Conductance Quantization
• One-dimensional ballistic/coherent transport
• Landauer theory
• The role of contacts
• Quantum of electrical and thermal conductance
• One-dimensional Wiedemann-Franz law
© 2010 Eric Pop, UIUC
ECE 598EP: Hot Chips
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“Ideal” Electrical Resistance in 1-D
• Ohm’s Law: R = V/I [Ω]
• Bulk materials, resistivity ρ: R = ρL/A
• Nanoscale systems (coherent transport)
– R (G = 1/R) is a global quantity
– R cannot be decomposed into subparts, or added up from pieces
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ECE 598EP: Hot Chips
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Charge & Energy Current Flow in 1-D
• Remember (net) current Jx ≈ x×n×v where x = q or E
J q  q nk vk  q gk f k  vk
k
Net contribution

k
k
J E   Ek nk v k   Ek  g k f k  v k
k
k
J q  q  g k f k v k dk
 J E   Ek g k f k v k dk
k
• Let’s focus on charge current flow, for now
• Convert to integral over energy, use Fermi distribution
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ECE 598EP: Hot Chips
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Conductance as Transmission
I
S
µ1
2q
f ( E )T ( E )dE
h 
D
µ2
J q  q  g k f k Tk v k dk
k
• Two terminals (S and D) with Fermi levels µ1 and µ2
• S and D are big, ideal electron reservoirs, MANY k-modes
• Transmission channel has only ONE mode, M = 1
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ECE 598EP: Hot Chips
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Conductance of 1-D Quantum Wire
I  q   g k f k  Tk vk dk
k
qV
q 1
I   f1 ( E )T ( E )dE
h 2
x
I
1D
k-space
V
0
1 dE
v
dk
x2 spin
gk+ = 1/2π
k
I q2
G 
V
h
quantum of
electrical conductance
(per spin per mode)
• Voltage applied is Fermi level separation: qV = µ1 - µ2
• Channel = 1D, ballistic, coherent, no scattering (T=1)
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ECE 598EP: Hot Chips
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Quasi-1D Channel in 2D Structure
van Wees, Phys. Rev. Lett. (1988)
spin
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ECE 598EP: Hot Chips
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Quantum Conductance in Nanotubes
• 2x sub-bands in nanotubes, and 2x from spin
• “Best” conductance of 4q2/h, or lowest R = 6,453 Ω
• In practice we measure higher resistance, due to
scattering, defects, imperfect contacts (Schottky barriers)
CNT
S (Pd)
D (Pd)
SiO2
L = 60 nm
VDS = 1 mV
G (Si)
Javey et al., Phys. Rev. Lett. (2004)
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ECE 598EP: Hot Chips
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Finite Temperatures
• Electrons in leads according to Fermi-Dirac distribution
• Conductance with n channels, at finite temperature T:
• At even higher T: “usual” incoherent transport (dephasing
due to inelastic scattering, phonons, etc.)
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ECE 598EP: Hot Chips
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Where Is the Resistance?
S. Datta, “Electronic Transport in Mesoscopic Systems” (1995)
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ECE 598EP: Hot Chips
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Multiple Barriers, Coherent Transport
•
Coherent, resonant transport
•
L < LΦ (phase-breaking length);
electron is truly a wave
• Perfect transmission through resonant, quasi-bound states:
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ECE 598EP: Hot Chips
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Multiple Barriers, Incoherent Transport
•
L > LΦ (phase-breaking length);
electron phase gets randomized
at, or between scattering sites
• Total transmission (no interference term):
2
Ttotal 
t1 t2
2
1  r1 r2
2
2
average mean
free path; remember
Matthiessen’s rule!
• Resistance (scatterers in series):
2
2

