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

Lee, Patrick A.

Matveev, Konstantin

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Chapter 2. Theory of Coulomb BlockadeChapter 2. Theory of Coulomb Blockade in SemiconductorDevicesAcademic and Research StaffProfessor Patrick A. Lee, Dr. Konstantin MatveevTechnical and Support StaffMargaret E. O'Meara2.1 Project DescriptionSponsorJoint Services Electronics ProgramGrant DAAH04-95-1-0038Project StaffProfessor Patrick A. Lee, Dr. Konstantin MatveevThe conductance through a small grain weaklycoupled to metallic leads shows periodic depend-ence on the voltage applied to a gate electrode.This phenomenon is observed in both metallic' andsemiconductor 2 devices, and is commonly referredto as the Coulomb blockade. This behavior of con-ductance is caused by electrostatic energy arisingdue to a change in the charge of the grain(quantum dot) by an electron tunneling through it.The Coulomb blockade is typically observed instructures with a well defined quantum dot, which isseparated from the leads by tunneling barriers.Recently, the experimental observation of Coulombblockade has become possible in more sophisti-cated devices. An experimental group at Harvardhas studied the Coulomb blockade in doublequantum dots.3 In a number of other experiments,4the phenomenon of Coulomb blockade wasobserved at unusually high temperatures T- 100K, where the transport may be due to the activationof electrons over the barriers. In the past year, ourresearch group worked on the theoretical problemsraised by these experiments.2.1.1 Transport through Double QuantumDotsIn these experiments, 3 the Coulomb blockade oscil-lations of conductance through a system of twoquantum dots (figure 1) were studied. The contactsbetween the quantum dots and the leads were inthe weak tunneling regime, with conductancesG,r << e2/h. On the other hand, the contactbetween the dots was tuned by adjusting the gatevoltage Vo in such a way that the correspondingtransmission coefficient To scanned the wholeregion from 0 to 1. As a result, a series of conduc-tance dependences on the gate voltage, at differentvalues of To, were measured.5 The most basic fea-tures of the experimental data, including the split-ting of the Coulomb blockade peaks when Toincreases from 0 to 1, were explained in our pre-vious work. 6 We have now extended the theoreticaldescription to include the effects of temperature andthe small asymmetry of the double dot system onthe heights and shapes of the Coulomb blockadepeaks observed in the experiments.3The experiments have demonstrated that small tun-neling between the dots gives rise to a small split-ting of the Coulomb blockade peaks. At non-zero1 H. Grabert and M.H. Devoret, Single Charge Tunneling, (New York: Plenum Press, 1992).2 M.A. Kastner, Rev. Mod. Phys. 64: 849 (1992).3 F.R. Waugh et al., Phys. Rev. Lett. 75: 705 (1995); F.R. Waugh et al., Phys. Rev. B 53: 1413 (1996).4 K. Murase et al., Microelectr. Eng. 28: 399 (1995); E. Leobandung et al., Appl. Phys. Lett. 67: 2338 (1995); E. Leobandung et al.,Appl. Phys. Lett. 67: 938 (1995).5 K.A. Matveev, L.I. Glazman, and H.U. Baranger, Phys. Rev. B 54: 5637 (1996).6 K.A. Matveev, L.I. Glazman, and H.U. Baranger, Phys. Rev. B 53: 1034 (1996).Chapter 2. Theory of Coulomb BlockadeFigure 1. Schematic view of the double quantum dot system. The dots are formed by applying negative voltage to thegates (shaded); the solid line shows the boundary of the 2D electron gas (2DEG). V! and V, create tunnel barriersbetween the dots and the leads while Vo controls the transmission coefficient through the constriction connecting thedots.temperatures, the splitting may or may not beresolved depending on the relation between thetemperature T and the charging energy Ec. Wehave studied the whole temperature dependence ofthe split peaks analytically; the results are pre-sented graphically in figure 2.In the experiments with double dots, one usuallytries to make the two quantum dots symmetric.However, a small asymmetry is unavoidable, and isreflected in the actual experimental data. Forexample, when the gate voltage is chosen in such away