Transport out of locally broken detailed balance

0.16 MB·

by  Rafael Sánchez

Chapter 1
Transport out of locally broken detailed balance
7
RafaelSa´nchez
1
0
2

n
a
J

1
1

]
l
l
a AbstractElectronsmovealongpotentialorthermalgradients.Inthepresenceofa
h globalgradient,appliede.g.tothetwoterminalsofaconductor,thisinduceselec-
-
s tricchargeandheatcurrents.Theycanalsoflowbetweentwoequilibratedterminals
e (atthesamevoltageandtemperature)ifdetailedbalanceisbrokeninsomepartof
m
thesystem.Aminimalmodelinvolvingtwometallicislandsinseriesisintroduced
. whoseinternalpotentialandtemperaturescanbeexternallymodulated.Thecondi-
t
a
tionsforafiniteelectricflowarediscussed.
m
-
d
n
1.1 Introduction
o
c
[
Anelectronicconductorrespondstoanonequilibriumsituationintheformofcharge

1 and heat currents. It can be due to the presence of electric or thermal gradients
v appliedtothetwoterminalsofthesystem,V −V andT −T .Transportishowever
L R L R
1
notrestrictedtothissituation.Theroleoffluctuationsintheintermediateregion(the
7
system)cannotbeoverlooked.Ifthetwoterminalsareinthermalequilibrium(for
0
3 beingatthesametemperatureandpotential)acurrentcanneverthelessflowdueto
0 rectifiednoise.
. TherelevanceofnoisystateswaspointedoutbyLandauerformodifyingtherela-
1
0 tiveoccupationofbistablepotentials[1].Thisisthecaseforinstanceiftemperature
7 is locally increased (with an ideal blowtorch) on one side of the barrier only [1].
1 These ideas were later applied to transportconfigurationsin a series of papers by
:
v Bu¨ttiker[2],vanKampen[3] andLandauer[4].Theypredictclassicalparticlesto
i flowalongsymmetricpotentialswhicharesubjecttostate-dependentnoise.Asan
X
example,particlesovercomeapotentialbarrierfromthesidewherethediffusionco-
r
a efficientishigher(duee.g.toalocallyincreasedtemperature,cf.Fig.1.1),leading
RafaelSa´nchez
InstitutoGregorio Milla´n,Universidad Carlos III deMadrid, 28911Legane´s, Madrid, Spain, e-
mail:[email protected]
1
2 RafaelSa´nchez
Fig. 1.1 Transport in the
presenceofstate-dependent
noise.Symmetricpotential
V(x)
and temperature distribu-
T(x)
tions give nevertheless a
finiteparticlecurrentiftheir
respective maximaaredis-
placed[2,3,4].
toabrokendetailedbalancesituation.Theresultingcurrentsdependonthesepara-
tionofthepotentialandtemperaturefields:symmetricstate-dependentpotentialand
temperaturedistributionsgivenotransport[2].OfcourseOnsagerrelationsarestill
fulfilled[5]ifonetakesintoaccounttheheatinjected(bytheblowtorch)inorderto
keepthetemperaturestationary.
Similarideashavemotivatedproposalsofthermalratchets[6,7]andthermoelec-
tricmetamaterials[9],andthemeasurementofradiationinducedcurrents[8,10]or
transverserectificationinsemiconductor2DEGs[11,12].Theyarecloselyrelated
totherecentfieldofmesoscopicthree-terminalthermoelectrics[13,14]whichdis-
cussestheconversionofaheatcurrentoutathermalreservoirintoachargecurrent
inanequilibriumconductor[15,16,17,18,19,20,21,22,23,24,25,26,27].Also
noisegeneratedeitherinacoupledconductor[28,29,30]orexternally[31,32]is
used for this purpose.Most of these cases use discrete levels in quantumdot sys-
tems where a temperature cannot be properly defined [33]. Another possibility is
a thermocouple configuration [34, 35, 36, 37, 38] where the current is due to the
separationofelectron-holeexitacionsatthetwocontactsofahotcavity.
