Coulomb oscillations in partially open quantum dots

Superlattices and Microstructures, Vol. 20, No. 3, 1996

Coulomb oscillations in partially open quantum dots C. H. Crouch, C. Livermore, R. M. Westervelt Division of Applied Sciences and Department of Physics, Harvard University, 9 Oxford St., Cambridge, MA 02138, U.S.A.

K. L. Campman, A. C. Gossard Materials Department, University of California at Santa Barbara, Santa Barbara, CA 93106, U.S.A.

(Received 20 May 1996) We report tunneling measurements of the Coulomb blockade in a single quantum dot at zero magnetic field and dilution refrigerator temperatures with weak tunneling from the dot to one lead (the ‘closed’ lead, conductance G closed ) and strong tunneling to the other lead (the ‘open’ lead, conductance G open ). We observe suppression of the Coulomb oscillations with G open ≈ 2e2 / h, and then see the oscillations return for G open > 2e2 / h. The oscillations show a strikingly lower threshold temperature at G open ≈ 2e2 / h than for greater or lesser G open . c 1996 Academic Press Limited

Key words: breakdown of Coulomb blockade, strong tunneling, changing effects.

1. Introduction Classical charging models [1] for the Coulomb blockade of transport through a quantum dot connected to its environment by two leads assume that the conductance of each lead to the dot is much less than g Q ≡ 2e2 / h, the conductance of a quantum point contact with one spin-degenerate transverse mode allowed in it. This amounts to assuming that tunneling between the dot and its leads is very weak and the system free energy is dominated by capacitive interactions. Recently, there has been much theoretical [2–5] and experimental [6–8] interest in the effect of strong tunneling on the Coulomb blockade. In particular, theory which models the dot-to-lead tunnel junctions as perfect one-dimensional channels [2] predicts that Coulomb blockade oscillations should vanish when the conductance of one lead to the dot equals or exceeds g Q . However, differing experimental results have been obtained; van der Vaart et al. [8] report that the Coulomb oscillations vanish when the conductance in the point contact connecting the dot to one lead exceeds g Q , whereas Pasquier et al. [7] observe persistence of Coulomb oscillations with dot-to-lead conductance up to several g Q . We measured Coulomb oscillations in a quantum dot connected to leads through separately adjustable quantum point contacts. Two nominally identical devices with dot lithographic area 500 × 800 nm2 were used. An SEM micrograph of one such device is shown in Fig. 1A. The device is defined by ten individual ˚ two-dimensional electron electrostatic gates on a high-mobility (5 × 105 cm2 V−1 s−1 ) near-surface (570 A) gas (2DEG) in a GaAs/AlGaAs heterostructure and was fabricated using electron beam lithography and 0749–6036/96/070377 + 05 $18.00/0

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B Vq1

Vg1

Vq2

dot 1

Vq1

Vg2

Vg1

Vq3

dot 2

Vq2

Vg3

Vq3

Fig. 1. A, SEM micrograph of the device used, two quantum dots in series formed in a GaAs/AlGaAs heterostructure. Only one dot at a time was used in the experiment and other gates were grounded to the electron gas. Lithographic size of each dot is 0.5 × 0.8 µm; scale bar is 1 µm. B, Schematic of device wiring; independent voltages are applied to each set of gates with a common label. The Vgi are side gates and the Vqi are quantum point contacts. The shaded gates represent the gates used for a single dot experiment (two neighboring quantum point contacts and the side gates in between); dot 1 could be used instead by energizing Vq1 , Vq2 , Vg1 , and Vg2 .

Cr/Au metallization. The gates were wired as shown in Fig. 1B, where common symbols denote a set of gates wired together. The voltage Vq on the pair of gates defining a quantum point contact controls the point contact conductance. The device is two quantum dots in series; one dot at a time was used by energizing two neighboring point contacts and the side gates in between, as indicated by shading in the figure. Thus the experiment was done with four individual quantum dots (two from each of two devices). The total capacitance of a single dot was measured to be C6 = 342 ± 15 aF, and electrostatic simulations [9] show that each dot contains about 700 electrons. Thus the dot charging energy e2 /2C6 ≈ 2.7 K and the single-particle level spacing 1E ≈ 400 mK. We measured the two-probe conductance of the dot using standard a.c. lock-in techniques and a drain-source a.c. voltage of 10 µV rms. The device was cooled in a dilution refrigerator with base temperature 15 mK. At base temperature the electron temperature Te was determined from peak widths to be 75 mK. Measurements were made at temperatures ranging from base to 1.5 K.

2. Results and discussion To study the dependence of the oscillations on the conductance between the dot and one of its leads, we fixed the conductance of one quantum point contact, G closed , to be much less than g Q , and varied the conductance of the other, G open , from ∼ 0 to 5g Q . We then performed two different conductance measurements. Figure 2 shows the conductance through the dot, G dot , at four different temperatures as a function of the voltage Vq on the gates which control the open quantum point contact conductance. Also shown is the open point contact conductance G open measured separately by energizing only the gates defining the point contact. We observe that indeed the Coulomb oscillations disappear when G open ≈ g Q , as predicted by theory. Surprisingly, at base temperature, the Coulomb oscillations return for G open > g Q and persist to G open > 5g Q . The period of the oscillations remains essentially constant over the entire range of G open . At T = 400 mK the oscillations return and persist only to somewhat below G open ∼ 2g Q , and for T = 600 mK the oscillations do not return above G open ∼ g Q . Figure 3 shows plots of G dot versus side gate voltage Vg for a series of temperatures at the four values of G open indicated. During each of these measurements Vq was held constant. We see that for G open greater or less

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Fig. 2. Medium weight traces: conductance measured through the quantum dot G dot as a function of the voltage Vq controlling the ‘open’ quantum point contact conductance G open at the four temperatures indicated. Traces are offset vertically for clarity. Heavy trace: conductance of ‘open’ point contact, G open , measured separately. Note the suppression of Coulomb oscillations when G open ≈ g Q and their subsequent return.

