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Coulomb Blockade in a Quantum Dot Coupled Strongly to a Lead Hangmo Yi and C. L. Kane Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104 (September21, 1995) We study theoretically a quantum dot in the quantum Hall regime that is strongly coupled to a single lead viaapointcontact. Wefindthatevenwhen thetransmission throughthepoint contact isperfect,importantfeaturesoftheCoulomb blockadepersist. Inparticular,thetunnelingintothe dot via a second weakly coupled lead is suppressed, and shows features which can be ascribed to elastic or inelastic cotunneling through the dot. When there is weak backscattering at the point contact, both the tunneling conductance and the differential capacitance are predicted to oscillate as a function of gate voltage. We point out that the dimensionless ratio ξ between the fractional oscillations inGandC isanintrinsicpropertyofthedot,which,inprinciple,canbemeasured. We 6 compute ξ within two models of electron-electron interactions. In addition, we discuss the role of 9 additional channels. 9 1 n a I. INTRODUCTION have a simple, well organizedstructure, which is insensi- J tive to the complicating effects of impurities and chaotic 6 The Coulomb blockade occurs in an isolated meso- electrontrajectories. Inthispaperweshallstudyaquan- scopicislandwhenthe capacitivechargingenergyto add tum dot in the integer quantum Hall regime which is 3 v a single electron suppresses the discrete fluctuations in strongly coupled to a lead via a quantum point contact 9 theisland’scharge.1Inthepastseveralyearsthisphysics with one (or a few) nearly perfectly transmitting chan- 3 has been studied extensively in both metallic systems2 nels. 1 andinsemiconductorstructures.3TheCoulombblockade In a recent paper, Matveev7 has considereda dot con- 9 ismosteasilyprobedbymeasuringtransportthroughan nectedtoasingleleadbyanearlyperfectlytransmitting 0 island which is weakly coupled to two leads. By varying point contact. When the transmission of the point con- 5 a gate voltage V , which controls the chemical potential tactisperfect(zerobackscattering),heshowedthateven 9 G / oftheisland,peaksintheconductanceareobservedeach in the presence of a substantial Coulomb energy U, the at timeanadditionalelectronisadded. Betweenthepeaks, differential capacitance C = dQ/dVG is independent of m theconductanceisactivated,reflectingtheCoulombbar- VG. Thus,theequilibriumchargeQontheislandismax- rier to changing the number of electrons on the dot. imally un-quantized, since charge is added continuously - d The coupling to the island via tunnel junctions intro- as the gate voltage is changed. He then showed that n ducesfluctuationsonthedotwhichrelaxthediscreteness the presence of weak backscatteringat the point contact o of its charge. When the conductance σ of the junctions leads to weak oscillations in the differential capacitance c : is verysmall, σ ≪e2/h, this effect is weak andgivesrise asafunctionofgatevoltagewithaperiodcorresponding v to the well known cotunneling effect, in which electrons to the addition of a single electron. These oscillations Xi may tunnel virtually through the Coulomb barrier.4,5 signal the onset of the quantization of the equilibrium Provided one is not too close to a degeneracy between charge on the island. r a differentchargestates,thisphysicsmaybedescribedsat- In addition to the capacitance measurements,7,8 the isfactorily within low order perturbation theory in the fate of the Coulomb blockade in an island connected to tunnel coupling. a single lead via a point contactcanbe probed by trans- For stronger coupling to the leads, the perturbative port,providedanadditionalleadispresent. Thisleadsus analysis is no longer adequate, and a quantitative de- totheinterestingpossibilityofsimultaneouslymeasuring scription of the problem becomes much more difficult. the conductance (a transport property) and the capac- Based on general arguments, the Coulomb blockade is itance (an equilibrium property) of a quantum dot. In expected to be suppressedwhenσ ≈e2/h,6 since inthat this paper, we build on Matveev’s work and compute, in regime, the RC decay time of the island leads to an un- addition to the differential capacitance, the tunnel con- certainty in the energy which exceeds the Coulomb bar- ductanceandI-V characteristicsforanislandconnected rier. However, this argument is unable to predict quan- to one lead with a nearly perfectly transmitting point titatively the nature of the suppression of the Coulomb contact and connected very weakly to another lead. blockade. Specifically, we consider the quantum dot depicted Aparticularlywellsuitedsystemto study this physics schematically in Fig. 1. The center region surrounded is a semiconductor quantum dot at high magnetic fields. by the gates forms a quantum dot, which is connected IntheintegerquantumHalleffectregime,thestatesnear to the leads on both sides. In the ν = 1 quantum Hall the Fermi energy of a quantum dot are edge states, and regime, there is a single edge channel going around the dot. The contact to the left lead is a tunnel junction, 1 characterized by a small tunneling amplitude t, which is theory of elastic cotunneling through a one dimensional controlled by the voltage on gate A. The right contact system. Evidently, the cotunneling theory is more gen- is a nearly perfectly transmitting point contact charac- eral than its derivation within weak coupling perturba- terized by a small backscattering amplitude v, which is tion theory suggests. In addition, when the temperature controlled by the voltage on gate