Hereaminimalmodelofamesoscopicconductorisproposedwhereallthenec-
essary ingredientscan be engineeredandcontrolledexperimentally.Itis based on
the discretization of a metallic conductor by means of three tunnel junctions, cf.
Fig. 1.2. This way an array of two small metallic islands are formed whose level
Fig.1.2 Sketchofoursys- Γ1L Γ21 ΓR2
tem. Two metallicislands T1 T2
aretunnel-coupledinseries ΓL1 gate1 Γ12 gate2 Γ2R
inatwo-terminalconfigura-
tion.Electronsareassumedto
thermaliseatadifferenttem- CL CC CR
peratureT ineachpieceof VL,T φ1,T1 φ2,T2 VR,T
i
theconductor.Theislandin-
ternalpotentialscanbetuned
bygatevoltagesVg,i.Thetwo Cg1 Cg2
terminalsareconsideredatthe
samevoltage,VL=VR=0, Vg1 Vg2
and temperature, T. The
epxiepceerwieinsceeCdobuylotmunbngealipngUji U1L U21 U2R
electronscanbemodulated
withthegatevoltagesV . L 1 2 R
gk
1 Transportoutoflocallybrokendetailedbalance 3
spacingismuchsmallerthanthethermalenergy,D E≪k T.Theelectron-electron
B
relaxationrateisfastsuchthattheyequilibratetoaFermidistributionfunction.This
way, each island has a well defined temperature,T. A physicalmesoscopic blow-
i
torchcanbedefinedthatmodulatesthetemperatureofthesystemlocally.Thiscan
bedoneeitherbyintroducingtimedependentdrivings[39,40,41],orbyusingeither
on-chiprefrigerators[42,43,44,45,46]orthenoisegeneratedinacoupledconduc-
tor[47,28].Inthefirstcase,thefrequencyofthedrivingmustbemuchlargerthan
the energyrelaxationrate suchthatelectronshave anincreasedeffectivetempera-
tureintheisland.Inthesecondcase,itisimportantthattherefrigeratorexchanges
heatbutnochargewiththeconductor[15].Theinternalpotentialofeachislandcan
beexternallymodulatedbymeansofgatevoltages.Thisway,bothtemperatureand
potentialprofilescanbespaciallyresolvedandtunedinasimpleconfiguration.
1.2 Two Coulombislandsinseries
Ourmodelisbasedonthewellknownphysicsofsingleelectrontunnelingatsmall-
capacitancetunneljunctions[48].Metallicislandsofamicrometersizecanbesep-
aratedbyinsulatingbarrierssuchthattheenergycostforaddingan extraelectron
(the charging energy, E) is larger than the thermal energy E ≫k T. This intro-
i i B
duceselectron-electroncorrelationsthatsuppressesthelowvoltagetransport,what
is called Coulomb blockade. The extra energy can be supplied by a gate voltage
V such that electrons can flow one by one throughthe island giving rise to con-
gi
ductance[49,50,51]andthermopower[52,53,54,55,56]oscillations.Electronic
coolingbasedonthesepropertieshasbeenrecentlyproposed[57]andrealizedex-
perimentally[58]insingleelectrontransistors.
