THERMOELECTRIC CONVERSION IN DISORDERED NANOWIRES

Complex Quantum Systems, Universität Regensburg

Cosimo Gorini

Mainz, 28 / 01 / 2016 | PAGE 1

Klaus Richter’s group

1-slidehystory ofthermoelectricity

q Thepioneers:Volta(1794),Seebeck (1821),Peltier (1834),Joule(1843),Thomson(1851;later LordKelvin).

q Theoretical formalisation:Onsager (1931).

q Thegolden age:ca.1930– 1960(Ioffe).

q Therebirth:1990’s(Hicks &Dresselhaus).

q Today:energy efficiency concerns.| PAGE 2

Thermoelectric effects: simultaneous charge & energy/heat flows

Siberia can listen to Stalin’s radio (USSR, 1948)Nanostructuration

Thermoelectricity:Rulesofthegame

HOT COLDelectrons

| PAGE 3

J1 = charge current

J2 = heat currentL i j = linear transport

coefficients

Seebeck(thermopower) Figure of merit

Max efficiencyCarnot efficiency Curzon-Ahlborn limit

Efficiency at max power

Thermoelectricity:Rulesofthegame

HOT COLDelectrons

Thermopower(orSeebeck coeff.):

Maximizetheefficiency i.e.thefigureofmerit:

…keepingareasonableelectricaloutputpower(powerfactor):

| PAGE 4

“… fundamental scientific challenges could be overcome by deeper understanding of charge and heat transport”

“… a newly emerging field of low-dimensional thermoelectricity, enabled by materials nanoscience and nanotechnology”

Dresselhaus et al: Adv. Mater. 2007

Majumdar: Science 2004

Whysemiconductornanowires?

SC nanowires

Reduced thermal conductancePhonon vs electrons mean free path, geometrical designs(Hochbaum 2008, Heron 2010)

Enhanced thermopowerField effect transistors (Brovman 2013, Roddaro 2013 & many others)

Scalable output powerArrays of parallel NWs (Pregl 2013, Stranz 2013)

| PAGE 5

Someexperimentalrealizations

Karg et al. (IBM Zurich), 2013

Fan et al. (Berkeley CA), 2008

Shin et al. (Seoul), 2011

Hochbaum et al. (Berkeley CA), 2008

Pregl et al. (TU Dresden), 2013

However à Imbalance between experimental / theoretical works !| PAGE 6

Outline

q Nanowires intheFieldEffect TransistorconfigurationThe1DAndersonModel

q Thermopower ofsingleNW:low Telastic regimeq Thermopower ofsingleNW:inelasticphonon-assisted regimeq Parallel nanowires:scalable power factor andhotspotcoolingq 3-Terminalthermoelectric ratchet (``all-in-one thermocouple’’)

q Conclusions andprospects| PAGE 7

Aller à l’idéal et comprendre le réel.

Jean Jaurès

Field-EffectTransistors(FET)

• Single (or array of) doped nanowire(s) in the FET configuration

• Substrate: Electrically and thermally insulating

• Gate: «back» or «top»

• Heater: for thermoelectric measurements

"Seebeck" configuration: thermal bias

"Peltier" configuration: voltage bias

Π = 𝑆𝑇: equivalent within linear response if time-reversal symmetrypreserved (Kelvin relation)

Goal: effect of gate modulation on the thermopower

Setup used by P. Kim (Columbia) (2013)

| PAGE 8

Vg Impurity band

1DAndersonModel

• 1Delectroniclatticewithon-site(uniform) disorder𝑉' ∈ (-W/2,W/2)

• Allelectronsarelocalizedwithlocalizationlengthξ (E)

• Statesdistributed withinanimpurityband2𝐸+≈4t+W

• Behaviorofthetypicalthermopowerwhenthegatevoltage𝑉, isvaried

µ

E

• Tight-binding Hamiltonian

GateVoltage

2EB

Prototypical model of localized system

| PAGE 9

B.Derrida&E.Gardner,J.Physique45,1283(1984)

"Bulk" formulas:

"Edge" formulas:

Analytical expressionsderived inweakdisorder limit

W=t:bandedge at2t+W/2 2.5t

1Ddensityofstatesν andlocalizationlengthξLocalizationlength

Density

ofstates

| PAGE 10

Outline

ü Nanowires intheFieldEffect TransistorconfigurationThe1DAndersonModel

q Thermopower ofsingleNW:low Telastic regimeq Thermopower ofsingleNW:inelasticphonon-assisted regimeq Parallel Nanowires:scalable power factor andhotspotcoolingq 3-Terminalthermoelectric ratchet

q Conclusions andprospects

| PAGE 11

Elasticregime:Thermopower

LowTemperatures+LinearResponseàMottformula:

Localizedregime:𝜏 decaysexponentiallywithlength

Typical𝜏dependsontheenergy viaξ(E)(localization length)

Theory:

Numerics: RecursiveGreenFunctioncalculationofS

Transportmechanism:elastic(coherent)tunnelling

| PAGE 12

R. Bosisio, G. Fleury, & J.-L. Pichard, New J. Phys. 16:035004 (2014)

Elasticregime:Summary

Enhancement of the thermopower atthe band edges (role of 𝜉(E))

Analytical description of the results

Sommerfeld Expansion (low T)Wiedemann-Franz law à Low S

Very low power factor 𝑄 = 𝐺𝑆4because of the exponential reductionof G at the band edges

Interest: Ultra-low T Peltier cooling?

OR à toward higher temperatures!| PAGE 13

Outline

ü Nanowires intheFieldEffect TransistorconfigurationThe1DAndersonModel

ü Thermopower ofsingleNW:low Telastic regimeq Thermopower ofsingleNW:inelasticphonon-assisted regimeq Parallel Nanowires:scalable power factor andhotspotcoolingq 3-Terminalthermoelectric ratchet

q Conclusions andprospects

| PAGE 14

VariableRangeHopping:TransportMechanism

Mott à competition between tunneling and activated processes

Phonon Bath

kT kTVg

𝛍

ener

gy

position

Variable Range Hopping: phonon-assisted transport à sequence of hopsof variable size

Optimal hop size: Mott hopping length

LM

or Mott hopping energy

i

j

| PAGE 15

TransportMechanisms

Low T:LM>>Là elastic transport

Increasing T: LMLà onset ofinelastic processes (VRH)

Increasing T: LM ξà simple activated transport (NNH)T

Cut-off required by ∆ 𝑇7 = 𝑘+𝑇7 ≤ 2𝐸+

(Ta :onset ofsimple activation in1D(Kurkijärvi 1973,Raikh&Ruzin 1989))

Elastic

VRH

NNH

Mott’s Hopping Energy:finiterange ofstates

contributing totransport

Relevant energy scaleforactivated transport

| PAGE 16

Between leadandlocalized states [Elastic tunnelingrates]

Between localized states[Inelastic hopping rates]

1. Transitionrates(Fermigoldenrule)

Inelastic(Phonon-Assisted)Regime:method

𝛼=L,R

Phonon Bath

𝛍α𝐢

𝐣

Essentialingredients tobuild &solve theRandom Resistor Network

Miller & Abrahams, PR (1960), Ambegaokar, Halperin & Langer PRB (1971), Jiang, Entin-Wohlman & Imry., PRB (2014)

ξ energy dependenceusually neglected!

| PAGE 17

Inelastic(Phonon-Assisted)Regime:method

2.Fermidistributionsatequilibrium (nobias)

3.Occupationprobabilities outofequilibrium

Jiang, Entin-Wohlman & Imry., PRB (2014)

4.Currents

5.Current conservationatevery node i(Kirchoff)Ncoupled equations inNvariables

à 𝛿𝑓'

| PAGE 18

Miller– AbrahamsResistorNetwork

6.Totalparticle/heat currents

7.Transport coefficients

Summing all terms flowingoutfromL(R)terminal

(Having assumed Peltier configuration: T constant everywhere)

Jiang, Entin-Wohlman & Imry., PRB (2014)

Conductance

Peltier coefficient

Thermopower

| PAGE 19

InelasticRegime:Typicalthermopower

Approachingthebandedge(increasingVg)

• Thermopower enhancement when thebandedges areapproached

• Rich behaviour oftheT-dependence ofthethermopower, "reflecting" theshape ofthedensity of statesandlocalization length