B. The backscattering T or the voltage bias eV exceeds ∆E, we find behav- at the point contact involves tunneling of electrons be- ior analogous to inelastic cotunneling. In particular, for tween the opposite moving edge channels at x = L and ∆E ≪ T ≪ U the tunneling conductance is suppressed x = −L, where x is a parameter specifying the spatial by(T/U)2. Moreover,for ∆E ≪eV ≪U,the tunneling coordinate along the edge channel. current varies as V3. Inelastic cotunneling behavior has also recently been discussed by Furusakiand Matveev10 for a quantum dot stronglycoupledto two leads. While similar in many re- spects, it is worthwhile to distinguish the present work from that in Ref. [10]. In that reference, the two point contacts are considered to be independent. An electron passingthroughonecontactcannevercoherentlypropa- gatetothesecondcontact. Elasticcotunnelingcanthere- L t fore not be describedwithin that framework. In general, 0 0 B v the elastic cotunneling will depend on the complicated -L transmission matrix of the sample. In contrast, in the quantum Hall regime, the edge channels have a particu- larly simple structure and directly connect the leads. If A B the phase coherence length can exceed the dot’s dimen- sions, then elastic cotunneling should be present. This physicsiscorrectlyaccountedforinouredgestatemodel. In addition, the beautiful results in Ref. [10] concerning the Coulombblockadefor spindegeneratesystemsis not FIG. 1. Schematic view of a quantum dot connected to applicable here, since the magnetic field destroys that leads on both sides. Negative voltage on the gates (shaded degeneracy. area)confineselectronsinthedot. Thicksolidlinesshowthe In addition to the limiting behaviors discussed above, edge channels formed by strong magnetic field B, where the we find that at low temperature T ≪ ∆E, there can directionoftheelectronmotionisdepictedbyarrows. Dashed be nontrivial structure in the I-V characteristic for V ≈ lines show tunneling paths, where the tunneling amplitude t ∆E. In particular, we find steps in the differential con- and thebackscattering amplitudev are controlled byvoltage ductance, dI/dV as a function of bias voltage. In the on gate A and gate B, respectively. weakcouplinglimit, the existence ofsuchsteps has been discussedbyGlattli.5 Theyareaconsequenceofinelastic A crude estimation of the Coulomb energy with the cotunnelingwhenthemanybodyeigenstatesofthequan- abovegeometrygives U =Ke2/ǫL,where K is a dimen- tum dot are discrete. Observation of such steps could sionless geometrical factor and ǫ is the dielectric con- provide a new spectroscopy of the low lying many body stant of the semiconductor material. Another geometry- states of a quantum dot. Remarkably, as we shall show dependent energy scale is the level spacing between in Sec. III, these steps can remain sharp in the strong edge states in an isolated dot, ∆E = π¯hv /L. While coupling limit, even though it becomes meaningless in F there are few reliable estimates of the edge state veloc- that limit to speak of discrete single particle states. ity v , the ratio ∆E/U is typically of the order 0.1 in When the transmission through the point contact is F GaAs/AlGaAs heterostructures in the integer quantum lessthanperfect,chargequantizationisintroducedinthe Halleffectregime.9Thesmallnessofthisratioisassumed dot. The rigidity ofthe quantizationgrowswith increas- throughout this paper. ing amplitude of the backscattering at the contact. For Thoughtheequilibriumchargeonthedotisnotquan- weakbackscattering,thisisreflectedinoscillationsinthe tized when the transmission at the point contact is per- differentialcapacitanceasafunctionofgatevoltage,with fect, we nonetheless find that important features of the an amplitude proportional to the backscattering matrix Coulomb blockade remain in the tunneling characteris- elementv. Wefindthatsimilaroscillations,proportional tics. Remarkably,mostofthesefeaturescanbeexplained to v, should be present in the conductance. A compar- within the usual weak coupling model. Specifically, we ison of these oscillations should therefore provide infor- find thatatzero temperature,the Ohmic conductance is mationabouttheintrinsicstructureofthequantumdot. suppressed by a factor of (∆E/U)2 below its noninter- We focus on the ratio of the fractionaloscillations in the acting (i.e. U = 0) value. This suppression of the tun- conductance to that of the capacitance, nel conductance is precisely ofthe formpredicted by the 2 G1/G0 In the absence of interactions, it is a simple matter ξ = . (1.1) C /C to describe this system in terms of the free electron 1 0 edge state eigenstates. Such a description is inconve- G0 andC0 aretheaveragecapacitanceandconductance, nient, however,for describing effects associated with the whereasG1 andC1 arethe amplitudes ofthe oscillations Coulomb blockade, which are due to the presence of a as a function of gate voltage. By considering the frac- Coulomb interaction. The bosonization technique, how- tionaloscillations,G1/G0 andC1/C0,thedependenceon ever, allows for an exact description of the low energy the tunneling matrix element t necessary to measure G physics in the interacting problem.12 andthe capacitiveleverarmη necessaryto measureC is At energies small compared to the bulk quantum Hall eliminated. Moreover, since in the weak backscattering energy gap, the many body eigenstates are long wave- limit both quantities are proportional to v, the depen- length edge magnetoplasmons. These may be described dence on v is eliminated by taking their ratio. Thus, ξ asfluctuationsintheonedimensionaledgedensity,n(x), measures an intrinsic property of the strongly coupled where x is a coordinate along the edge. Following the quantum dot which is independent of the details of the usual bosonization procedure, we introduce a field φ(x), tunneling matrix elements. We have computed ξ within such that n(x) = ∂ φ/2π. The Hamiltonian, which de- x a constant interaction model, in which the Coulomb in- scribesthe compressibilityofthe edge maythenbe writ- teraction couples only to the total number of electrons ten, on the dot. We find that ξ ≈1.59 for T,∆E ≪U, independent of T, ∆E and U. More generally, ξ will depend H = dxvF(∂ φ)2. (2.1) on the specific formof the electron-electroninteractions. 