Consideranarrayofthreetunneljunctionsformingtwoislands[59,60,61,62],
assketchedinFig.1.2.EachbarrierisdescribedbyacapacitanceC andaresistance
i
R in parallel. The former determine the electrostatic potential of each island, f ,
i i
which is obtainedself-consistentlyfromthe equationsfor the chargeaccumulated
ineachisland:
Q =C (f −V )+C (f −f )+C V (1.1)
1 L 1 L C 1 2 g1 g1
Q =C (f −V )+C (f −f )+C V . (1.2)
2 R 2 R C 2 1 g2 g2
The electrostaticpotentialin the coupledsystem isU(Q ,Q )=(cid:229) dQf . Itde-
1 2 i i i
fines the charging energies Ei = e2/2C˜i, with C˜i = CS 1CS 2−CC2 R/CS i¯ (where i¯
denotes the island next to i), in terms of the total g(cid:0)eometric cap(cid:1)acitance CS i =
Ca i+CC+Cgi of each island i (with a 1 =L and a 2 =R), the centre capacitance
C ,andtheonecouplingtothegate,C .Thereisacrossedchargingenergyrelated
C gi
totheoccupationofthenextisland,parametrizedasJ=2e2CC/ CS 1CS 2−CC2 .It
correspondstotherepulsionofelectronsindifferentislands.The(cid:0)internalpoten(cid:1)tial
oftheislandsisfurthermoretunedbythecontrolparameters
4 RafaelSa´nchez
1 4E E J
n = 1 2 Qext− Qext , (1.3)
gi eJ2−4E E (cid:18) i 4E i¯ (cid:19)
1 2 i
withtheexternallyinducedcharges[50]Qext=C V +C V andQext=C V +
1 L L g1 g1 2 R R
C V .Hereweareinterestedinthecurrentgeneratedinanunbiasedconfiguration,
g2 g2
such that V =V =0. Hence, n depends only on the gate voltages. Note that
L R gi
each gate affectsthe two islands. Single electron pumpsbased on time dependent
modulationofthegatesachievemetrologicalprecisions[60,61].
AllthesecanbecastintotheelectrostatictermoftheHamiltonianofthesystem:
H=(cid:229) E(n −n )2+J(n −n )(n −n ), (1.4)
i i gi 1 g1 2 g2
i
wheren =Q/eisthenumberofelectronsineachisland.Letusrestrictheretothe
i i
simplest configurationwith up to one extra electron in the system, i.e. the charge
configurationofthesystemisdescribedbythethreestates(n ,n ):(0,0),(1,0)and
1 2
(0,1).
TunnelingeventsarecharaterizedbytheenergycostU (theCoulombgap)for
ji
electronstunnelingfromregionito j,whichread:
1
U = 2E −n −Jn (1.5)
1L 1 g1 g2
(cid:18)2 (cid:19)
1
U = 2E −n −Jn (1.6)
2R 2 g2 g1
(cid:18)2 (cid:19)
U = E −E (1.7)
21 2R 1L
in the absence of a bias voltage. Obviously, U =−U . We emphasize that the
ij ji
energetics of the mesoscopic junction is fully tunable with the gate voltages. The
ratesforthecorrespondingtunnelingtransitionsaregivenby:
1
G = dEf(E,T)[1−f(E−U ,T )], (1.8)
ji e2RjiZ i ji j
whereR =R ,R orR ,dependingontheinvolvedjunction.Notethattheresis-
ji L C R
tancesRiareenergyindependentparameters.Here f(E,T)= 1+eE/kBT −1isthe
Fermi-DiracdistributionforaregionattemperatureT. (cid:0) (cid:1)
Withthese,onecanwritetherateequations:
P˙ =G P +G P −(G +G )P
(0,0) L1 (1,0) R2 (0,1) 1L 2R (0,0)
P˙ =G P +G P −(G +G )P (1.9)
(1,0) 1L (0,0) 12 (0,1) L1 21 (1,0)
P˙ =G P +G P −(G +G )P ,
(0,1) 21 (1,0) 2R (0,0) 12 R2 (0,1)
forthedynamicsoftheoccupationprobabilityofeachstate,Pa (t).
ThestationaryoccupationsareobtainedbysolvingP˙=0,giving:
P =G −2(G G +G G +G G )
(0,0) T 21 R2 L1 12 L1 R2
1 Transportoutoflocallybrokendetailedbalance 5
P =G −2(G G +G G +G G ) (1.10)
(1,0) T 1L R2 12 1L 12 2R
P =G −2(G G +G G +G G ),
(0,1) T L1 2R 21 1L 21 2R
withG T2givenbythenormalizationcondition,(cid:229) a Pa =1.Thestationarystateobeys
detailed balance if tunneling transitions satisfy: G P =G P , for the left
1L (0,0) L1 (1,0)
G P =G P ,forthecenter,andG P =G P ,fortherightjunction.
