NANoSTRUCTURED MATERIALS VOL. 3, PP. 283-291, 1993 0965-9773/93 $6.00 + .00 COPYRIGHT ~)1993 PERGAMON PRESS L'm. ALL RIGHTS RESERVED. PRINTED IN THE USA

COULOMB BLOCKADE IN SMALL QUANTUM DOTS

Cd.B. Ford, P.J. Simpson, M. Pepper, D. Kernf, J.E.E Frost, D.A. Ritchie and G.A.C. Jones

Cavendish Laboratory, Madingley Road, Cambridge, CB30HE, UK tIBM T.J. Watson Research Center, PO Box 218, Yorktown Heights, NY 10598, USA

Abstract-We have studied Coulomb blockade effects in semiconductor quantum dots connected to leads via adjustable tunnel barriers. The single-particle energy level spacing becomes non-negligible compared with the Coulomb charging energy. The oscillations become irregularly spaced as the dot is made very small.

INTRODUCTION

Recent technological advances in the growth of semiconductor crystals and in lithography have made it possible to create structures smaller than the elastic and inelastic mean free paths of the charge carriers (at low enough temperatures). Surface gates can be used to form adjustable barriers through which electrons must tunnel in order to reach the rest of the sample. The carrier concentration can be so low that only a few Fermi wavelengths fit within the structure, in all directions; the number of electrons within the device is then also small, for example less than a few hundred, and the capacitive coupling to the rest of the sample is very low. We thus have a"quantum dot" in which the quantum -mechanical states are discrete, but in which the states are perturbed very significantly by adding even one extra electron to the dot, due to the Coulomb interaction.

The starting material for such studies is usually a GaAs-AIGaAs heterostructure grown by Molecular Beam Epitaxy. Extremely pure, single-crystal layers may be grown with atomic precision. A typical structure is shown in Figure 1. An undoped layer of GaAs is grown on a GaAs substrate, followed by an undoped layer ofAlxGal.xAs (x = 0.3). This "spacer layer" is usually 20- 40 nm thick. Silicon donors are then incorporated into more AlxGal.xAs, and the sample is capped by a thin protective layer of GaAs. Electrons from the donors then accumulate in a thin layer at the conduction band discontinuity at the interface between the GaAs and the A1GaAs. This technique of modulation doping produces a high-mobility two-dimensional electron gas (2DEG), since the spacer layer separates the electrons from the ionized impurities which gave rise to them, dramatically reducing the scattering, provided that the temperature is low (e.g. 4K). The potential well confining the electrons is only about 10 nm wide, so usually only one quantum-mechanical subband is occupied in the vertical direction.

283

284 C_,dB F~o ET At..

300rim

lOOnm

20rim undoped GaAs

40rim AIGaAs (Si doped)

40nm undoped AIGaAs

. . . . . . . . . . . . . . 2DEG

- 0.5pro undoped GaAs

Figure 1. The cross-section of the heterostructure used for the device described in this paper. The shaded regions depict a split-gate on the surface.

We can confine the electrons to fewer than two dimensions using gates on the surface. A metal gate deposited on the GaAs forms a Schottky barrier, so that negative voltages may be applied to the gate, progressively depleting out the electrons beneath. I fa gap is left between two such gates, the electrons are restricted to the region beneath the gap, forming a narrow channel (see Figure 1) (1). The geometry of the channel can be whatever shape is required, since it resembles that of the lithographically-defined gap. Electron beam lithography is normally used to define the fine features of such patterns. Split gates have several advantages over devices made by other fabrication methods. They do not involve damaging the material by etching or implanting into it. Furthermore, the width of the channels is continuously variable, from a little less than the lithographic width, down to nothing. A consequence of the depletion is that the carrier concentration in the narrow regions is reduced. One-dimensional (ID) subbands are formed, and these are pushed up through the Fermi energy as the gap is squeezed with increasingly-negative gate voltage Vg.

