Charge transport in a Tomonaga-Luttinger liquid effects of p(6)
发布时间:2021-06-08
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)
(ω
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.