Conductance Quantization
description
Transcript of Conductance Quantization
© 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 1
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 2
“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
© 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 3
• Remember (net) current Jx ≈ x×n×v where x = q or E
• Let’s focus on charge current flow, for now• Convert to integral over energy, use Fermi distribution
q k k kk
J q g f dk v
Charge & Energy Current Flow in 1-D
q k k k k kk k
J q n q g f v v
E k k k k k k k E k k k kk k k
J E n E g f J E g f dk v v v
Net contribution
© 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 4
Conductance as Transmission
• 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
Sµ1
Dµ2
2 ( ) ( )qI f E T E dEh
q k k k kk
J q g f T dk v
© 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 5
Conductance of 1-D Quantum Wire
• Voltage applied is Fermi level separation: qV = µ1 - µ2
• Channel = 1D, ballistic, coherent, no scattering (T=1)
qV
1 dEvdk
01Dk-space
k
x
VI
2I qGV h
k k k kk
I q g f T v dk
1
21( ) ( )qI f E T E dE
h
quantum ofelectrical conductance(per spin per mode)
x2 spingk+ = 1/2π
© 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 6
Quasi-1D Channel in 2D Structurevan Wees, Phys. Rev. Lett. (1988)
spin
© 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 7
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)
S (Pd) D (Pd)SiO2
CNT
G (Si)
Javey et al., Phys. Rev. Lett. (2004)
L = 60 nmVDS = 1 mV
© 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 8
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.)
© 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 9
Where Is the Resistance?S. Datta, “Electronic Transport in Mesoscopic Systems” (1995)
© 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 10
Multiple Barriers, Coherent Transport
• Perfect transmission through resonant, quasi-bound states:
• Coherent, resonant transport• L < LΦ (phase-breaking length);
electron is truly a wave
© 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 11
2 21 2
2 22 21 2
Resistance 1 12 2
r rh h Le et t
Multiple Barriers, Incoherent Transport
• Total transmission (no interference term):
• Resistance (scatterers in series):
• Ohmic addition of resistances from independent scatterers
• L > LΦ (phase-breaking length); electron phase gets randomized at, or between scattering sites
2 21 2
total 2 21 21
t tT
r r
average mean
free path; rememberMatthiessen’s rule!
© 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 12
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. . . . . .
1 1 1 1 1 1
op phon ac phon imp def e e
© 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 13
1D Wiedemann-Franz Law (WFL)• Does the WFL hold in 1D? YES• 1D ballistic electrons carry energy too, what is their
equivalent thermal conductance?
2 222 2
23 3B
th eB eG L T k kT
hhT
e
(x2 if electron spin included)
Greiner, Phys. Rev. Lett. (1997)
0.28thG nW/K at 300 K
© 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 14
Phonon Quantum Thermal Conductance• Same thermal conductance quantum, irrespective of the
carrier statistics (Fermi-Dirac vs. Bose-Einstein)
>> syms x;>> int(x^2*exp(x)/(exp(x)+1)^2,0,Inf)ans =1/6*pi^2
Matlab tip:
Phonon Gth measurement inGaAs bridge at T < 1 KSchwab, Nature (2000)
SWNT
suspended
over trench
Pt
Pt 1 μm Single nanotube Gth=2.4 nW/K at T=300K
Pop, Nano Lett. (2006)
nW/K at 300 K2 2
0.283t
Bh
k TGh
© 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 15
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
© 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 16
• Single energy barrier – how do you get across?
• Double barrier: transmission through quasi-bound (QB) states
• Generally, need λ ~ L ≤ LΦ (phase-breaking length)
Back to the Quantum-Coherent Regime
EQBEQB
thermionic emission
tunneling or reflection
fFD(E)E
© 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 17
Wentzel-Kramers-Brillouin (WKB)
• Assume smoothly varying potential barrier, no reflections
tunneling only
fFD(Ex)Ex
2.
20.
exp 2 ( )L
trans
incid
T k x dx
A BE||
#incident states ( ) 1A B A A x B BJ f g T E f g dE
k(x) depends onenergy dispersion
*
2 3 ( )2A B B A A B A BqmJ J J f f g g T E dE
E.g. in 3D, the net current is:
0 L
Fancier version ofLandauer formula!
© 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 18
Band-to-Band Tunneling
• Assuming parabolic energy dispersion E(k) = ħ2k2/2m*
• E.g. band-to-band (Zener) tunnelingin silicon diode
* 3/24 2( ) exp
3x x
x
m ET E
q F
F = electric field
3 * 3/2*
3 2
4 22 exp4 3
eff x GBB
G
q FV m EmJE q F
See, e.g. Kane, J. Appl. Phys. 32, 83 (1961)