Contains a description of one research project, reports on two research projects and a list of publications.Joint Services Electronics Program Grant DAAH04-95-1-003

Furusaki, Akira

Lee, Patrick A.

Matveev, Konstantin

O'Meara, Margaret E.

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Chapter 3. Coulomb Blockade In Single And Double Quantum DotsChapter 3. Coulomb Blockade In Single And Double QuantumDotsAcademic and Research StaffProfessor Patrick A. Lee, Dr. Akira Furusaki, Dr. Konstantin MatveevTechnical and Support StaffMargaret O'Meara3.1 Project DescriptionSponsorJoint Services Electronics ProgramGrant DAAH04-95-1-0038A few years ago, Professor Kastner's researchgroup discovered that the conductance through aquantum dot weakly coupled to the leads showsperiodic peaks as a function of the voltage on agate electrode. 1 This behavior was attributed to thephenomenon of Coulomb blockade: to add an elec-tron to the dot, the system must have an energyequal to the electrostatic energy of a dot chargedby one electron. Most of the theoretical and exper-imental work on Coulomb blockade is devoted tothe case when the quantum dot is separated fromthe leads by very high tunnel barriers, which ensurethe discreteness of charge in the dot. A recentexperiment2 indicates that as the height of the bar-riers is lowered, the Coulomb blockade peaks aresmeared and transformed into small oscillations.Apparently, as one lowers the barriers between thedot and the leads, the discreteness of the charge ofthe dot is smeared, and the Coulomb blockade issuppressed. An important theoretical problem is tofind the conditions at which a dot strongly coupledto the reservoirs can exhibit the Coulomb blockade.Our preliminary studies3 showed that a quantum dotconnected to a lead by a single contact shoulddemonstrate the Coulomb blockade oscillations ofthe dot's charge as long as the transmission coeffi-cient of the contact T is less than unity. Exper-imentally, it is difficult to measure charge and in thepast year our research group worked on the theoryof the transport through quantum dots.3.2 Transport Through a SingleQuantum DotThe system we study is shown in figure 1.4 Thequantum dot is coupled to the reservoirs (leads) bytwo quantum point contacts controlled by the corre-sponding gate voltages. Inside the electrostaticallycreated constrictions, the potential changesadiabatically as the electrons move along thechannel, which enables us to describe the transportby a one-dimensional model. We study the linearconductance through the quantum dot as a functionof the gate voltage in the regime where the trans-mission through one or both contacts is almostperfect: the transmission coefficient T is close to 1.The results are as follows:1. In agreement with the experiments, we foundthat the Coulomb blockade oscillations of theconductance disappear when the transmissionthrough at least one contact is perfect, T=1. Ifneither of the contacts is in the regime ofperfect transmission, the conductance showsweak periodic oscillations as a function of thegate voltage.2. An unexpected result was obtained: unlike theoscillations of the average charge of the dot,the amplitude of the oscillations of conductancegrows as the temperature is decreased (figure2). At very low temperatures, the oscillationsare no longer weak, and to find the conduc-1 M.A. Kastner, Rev. Mod. Phys. 64: 849 (1992).2 N. C. van der Vaart et al., Physica 189B: 99 (1993).3 K.A. Matveev, Phys. Rev. B 51: 1743 (1995).4 A. Furusaki and K.A. Matveev, Phys. Rev. Lett. 75: 709 (1995); A. Furusaki and K.A. Matveev, Phys. Rev. B 52: 16676 (1995).Chapter 3. Coulomb Blockade In Single And Double Quantum DotsFigure 1. Schematic view of a quantum dot connectedto two bulk 2D electrodes. The dot is formed by applyingnegative voltage to the gates (shaded). Solid line showsthe boundary of the 2D electron gas (2DEG). Electro-static conditions in the dot are controlled by the voltageapplied to the central gates. Voltage VL,R applied to theauxiliary gates controls the transmission probabilitythrough the left and right constrictions.tance a non-perturbative solution is required.Such a solution was found for several cases,and we discovered that in the low-temperatureregime the weak oscillations of conductanceare transformed into sharp peaks.3. The conductance in the centers of the peaks isof the order of the conductance quantum e2/h ifthe barriers are identical, and proportional tothe temperature in the asymmetric case.Between the peaks, the conductance is alwaysproportional to the square of the temperature.Our theoretical description of the problem is appli-cable in a wide range of temperatures between thecharging energy of the dot and the quantum levelspacing. We believe that the results can be easilytested in the experiments.3.3 Transport Through DoubleQuantum DotsIn a recent experiment,5 the Coulomb blockadeoscillations of conductance through a system of twoquantum dots (figure 3) were studied. 