# Superconducting single-electron transistor coupled to a two dimensional electron gas: Transmission lines, dissipation, and charge averaging

###### Abstract

We have developed a novel system consisting of a superconducting single-electron transistor (S-SET) coupled to a two-dimensional electron gas (2DEG), for which the dissipation can be tuned in the immediate vicinity of the S-SET. To analyze our results, we have developed a model of the environment for S-SET/2DEG systems that includes electromagnetic fluctuations coupled both through the S-SET leads and capacitively to the S-SET central island. We analyze this model, treating the leads as finite transmission lines, to find the probability function for exchanging energy with the environment. We also allow for the possibility of low-frequency fluctuations of the S-SET offset charge. We compare our calculations with measurements of SET conductance versus 2DEG conductance and find good agreement for temperatures , while unexplained discrepancies emerge for lower temperatures. By including the effects of charge averaging we are also able to predict the shape and evolution of - curves as the 2DEG in the vicinity of the S-SET is changed.

###### pacs:

74.50.+r,73.23.Hk,74.40.+k## I Introduction

The effects of the electromagnetic environment on electric transport has been a subject of extensive theoretical and experimental interest in recent years. The reasons for interest are varied, as are the systems for which studies of the effects of the environment have been performed. Recent interest in quantum computationMakhlin et al. (1999); Nakamura et al. (1999); Mooij et al. (1999); Vion et al. (2002) has prompted interest in the effects of dissipation on decoherence rates in superconducting qubits.Shnirman et al. (1997); Makhlin et al. (2001) Double quantum dots have been used to study the effects of the environment on inelastic tunneling rates,Fujisawa et al. (1998) and have been proposed as detectors of high-frequency noise produced by mesoscopic devices.Aguado and Kouwenhoven (2000) Finally, interest in quantum phase transitionsSohdhi et al. (1997) has prompted study of the effects of dissipation on superconducting systems such as thin filmsMason and Kapitulnik (1999); Kapitulnik et al. (2001) and Josephson junction arrays.Rimberg et al. (1997)

Given the possibility of using a single Cooper pair boxBouchiat et al. (1998); Nakamura et al. (1999) as a qubit and its similarity to the superconducting single-electron transistor (S-SET),Grabert and Devoret (1992) it seems logical to use the S-SET as a model system for studying the effects of environmental dissipation on coherence and transport in small tunnel junction systems. Recently, there have been several such attempts, motivated by experiments in which a Josephson junction array was fabricated in close proximity to a two-dimensional electron gas (2DEG) in an GaAs/AlGaAs heterostructure,Rimberg et al. (1997) which can be used as a tunable source of dissipation. (While the effects of a mechanically tunable environment were earlier studied in the macroscopic quantum tunneling regime,Turlot et al. (1989) use of a 2DEG allows more flexible tuning of the environment over a larger impedance range.) In place of an array, we and the Berkeley group have instead used similar fabrication techniques to couple an S-SET to a 2DEG.Kycia et al. (2001); Lu et al. (2002, ) In one instance, the focus was on transport at higher biases, in the regime of the Josephson-quasiparticle cycle.Lu et al. (2002) In the others, the focus instead was on the low bias regime and the tunneling rates of Cooper pairs.Kycia et al. (2001); Lu et al. Theoretical work aimed at an explanation of the results of the Berkeley group was undertaken by Wilhelm, et al.Wilhelm et al. (2001)

The primary experimental difference between our own work and that of the Berkeley group lies in the way in which the environment is varied. The Berkeley group followed an approach developed earlier for the study of junction arrays,Rimberg et al. (1997) using a gate on the back side of the substrate to vary the sheet density of the 2DEG, and therefore its resistance per square . Such a change is global, and affects the 2DEG not only immediately beneath the S-SET, but beneath the macroscopic leads used to measure it as well. In our own work, by placing Au gates on the surface of the sample near the S-SET itself, we were able to vary the dissipation in the 2DEG locally, while leaving 2DEG beneath the leads virtually unchanged.Lu et al. (2000, )

The Berkeley group compared their results to the theory of Wilhelm, et al., which predicted that within linear response the conductance of the S-SET would scale with the ground plane conductance and temperature as . While the Berkeley group did observe power law behavior, their measured exponents were not in quantitative agreement with theory. Furthermore, the measured depended on and on , calling the scaling form into question.