S0 vs Temperature

R. Bosisio, C. Gorini, G. Fleury, & J.-L. Pichard, New J. Phys. 16:095005 (2014)

Vg=1.5 t

Vg=2.3 t

T-1

| PAGE 20

Inelasticregime:theory

Theory:Percolation approach to solve the RRN:

Ambegaokar, Halperin, Langer à conductance (1971)Zvyaginà thermopower (1973)

thermopower = energy averaged over the percolating path

~𝑇AB:simple activation:energy to «jump»

toward 𝜖E

(inside)

(outside)

~𝑇AB/4:Variable Range Hopping

(Mott)

Integrate between (μ-∆; μ+∆)

| PAGE 21

InelasticRegime:Typicalthermopower

S0 vs Temperature

S0 vs Gate Voltage

R. Bosisio, C. Gorini, G. Fleury, & J.-L. Pichard, New J. Phys. 16:095005 (2014)

Electrical Conductance

(inside)

(outside)

| PAGE 22

µ

2∆

InelasticRegime:Mottenergy

Mott’s HoppingEnergy:finiterange ofstates contributing totransport

Sdepends ontheasymmetryofthestates

aroundμwithin[μ-∆, μ+∆]

Integrationinside[μ-∆, μ+∆]

| PAGE 23

Outline

ü Nanowires intheFieldEffect TransistorconfigurationThe1DAndersonModel

ü Thermopower ofsingleNW:low Telastic regimeü Thermopower ofsingleNW:inelasticphonon-assisted regimeq Parallel Nanowires:scalable power factor andhotspotcoolingq 3-Terminalthermoelectric ratchet

q Conclusions andprospects

| PAGE 24

Arraysofparallelnanowires

Suspended Deposited

• Neglect inter-wire hoppingà independent nanowires

• Transport through each NW: VRH / NNH regime (same treatment as before)

Fan et al., 2008Shin et al., 2011

| PAGE 25

Parallelnanowires:powerfactorandfigureofmerit

Rescaled power factor Electronic figure of merit

Scalable Power Factor(without affecting the electronic figure of merit)

Iso-curve S0=2 kB/e

Band edge

Motttemperature

Parameters:

R. Bosisio, C. Gorini, G. Fleury, & J.-L. Pichard, Phys. Rev. Applied 3, 054002 (2015)

P μW for 105 NWs (1 cm) and δT10 K| PAGE 26

Parallelnanowires:powerfactorandfigureofmerit

Rescaled power factor Electronic figure of merit

Estimation of the parasitic phononic contribution to ZT

(For doped Si-NWs and SiO2 substrate)

| PAGE 27

Parallelnanowires:HotSpotcooling

Hopping heat current through each localized state i

Ei randomly distributed à Local fluctuations

Λph: inelastic phonon mean free path = thermalization length in the substrate

kT kT

𝛍

| PAGE 28

R. Bosisio, C. Gorini, G. Fleury, & J.-L. Pichard, Phys. Rev. Applied 3, 054002 (2015)

Parallelnanowires:HotSpotcooling

Vg=0: band center Vg=2.25 t: band edge

Parameters: (Heat currents in unit of )

Formation of hot and cold spots near the boundaries of the

substrate

Top view of the substrate

| PAGE 29

R. Bosisio, C. Gorini, G. Fleury, & J.-L. Pichard, Phys. Rev. Applied 3, 054002 (2015)

Parallelnanowires:HotSpotcooling

Estimation of the cooling power density:

10 W/cm2

at δμ=1mV 7.7 x 10-2 tand T = 75K 0.5t

(still linear response)

Opportunities for heat management in microstructures

E.g.: hot spot cooling in microelectronics: transferring heat some microns away

| PAGE 30

10 μm

R. Bosisio, C. Gorini, G. Fleury, & J.-L. Pichard, Phys. Rev. Applied 3, 054002 (2015)

Microprocessors ~ 100 W/cm2

Outline

ü Nanowires intheFieldEffect TransistorconfigurationThe1DAndersonModel

ü Thermopower ofsingleNW:low Telastic regimeü Thermopower ofsingleNW:inelasticphonon-assisted regimeü Parallel Nanowires:scalable power factor andhotspots coolingq 3-Terminalthermoelectric ratchet

q Conclusions andprospects

| PAGE 31

3-Terminalthermoelectricratchet

| PAGE 32

Bosisio, Fleury, Pichard & Gorini, arXiv (2015)

q Microscale heat harvester or coolerq Non – local thermal machine (``all-in-one thermocouple’')q Simple (ideally Si – based)