Gopen (2e2/h) ~ 0.05 8

A

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Gopen ~ 1.0 C

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6 Gdot (10–3 e2/h)

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0 –0.76

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Side gate voltage, Vg (V) Fig. 3. A–D, Temperature dependence of Coulomb oscillations for the four ‘open’ point contact conductances G open indicated. Note the strong suppression of the oscillations at G open ≈ g Q and the return of oscillations for greater G open . The side gate swept corresponds to gate Vg3 of Fig. 1B.

than g Q , well-defined Coulomb oscillations persist to temperatures above 300 mK; however, for G open ∼ g Q , the oscillations are significantly suppressed at T = 150 mK. The temperature dependence of the Coulomb oscillations is summarized in Fig. 4. We define the threshold temperature for the oscillations as the temperature at which the amplitude of the oscillations is less than 2 × 10−4 e2 / h. (This threshold amplitude was chosen to be slightly above the minimum conductance of the measurements, 5 × 10−5 e2 / h; see Fig. 3A.) We see that

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Threshold temperature (K)

1200 1000 800 600 400 200 0 0.0

0.5

1.0

1.5

2.0

Gopen Fig. 4. Threshold temperature (defined in the text) of Coulomb oscillations as a function of G open . Symbols with error bars are data points; the line drawn through them is a guide to the eye.

the threshold temperature for the oscillations decreases slowly and smoothly from 1.2 K for the well-defined dot (very low G open ) to 300 mK for G open ∼ 2.1g Q , except for a sharp dip at G open ≈ g Q . We interpret the disappearance of Coulomb blockade for G open ∼ g Q and its subsequent reappearance as indicating that at G open ∼ g Q , the constriction from dot to lead is perfectly transmitting for electrons in the lowest subband, but as the conductance of that constriction is further increased, reflection of electrons in the lowest subband from the constriction actually increases. Using the formalism of Ref. [3], we would say that r , the probability of reflection for electrons incident on the constriction is zero for G open = g Q , but as G open is increased above g Q , r increases. We consider two possible explanations for this surprising increase of r . In a single-particle model, as the constriction conductance is increased, introducing the second subband into the channel, intersubband scattering due to disorder is increased abruptly due to the increase in the density of states [10]. In this model, however, we do not expect the reflection coefficient to go to zero at G open = g Q . In a many-body model, electron–electron scattering within the channel could give both r = 0 at G open = g Q and r > 0 for G open > g Q if the presence of higher subbands enhanced electron–electron scattering in the lowest subband. Further experiments are needed to distinguish between these mechanisms. The oscillation amplitude temperature dependence suggests that the reflection mechanism is temperaturesensitive. Acknowledgements—The authors wish to thank J. M. Golden, B. I. Halperin, K. A. Matveev, and M. Stopa for helpful discussions, and A. S. Adourian, R. G. Beck, M. A. Eriksson, J. M. Hergenrother, J. A. Katine, and J. G. Liu for experimental assistance. This work was supported at Harvard by ONR grants N0001495-1-0866 and N00014-95-1-0104 and NSF grants DMR-95-01438 and DMR-94-00396, and at UCSB by AFOSR F49620-94-1-0158. C.H.C. and C.L. acknowledge support from the NSF Graduate Fellows program and C.H.C. from the AT&T Graduate Research Program for Women.

References [1] See, for example, H. Grabert and M. H. Devoret, eds. Single Charge Tunneling. NATO ASI Series B, Physics. Vol. 294. Plenum Press, New York (1992). [2] K. A. Matveev, Phys. Rev. B 51, 1743 (1995). [3] A. Furusaki and K. A. Matveev, Phys. Rev. Lett. 75, 709 (1995).

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K. Flensberg, Phys. Rev. B 48, 11156 (1993). K. Flensberg, Physica B 203, (1994). L. W. Molenkamp, K. Flensberg and M. Kemerink, Phys. Rev. Lett. 75, 4282 (1995) C. Pasquier, U. Meirav, F. I. B. Williams and D. C. Glattli, Phys. Rev. Lett. 70, 69 (1993). N. C. van der Vaart, A. T. Johnson, L. P. Kouwenhoven, D. J. Maas, W. de Jong, M. P. de Ruyter van Steveninck, A. van der Enden, C. J. P. M. Harmanss and C. T. Foxon, Physica B 189, 99 (1993). [9] M. Stopa, personal communication. [10] A similar effect on transport in arrays of narrow Si inversion lines was calculated in S. Das Sarma and X. C. Xie, Phys. Rev. B 35, 9875 (1987).