A non-invasive voltage probe to measure Coulomb charging

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Surface Science 305 (1994) 553-557

A non-invasive voltage probe to measure Coulomb charging C.G. Smith *, M. Field, M. Pepper ‘, D.A Ritchie, J.E.F. Frost, G.A.C. Jones, D.G. Hasko Cavendish Laboratory, University of Cambridge, Cambridge CBS OHE, UK

(Received 19 April 1993; accepted for publication 15 July 1993)

Abstract We have fabricated a quantum box device in close proximity to a one-dimensional (1D) ballistic channel, but in a separate circuit. This is achieved using electron beam lithography to define sub-micron gates on a modulation doped GaAs/GaAlAs heterostructure containing a high mobility electron gas. When the 1D channel is biased in the tunneling regime it is extremely sensitive to neighbouring electric fields, and therefore the potential variation in the quantum box. The resistance variation with voltage in this channel is calibrated by applying a known voltage to the quantum box and in this way the Coulomb charging energy is measured in a non-invasive manner. The Coulomb charging caused by removal of electrons from the dot is detectable while immeasurably small currents flow through the dot. The measured charging energy is 500 PeV which compares well with that calculated from a measurement of the total capacitance between all the surrounding conducting regions and the quantum box.

1. Introduction Theoretical suggestions for non-invasive measurements of voltage and chemical potential in sub-micron devices have been discussed in the literature for some time [ll. To measure voltage a capacitance technique is required, while to measure chemical potential one would require a weakly coupled voltage probe. In this paper we will discuss the realization of a technique to measure the single electron charging energy in a semiconductor quantum box in a non-invasive manner [2]. The charging voltage on a metal

* Corresponding author. ’ Also at: Toshiba Cambridge Research Centre, 260 Cambridge Science Park, Milton Road, Cambridge CB4 4WE, UK.

island has been investigated using the known properties of a similar metal CB device fabricated nearby [31. Recently there has been a great deal of work in the field of single electron charging or Coulomb blockade (CB). Such effects were first observed in granular metal films [41 and later in sub-micron metal-oxide-metal junctions [5]. In the light of the initial transport measurements through a lateral quantum dot defined using Schottky gates above a high mobility GaAs/GaAlAs heterostructure [6], it was suggested [7] that such a structure should show CB oscillations in the conductance with changing gate voltage. These CB oscillations were subsequently observed in such structures [S]. In order to make a non-invasive measurement of the electrostatic potential on the quantum dot in the CB regime ‘we nee’ded to make use of

0039.6028/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved SSDI 0039-6028(93)E0588-L

capacitance coupling so that no charge flows from the dot. Typically a 2DEG area of 0.5 Frn square has a capacitance of IO-‘” F while the capacitance coupling to the dot of a 2DEG 0.1 pm away is of the order lo-” F. Thus the addition of one electron onto the dot would put l/10 of an electron onto the neighbouring 2DEG area. A voltmeter connected to this region would be unable to measure this voltage as a charge of greater than l/10 of an electron would be required to make such a measurement. What was required was a method of miniaturising our voltmeter and putting this next to the quantum box. The solution was to use a ballistic point contact next to the quantum box, but in a separate circuit. When such a channel is biased in a tunneling regime it is very sensitive to surrounding voltages, so the voltage change on the dot due to the addition of single electrons then causes the resistance of this channel to alter.

2. Measurement

of charging

energy

Fig. 1 is an electron micrograph of the gate pattern defined by electron beam lithography on our device. The bar down the middle separates two circuits, one with a quantum box defined by the three gates on the right-hand side of the bar, and the other with a ballistic point contact defined between the gate on the left and the bar. The gates were fabricated on top of a two-dimensional electron gas (2DEG) with a mobility of 129 m? VI spI and a carrier concentration of 3.61 X 1O’5 ml. The 2DEG resides 70 nm below the surface. Voltage probes on either side of the box and the ballistic point contact made it possible to measure their conductance simultaneously. By putting a negative voltage of - 0.8 V on the bar down the middle (Gl in Fig. 2) the two circuits can be electrically isolated so that even when a DC bias voltage of more than 100 mV is applied a current of less than 1 pA flows between them. By adjusting the gates G3, G4 and G5 on the right of the bar, CB oscillations in the conductance measured from top to bottom are observed. The conductances are measured using a 10 PV AC applied voltage and a current ampli-

Fig. I. An clcctron micrograph of sample showing the gates Cl. G2. G.3. G‘l and G5.

ficr. All the mcasurcmcnts wcrc performed in a dilution refrigerator with a base temperature 7‘ c 20 mK. The gates GS and G3 form two barriers and the gate G4 is used as a plunger to change the area of the confined 2DEG dot. From Aharonov-Bohm-like oscillations [9] when the conductance is greater than 2c’/h we deduce that the area of the dot is approximately O.hS x 0.24 pm’. With the conductance less than 2r’/lr. sweeping the plunger G4 results in periodic oscillations of period 12.3 mV. These oscillations are independent of magnetic field indicating that they are due to single electron charging. In parallel with this measurement the resistance of the channcl between the gates Gl and G2 is measured with the resistance set greater than h/2c’ ttypitally 100 k0). We chose a resistance greater than the quantized value, because then the transmis-

