-df0/dE)qVD (-df0/dE)(m2-m1) What does df0/dE look like?

Maximum Conductance ? µ1 µ2 E f1 f2 1 - 0

Maximum Conductance ? E µ1 µ2 f1 f2 Conductance seems unlimited Experiment and calculation say otherwise !

Conductance quantization in gold nanowires EXPT Halbritter PRB ’04 G0 = 2q2/h = 77 mA/V Minimum resistance of a conductor (h/2q2 = 12.9 kW) Different from Ohm’s Law R = rL/A Modified Ohm’s Law R = r(L + L0)/A

Conductance quantization in gold nanowires EXPT Halbritter PRB ’04 G0 = 2q2/h = 77 mA/V Ohmic IV, quantized slope, fundamental constants How would this come from our equations?

Solution: Broadening limits conductance µ1 µ2 E f1 f2 For g1=g2, max conductance “Spillage” Only fraction of level lies in m1-m2 window

Why do levels broaden? P(t) t Fourier Transform t E Occupation probability Isolated channel Density of states (Sharp level) E=hn for quantum particles like electrons, photons Thus (t,E) form Fourier pairs

Why do levels broaden? P(t) e-t/t e t Fourier Transform t E Occupation probability Channel coupled to contacts Density of states (Broadened level) D(E) = g/2p[(E-e)2 + (g/2)2] g = g1 + g2 = ħ/2t Fourier transform of an exponent  Lorentzian peak value 2/pg, width ~ g ∫D(E)dE = 1 (Sum rule  Can still hold 1 electron)

Current with Broadening µ1 µ2 T(E) Landauer theory (“Conduction is Transmission”)