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Conductance Quantization. One-dimensional ballistic/coherent transport Landauer theory The role of contacts Quantum of electrical and thermal conductance One-dimensional Wiedemann-Franz law. “Ideal” Electrical Resistance in 1-D. Ohm’s Law: R = V/I [ Ω ] - PowerPoint PPT Presentation

ECE 598 EP

Conductance Quantization One-dimensional ballistic/coherent transport Landauer theory The role of contacts Quantum of electrical and thermal conductance One-dimensional Wiedemann-Franz law1 2010 Eric Pop, UIUCECE 598EP: Hot ChipsIdeal Electrical Resistance in 1-DOhms Law: R = V/I []Bulk materials, resistivity : R = L/ANanoscale systems (coherent transport)R (G = 1/R) is a global quantityR cannot be decomposed into subparts, or added up from pieces2

2010 Eric Pop, UIUCECE 598EP: Hot ChipsRemember (net) current Jx xnv where x = q or E

Lets focus on charge current flow, for nowConvert to integral over energy, use Fermi distribution

Charge & Energy Current Flow in 1-D3

Net contribution 2010 Eric Pop, UIUCECE 598EP: Hot ChipsConductance as TransmissionTwo terminals (S and D) with Fermi levels 1 and 2S and D are big, ideal electron reservoirs, MANY k-modesTransmission channel has only ONE mode, M = 14

S1D2

2010 Eric Pop, UIUCECE 598EP: Hot ChipsConductance of 1-D Quantum WireVoltage applied is Fermi level separation: qV = 1 - 2Channel = 1D, ballistic, coherent, no scattering (T=1)5

qV

01Dk-spacekxVI

quantum ofelectrical conductance(per spin per mode)x2 spingk+ = 1/2 2010 Eric Pop, UIUCECE 598EP: Hot ChipsQuasi-1D Channel in 2D Structure6

van Wees, Phys. Rev. Lett. (1988)spin 2010 Eric Pop, UIUCECE 598EP: Hot ChipsQuantum Conductance in Nanotubes2x sub-bands in nanotubes, and 2x from spinBest conductance of 4q2/h, or lowest R = 6,453 In practice we measure higher resistance, due to scattering, defects, imperfect contacts (Schottky barriers)7

S (Pd)D (Pd)SiO2CNTG (Si)

Javey et al., Phys. Rev. Lett. (2004)L = 60 nmVDS = 1 mV 2010 Eric Pop, UIUCECE 598EP: Hot ChipsFinite TemperaturesElectrons 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.)8

2010 Eric Pop, UIUCECE 598EP: Hot ChipsWhere Is the Resistance?9

S. Datta, Electronic Transport in Mesoscopic Systems (1995) 2010 Eric Pop, UIUCECE 598EP: Hot ChipsMultiple Barriers, Coherent TransportPerfect transmission through resonant, quasi-bound states:10

Coherent, resonant transportL < L (phase-breaking length); electron is truly a wave 2010 Eric Pop, UIUCECE 598EP: Hot Chips

Multiple Barriers, Incoherent TransportTotal transmission (no interference term):

Resistance (scatterers in series):

Ohmic addition of resistances from independent scatterers11

L > L (phase-breaking length); electron phase gets randomized at, or between scattering sites

average meanfree path; rememberMatthiessens rule! 2010 Eric Pop, UIUCECE 598EP: Hot ChipsWhere Is the Power (I2R) Dissipated?Consider, e.g., a single nanotubeCase I:L > R ~ h/4e2(1 + L/)Power I2R ?

Remember12

2010 Eric Pop, UIUCECE 598EP: Hot Chips1D Wiedemann-Franz Law (WFL)Does the WFL hold in 1D? YES1D ballistic electrons carry energy too, what is their equivalent thermal conductance?13

(x2 if electron spin included)Greiner, Phys. Rev. Lett. (1997)

nW/K at 300 K 2010 Eric Pop, UIUCECE 598EP: Hot ChipsPhonon Quantum Thermal ConductanceSame thermal conductance quantum, irrespective of the carrier statistics (Fermi-Dirac vs. Bose-Einstein)14>> syms x;>> int(x^2*exp(x)/(exp(x)+1)^2,0,Inf)ans =1/6*pi^2Matlab tip:

Phonon Gth measurement inGaAs bridge at T < 1 KSchwab, Nature (2000)

Single nanotube Gth=2.4 nW/K at T=300KPop, Nano Lett. (2006)nW/K at 300 K

2010 Eric Pop, UIUCECE 598EP: Hot ChipsElectrical vs. Thermal Conductance G0Electrical experiments steps in the conductance (not observed in thermal experiments)In electrical experiments the chemical potential (Fermi level) and temperature can be independently variedConsequently, at low-T the sharp edge of the Fermi-Dirac function can be swept through 1-D modesElectrical (electron) conductance quantum: G0 = (dIe/dV)|low dVIn thermal (phonon) experiments only the temperature can be sweptThe broader Bose-Einstein distribution smears out all features except the lowest lying modes at low temperaturesThermal (phonon) conductance quantum: G0 = (dQth/dT) |low dT15 2010 Eric Pop, UIUCECE 598EP: Hot ChipsSingle energy barrier how do you get across?

Double barrier: transmission through quasi-bound (QB) states

Generally, need ~ L L (phase-breaking length)

Back to the Quantum-Coherent Regime16EQBEQB

thermionic emissiontunneling or reflectionfFD(E)E 2010 Eric Pop, UIUCECE 598EP: Hot ChipsWentzel-Kramers-Brillouin (WKB)Assume smoothly varying potential barrier, no reflections17

tunneling onlyfFD(Ex)Ex

ABE||

k(x) depends onenergy dispersion

E.g. in 3D, the net current is:0LFancier version ofLandauer formula! 2010 Eric Pop, UIUCECE 598EP: Hot ChipsBand-to-Band TunnelingAssuming parabolic energy dispersion E(k) = 2k2/2m*

E.g. band-to-band (Zener) tunnelingin silicon diode18

F = electric field

See, e.g. Kane, J. Appl. Phys. 32, 83 (1961) 2010 Eric Pop, UIUCECE 598EP: Hot Chips