r
r
h
Resistance  2 1  1 2  2 2 
2e 
t1
t2
 h  L
  2 1  
 2e   

• Ohmic addition of resistances from independent scatterers
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ECE 598EP: Hot Chips
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Where Is the Power (I2R) Dissipated?
• Consider, e.g., a single nanotube
• Case I: L << Λ
R ~ h/4e2 ~ 6.5 kΩ
Power I2R  ?
• Case II: L >> Λ
R ~ h/4e2(1 + L/Λ)
Power I2R  ?
• Remember
© 2010 Eric Pop, UIUC
1
1
1
1
1
1





 op. phon.  ac. phon. imp.  def . ee
ECE 598EP: Hot Chips
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1D Wiedemann-Franz Law (WFL)
• Does the WFL hold in 1D?  YES
• 1D ballistic electrons carry energy too, what is their
equivalent thermal conductance?
  2 kB2  e2 
 2 kB2T
Gth  L eT   2   T 
3h
 3e   h 
(x2 if electron spin included)
Gth  0.28 nW/K at 300 K
Greiner, Phys. Rev. Lett. (1997)
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Phonon Quantum Thermal Conductance
• Same thermal conductance quantum, irrespective of the
carrier statistics (Fermi-Dirac vs. Bose-Einstein)
Phonon Gth measurement in
GaAs bridge at T < 1 K
Schwab, Nature (2000)
Gth 
1 μm
 2 k B2T
3h
 0.28 nW/K at 300 K
Pt
Single nanotube Gth=2.4 nW/K at T=300K
Pop, Nano Lett. (2006)
SWNT
suspended
over trench
Matlab tip:
Pt
© 2010 Eric Pop, UIUC
>> syms x;
>> int(x^2*exp(x)/(exp(x)+1)^2,0,Inf)
ans =
1/6*pi^2
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ECE 598EP: Hot Chips
Electrical vs. Thermal Conductance G0
• Electrical experiments  steps in the conductance (not
observed in thermal experiments)
• In electrical experiments the chemical potential (Fermi
level) and temperature can be independently varied
– Consequently, at low-T the sharp edge of the Fermi-Dirac
function can be swept through 1-D modes
– Electrical (electron) conductance quantum: G0 = (dIe/dV)|low dV
• In thermal (phonon) experiments only the temperature
can be swept
– The broader Bose-Einstein distribution smears out all features
except the lowest lying modes at low temperatures
– Thermal (phonon) conductance quantum: G0 = (dQth/dT) |low dT
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ECE 598EP: Hot Chips
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Back to the Quantum-Coherent Regime
• Single energy barrier – how do you get across?
E
thermionic emission
fFD(E)
tunneling or reflection
• Double barrier: transmission through quasi-bound (QB) states
EQB
EQB
• Generally, need λ ~ L ≤ LΦ (phase-breaking length)
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ECE 598EP: Hot Chips
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Wentzel-Kramers-Brillouin (WKB)
Ex
E||
fFD(Ex)
A
B
tunneling only
0
L
• Assume smoothly varying potential barrier, no reflections
T
 trans.
2
 incid .
2
 L

 exp  2 k ( x) dx 
 0

k(x) depends on
energy dispersion
J A B   #incident states   f A g AT ( Ex ) 1  f B  g B dE
E.g. in 3D, the net current is:
J  J AB  J B A
© 2010 Eric Pop, UIUC
qm*
 2 3
2
  f A  f B  g A g BT (E )dE
ECE 598EP: Hot Chips
Fancier version of
Landauer formula!
17
Band-to-Band Tunneling
• Assuming parabolic energy dispersion E(k) = ħ2k2/2m*
 4 2m* E 3/2 
x x

T ( Ex )  exp  

3q F 


F = electric field
• E.g. band-to-band (Zener) tunneling
in silicon diode
J BB 
q3 FVeff
4 3
2
 4 2m* E 3/2 
2m*
x G

exp  

EG
3q F 


See, e.g. Kane, J. Appl. Phys. 32, 83 (1961)
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ECE 598EP: Hot Chips
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