0 4π x Z Withafewmodifications,theaboveconsiderationscan ThedynamicsoftheedgeexcitationsfollowfromtheKac be extended to the case in which there are more than Moody commutation relations obeyed by φ, onewelltransmittedchannels. We find thatthe lowbias linear conductance is less suppressed as the number of ∂ φ(x) channels N increases. It has been recently shown that x ,φ(x′) =iδ(x−x′). (2.2) 2π thesuppressionfactorbecomes(∆E/U)2/N inaconstant (cid:20) (cid:21) interaction model.11 However, if we take it into consid- Using (2.1) and (2.2) it may easily be seen that ∂ n = t erations that the interaction strength may differ within v ∂ n,sothatthe edgeexcitationspropagateinasingle F x thesamechannelandbetweendifferentchannels,wefind directionatvelocityv alongthe edge. Inthis language, F that the factor still has the form (∆E/U)2. In addi- theelectroncreationoperatorontheedgemaybewritten tion, the ratio ξ depends sensitively on the form of the as ψ†(x)=eiφ(x). electron-electron interactions between different channels Equations (2.1) and (2.2) are an exact description of on the dot. It is equal to zero if different channels do a single edge channel of noninteracting electrons. We not interact and grows with increasing strength of the may easily incorporateinteractions into this description. inter-channel interactions. Thus, ξ is a measure of the Specifically, we consider a “constant interaction model” inter-channel interaction strength. in which the Coulomb interaction couples to the total This paper is organized as follows. In Sec. II, we de- number ofelectronsN onthe dot, whichdepends onthe scribe the model and derive the I-V characteristicequa- edge charge between x=−L and x=L, tion. In Sec. III we consider a single channel system 1 withaperfectlytransmittingpointcontact. Wecompute N = [φ(L)−φ(−L)]. (2.3) the current and discuss various features of the result. In 2π Sec.IV,weconsidertheeffectoftheweakbackscattering The self capacitance and the coupling to a nearby gate to the conductance in connection with the effect to the can then be described by the Hamiltonian, capacitance. We compute ξ within two specific models ofelectron-electroninteractions. InSec.V,wegeneralize U H +H = N2+eηV N, (2.4) the results of Sec. III and Sec. IV to multiple channel U G 2 G systems. Finally, some concluding remarks are given in where η is a “lever arm” associated with the capaci- Sec. VI. tance coupling to the gate. Since the interaction is still quadratic in the boson fields, it may be treated exactly in this representation. II. EDGE STATE MODEL Nowweconsidertunnelingbetweentheedgechannels. We consider the left lead in Fig. 1 to be a Fermi liquid. We begin in this section by describing our edge state Without loss of generality, we model it as another ν =1 modelofaquantumdotstronglycoupledtoasinglelead. quantumHalledge,characterizedbyabosonfieldφ with l Herewewillconsiderthecaseinwhichonlyasinglechan- a Hamiltonian H in (2.1). Tunneling from the left lead 0 nel is coupled to the lead by a nearly perfectly transmit- into the dot at x=0 is then described by the operator ting point contact. Later, in section V we will consider the case where more than one channel is transmitted. T =ei[φ(0)−φl(0)] ≡eiθ. (2.5) → 3 wherewedefineθ ≡φ(0)−φl(0). Forthereverseprocess, I = πet2 e−Em/kT |hn|T |mi|2δ(E −E −eV) T =exp−iθ. The left point contactmay be character- 2h¯ → n m ← ized by a tunneling Hamiltonian, Xm,n h −|hn|T |mi|2δ(E −E +eV) , (2.12) ← n m H =tcosθ (2.6) t i where |mi is an eigen state of the unperturbed Hamil- where t is the tunneling matrix element. tonian with energy E . Since the sum on n is over a m Similarly,tunneling atthe rightpointcontactinvolves completesetofstates,wecanre-expressthe aboveequa- transfer of electrons between x = −L and x = L. The tion as Hamiltonian describing these processes canbe expressed as πet2 I = P(eV) (2.13) 2h H =vcos[φ(L)−φ(−L)]=vcos2πN, (2.7) v where where v is the backscattering matrix element. It is convenient to eliminate the linear gate voltage P(E)= dteiEt eiθ(t),e−iθ(0) . (2.14) termin (2.4)by the transformationN →N−N0, where Z Dh iE N =eηV /(∆E+U)istheoptimalnumberofelectrons 0 G In order to compute P(E), it is useful to consider the on the dot. Our model Hamiltonian describing a quan- imaginary time ordered Green’s function, which may be tumdotwithasinglechannelcoupledtoaleadmaythen readily computed using path integral techniques. be written H =H [φ ]+H [φ]+ UN2 P(τ)≡ Tτeiθ(τ)e−iθ(0) . (2.15) 0 l 0 2 D E +tcosθ+vcos2π(N −N ). (2.8) The real time correlation function may then be deduced 0 by analytic continuation. The two terms in the commu- We will find it useful in our analysis to represent the tater in (2.14) lead to partition function as an imaginary time path integral. P(E)=P>(E)−P<(E) (2.16) The action corresponding to H is then given by 0 1 with S = dxdτ ∂ φ(v ∂ φ+i∂ φ). (2.9) 0 x F x τ 4π Z P>,<(E)= dteiEtP(τ →it±0+) (2.17) Since the remaining terms in the Hamiltonian depend Z only on θ and N, it is useful to integrate out all of the Limiting behaviorofthe tunneling currentmaybe de- otherdegreesoffreedom. Theresultingaction,expressed ducedanalyticallyfromtheasymptoticbehaviorofP(τ). in terms of θ(τ) and N(τ), is then given by In particular, at zero temperature, the Ohmic conductance is proportional to the coefficient of the 1/τ2 term. 1 θ(ω ) S = θ(−ω ) N(−ω ) G−1 n In addition, it is possible to compute P(E) numerically, tot 2Xiωn (cid:2) n n (cid:3) (cid:20) N(ωn)(cid:21) ainsiaspdpeesncdriibxeAd.inthefollowingsectionandinmoredetail + dτ[tcosθ+vcos2π(N −N )] (2.10) 0 Z where ω is a Matsubara frequency. G−1 is the inverse III. POINT CONTACT WITH PERFECT n TRANSMISSION of the Green’s function matrix and can be explicitly expressed as In this section, we will consider the case where the 1 |ωn|(1+e−π∆|ωEn|) −ωn transmission through the right point contact is perfect. G−1 = T  4π ω2n 1−eπ−|ωπn∆|ω|En2| +U , (2.11) Itnhetdhoist.liSminitc,etchhearregeismnaoyqfluoawntciozanttiionnuooufsltyhethcrhoaurgghetohne   point contact, there is no preferred integer value for the where ∆E ≡πv /L. charge. This may be seen clearly from the Hamiltonian F We now briefly develop the framework for our calcu- in equation (2.8), where for v =0 the dependence on N 0 lation of the tunneling current. The I-V characteristic is absent. (or equivalently the tunneling density of states) may be It follows that as the gate voltage is varied that there computed using the action S in (2.10). Working per- should be no oscillations in either the differential capac- tot turbatively in the tunneling matrix element t, we may itance or the conductance. However, we will show below compute the tunneling current in the presence of a DC that tunneling through a large barrier onto the dot is bias V using Fermi’s golden rule. still blocked by the charging energy. This blockade is a 4 result of the fact that the tunneling electron has a dis- dotisaonedimensionalsystem(asitisforquantumHall crete charge, which cannot immediately be screened. edge states), the result has been shown to be To compute the tunneling current, we evaluate P (τ), 0 where the subscript 0 indicates that v = 0. Since the e2 ∆E 2 G∝ T T . (3.8) action (2.10) is quadratic, we may write, h L R U (cid:18) (cid:19) P (τ)=e−12hTτ[θ(τ)−θ(0)]2i (3.1) Evidently, the ∆E/U suppression predicted in (3.7) re- 0 mains valid all of the way up to T =1. R =exp − 1−eiωnτ G (3.2) At finite temperatures or voltages, ∆E ≪eV,T ≪U, θθ " # the lower limit of the integral in (3.6) is cut off by T Xωn (cid:0) (cid:1) and eV. The resulting tunneling current may thus be where G is the topleft elementofthe matrixG defined obtained by setting ∆E =0 and written, θθ in (2.10). This may be rewritten as e2 (eV)2+4π2T2 I =c T V, (3.9) πτ /β 2 f(ω ) 2 h L U2 P (τ)= c exp −2πT (1−eiωnτ) n 0 (cid:18)sinπτ/β(cid:19) " Xωn |ωn| # with c2 = 2π2e−2C/3 ≈ 2.07 where C is Euler’s constant. This result is, again, exactly in accordance with (3.3) the theory of inelastic cotunneling, setting T =1. R Analternativeinterpretationofthissuppressionofthe where τ is the short time cutoff (which is of order the c tunnelingcurrenthasbeenpointedoutinRef.[13]. Sup- inverse of the cyclotron frequency) and pose that an electron tunnels into the dot. The dot U 1−e−π∆|ωE| 2 would minimize its electrostatic energy by discharging exactly one electron. According to Friedel sum rule, the f(ω)= . (3.4) 2π|ω|+(cid:16)U 1−e−(cid:17)2∆π|Eω| number of added electrons, which is −1 in this case, is equal to δ/π where δ is the scattering phase shift of the (cid:16) (cid:17) one dimensional channel. As in Anderson orthogonal- Thefirsttermin(3.3)describestheresponsefornonin- itycatastrophe,14 thesuppressionfactorinthetunneling teractingelectrons,U =0. This givesus apurelyOhmic rate is related to the phase shift by tunneling current I = G V. G is related to the U=0 U=0 transmission probability of a free electron through the dI left barrier, GU=0 =(e2/h)TL, where TL ∝t2. dV ∝εγ (3.10) Inthepresenceofinteractions,thelowbiaslinearcon- ductancemaybededucedfromthelongtimebehaviorof where ε=max(∆E,T,eV) is the low energy cutoff and P (τ). Inthe stronginteractionlimit, U ≫∆E, wemay 0 2 estimate the limiting behavior of the exponential factor δ γ =2 in (3.3) by noting that f(ω) is approximately a constant π (cid:18) (cid:19) in each of the following three limits: =2(−1)2 =2. (3.11) 0 if ω ≪∆E, Combining(3.10)and(3.11),wemayreproduce(3.7)and f(ω)≈ 1 if ∆E ≪ω ≪U, (3.5) (3.9). Notethat(3.11)differsfromtheusualorthogonal-   0 if U ≪ω. ity exponent, (δ/π)2, by a factor of two. This is because we are tunneling into the middle of a “chiral” system At zero temperature, we thus have, to logarithmic accu- consisting only of right moving electrons (or equally the racy, end of a one dimensional normal electron gas.) Finally, we note that in the high bias limit, eV ≫ U, τ 2 U dω P (τ)≈ c exp−2 (3.6) we recover the linear I-V characteristic with an offset 0 (cid:16)τ (cid:17) Z∆E ω characteristic of the Coulomb blockade, in the long time limit (τ ≫∆E−1). It then follows that e2 U I = T V − . (3.12) the linear conductance is suppressed. Performingthe in- L h 2e tegral in (3.3) exactly, we find (cid:18) (cid:19) Thisoffsetisaconsequenceofthefactthatatshorttimes, e2 ∆E 2 theelectronwhichtunnelscannotbeeffectivelyscreened G=c T . (3.7) 1 h L U by the point contact. (cid:18) (cid:19) In addition to the limiting behaviors described above, withc ≈3.11. Thisshouldbecomparedwiththetheory we have computed the I-V