12 (1,0) 21 (0,1) 1R (0,0) R1 (0,1)
Hence,thecurrentintherightterminal:
I =e G P −G P , (1.11)
R R2 (0,1) 2R (0,0)
(cid:0) (cid:1)
gives a measure of the deviation from detailed balance for processes at the right
barrier.Injectedcurrentfromtheterminalisdefinedaspositive.UsingEqs.(1.10)
resultsintheexpression:
e
I = (G G G −G G G ). (1.12)
R G 2 1L 21 R2 L1 12 2R
T
Note that the first term in the right hand side of Eq. (1.12) is proportionalto the
rateforanelectrontobetransportedfromtherighttotheleftterminal.Thesecond
termisproportionaltotherateoftheoppositeprocess.Itisthenclearthatthecase
ofhavingtransitionssatisfyinglocaldetailedbalanceateveryjunctionresultsinno
netcurrent.In the unbiasedand isothermalconfiguration,this translatesto having
tunnelingratesrelatedbyG ij=G jieUij/kBT.Thisisnotthecaseifoneoftheleadsor
islandsisatadifferenttemperature,asdiscussedinthenextsection.
1.2.1 Brokendetailedbalance. Linear response
Letusfirst emphasizetheimportanceofthe Coulombgapintroducedbythe elec-
tronicconfinementintheisland.IfU =0,tunnelingiselectron-holesymmetricand
ji
atemperaturedropacrossthejunctionisnotsufficienttobreakdetailedbalance.It
canbeeasilyshownfromsymmetriesoftheFermifunctionthat:
¥ ¥
dEf(E,T)[1−f(E,T′)]= dEf(E,T′)[1−f(E,T)]. (1.13)
Z−¥ Z−¥
HenceG =G ,forU =0,independentlyofthetemperaturesT andT .
ji ij ji i j
Inthefollowing,thecasewithfiniteU isassumed,unlessexplicitelymentioned.
ji
SomeoftheratesEq.(1.8)canbeanalyticallycalculatedbyusingtherelation:
E E′
f(E,T)[1−f(E′,T′)]=n − [f(E′,T′)−f(E,T)], (1.14)
B(cid:18)k T k T′(cid:19)
B B
wheren (x)=(ex−1)−1istheBose-Einsteindistributionfunction.Fortransitions
B
betweenregionsatthesametemperatureonegets:
6 RafaelSa´nchez
U U
G (0)=− ji n ji eUji/kBT (1.15)
ji e2R B(cid:18)k T(cid:19)
ji B
and,ontheotherhand:
U U
G (0)=− ji n ji , (1.16)
ij e2R B(cid:18)k T(cid:19)
ji B
forthereversedprocess.Itisstraightforwardtocheckthatlocaldetailedbalanceis
satisfiedatsuchajunction.Iffurthermorethegatesaretunedsuchthatthereisno
energycost,U =0,tunnelingisgovernedbythermalfluctuations:G (0)=G (0)=
ji ji ij
kBT .
e2Rji
Thisisnotthecasewhenthejunctionseparatestwopiecesofthemetalwhichare
atdifferenttemperatures.Thiseffectshowsupin thelinearregime.Consideringa
smalltemperaturedifferenced T =T −T,onecanlinearizetheinvolvedtunneling
j i
rates,G ≈G (0)+d TG (1)withG (1)= 1 J(U /k T)and:
ji ji T ji ji e2Rji ji B
p 2
J(x)=ex[n (x)]2 +x2+Li (−ex)+Li −e−x , (1.17)
B 2 2
(cid:20) 6 (cid:21)
(cid:0) (cid:1)
withthe dilogarithmfunctionLi (z)=(cid:229) ¥ zk.Forthesame gradientandtheop-
2 k=1k2
positetransition:G (1)=G (1).Hence:
ij ji
G d T
G ji ≈eUji/kBT+ T F(Uji), (1.18)
ij
doesnotsatisfydetailedbalance,whereF(U )=U−1J(U /k T)[n (U )]−2.Note
ji ji ji B B ji
that the effect appearsin the simultaneousoccuranceof a temperaturedrop and a
Coulombgap.