Sànchez & Büttiker, PRB (2011) à Low T, coherent, feasible but extremely low power

3-Terminalthermoelectricratchet

| PAGE 33

Theend

| PAGE 34

q Variable Range Hopping à inelastic, large window for e-h asymmetry

q Parallel nanowires à high Seebeck AND good power factor Qq Gate-controlled local heat exchanges (hot spots cooling)q 3-Terminal thermoelectric ratchet:

1. Simple setup, energy harvester/cooler2. Good power factor Q and electronic ZT (0.1 - 100)

Riccardo Bosisio Jean-Louis PichardGeneviève Fleury

Aller à l’idéal et comprendre le réel.

Jean Jaurès

Thefuture

Wishlist

• Rigorous treatment of e-ph coupling

• Combining DFT calculations of γe and γep

• Non-linear regime (higher output powers), multi-terminal setups

• Heat-to-spin conversion I: magnetism and/or spin-orbit in wires

• Heat-to-spin conversion II: hot magnon bath

• Time-dependent configurations.

| PAGE 35

InelasticRegime:Thermopower fluctuations

R. Bosisio, C. Gorini, G. Fleury, & J.-L. Pichard, New J. Phys. 16:095005 (2014)

Vg = 0

Vg = 2.2t

Numerical simulationsfor kT=t and W=t

Bulk of the band Band Edge

| PAGE 36

Parallelnanowires:HotSpotcooling

| PAGE 37

ParasiticPhononContributiontoZT

(For doped Si-NWs and SiO2 substrate)

| PAGE 38

Parallelnanowires:addenda

| PAGE 39

DensityofstateswithCoulombgap

∆ ?

| PAGE 40

Thermopower Enhancement:experimentalresults

GroupofPhilipKim(preprint2013)SimilardatainthegroupofF.Giazotto orH.Riel

Largelyenhancedthermopower nearthebandedgeofthenanowire

(Fieldeffect transistordevice configuration)

(ExperimentsmainlycarriedoutatRoomTemperature)| PAGE 41

Linearresponse:Onsagerformalism

(Two-terminal) Coupled 1D charge and heat transport

Fluxes

Affinities (forces)

Positivity of entropy production Onsager relation (TRS)

Transport coefficients & Onsager matrix elements

| PAGE 42

ElasticRegime:validityofSommerfeld expansion

Validity of Sommerfeld Expansion

Q: Validity of the Mott’s formula for S

Wiedemann-Franz (WF) law, Mott’sformula

Sommerfeld temperature isproportional to the mean energy

level spacing in the system:

(Proportionality constant dependson required precision)

Result for a tunnel barrier:

Estimation for Si nanowire: 100 mK| PAGE 43

ElasticRegime:Typicalthermopowerofatunnelbarrier

In the basis

σ = nonvanishingpart of Σ

| PAGE 44

Three-terminalquantumthermalmachines

Efficiency: depends on the sign of the heat currents

Carnot limit

Not a function of the temperatures only !!

Efficiencies at maximum power

Generalized figures of merit

F. Mazza, R. Bosisio, G. Benenti, V. Giovannetti., R. Fazio & F. Taddei, New J. Phys. (2014)| PAGE 45

GATE MODULATED CARRIER DENSITY

ln G(V gate) at three temperatures in a large gate voltage range(details of the conductance pattern are not seen for this gate voltage sampling)

Inset: the same curve in a linear scale. Note the linearity at voltages above the transition.

ConductorAnderson Insulator

Edge of the impurity band = -2,5 V (Complete depletion of the disordered nanowire)

| PAGE 46

CONDUCTANCE FLUCTUATION S INDUCED BY VARYING THE GATE VOLTAGE

Bulk of the impurity band Edge of the impurity band

Mott Formula :

Larger thermopower at the band edges

𝑺 ≈𝝏𝐥𝐧(𝑮)𝝏𝑽𝒈

| PAGE 47