characteristic numerically at 1 ofelasticcotunneling,whichisderivedinthecaseofweak zero temperature, as explained in appendix A. Fig. 2 tunneling through both barriers, T ,T ≪1. When the showsthe differentialconductancedI/dV. Whatismost L R 5 striking is that there are sharp steps whose sizes are ap- are the energies of the particle and the hole respectively. proximately ∆E. For a nearly isolated dot, a similar The electroneigenstatesin the leadare labeledby k and phenomenon has been pointed out by Glattli,5 and can k′, and V and V are matrix elements of the coupling 1k 2k′ be understoodto be a consequence of inelastic cotunnel- Hamiltonian. If ǫ −ǫ ≪ U/2, the above equation can 1 2 ing through a dot with a discrete energy level spectrum. be approximated 32π |V|4 ǫ −ǫ τ−1 = 1 2 1.0 ¯h ∆2 U2 lead (b) ∆ E =.10U = 4 (ǫ −ǫ ) ∆E 2T2, (3.14) 0.8 .3U π2h 1 2 U R (cid:18) (cid:19) where ∆ is the level spacing of the lead and T ≈ 0.6 lead R |2πV|2/∆E∆ is the transmissionprobability. We as- V lead dI/d sume |V1k|2 ≈ |V2k′|2 ≈ |V|2 is constant in the given 0.4 range. Inourmodel,asimpleconsiderationoftheI-V charac- teristicequationin(2.13)and(A1)showsthatthewidth 0.2 of the step risers is proportional to (∆E/U)2. Since the width of the risers is directly proportional to the line 0.0 width of the energy levels, it is evident that Eq. (3.14) 0.0 0.2 0.4 0.6 0.8 1.0 eV/U is valid even up to TR = 1 with a possible numerical factor. We thus conclude that even in the presence of FIG. 2. Differential conductance dI/dV as a function a perfectly transmitting contact, there can be long-lived of scaled bias voltage eV/U, computed in the absence of excited states in the dot, whose decay is suppressed by backscattering (TR = 1) for various ∆E. Curves are scaled the Coulomb blockade. sothatdI/dV =1inthehighbiasvoltagelimit. Itiseasyto So far we have assumed ∆E is constant. If we allow see that thecurvesconsist of steps of approximatesize ∆E. non-uniformlevelspacing,degeneracyintheparticle-hole excitation energy is lifted and all steps but the first one split into several sub-steps. As the degree of degener- An inelastic cotunneling process leaves a particle-hole acy increases with the energy, more splittings occur at excitation in the dot. As the bias voltage increases, the higherbiasvoltages,finallymakingithardtodistinguish number of available particle-hole combinations also in- between steps. creases. Because of the discrete nature of the energy Before we close this section, let us consider other re- spectrum of the dot, this increase in number occurs dis- laxation processes. It is only when all relaxation rates continuously at every∆E/e ofthe bias voltage,which is are less than ∆E that it is possible to experimentally manifested in the tunneling density of states or the dif- observe the steps. This criterion is equivalent to saying ferential conductance. However, the above explanations that the inelastic scattering length, l , is much longer are not fully adequate in our model because the dot is φ than the circumference of the dot. It is known that in- strongly coupled to the lead. For perfect transmission, elastic scattering is strongly suppressed in the quantum the line widthofa singleparticle energylevelis approxi- Hallregime15 mostlikelyduetothedifficultyofconserv- matelyΓ∼∆E. Itmeansthatthe levelsareasbroadas ing both energy and momentum when scattering occurs the level spacing and the steps are expected to be wiped in a one dimensional channel. These steps may thus be out altogether. The reasonfor the apparent discrepancy observable in the quantum Hall regime. is that the lifetime of a many body excited state (i.e. a particle-holepair)canbemuchlargerthanthenaivesin- gle particle lifetime. IV. POINT CONTACT WITH WEAK If the dotis weaklycoupledto the lead,the lifetime of BACKSCATTERING aparticle-holeexcitationmaybeeasilycalculated. First, weassumetheexcitationisrelaxedonlythroughthepro- In this section, we consider the case where there is cess in which both the particle and the hole tunnel out of the dot. In analogy with the theory of cotunneling,4 weak backscattering at the right point contact. As the contact is pinched off, fractional charge fluctuations in we use Fermi’s golden rule to estimate the decay rate, the dot are hampered and the discreteness of charge be- 2 2π 1 1 comes important. For weak backscatteringv, there is an τ−1 = V V + ¯h Xk,k′(cid:12)(cid:12) 1k 2k′"ǫk−(ǫ1− U2) ǫ2+ U2 −ǫk′#(cid:12)(cid:12) feonrerngoyn-cionstte,grparlocphoarrtgioencaolntfiogvuriantitohnes.HTahmisiltgoivneiasnri(s2e.7to) (cid:12) (cid:12) ×δ(ǫ1−(cid:12)ǫ2−ǫk+ǫk′)θ(ǫk−µ)θ(µ−ǫk′), (3.1(cid:12)3) oscillationsinphysicalquantitiessuchasthecapacitance (cid:12) (cid:12) and the conductance as a function of gate voltage. The whereµisthechemicalpotentialoftheleadandǫ andǫ 1 2 6 period of these oscillations corresponds to changing the G1 =vG0e−2π2hN(0)2i0 ×2 dτ′[1−cosh2πGθN(τ′)], optimal number of electrons on the dot N by one. 0 Z Fornearlyperfecttransmissionthroughthepointcon- (4.8) tact,wethusexpecttheconductanceandthe differential capacitance C =−edN/dVG to have the form, where G0 is the zeroth order linear conductance. Using (4.4), (4.5), and (4.8), we obtain an exact ex- C =C +C cos2πN (4.1) 0 1 0 pression for the ratio ξ, G=G +G cos2πN , (4.2) 0 1 0 ∆E+U ξ = ×2 dτ′[1−cosh2πG (τ′)]. (4.9) wherethe oscillatorycomponents C andG arepropor- 4π2 θN 1 1 Z tional to v. An intrinsic quantity, which is independent In the limit T,∆E ≪U, the integral approaches a finite of the backscattering amplitude v, the tunneling matrix value which depends only on U. In this limit we find elementt,andthecapacitiveleverarmη associatedwith the gate, is the ratio ξ ≈1.59. (4.10) G /G ξ ≡ 1 0. (4.3) Thecancellationofinputparameterslikev,U, and∆E C /C 1 0 may tempt us to suspect ξ be a universal number being constant for all samples. As will be shown below, how- Using the model we havedeveloped so far,we cancal- ever,Eq.