In our own work,Lu et al. we examined a somewhat more complex model for the environment than that considered by Wilhelm, et al. Specifically, we allowed for coupling of electromagnetic fluctuations to tunneling Cooper pairs due simultaneously both to the S-SET leads and to the 2DEG in the immediate vicinity of the S-SET, which is coupled to the S-SET central island via a capacitance . By also allowing for averaging of the S-SET offset charge, we were able to obtain good agreement between our measurements and calculations. Here we examine our model of the environment in somewhat more detail, provide additional experimental data which supports our earlier analysis, and also give some additional details of the calculation.

## Ii Sample and Environmental Calculations

### ii.1 Sample Design and the Environmental Model

#### ii.1.1 Sample Design

Our samples consist of an -based S-SET fabricated in close proximity to a 2DEG formed in a GaAs/AlGaAs heterostructureLu et al. (2000) as shown in Fig. 1 below. We begin by fabricating six Au gates which can be used to deplete the electrons beneath them by application of a negative gate voltage . At the center of the Au gates we then fabricate our S-SET, as can be seen in the electron micrograph (expanded view in Fig. 1). Note that as shown in the larger diagram the S-SET leads extend over the 2DEG to macroscopic contact pads. For the vast majority of their length they are well away from the Au gates and the 2DEG beneath them is independent of . When we apply a gate voltage to all six Au gates, the electrons immediately beneath them are depleted, leaving a small pool of electrons beneath the S-SET. This pool is connected to the rest of the 2DEG (held at ground) only by two quantum point contacts (QPCs) with conductances (assumed equal) as shown in the micrograph. It is also capacitively coupled to the S-SET island through a capacitance as shown in the lower right inset to Fig. 1. When all six Au gates are energized as described above, we say that the electrons are confined in the “pool” geometry. We do not refer to the pool as a quantum dot since for these experiments the QPCs are sufficiently open that no Coulomb oscillations are detected in the pool and discrete energy levels have not formed.

Because the Au gates can be biased independently, we can also apply to only the four outermost gates which form the QPCs. As before, the electrons beneath the S-SET are coupled to ground through the QPCs. In addition, however, they are now coupled through a resistance to two large reservoirs of electrons located between the four outermost gates, as can be seen in Fig. 1. The reservoirs are in turn coupled to ground only through a capacitance . When only the four outer gates are energized, we say that the electrons in the 2DEG are confined in the “stripe” geometry. We observe significant differences between the measured S-SET conductance versus applied gate voltage for the two different geometries, as will be discussed below.

Regardless of the gate configuration used, we can apply a single model of the environment to our results, as shown in the lower left in Fig. 1. The SET island is connected to its leads through junctions with resistance and capacitance . We assume that the S-SET leads present an impedance to the SET while the 2DEG electrons have a total impedance to ground which is coupled to the SET through the capacitance . Nearly the entire length of the SET leads is far from the Au gates, so that is almost completely unaffected by the gate voltage . The electrons immediately beneath the SET are strongly affected by so that will in general be a function of , and may also depend on the configuration of gates used (i. e., on the pool versus stripe geometry). Finally, the SET is also coupled to the Au gates by a capacitance . We neglect the possibility of a substantial impedance on the gate lines largely because is by far the smallest capacitance in the problem. Furthermore, any gate impedance would be substantially reduced since there are six gates whose impedances would combine in parallel. This general model (excluding the small gate capacitance ) has been investigated previously,Ingold et al. (1991); Odintsov et al. (1991) but without considering any particular form for the impedances and .