In very short split-gate constrictions, electrons can travel ballistically (i.e. without scatter- ing), and the conductance turns out to be the same for each ID subband, due to a cancellation between the density of states and the velocity in ID. Thus the total conductance G rises in a series ofquantized steps as the channel is made wider, i.e. G = 2ie2/h, where i is the number of subbands (2,3). More complicated structures can be made, such as a cavity where a pool of electrons is connected to the wider 2DEG via two or more constrictions, called a "quantum dot" (4,5). Multiply-connected structures are slightly harder:, an insulating layer is required to keep the metal away from the surface except where the gates are required. Aring structure made in this manner shows interference between the electron waves propagating between the leads around the two sides of the ring (6). The resistance oscillates as the magnetic flux through the ring changes, due to the Aharonov-Bohm effect, with period h/e in the flux. This requires phase coherence around the ring, and therefore even lower temperatures than for ballistic effects, namely < IK.

COULOMB BLOCKADE IN SMALL QUANTUM DOTS 285

When a quantum dot becomes isolated from the leads by constrictions which are just past pinch-off, electrons can only enter and leave by tunneling through the potential barriers at those constrictions. The barriers act like leaky capacitors with a total capacitance C to the external circuit and the gates. There is a charging energy e2/C associated with increasing the charge on the dot by one electron, ignoring the quantum single-particle energy levels. At low applied bias (<e/C), this energy is greater than that provided by the bias, and tunneling into the dot is blocked unless the highest occupied level in the dot lies e2/C below the Fermi energy EF in the leads. If we include the single-particle levels EN, with the simplifying approximation that their spacing is independent of the number N of electrons in the dot, then the energy required becomes AEN = E/v+ 1 - EN + e2/C (7). As the dot size is increased, each time the electrochemical potential in the dot falls to a value AEN below EF, an extra electron will be able to enter the dot, making the dot neutral again, and the process will repeat. Thus the conductance through the dot should oscillate. This effect is referred to as the Coulomb blockade of single-electron tunneling (8). It has been studied extensively in metals, where there are many electrons in the dot. However, it has only recently been possible to make semiconductor dots, containing only a few electrons (9-11). If, as is the case in all existing devices, the single-particle level spacing is small compared to the charging energy, then the oscillations will be approximately periodic, with period e/Cg, where Cg is the capacitance to the gate being swept. Any distortion of the periodicity should be due to the single-particle levels. We have tried to move into the regime where EN+I - EN is not negligible in comparison with e2/C, by squeezing the dot as much as possible. We will show results where distorted oscillations are indeed seen, and will discuss whether they can all be explained in such a way. It is not necessarily a good approximation simply to add the charging energy to the single particle energies EN (12); it may sometimes be necessary to consider all the interactions for an N particle system (13).

0.4

0

G1

G,I

G3

(;2

• 41.45 -0.4 -0..-35 VsfV)

Figure 2. The conductance through the cavity shown schematically in the inset. The gates are shown in black, except for the gate being swept, which is shown in outline.

286 GJB FORD ET AL

Cavities were fabricated on a GaAs-A1GaAs heterostructure with mobility -100 m2/Vs, and sheet carrier concentration - 1.5 x 1015 m -2. Titanium/gold was used for the gate metallization. The layout of the device is shown schematically in the inset to Figure 2. It consists of a set of narrow gates, allowing approximately independent control of each constriction and of the size of the dot. Some gates have such small gaps between them that electrons there have been depleted out by the time the gates are fully defined. These gaps can be ignored but they allow the gates to be biased independently. The constriction widths are 0.4 p.m, and the distance between constrictions is about 0.5 l.tm.

RESULTS AND DISCUSSION

The conductance through the dot is shown in Figure 2, as a function of the gate shown in outline in the inset. This cenlral"plunger" gate changes the size of the dot without greatly affecting the two constrictions defined by the other gates. The magnetic field B (perpendicular to the surface) was zero in this case. All measurements were carried out using a low-frequency ac voltage ofconstantamplitude 101aV, at temperatures below I00 mK.Theoscillations haveaperiodof7mV, independent of B even at the highest fields used (up to l iT) (11). Although the field increases the confinement in the constrictions, raising the tunnel barriers so that they pinch off sooner, the capacitance between the dot and the rest of the sample (gates and leads) is largely unchanged. If these oscillations were due to resonant tunneling through a ladder of single-particle states, these states would have become spin-split by such large fields and would have changed their spacing as the magnetic confinement took over from the electrostatic confinement. The periodicity agrees with the capacitance estimated for such a gate, if the number of electrons in the dot changes by one in each period, and it scales with the number of gates (11), although individual gates may suffer from imperfect lithography giving more or less coupling to the dot. Thus we can attribute the oscillations to Coulomb blockade (CB) as described in the introduction. We estimate the charging energy e2/C to be 0.5-1 meV.

Figure 3(a) shows more CB oscillations for the same sample (14). The periodicity is much the same as in Figure 2, but we have tried to squeeze the dot as much as possible in order to increase the single-particle energy level spacing. The oscillations become much less regular near pinch- off, with some peaks moving apart or together (such as the double peak labeled"D"). Additionally, some peaks seem to split (e.g. the peak labeled "S"), although they are still so close together that it seems unlikely that they are peaks due to two different electrons. Similar results have been reported by Staring et al. (15). The structure of each peak changes gradually as B is incremented, but there are few clear trends which can be associated with the increasing dominance of magnetic confinement or with spin splitting (14). Figure 3(b) shows the irregular oscillations at a number of widely-spaced values of B. As the field increases, the zeros become better defined and the peaks become sharper, but some irregularity remains, even at B = 2.5 T.

There are various possible explanations for the split and double peaks. A simple reason could be that the dot may break up into two parts, effectively forming two dots in series, each exhibiting Coulomb blockade. Then a zero of conductance in either dot would give a zero for the pair, causing some peaks to appear split when a zero of one lies near the middle of a peak of the other. Figure 4 shows what happens when a device is made intentionally with two adjacent regions, both of which are set to give Coulomb blockade. There are two periods visible: a very short period, corresponding to CB in the dot next to the gate which is being swept. However, the conductance

COULOMB BLOCKADE IN SMALL QUANTUM DOTS 287

r~

0.4

0.2

0

1.5

1

0.5

O1 -0.95 -0.9 -0.85 v oo

Figure 3. The conductance as a function of the plunger gate's voltage. (a) B = 0. (b) B between 0.5 T and 2.5 T, for similar voltages on the other gates to those in (a).

288 CJB FORD ET AL

is modulated by another, long period, oscillation (the three large peaks spread across the graph). This is CB in an adjacent region, which is only affected slightly by the swept gate, hence the large change in voltage required to change the number of electrons there by one. It is possible that in our single dot, the plunger gate starts to pinch offto the opposite side of the dot, forming two little dots, but this would be unlikely to produce the simple splittings and distorted peaks that we see.

Another explanation for the split peaks has recently come from measurements of the effects of an applied dc source-drain bias (16,17). Regular CB peaks become broadened as the bias is increased, and structure appears, each peak developing a number of bumps on it, the number being the same for each peak and increasing with bias. This can be understood if the zero-dimensional (0D) single-particle level spacing is an appreciable fraction of the charging energy. A dc bias increases the chemical potential ~tL in one lead (say the left one), with respect to that in the right lead, laR (see Figure 5). At low bias, when the electrochemical potential forN electrons in the dot, lad(N), lies between [.tL and ~tR, eleclrons can flow from left to right, occupying the Nth 0D level EN while in the dot and giving a simple CB peak (Figure 5(a)). However, if the bias is a little greater than the splitting of the 0D levels, i.e. ~L- [tR > EN+I - EN, then when both EN andEN+] lie between ~tL and l.tR, tunneling can also proceed via the first excited state,EN+ ] (Figure 5(b)). There are thus two states available through which to tunnel, and so the conductance is increased. As the dot size is decreased, EN+I will exceed IlL so that only one state remains between ~tL and ~tR. Thus the conductance drops, rising again when it becomes possible to tunnel out via either the Nth or (N- l)th level. It then drops to zero as EN moves out of the range, until the next CB peak is reached. Similarly when the bias just exceeds n level spacings, there will be n + 1 bumps on the peak.