Using the above relations in the expression for the charge current, Eq. (1.12),
onefindsthatbreakingdetailedbalanceinasinglebarrierisenoughtohaveafinite
current,evenifthetwoterminalsareatthesamevoltageandtemperature.
To be more specific, let us consider the case where only the first island is at a
differenttemperature,T =T+d T,withT =T.Thenacurrentisgenerated:
1 2
d T
I (cid:181) e−U21/kBTF(U )−eU1L/kBTF(−U ) , (1.19)
R 21 1L
T
h i
intermsoftheenergycostfortunnelingtothehotislandfromtheleftleadandfrom
theotherisland,U and−U ,respectively.Theavoidedprefactordependsonthe
1L 21
equilibriumrates,G (0).Itisclearfromtheaboveexpressionthatnocurrentwillflow
ji
ifdetailedbalanceisbrokensymmetricallyatthetwobarriersofthehotisland,i.e.
iftheenergycostisequal:U =−U . Thisway,tuningthegatevoltagesallows
1L 21
onetocontrolthecurrent.
1 Transportoutoflocallybrokendetailedbalance 7
(a) (b) 5
I[10−4(e2R0)−1]
-5 0 5
−1]
1 )
R0 0
2e
(
−410 T1100[mK]:
[
I 150
C -5 200
250
ng2 0.5 300
B 0 0.5 1
ng2
A A:
B:
0
0 0.5 1 C:
ng1
Fig.1.3 Transportfromahotspot.(a)Currentasafunctionoftheislandcontrolparameters,n
gi
foranunbiasedconfigurationVL=VR=0andT =T2=200mK.Thetemperatureofisland1is
increasedatT1=150mK,withRL=RC=RR=R0,E1=E2=0.15meV,andJ=0.05meV.The
configurationsinpointsA,BandCaresketchedontherightbottomside.(b)Cutalongthewhite
dashedlinein(a)asafunctionofng2.Island1istunedsuchthatU1L=0.015meVisconstant.
ReversingthetemperaturegradientorthesignofU changesthesignofthecurrent.
2R
1.3 Transport from a hotspot
As discussed in the previous section, broken detailed balance occurs in tunneling
throughajunctionseparatingregionsatdifferenttemperatures.Thisishowevernot
a sufficient condicion to generate transport in a conductor in global equlibrium.
Forexample,nocurrentwilloccurinasystemconsistingonasinglehotisland,as
brokendetailedbalanceissymmetricinthetwojunctions.Inordertohaveacurrent,
anasymmetryneedstobeintroduced,e.g.bymakingtheenergycostfortunneling
throughthetwobarriersdifferent.Inourcase,thisistheroleofthesecondisland.
ItschargingenergyliftstheasymmetrymakingU andU different.Furthermore,
1L 12
thisasymmetrycanbetunedwithgatevoltagesV andV ,asdiscussedabove.If
g1 g2
thesecondislandisatthesametemperatureasthetwoterminals,T =T,detailed
2
balanceissatisfiedatthethirdjunction,withG 2R=G R2eU2R/kBT.
ThecurrentgeneratedinsuchaconfigurationwithT 6=T isplottedinFig.1.3(a)
1
asafunctionofthecontrolparametersn andn .Fixingthegatevoltageinisland
g1 g2
1,thecurrentchangessignwhenthegatevoltageofisland2istuned.Asdepicted
in the inset, the sign of the current depends on the sign of U +U : electrons
1L 21
tunneling out from the hot island into the region i giving the largest U have a
1i
largerrate.ForU ≫k T,thedifferenceofthetworatesisexponentiallysmalland
1i B
thereforecurrentissuppressed.