In Table 1 below we give the relevant sample parameters for the two samples S1 and S2 considered here. The parameters were determined from electrical measurement and simulations as discussed elsewhere.Lu et al. (2002)

sample | |||||||
---|---|---|---|---|---|---|---|

S1 | 181 | 120 | 20 | 356 | 118 | ||

S2 | 375 | 260 | 20 | 382 | 77 |

#### ii.1.2 Tunneling Rates

In general for our samples the charging energy where satisfies , where is the Josephson energy of junction given by the Ambegaokar-Baratoff relationAmbegaokar and Baratoff (1963) and is the superconducting gap. Under these circumstances the S-SET island charge is well defined, so that charge states can be used as the basis for calculating the tunneling rates.Ingold and Nazarov (1992); Schön (1998) We will also be concerned with transport at sufficiently low bias voltages and temperatures , that we need only consider the tunneling of Cooper pairs, for which the sequential tunneling rate through junction is given byAverin et al. (1990)

(1) |

which is valid for . Here is the change in free energy associated with the tunneling event, and the function describes the probability of the Cooper pair exchanging an energy with the electromagnetic environment during the tunneling process. Following the usual environmental theory,Ingold and Nazarov (1992) can be expressed in terms of the real part of the total impedance seen by the tunneling electron first through a kernel

(2) | |||||

and then through the Fourier transform

(3) |

A calculation of the tunneling rate must therefore begin with a clear understanding of the impedance presented to the tunneling electrons by the environment.

#### ii.1.3 Model of the Environment

Given the circuit model shown in Fig. 1, one can use standard network analysisGrabert et al. (1991) to calculate the impedance seen by an electron tunneling through junction in terms of the impedances and capacitances shown in Fig. 1. The result is given byIngold et al. (1991)

(4) |

where

(5) |

where for and

(6) |

ignoring terms of order . To proceed, we need accurate models of and ; we begin by considering .

#### ii.1.4 Model of

Since our leads are fabricated above the 2DEG, which acts as a ground plane, it is appropriate to model them as transmission lines.Pozar (1998) The most general form for the impedance of a lossy transmission line terminated in a load is given by

(7) |

where is the characteristic impedance of the line, its complex propagation constant and its length. At the relatively low frequencies () considered here, it is reasonable to ignore the inductive reactance of the line and treat it as a simple line with a resistance and capacitance per unit length and . Doing so, we have that and . Looking out at the line from the sample, the line termination is provided by the bias circuitry, which typically presents a low impedance . For simplicity we therefore take in Eq. (7), and obtain the resulting approximation

(8) |

which we take as the basic form for the impedance of a finite line. This form has been considered previously in the context of incoherent tunneling of Cooper pairs in individual Josephson junctions.Kuzmin et al. (1991)

While this is likely a fairly accurate description of the impedance of a section of our leads, when used in evaluating the kernel in (2) it leads to integrals which are analytically intractable. Fortunately a further simplification is possible. We are interested in the low energy part of , which we expect from (3) to be dominated by the long time behavior of , which is in turn dominated by the low frequency part of the impedance . We therefore expand Eq. (8) around to obtain

(9) |

as a reasonable approximation to in the interesting limit.

Another common treatmentSchön (1998) of the transmission line problem is to consider an infinite line, whose impedance is given by . Unlike the finite line, for which the impedance approaches a constant at , the infinite line has a singularity at which dominates the long-time limit of and therefore . The kernel for the infinite line, as well as , can be calculated exactly in the limit. At non-zero temperatures, a high-temperature expansion must be performed instead.Wilhelm et al. (2001)

#### ii.1.5 Model of

Having developed a model for , we now consider a model for . The particular model will depend on the geometry we choose. For an unconfined 2DEG, the simplest choice is that is ohmic with an impedance related to of the 2DEG: . When the electrons are confined in the pool geometry, they are coupled to the remaining 2DEG by two QPCs with conductance (assumed equal), which appear in parallel from the vantage point of the SET. There is likely some shunt capacitance as well, but the associated roll-off frequency is typically large () and we therefore neglect it. So for the pool geometry, we take

(10) |

The stripe geometry is more complex. Here, in addition to the QPC conductances, the electrons beneath the SET are coupled to two large electron reservoirs with resistance located between the outermost Au gates. At their narrowest, the reservoirs are wide, but broaden in five sections to a width of . Each section contributes roughly , so that . These reservoirs are in turn coupled to ground capacitively through a capacitance , which we estimate from the size of the reservoirs to be on the order of . Using where we find

(11) |

where . For , then, approaches , while for , approaches . The imaginary part of in the stripe geometry is nonnegligible only in the vicinity of , so for our purposes we neglect it. In general then, at low frequencies is kept finite by the presence of the QPCs, and at higher frequencies is dominated by the smaller of and .