This effect could give rise to the split peaks we observe, if the -10 laeV provided by the ac bias were larger than the level spacing. However, this seems unlikely since we estimate the spacing to be at least 100 ~teV. We would also expect each peak to be split in a similar way, which is not the case. There might nevertheless be an unintentional dc voltage present due to some thermoelec- tric emf, for example.

0.04

0.02

m i l l

ltkr

-0.85 -0.8 -0.75 -0.7 VgfV)

Figure 4. The conductance through two quantum dots in series, sweeping the gate shown in outline in the inset. The barriers are set to give CB in both dots.

COULOMB BLOCKADE IN SMALL QUANTUM DOTS 28~

PL

(a)

PR

V

(b)

L j L

Figure 5. Schematic diagrams of the conduction band and levels in the dot for (a) low bias and (b) slight de bias such that E N . 2 - EN > ~tt, - P4¢ > EN+ I - EN.

Very recently, Kouwenhoven et al. have found similar peak splitting in a very small dot at low bias (18). They estimate the number of electrons to have been about 10, with a 0D level splitting of about 300 I~eV. Each CB peak consists of a series of narrow peaks, with good zeros between them. The CB energy e2/C is larger than in our dot, making it easier to separate the two types of peak. Consideration of the detailed behavior of the oscillations enabled them to rule out most of the plausible explanations, including two dots in series, and most forms of electron correlation effects. We cannot tell whether the split or double peaks that we observe are related to these multiple peaks. Measurements of more such structures are required to clarify the matter.

The double peak "D" of Figure 3(a) comes and goes as the tunnel barrier heights are varied (Figure 6). Each of the three sections is at a different voltage on the gate controlling the left-hand barrier. Within each section, the other barrier is incremented. It appears in Figure 6(a) that the last peak before pinch-off is the usual distance from its neighbor. The peaks can then be followed as the left-hand barrier is raised, through Figure 6(b) to 6(c), where the last peak only appears for intermediate values of the right-hand barrier. Surprisingly, it is now considerably closer to its neighbor than are all the other peaks. The reduction in size of the dot due to the raising of the barriers might be expected to reduce the capacitance between the dot and the gates and leads. This would, however, cause the spacing to increase . There is not enough data as yet to reach any particular conclusion, but it is possible that the 0D levels are now sufficiently widely spaced to shift the peaks noticeably. Also, it may be significant that only the last peak before pinchoff behaves strangely. It is very hard to know how many electrons are left in the dot at this point, because the first thing to pinch-off may be one or other of the barriers, or the dot itself. If the dot were about to deplete out completely, then the shift might be due to the lack of screening in a dot containing only a few electrons. Adjusting the tunnel barriers may have symmetrized them so that they both pinch off together, allowing the states in the dot to be probed for as few electrons as possible.

2g0 CJB FORD El" AL

-0.96 -0.94

Figure 6. The conductance through a single dot for various values of the barrier heights, showing the behavior of the last (left-most) peak. In each graph, the right-hand gate is

incremented from -0.840 V to -0.849 V in lmV steps, and the curves are offset upwards for clarity. Each graph has a particular voltage on the left-hand gate: (a) -0.731 V, (b) -0.733 V

and (c) -0.737 V. The voltage on the upper gate is -0.75 V.

CONCLUSIONS

In conclusion, we have measured gated samples in which a few (between 0 and 150) electrons are confined by tunnel barriers. Coulomb Blockade oscillations are seen, and these become irregularly spaced when the dot is small. Some of the irregular structure may be due to the

COULOMB BLOCKADE IN SMALL QUANTUM DOTS 291

dot splitting into two smallerregions, but the rest may indicate that single-particle level separations are no longer negligible compared with the charging energy, or that there are so few electrons in the dot that interactions between them cause the charging energy to depend on the exact number of particles.

ACKNOWLEDGEMENT

We would like to acknowledge financial assistance from the SERC.

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