C. G. Smith et al. /Surface

6

sI

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Fig. 2. (i) The CB oscillations in the dot against voltage, with the detector resistance also shown. calibrated dot potential variation.

plunger (ii) The

sion probability must be less than one. In other words the electrons are tunneling through a barrier. As we are measuring the resistance of the detector with a constant current of 1 nA and the detector resistance is around 100 kR we have a bias voltage of around 100 PV on the detector, a value comparable with the tunneling barrier height. This level of current was required to ensure a sensible signal-to-noise ratio. As the screening in the plane of the 2DEG is not perfect the potential seen by the dot is sensitive to all the surrounding gates including G2. The capacitance between all these gates and the regions of 2DEG can be deuced from the period of the CB oscillation with an applied DC voltage to each gate or area of 2DEG. The total capacitance of the dot is then the sum of all these values and comes to C, = 2.9 X lo-l6 F. This implies that the charging energy of the dot AE = e2/CX = 550 I*.eV. In Fig. 2 the conductance of the dot and the resistance of the detector are shown. The conductance in the quantum box oscillates from 0.2 to 6 PS while the resistance of the detector rises from 90 to 120 kR. On this background there are dips

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which correspond with the peaks in the conductance in the neighbouring circuit. The rising background is due to direct coupling between the gate G4 (plunger) and the detector. Although it is useful to measure relative potentials, it is far more useful to measure the absolute change in potential. To obtain this information the detector had to be calibrated. To achieve this the voltage on the plunger was removed so that a voltage could be applied directly to the 2DEG region between the tunnel barriers of the CB dot. The variation in resistance with voltage applied to the 2DEG is then recorded and was found to be nearly linear and not exponential. This occurs because the bias voltage across the barrier is large compared to the barrier in the detector; however, the exact form of the relation is unimportant. Although in this calibration procedure the leads connected to the dot also float up to the applied voltage, this modification to the resistance of the detector is small because of the greater distance of these regions. The rising background due to the plunger G4 is corrected for by raising the temperature of the device until there is no CB structure in the detector. When this high temperature background is subtracted from the low temperature curve and the calibration with the applied 2DEG potential is used we get a non-invasive measurement of the potential on the dot. As the bias voltage across the detector is of the order of 100 PV the temperature dependence of the detector is weak for temperatures less than 1 K. It is worth noting that the structure in the detector is washed out by a temperature of 500 mK while the CB in the dot itself is lost at around 1.2 K. The signal in the detector is being lost when the thermal smearing is comparable to the change in the barrier height due to the CB potential in the box. This implies a value of 40 PeV for the detector modulation. Returning to Fig. 2, the bottom curve shows the amplitude of the potential oscillations in the quantum dot. The first thing to notice is that the amplitude of the oscillations in potential are of the order 500 PeV in good agreement from that calculated from our estimate of e2/CZ. This is around 12 times larger than the modulation it causes in the detector. This value fits nicely with

W, is the energy difference between the highest states in the dot to the first available state in the leads. The total conductance is given by

the ratio of the total capacitance, C,, to the capacitance from the dot to the detector, which comes to 14. The second obvious feature of this graph is that peaks in the conductance through the box correspond to the points of inflection in the potential. This is to be expected, because we are measuring the time averaged potential on the dot as a current of 0.6 nA flows at the peaks in the conductance. This corresponds to 600 million electrons per second passing through the dot making it oscillate from +e2/C, to -e’/CX every nano-second. As the time constant in the detector circuit is 300 ms no potential is measured. Changing the gate voltage off a peak in conductance, means that the probability of finding an electron in the dot or outside the dot is no longer identical. If we are on the left of the peak the voltage on the plunger is more negative so the electron spends less time on the dot and the detector becomes more positive. The potential of the dot reflects the difference between the charge on the dot that would minimise the charging energy and the percentage time an electron is on the dot. In the case of extremely narrow conductance peaks, this can lead to a sawtooth wave form for the dot potential [lo]. As a check of the charging energy one can measure the temperature dependence of the conductance troughs in the quantum box when showing CB. From a plot of the log of these conductance minima against l/T, the two activation energies W, and W, can be deduced. Where W, is the difference between the Fermi energy in the leads and the next available state in the dot, and

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G = G, exp( - W,/kT)

+ G, exp( - W,/kT),

through the dot where G, 7 are the conductances due to the two levels. The Arrhenius plot of In(G) versus l/T gives a straight line with an intercept of 11.4 PS and a slope with an activation energy of 50 PeV. We would expect a value of e’/2C, = 2.50 PeV for the activation energy and a value of twice the conductance in the peaks for the intercept of 12 pS. This discrepancy between e’/CZ: and the activation energy has been noted before [S], and may be due to co-tunneling. The charging energy being reduced by an electron being ejected from the far end at the same time as an electron enters. The reduction in the energy will then be given by the ratio of the tunneling time through one barrier to the RC time constant for the dot.

