Charge transport in a Tomonaga-Luttinger liquid effects of p(6)

发布时间:2021-06-08

We study the current produced in a Tomonaga-Luttinger liquid by an applied bias and by weak, point-like impurity potentials which are oscillating in time. We use bosonization to perturbatively calculate the current up to second order in the impurity potent

Theaboveexpressionsuggeststhatthecurrentwillbemaximizedifeither2kFxp+φpor2kFxpallp.Thismeansthatthe φphasthesamevalueforpotentialsinEq.(3)shouldbeoftheformU

pcos(ωt 2kFxp)orUpcos(ωt+2kFxp);thisdescribesapotentialwavetravelingtotherightortotheleft.Suchawavehasbeenstudiedextensivelyforthecaseofnon-interactingelectrons;seeRefs.[20,21,22,31,32,33,34,38]and[40,41,42,43].

Anunusualphenomenonoccursiftheinteractionsaresu cientlystrong,i.e.,ifK<1/2.Ifthereisnobias,theDCpartofthecurrentgenerallygoesasω2K 1whichincreasesasωdecreases.However,itisclearthatifωwasexactlyzero(time-independentimpurities),thenthecurrentwouldalsobezero.Thesetwostatementsimplythatthecurrentmustbeanon-monotonicfunctionofω,andmusthaveatleastonemaximumatsomevalueofω.FindingthelocationofthemaximumrequiresustogobeyondthelowestorderperturbativeresultsofthisFIG.1:(Coloronline)DCpartofthebackscatteredcur-rentasafunctionofthebiasω0forseveralimpurities,whenω±aresmall.Thered(dot),magenta(dash),blue(dash-dot)andblack(solid)linesshowtheresultsforK=1/4,1/2,3/4and1respectively.Wehave|Ptaken+φpUpei(2kFxpp)|2/|P

i(2kpUpeFxp φp)|2=2:1.

Figure1showsthedcpartofthebackscatteredcur-rentasafunctionoftheappliedbias,fora xednon-zerovalueofthepumpingfrequencyω,assumingthatωandω0aresmall.WehaveusedtheexpressioninEq.(31)toplotthevalueofIbs,dc(ω0)/Ibs,dc(ω0=0)asafunctionofω0/ω,forfourdi erentvaluesoftheparameter| pUpe

i(2kK=1/4,1/2,3/4Fxp+φp)2

example.ForK=1|/|/4, and1,takingtheratiothepUpei(2kFxp φp)currentdiverges|2=2:1asanatω0=±ωasmentionedabove.Wealsonotethelinearandpiece-wiseconstantdependencesofthecurrentonω0forK=1

6

and1/2respectively;thisisdiscussedinSubsec.III.Ebelow.

Ifwerelaxtheassumptionsthatωxrp/vandω0xrp/varesmall,thentheexactexpressions(uptosecondorderintheimpuritypotentials)fortheacanddccomponentofthebackscatteredcurrentaregivenby

Ipr=

qUpUr

√bs,dc

4πv2Γ(K)

αv

×[sgn 1/2 K

+)|ω+|K 1/2JK 1/2(|ω+xrp×+sgncos(2(ωk|/v)Fxrp+φrp)

k )|ω |K 1/2JK 1/2(|ω×cos(2 xrp|/v)Fxrp φrp)],

(32)

Iprbs,ac

=

qUpUr

√4πv2Γ(K)cos( αv

×[sgn(ω πK)

1/2 K

cos(2kFxrp)

+)|ω+|K 1/2JK 1/2(|ω+xrp×+sgncos(2(ωωt+φ|/v)p+φr+sgn(ω+)πK)

)|ω |K 1/2JK 1/2(|ω+φ xrpr+×{|cos(2ωωt+φ sgn(ω|/v)p )πK)

+|K 1/2J1/2 K(|ω+xrp|/v)

sin(2 |ωtω +|K 1/2J1/2 K(|ω×φ xrp|/v)}p+φr)].(33)

BesselfunctionJisdiscussedintheAppendix;us-apowerseriesexpansiongiventhere,wecanshowEqs.(32-33)reducetoEqs.(29-30)inthelimitrp/v→0.WenotethattheexpressionsinEqs.(27-donotchangeifwerelaxtheassumptionsthatωand0aresmall.

Figure2showsthedcpartofthebackscatteredcurrentasafunctionoftheappliedbiasforthecaseoftwoim-purities,labeled1and2,takingU1=U2,2kFx12=π/2,φ12= π/4,andωx12/v=1;thusωandω0arenotsmall,incontrasttothecaseshowninFig.1.WehaveusedtheexpressionsinEqs.(27)and(32)toplotthevalueofIbs,dc(ω0)/Ibs,dc(ω0=0)asafunc-tionofω0/ω,forfourdi erentvaluesoftheparameterK=1/4,1/2,3/4and1.WeseesomeoscillationsinFig.2duetotheappearanceoftheBesselfunctionsinEq.(32).ForK=1/4,weagainseedivergencesatω0=±ω.

E.

K=1and1/2

WenowdiscussthespecialcasesK=1and1/2wheretheexpressionsforsomepartsofthecurrentssimplifyconsiderably.

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