8 RafaelSa´nchez
Fig.1.4 Anarrayofislandsofdifferentsizeinthepresenceofalocaltemperatureincrease.The
chargingenergyofeachislanddependsonitssize,henceintroducinganspacialmodulationofthe
Coulombgap(blacksolidlines).Thepiece-wisetemperaturedistribution(reddottedlines)allows
forawiderextensionduetoheatleakingtoneighbourislands.
Reversing the sign of the local temperature gradient, T −T, changes the sign
1
of the current, cf. Fig. 1.3(b). Being an obvious statement, it has practical conse-
quences:ratherthanbyheatingoneoftheislands,theeffectcanbeexperimentally
detectedbycoolingit.Thiscanbedonelocallyandnon-invasivelywithinnowadays
technology[44,46].
Aparticularlyinterestingconfigurationiswhenn istunedsuchthatU =0.
g2 2R
ThisisconfigurationBinFig.1.3.Then,G =G =k T/(e2R )onlycontribute
R2 2R B R
to the prefactor of the current and the second island plays no role. In this case,
U =U , i.e. the temperaturegradientand the energycost are the same for tun-
1L 12
nelingthroughbothjunctionsofisland1.Henceelectronsinthehotislandhaveno
preferreddirectiontotunnelout.Thusdetailedbalanceissymmetricallybrokenin
the two junctions, with the overall rate through the island being the same in both
directions:G G =G G ,andthecurrentiszero,seeEq.(1.12).
1L 21 12 L1
1.4 Multijunction arrays
Onecanenvisiontoextendtheaboveresultstolongerarrays,cf.Fig.1.4.Thisre-
laxestherequirementtoincreasethetemperatureinamicrometer-sizesingleisland.
Thesizeoftheislandscanbecontrolledinthesamplegrowth,whichintroducesa
natural way to spacially modulate the Coulomb gap in the device. This way the
needtogatethesystemcanalsobeavoided.Thecurrentistheninducedbythein-
terplayoflocalnon-equilibriumandelectron-electroninteractions,emphasizingthe
mesoscopicnatureofthedevice.
1.5 Summary
Asimplemodeloflocalbrokendetailedbalancegivingrisetotransportinanelec-
tronic conductoris presented.A system of two metallic islands, one of which ex-
periences a differenttemperature,can be interpreted as a mesoscopic analogue of
transportinstate-dependentdiffusionatasinglescatterer.Partitioningtheconductor
intoanarrayofmetallicislandsallowsforthelocalcontrolofvoltagesandtemper-
1 Transportoutoflocallybrokendetailedbalance 9
atures.ThecooperativeoccurranceofalocaltemperaturegradientandaCoulomb
gapintroducesapreferreddirectionfortunnelingelectrons.Thisasymmetry,which
determinesthesignofthecurrent,canbefurthermoretunedbymeansofgatevolt-
ages applied to each island. Broken detailed balance is of relevance e.g. for the
investigationoftheJarzynskiequalityandfluctuationtheorems[40,28,63,64,65],
andcanbedetectedinthefullcountingstatistics[66].
Theelectron-holeasymmetryrequiredforhavingathermoelectricresponseisin-
troducedbyinhomogeneoustunnelingchargingenergies.Thusenergy-dependence
of the barriers is not necessary. The involved technology is within nowadays
reach[60,61,62,44,40,46]andcouldreadilybetested.Thisproposalcontributes
tothefieldofinteraction-inducedthermoelectricproperties[57,58]andthecontrol
ofthermalflows[67,68]inlow-dimensionalmetallicconductors.
Acknowledgements I acknowledge Markus Bu¨ttiker and Jukka P. Pekola for discussions and
commentsonanearlierversionoftheresultspresentedhere.WorksupportedbytheSpanishMin-
isteriodeEconom´ıayCompetitividadviagrant No.MAT2014-58241-P. IalsothanktheCOST
ActionMP1209“Thermodynamicsinthequantumregime”.
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