3. The detector signal the box is small

when the conductance

in

By continually sweeping the plunger voltage to more negative values the capacitance effect on the barriers reduces the conductance through the dot, until it can no longer be measured. At the same time the oscillations in the detector become sharper and can be followed for a further 110 periods, until the voltage is -4.05 V. After this point the signal vanishes. This is shown in Fig. 3. The detector has to be reset slightly to take into

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Fig. 3. The detector resistance when the dot conductance is immeasurably small. The detector resistance is reset to 150 kl2 whenever it is driven above 360 kR by the capacitance effect of the plunger on the detector. The insert shows the development of the sawtooth shape.

C.G. Smith et al. /Surface

account the effect of the plunger on the detector. At more negative voltages the oscillations disappear and the effect of the plunger on the detector becomes greater suggesting the dot is no longer screening it. To see if this is the last electron in the dot we must estimate the likely number of electrons to start with. To do this we can use the Aharonov-Bohm-like oscillations seen when the conductance is greater than 2e2/h. These occur when a magnetic field is applied such that Landau levels are formed in the 2DEG, which results in edge states circulating round the dot. The conductance then oscillates periodically in B with a period of h/d, where A is the area of the box. The measured period is 20 mT, which corresponds to an area of 2.1 x lo-l3 m2 [9]. At a constant magnetic field similar oscillations are seen as the plunger voltage V,, is changed altering the area by d A. As we have a value for dV,,/dA (3.3 x 10’” V rnp2 measured from the Aharonov-Bohm-like oscillations) and dl/,,/dN (12.2 mV measured from the Coulomb blockade oscillations) we therefore can calculate dN/dA, the carrier concentration in the dot. This calculation results in a value of 2.57 X 1015 rnp2, which is 20% lower than the sheet carrier concentration in the 2DEG away from the gates. When this is combined with the calculated area of the dot one obtains 532 for the number of electrons in the dot. From these estimates we can deduce that the loss of signal in the detector is not due to the removal of the last electron, but must be due to an increase in the barriers to the dot by such an extent that no electrons can flow within the time constant of the measurement of the detector. It is worth noting that as the barriers become higher the potential measured on the detector becomes more sawtooth in nature reflecting the narrower, but immeasurably small amplitude conductance peaks.

4. Conclusion By defining a quantum box in one circuit and a ballistic constriction in another nearby circuit, we

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have been able to use the constriction biased in the tunneling regime as a non-invasive voltage probe of single electron charging on the dot. The constriction can be calibrated to an accuracy of 50 pV, giving a value of 500 PV for the charging energy on the dot. The detector can still be used to show the sawtooth shape of the potential, even though the current through the dot is immeasurably small. The activation energy of the dot is a factor of five smaller then the measured, and calculated value of the charging energy in the dot. This may be a result of co-tunneling of electrons through the dot.

5. Acknowledgements

This work was supported by the Science and Engineering Research Council and Esprit Project No. BRA6536.

6. References [l] R. Landauer, Z. Phys. B (Condensed Matter) 68 (1987) 217. [2] M. Field, C.G. Smith, M. Pepper, D.A. Ritchie, J.E.F. Frost, G.A.C. Jones and D.G. Hasko, Phys. Rev. Lett. 70 (1993) 1311. H. Pothier, E.R. Williams, D. Esteve, C. 131 P. Lafarge, Urbina and M.H. Devoret, Z. Phys. B (Condensed Matter) 85 (1991) 327. [41 I. Giaever and H.R. Zeller, Phys. Rev. Lett. 20 (1968) 1504. [51 D.V. Averin and K.K. Likharev, in: Mesoscopic Phenomena in Solids, Eds. B.L. Al’tshuler, P.A. Lee and R.A. Webb (Elsevier, Amsterdam, 1991). [61 C.G. Smith, M. Pepper, H. Ahmed, J.E.F. Frost, D.G. Hasko, D.C. Peacock, D.A. Ritchie and G.A.C. Jones, J. Phys. C 21 (1988) L893. and R.I. Shekhter, J. Phys. (Condensed 171 L.I. Glazman Matter) 1 (1989) 5811. [81 U. Meirav, M.A. Kastner, M. Heiblum and S.J. Wind, Phys. Rev. B 40 (1989) 5871. H. Ahmed, J.E.F. [91 R. Brown, C.G. Smith, M. Pepper, Frost, D.G. Hasko, D.C. Peacock, D.A. Ritchie and G.A.C. Jones, J. Phys. (Condensed Matter) 1 (1989) 6291. N.C. van der Vaart, A.T. Johnson, [lOI L.P. Kouwenhoven, W. Kool, C.J.P.M. Harmans, J.G. Williamson, A.A.M. Staring and C.T. Foxon, Z. Phys. B (Condensed Matter) 85 (1991) 367.