Minimal Model for Transport + Broadening D(E) g1f1 + g2f2 N = dE D(E ) g1 + g2 I1 = dE D(E ) [f1-f2] 2q g1g2   h     D(E) g1f1 + g2f2 N = dE D(E-U) g1 + g2 I1 = dE D(E-U) [f1-f2] 2q g1g2   h     + Electrostatics U (Self-consistent Field) g1f1 + g2f2 N = g1 + g2 I1 = [f1-f2] 2q g1g2   h   Rate equation g1,2, f1,2 VG VD CHANNEL INSULATOR DRAIN SOURCE I Poisson: U = UL + U0(N-N0) µ2 g1 g2 µ1 Silicon / Nanotubes / Molecules (FetToy, CNTbands, MolCtoy)

What can we capture with this model?

Classical Theory of MOSFETs g1f1 + g2f2 N = dE D(E-U) g1 + g2 I1 = dE D(E-U) [f1-f2] 2q g1g2   h     I = mCoxW L [(VG-VT)VD-VD2/2] g1 = g2 = ħv/L Ballistic FETs: v determined by bandstructure alone Classical FETs: v = mdV/dx (limited by scattering) Ballistic FETs: occupancy determined by D(qV) Classical FETs: CoxA

Deriving Ohm’s Law need 1/R  A/L ! 1/R = G = (2q2/h)T  1/L T  gD (longer channel, slower escape into leads)  1/R  A Still missing 1/L (R indep. of L here)! T  gD D  AL (volume)  1/R  gAL Not there yet ! Where does extra 1/L come from? Missing piece: Scattering inside channel

We can reproduce all classical theories ‘bottom-up’ We can also capture physics not describable by classical models

HW1.1: MOSFET theory intel.com

HW 1.2: ThermoEMF E µ1 µ2 f1 f2 m1 = m2 (no applied bias) T1 >> T2 Which way would current flow?

HW 1.3: Silicon-molecule-metal systems f1 E f2 Gap Put positive bias on tip, assume levels float halfway What happens to I-V when level enters bandgap?

Beyond Minimal Model 1. Interference 2. Dephasing 3. Correlation

1. Interference Between Levels Oscillations in magneto- Conductance (‘Shubnikov De-Haas’) Interference between a dot and a channel (‘Fano’) D(E) is an ‘independent level’ model To capture interference, need a matrix version We will see that later (NEGF)

Must go beyond minimal model 1. Interference (Rest of the book!) µ1 µ2 H + U Numbers (e,g,U)  Matrices (H, S, U) Rate equations  NEGF formalism

2. Dephasing Dissipation Mostly in the contacts Where does dissipation occur? (I2R) ‘Hot’ hole ‘Hot’ electron

2. Dephasing Vibrational ‘fingerprints’ Electron can lose energy by setting molecule vibrating Current picks up signatures of these vibrations (Inelastic Electron Tunneling Spectroscopy) Expt. Mark Reed (Yale)

3. Correlation U = UL + U0(N-N0) Adding an electron to a channel raises all its levels by U But an electron should not feel itself !! This should split conductance levels (Coulomb Blockade)

3. Correlation El-El interactions µ1 µ2 “ Coulomb Blockade “ Levels split for large U0 Metal-insulator transition

3. Correlation When electrons cooperate Antiferromagnetism Ferromagnetism Superconductivity Superfluidity Quantum Hall Effect Myriad other effects… The very notion of a ‘potential’ U questionable (Ch 4)

To summarize Minimal Model for current conduction Ingredients can be measured or calculated Complications due to quantum interference, scattering and correlation ‘Minimal’ model already good enough to describe most transport experiments!