1. 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)

2. What can we capture with this model?

3. 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

4. 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

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

6. HW1.1: MOSFET theory
intel.com

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

8. HW 1.3: Silicon-molecule-metal systemsf1 E f2
Gap
Put positive bias on tip, assume levels float halfway
What happens to I-V when level enters bandgap?

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

10. 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)

11. 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

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

13. 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)

14. 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)

15. 3. Correlation El-El interactionsµ1 µ2
“ Coulomb Blockade “
Levels split for large U0
Metal-insulator transition

16. 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)

17. 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!