(4.10)istrueonlywithinaconstantinteraction culate the capacitance and the linear conductance per- model, in which the dependence of the interaction on turbatively in the backscattering matrix element v. The the spatial separation is ignored. In order to see how differential capacitance is given by C =Td2lnZ/dV2/η, G ξ changes with different models, let us consider a more whereZ isthepartitionfunction. AsshownbyMatveev, generalmodelwhose interactionHamiltonianis givenby to leading order in v the differential capacitance has the form(4.1)withaverageandoscillatorycomponentsgiven 1 by HU′ = 8π2 dxdx′∂xφ(x)U(x,x′)∂xφ(x′). (4.11) Z ηe2 The constant interaction model is regained by assuming C = , (4.4) 0 ∆E+U U(x,x′) = U to be uniform. It is sufficient for our pur- C1 =vηe−2π2hN(0)2i0 2πe 2. (4.5) ploocsaeltinotceornacsitdioenrjUu(sxt,axn′o)t=he2rLexUaδm(xp−le.x′W)θe(Lca−n|xth|)i,nwkhoifcha ∆E+U (cid:18) (cid:19) is certainly an extreme limit to the other direction from The averageh···i is with respect to the ground state of theconstantinteractionmodel. TheappropriateGreen’s 0 the unperturbed action S in (2.10). function is given by tot In order to compute the conductance, we must calcu- lateP(τ)inthepresenceoftheperturbationvcos2π(N− GθN(ωn)= T 1−eπ|Uωn| , (4.12) ω N ). To the first order in v, we find that P(τ) = n 0 (cid:16) (cid:17) P (τ)+P (τ)cos2πN ,whereP (τ)isgivenin(3.1)and 0 1 0 0 and then using (4.9) we get P (τ)=− T ei[θ(τ)−θ(0)]S ξ =1. (4.13) 1 τ v 0 D E + T ei[θ(τ)−θ(0)] hS i (4.6) Nowitisclearthatξdependsontheformoftheelectron- τ v 0 0 electron interaction. It is, however, noteworthy that the D E values of ξ computed in two extreme limits are of the whereS = βdτ′vcos2πN(τ′). Thismaybe writtenas v 0 same order of magnitude. P1(τ)=vP0R(τ)e−2π2hN(0)2i0 V. MULTIPLE CHANNEL SYSTEMS × dτ′ {1−cosh2π[G (τ −τ′)−G (τ′)]}, θN θN Z (4.7) In this section we generalize the considerations in the previous sections to the systems of which right contact where G (τ) is the off diagonal element of the Green’s (nearly) perfectly transmits more than one channel. As θN function defined in (2.10), which may be computed ex- the conductance of the contact increases with the num- plicitly using (2.11). berofwelltransmittedchannelsN,theshorterRC decay In order to compute the linear conductance, we must time allows a higher uncertainty in the energy of the is- compute the large τ limit of (4.7). For Uτ ≫ 1 G (τ) land. Therefore it is natural to expect that the effect θN decays as (Uτ)−1, so that the integral is independent of of the Coulomb blockade become weakened in multiple τ. We thus find channel systems, which will be confirmed below. 7 It turns out that most of the qualitative considera- along the same lines as in Sec. III. In this case, the tions for the single channel systems can directly be ap- function f(ω) defined in (3.3) is given by plied to multiple channel systems. Similar calculations as in Sec. III show that in the absence of backscatter- U 1−e−π∆|ωE| 2 ing, the low bias linear conductance is still suppressed f(ω)= . (5.3) below its noninteracting value, although the suppression 2π|ω|+(cid:16)NU 1−e(cid:17)−2∆π|Eω| islessstrongifN isbigger. Ontheotherhand,thereare (cid:16) (cid:17) new features arising from the introduction of additional Note that the limiting values of f(ω) are channels, on which we will focus in this section. For integer quantum Hall states with ν > 1, the edge 0 if ω ≪∆E, channels tend to be spatially separated. The tunneling f(ω)≈ 1 if ∆E ≪ω ≪NU, (5.4)  N will be dominated by the coupling to the nearest edge 0 if NU ≪ω.  channel. As indicatedinFig.3,this means thatthe tun- nelingandbackscatteringwilloccurindifferentchannels. It immediately follows that Then the Hamiltonian for N channels is represented by dI e2 ε 2 ∝ T N , (5.5) L H =H [φ ]+ H [φ ]+H dV h U 0 l 0 i U (cid:16) (cid:17) i X where ε = max(∆E,eV,T). The exponent 2/N has + tcosθ +vcos2π(N −N ), (5.1) 1 N 0 been derived in some other papers in several different contexts.11,13 It is clear from (5.5) that the conductance where the notation is similar to that in Sec. III with the is less suppressed if there are more channels. subscript denoting the channel number except that φ is l However, it turns out that the nonanalytic behavior the boson field of the left lead. We have redefined N 0 withexponent2/N iscorrectonlyto the extentthatthe as the optimal number of electrons in channel N alone, constant interaction model is valid. As explained below, which in general depends on lever arms for all channels. this is due to a special symmetry of the charging energy withrespecttoredistributionofchargeamongthediffer- entchannels. Weconsideraneffective-capacitancemodel which is one step more general and has been introduced and developed by several authors to remedy some prob- lems with the constant interaction model.16 This model assumes that the edge channels are capacitively coupled metal bodies and the Coulomb interaction energy de- L pends on the number of electrons in each channel. The t 0 0 B v CoulombinteractionpartofthenewHamiltoniancanbe -L written 1 H = N U N , (5.6) U i ij j 2 ij X A B where U is an N ×N matrix which can be determined experimentally. In order to get a clear understanding of the effect