that an electron can tunnel to the first dotwithout changing its energy, tunneling to the otherdot can still cost some electrostatic energy. Thisenergy itself depends on the gate voltage X, andleads to a periodic in X suppression of the peakheights. Our results are summarized in figure 3,which strongly resembles the actual experimentaldata.70.20.110 -8 -6 -4 -2 0 2 4Gate voltage X6 8 10Figure 2. The evolution of the split peaks with temper-ature. The gate voltage X is plotted in units of the dis-tance between splitted peaks. The peak splitting isobservable at a sufficiently low temperature, /3 Ec/T > 2.7 F.R. Waugh et al., Phys. Rev. Lett. 75: 705 (1995); F.R. Waugh et al., Phys. Rev. 6 53: 1413 (1996).54 RLE Progress Report Number 139- = 0.6--- = 1.2/ - 2.5/ \ .......... = 5/ ./\/ / \-- - .: .\- -Chapter 2. Theory of Coulomb Blockade1.0p)0.8S0.6- 0.40.20.0--5 0 5 10XFigure 3. The conductance as a function of dimensionless gate voltage X in the case of 15 percent asymmetry betweenthe dots. Note the correlation between the modulation of the peak height and the separation of adjacent peaks: when thepeaks are high the splitting is small, while when the peaks are small they are well separated.2.2 Coulomb Blockade in the Regime ofActivated ConductionIn recent experiments,8 the periodic oscillations ofconductance through a quantum dot were observedat temperatures exceeding 100 K. In this regime, itis not obvious that the transport through the barriersis due to tunneling rather than activation of elec-trons over the barriers. A simple theoretical investi-gation shows that for any barrier there is acharacteristic temperature Tx determined by theshape of the barrier, such that at temperaturesexceeding Tx the transport is activated. Though itdoes not seem possible to determine Tx for the bar-riers used in the experiments,8 the unusually hightemperatures at which the conductance oscillationswere observed raised the fundamental problem ofthe Coulomb blockade of activated conduction.Our studies 9 show that the Coulomb blockade inthis regime is still possible, but there are severalfeatures which distinguish it from the standard tun-neling regime. In particular, we have found that onehas to consider the dependence of the heights ofthe barriers on the electric field created by thecharge in the dot. Unlike in the tunneling case,without the effect of the charge of the dot on barrierheights, there is no Coulomb blockade.When the effects of the charge on the barriers areincluded, the Coulomb blockade is restored. Theshape of conductance peaks is given byeEcAX/G(N) oc eeE T + e- EcX/T'where X is a dimensionless gate voltage, and A isa geometrical parameter depending on the sym-8 K. Murase et al., Microelectr. Eng. 28: 399 (1995); E. LeobandungAppl. Phys. Lett. 67: 938 (1995).9 K.A. Mateev and L.I. Glazman, Phys. Rev. B 54: 10339 (1996).et al., Appl. Phys. Lett. 67: 2338 (1995); E. Leobandung et al.,Chapter 2. Theory of Coulomb Blockademetry of the barrier, -1 < A < 1. For symmetric bar-riers A = 0, and the peaks are similar to those1 o ofthe blockade of tunneling conductance. It is inter-esting, however, that the activation energy EcX forthe conductance (1) at off-peak values of the gatevoltage is smaller than in the tunneling case 10 by afactor of two.For an asymmetric barrier A * 0, and the peaks (1)have an unusual asymmetric shapes. This effect isspecific to the activated regime. Thus, by mea-suring the Coulomb blockade peaks, one may beable to identify the mechanism of transport throughthe quantum dot.2.3 PublicationsMatveev, K.A., L.I. Glazman, and H.U. Baranger."Coulomb Blockade of Tunneling Through aDouble Quantum Dot." Phys. Rev. B 54: 5637(1996).Matveev, K.A., and L.I. Glazman, "CoulombBlockade of Activated Conduction." Phys. Rev.B 54: 10339 (1996).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).10 L.I. Glazman and R.I. Shekhter, J. Phys. CM 1: 5811 (1989).56 RLE Progress Report Number 139

Theory of Coulomb Blockade in Semiconductor Devices

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Theory of Coulomb Blockade in Semiconductor Devices

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