1. 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?

2. 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

3. 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]

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

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

6. 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)

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

8. 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 JK-1
Room Temperature T=300K, Thermal Energy kBT = 4.14 x J
(To get eV, divide by q=1.6 x 10-19C), so Room temperature ~ 25 meV

9. 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

10. 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

11. 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)

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

13. 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

14. 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)

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

16. What drives current? µ1 µ2 VD splits Fermi levels in source and drainVG 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

17. 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

18. 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

19. 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 Js
g1 = 1 meV  g1/ћ ≈ 1012/s = 1 THz

20. 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)

21. 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?

22. Maximum Conductance ? µ1 µ2 E f1 f2
1 - 0

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

24. 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

25. 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?

26. Solution: Broadening limits conductanceµ1 µ2 E f1 f2
For g1=g2, max conductance
“Spillage”
Only fraction of level lies in m1-m2 window

27. 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

28. 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)

29. Current with Broadeningµ1 µ2
T(E)
Landauer theory (“Conduction is Transmission”)