of this generalization, let us consider a simple FIG. 3. Schematic view of a quantum dot analogous to specific example of the electron-electroninteraction, i.e., Fig. 1, with two well transmitted channels. The symbols are the same as in Fig. 1. Note that the tunneling through the U =u[Naδ +(1−a)]. (5.7) ij ij leftcontactoccursintheouterchannelwhereasthebackscat- tering at theright contact occurs in theinner channel. The diagonal component (Na + 1− a)u is the magnitude of the interaction strengthwithin each channeland the off-diagonalcomponent (1−a)u is that between dif- Oneofthesimplestmodelstostudyamultiplechannel ferent channels. This matrix is chosen such that if the system is a constant interaction model in which the in- lever arms are all equal to unity, the total capacitance teraction Hamiltonian depends only on the total charge C ≡ dQ /dV = e2 U−1 = e2/u in the limit in the island. The interaction may be explicitly written tot tot G ij ij ∆E = 0, independent of a. Note that we regain a con- P stant interaction model if a = 0. As a grows we move 2 U away from the model, finally reaching an independent H = N . (5.2) U i 2 ! channel model at a=1, where different channels do not i X interact. As in (3.5), the limiting behavior of f(ω) de- The calculation of the tunneling conductance proceeds fined in (3.3) is given by 8 0 if ω ≪∆E, where δ is the phase shift in channel i. It needs only i 1 if ∆E ≪ω ≪Nau, a little consideration of electrostatics to figure out δ . f(ω)≈ (5.8) i Na+1−a Nau≪ω ≪Nu, FollowingtheargumentinSec.III,letussupposethatan  0 N if Nu≪ω, electronhasjusttunneledintochannel1throughtheleft contact. The number of electrons discharged from each Itisclearfromtheaboveequationthatifa→0(constant channel −δ /π depends on the form of the interaction, i interaction), (5.4) is restored and we get the exponent provided they satisfy the constraint δ /π = −1. If i 2/N, as we discussed earlier. When a is not small, there we work in a constant interaction model, because the is no appreciable range in which f(ω) = 1/N, and the Hamiltonian depends only on the totaPl charge, from the differential conductance has a different exponent, 2, i.e. symmetry δ /π =−1/N for all i. Therefore i dI e2 ε 2 N 2 ∝ T (5.9) 1 2 dV h L u γ =2 − = . (5.12) N N (cid:16) (cid:17) i=1(cid:18) (cid:19) provided ε=max(∆E,eV,T)≪Nau. X Theaboveconsiderationscanbegeneralizedtoamodel On the other hand, if we use an effective-capacitance with a generic matrix U. Even though f(ω) is a compli- model and the system is safely away from the constant cated function which depends on all matrix elements of interaction limit (Nu˜ ≫ ∆E), it is always energetically U,thereexistsanenergyscaleu˜whichcorrespondstoau favorable to take a whole electron from channel 1 and in the above special model, such that have the exactly same ground state charge configuration as before. Then δ /π = −1 and δ = 0 (i = 2...N), so 1 i f(ω)≈1 if ∆E ≪ω ≪Nu˜. (5.10) that Then Eq. (5.9) is valid if ε ≪ Nu˜, except for the factor γ =2(−1)2 =2. (5.13) depending on a. In general, u˜ = 0 for a constant interaction model and it measures how far the used model is Then Eqs. (5.9) and (5.5) are readily reproduced from awayfromtheconstantinteractionmodel. VanderVaart (3.10) and (5.11). The physical distinction between the et al.18 have measured the matrix elements of U for two energy scales u and u˜ is thus clear. When an electron Landau levels confined in a quantum dot. Even though is added to the dot, (Nu)−1, which corresponds to the both contacts were nearly pinched off in the reference as RC decay time, sets the time scale for the total charge opposed to our model, it is suggestive to estimate the of the dot to return to its original value. However, even magnitude of u˜ using the experimental data. With their after the total charge has been screened, there may be particular setup, they got U = 800, U = 1175, and someimbalanceinthedistributionofchargebetweenthe 11 22 U = 650 (all in µeV). A simple estimation with these channels. (Nu˜)−1 setsthe scaleforthe relaxationofthis 12 numbers gives Nu˜ ∼ 260µeV ≫ ∆E,T,eV, which sug- imbalance. Intheconstantinteractionmodel,thereisno gests that (5.9) must be used rather than (5.5) in this Coulomb energy cost for such an imbalance, so u˜→0. case. It has to be admitted that this is a naive estima- Now let us consider the effect of weak backscattering. tion considering the difference between their experimen- As in a single channel model, the introduction of weak tal setup andour theoreticalmodel. Opening up a point backscatteringv resultsinoscillationsinthecapacitance contactwouldin generalreduce the strengthofelectron- and the conductance. However, an important difference electron interactioins in the dot and it would change arisesfromthefactthatthetunnelingandbackscattering the capacitance appreciably. However, even though the occur in different channels. above estimation of u˜ may be merely speculative at its It has been shown both theoretically16 best, we expect (5.9) has to be true if different channels andexperimentally17 thatthe periodofthe conductance are weakly coupled, which seems more general than the andthecapacitanceoscillationsincreaseswithincreasing constant interaction limit, in practice. numberofwelltransmittedchannels. Thisisbecausethe It is now evident that the exponent is 2/N only for oscillations arise only from the quantization of NN, the the constant interaction model. This is due to a special number of electrons in the backscatteredchannel. When symmetry of the constant interaction model, i.e., the in- there are many perfectly transmitting channels, many teractionpartoftheHamiltonian(5.1)isinvariantunder electrons must be addedto the dot to increase NN by 1. redistribution of total charge among the different chan- The analysis of the amplitude of the oscillations is a nels. The effect of the symmetry to the exponent can be little morecomplicated. We will againfocus onthe ratio mosteasilyunderstoodintermsofAndersonorthogonal- ξ defined in (4.3), using the model interaction in (5.7). itycatastrophe.13Eq.