6 In the exper-iment the contacts between the quantum dots andthe leads were in the weak tunneling regime, withtransmission coefficients T,r < 1. On the otherFigure 2. Conductance through a quantum dot as afunction of the dimensionless gate voltage N for the casewhen the transmission coefficients for both contacts are0.96. The dashed line represents the conductance in thehigh-temperature limit. The curves a) and b) are for thetemperature equal to 0.04 and 0.005 in units the chargingenergy.hand, the contact between the dots was tuned byadjusting the gate voltage Vo in such a way that thecorresponding transmission coefficient To scannedthe whole region from 0 to 1. As a result, a seriesof the dependences of the conductance on the gatevoltage at different values of To were measured.The experiment showed that as the transmissionthrough the constriction between the dots isincreased, the periodic peaks in the conductancesplit into doublets. The distance between the com-ponents of the doublets grows with To, and at To = 1the periodicity is restored, but with a period which issmaller by a factor of two.Some features of the experiment are easilyexplained on the basis of conventional theories ofthe Coulomb blockade. For instance, the change ofthe period by the factor of 2 is due to the fact thatwhen the two dots a strongly coupled, they form alarge single dot with a doubled capacitance. On theother hand, the splitting of the peaks when thetransmission coefficient To is between 0 and 1cannot be explained with the usual electrostaticapproach. The goal of this project is to develop aquantum theory of transport through the double dotsystem.5 F.R. Waugh et al., Phys. Rev. Lett. 75: 705 (1995).6 K.A. Matveev, L.I. Glazman, and H.U. Baranger, Phys. Rev. B 53: 1034 (1996).56 RLE Progress Report Number 138Chapter 3. Coulomb Blockade In Single And Double Quantum DotsFigure 3. Schematic view of the double quantum dotsystem. The dots are formed by applying negativevoltage to the gates (shaded); the solid line shows theboundary of the 2D electron gas (2DEG). V, and V,create tunnel barriers between the dots and the leadswhile Vo controls the transmission coefficient through theconstriction connecting the dots.We start with the theory of the positions of thepeaks in conductance as a function of the gatevoltage. When the tunneling between the quantumdots is weak, an electron added to the double dotsystem can occupy two energetically equivalentpositions on either the left or right dot. As a result,there is a double degeneracy of the peak positions.As the barrier between the two dots becomes moretransparent, the degeneracy of the two states islifted due to the hybridization of the two quantumstates. Thus, the splitting of the peaks at To > 0 isproportional to the decrease of the ground stateenergy of the double dot system due to thequantum fluctuations of the charge between thedots. We have been able to find the peak splittingin the both limiting cases of the transmission coeffi-cient To close to 0 and 1. At small To, the splittingof the peaks is linear in To; at To near 1, the devi-ation of the peak positions from the equidistantpositions is proportional to (1 - To) In(1 - To). Theresults are summarized and compared with theexperiment in figure 4.We have also investigated the temperature depend-ence of the peak heights. In the conventional theo-ries of the Coulomb blockade, the heights of thepeaks in conductance are temperature-independent. This result applies to the double-dotsystem in the regime of very weak couplingbetween the dots. However, as the couplingbetween the dots becomes stronger, the peaksacquire a power-law temperature dependence. Theorigin of this temperature dependence is similar to1.0p 0.8S0.6= 0.4Eo 0.20.00.0 0.2 0.4 0.6 0.8 1.0Go/(2e 2/h)Figure 4. The normalized splittingblockade peaks as a function of thetance. Our theoretical results (dashedof the Coulombinter-dot conduc-lines) are in goodagreement with the experiment (crosses and pluses).The splitting is normalized by the period of the peaks inthe strong tunneling limit.the well known "orthogonality catastrophe." Whenan electron tunnels from the left lead into the leftdot, the right dot is also affected: to achieve thelowest possible energy, one-half of the charge ofthe added electron must be transferred to the rightdot. Thus, the system must significantly modify itsground state in order to accommodate an additionalelectron. This effect shows up as a power-law sup-pression of the tunneling rates at low temperatures.In the case of nearly symmetric dots, we found thatthe conductance peaks are suppressed as power5/4 of the temperature. We expect that this resultcan be easily tested in our experiments.3.4 PublicationsFurusaki, A., and K.A. Matveev. "CoulombBlockade Oscillations of Conductance in theRegime of Strong Tunneling." Phys. Rev. Lett.75: 709 (1995).Furusaki, A., and K.A. Matveev. "Theory of StrongInelastic Cotunneling." Phys. Rev. B 52: 16676(1995).Matveev, K.A., L.I. Glazman, and H.U. Baranger."Tunneling Spectroscopy of Quantum ChargeFluctuations in the Coulomb Blockade." Phys.Rev. B 53: 1034 (1996).-full C matrixfull C matrix-- N. N. approx.10 10-2L Uk' o-4 ,, 4---. ", ,",, ., , i ,h "/  " .I I'10-6 r ,/10- k' IL % L1 U -6 io I:I I'I 5V,7 ..,"Soliton" potential distribution in a 1 lx11 2D quantum-dot(solid line mesh) and the nearest neighbor approximationcenter of the array (x = y = 0). (see section 4.6)array, computed using the full capacitance matrix(dashed line mesh). The soliton is located at the58 RLE Progress Report Number 138x/a

Coulomb Blockade in Single And Double Quantum Dots

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Coulomb Blockade in Single And Double Quantum Dots

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