(3.10)canbedirectlyusedwithan We assume ∆E,T ≪ Nau and all lever arms are taken appropriately generalized definition of γ, i.e., to be unity. One may include the lever arms explicitly, but it does not change the result qualitatively. Along δ 2 the same lines as in Sec. IV, the fractional capacitance i γ =2 , (5.11) oscillation may be written π i (cid:18) (cid:19) X 9 2 C1 =ve−2π2hNN(0)2i0 2π 1. (5.14) consequentlya weakereffect of the backscatteringto the C N u conductance. At a = 1 (independent channels), we can- 0 (cid:18) (cid:19) not use the above equation, but we know that the con- The fractional conductance oscillation may also be writ- ductanceoscillationwouldeventuallyvanishbecausethe ten backscatteringpotential does not affect the conductance at all, and therefore ξ = 0. At a = 0, (constant inter- G1 =vG0e−2π2hNN(0)2i0 ×2 dτ′[1−cosh2πGθN(τ′)], action) G1/G0 diverges, and so does ξ. It is because 1 Z G1/G0 diverges as (u/ε)N if the low energy cutoff ε (5.15) is small, suggesting that the perturbation theory break down. Without detailed calculations, one might have where the Green’s function is given by been able to infer it from the following physical argument. In a constant interactionmodel, the totalnumber G (τ)=hT θ (τ)N (0)i θN τ 1 N of electrons in the dot N is the only gapped mode i i =− dω e−iωnτ and there are N −1 combinations of Ni whose fluctua- n P tions are not bounded, leading to divergences in individ- Z (1−a)usgn(ω ) ual terms in the perturbation expansion. Therefore, we n × . (5.16) (2π|ω |+Nau)(2π|ω |+Nu) need to sum up all higher order terms in order to obtain n n a correct result. In a series of recent papers, Matveev WemayeasilycomputeG (τ)inseverallimits,namely, and Furusaki7,10 have calculated both the conductance θN and the capacitance oscillations nonperturbatively in a 0 if τ =0, spin-degenerate two-channel model, which they related GθN(τ)≈−iπ(1N−a) if N2πu ≪τ ≪ N2aπu, (5.17) to the multichannel Kondo problem. Their calculations 0 if τ →∞, show that the oscillations are no longer sinusoidal and  the period becomes N times smaller so that the maxi- and it is monotonically interpolated in between. The mum occurseachtime anelectronis addedto the dot as above equation is not helpful if a ∼ 1, but it is suf- a whole (not channel N alone). Such results, however, ficient for our purpose which is to see how ξ changes clearly apply only in the case where the degeneracy is as the system moves away from the constant interaction guaranteed by a symmetry and hence should not apply limit. Since GθN(τ) measures the response of NN, a pe- in this quantum Hall system. riod of time τ after an electron is added into channel 1, Without qualitative changes,the aboveconsiderations thephysicalinterpretationoftheabovelimitingbehavior canbegeneralizedtoaneffective-capacitancemodelwith is clear. It takes a time period of order 2π/Nu for the a generic matrix U. As in the discussions of the differen- total charge of the dot to return to its original value. tial conductance dI/dV earlier in this section, anenergy Channel N contributes to this process by discharging scale u˜, which plays the role of au, can be determined (1−a)/N of an electron, which can be read from the fromthegivenmatrixU. Inmostrealsituationsofquan- second line of (5.17). The reason it is proportional to tumHalleffectedgechannels,ξ isafinitequantitywhich 1−aisthattheinter-channelinteractionstrengthispro- canbenumericallycalculatedintheeffective-capacitance portionalto1−a. After atime periodoforder2π/Nau, model if all matrix elements of U are known. the charge in each channel returns to its original value, which is reflected in the vanishing G (τ) in the long θN time limit. In order to estimate the integral in (5.15), VI. CONCLUSIONS we may make a crude approximation by substituting a squarefunctionforG (τ), i.e.,G (τ)=−iπ(1−a)/N θN θN Inthispaperwehaveshownthatcharacteristicsofthe if 2π < τ < 2π , and G (τ) = 0 otherwise. Then we Nu Nau θN Coulomb blockade, which are normally associated with get the weak coupling limit, persist to strong coupling to a lead via a single channel point contact. In particular,(i) G1 ≈ve−2π2hNN(0)2i08π(1−a)sin2 π(1−a), (5.18) we find the analogies of elastic and inelastic cotunneling G Nau 2N 0 inthe(ε/U)2 suppressionofthetunnelconductance. (ii) and finally We find that particle-hole excitations on the dot can ac- quire a long lifetime due to a “Coulombblockade” to re- 2N 1−a π(1−a) laxation. Thisinprinciple couldleadto observablesteps ξ ≈ sin2 . (5.19) π a 2N in the low bias differential conductances as a function of bias voltage. (iii) The high bias behavior of the I-V This is a good approximation if a ≪ 1. This is a mono- characteristic has an offset, indicating the presence of a tonically decreasingfunction ofa andasis explainedbe- Coulomb gap. We find similar conclusions when multi- low, it is a consequence of the fact that the tunneling plechannelsaretransmittedthroughthecontact,though and the backscattering occur in different channels. A thesuppressionoftheOhmicconductanceisreduced. In bigger a implies weaker inter-channel interactions and 10