#### ii.1.6 Decomposition of

While the form for given by Eqs. (4)–(6) is complete, it is generally too complex to make significant headway in calculating . Fortunately, significant simplification is possible. For typical values of – and typical line lengths –, for small . In contrast, for sufficiently large , becomes quite small () and the condition is usually satisfied. It then becomes possible to decompose in Eq. (6) into a low part dominated by and a high part dominated by .

For small , as long as , we can safely neglect the terms in Eq. (6) involving . Furthermore, for such that we can ignore terms in Eqs. (4) and (6) that depend explicitly on . Making these simplifications, we have that in the small , large limit

(12) |

For large , we drop terms of order , and find that we can neglect the explicit frequency dependence in the denominator of (6) for . In contrast, we cannot necessarily neglect the explicit frequency dependence in the numerator, and find

(13) |

We combine this with (4) to find in this limit

(14) | |||||

Combining this result with (12), we obtain for the real part of

(15) |

which we take as our basic model for the real part of the impedance seen by an S-SET fabricated above a 2DEG ground plane. We believe this model should be applicable not only to our own system, but to that of the Berkeley group as well.Kycia et al. (2001)

To illustrate the degree of approximation associated with Eqs. (9) and (15), we show in Fig. 2(a) for three separate models of the environment; all are based on an line with and . The solid line shows based on Eqs. (4)–(6) and Eq. (8), i. e., on the impedance of a finite line coupled to a ground plane with impedance , using the full form for . The dotted line is calculated using the low-frequency version of given in (9) and using the decomposition (15) of , while the dashed line is the impedance of an infinite line using the same values of and (with no ground plane). We have not included a curve using the decomposition (15) and the exact form for a finite line in (8) since it is virtually indistinguishable from the full shown.

We note that the impedance of the infinite line rises above that of the more realistic forms for frequencies below a few MHz, so that in general it gives more weight to the low frequency modes and may be expected to give a more sharply peaked . More importantly, the low-frequency approximation to , while agreeing quite well below with the exact finite line result, significantly underestimates it for intermediate frequencies below . The approximation may therefore be of limited use for larger bias voltages; for low biases, however, it is likely to be more accurate than a model based on an infinite transmission line, which overestimates the impedance at low frequencies. Finally, at sufficiently high frequencies, the approximate and exact forms for converge.

### ii.2 Calculation of and

#### ii.2.1 Calculation of

Having produced a tractable form for , we can now proceed to a calculation of and through Eqs. (2) and (3). We begin by noting that when using the low frequency form for in (9), both parts of have the same form, namely

(16) |

for appropriate and . Calculations for and for this form have been given in detail elsewhere,Grabert et al. (1998); Grabert and Ingold (1999) but emphasize a different range for and result in different forms for . Using

(17) |

and we find that

(18) | |||||

where and are the Fourier cosine and sine transforms of respectively, taken to be functions of . We have ignored terms in independent of ; since for to satisfy the normalization condition we must have ,Ingold and Nazarov (1992) we will later ensure normalization by adding an appropriate constant in any case.

Of the four terms in curly braces in Eq. (18) for , the first three can all be evaluated analytically.Wilhelm et al. (2001) An analytic form for the entire kernel has also been found,Grabert et al. (1998) and analyzed for the overdamped case such that is large compared to the Josephson frequency . However, the range of and in which we are interested was not investigated. Nevertheless, we have made some progress in certain limits. We note first that the fourth term in (18) depends on the temperature only through the dimensionless combination , and write

(19) |

For zero temperature (), we have

(20) |

where is a Meijer G function. In the long time limit, this result goes as .

More generally, for , we find that it is important to consider the relative importance of the terms in Eq. (18). If we evaluate the integrals which we can treat analytically, we have

(21) | |||||

where is Euler’s constant. In order to compare the relative size of the terms, we evaluate numerically. We find that for , (either low temperature or small ), decays slowly with time. For long times then the term going as is by far the smallest, and can be neglected. Of the remaining terms, in the long time limit the logarithmic term dominates over and we write the kernel as

(22) | |||||

where the constant term will allow to be approximately normalized. This is essentially the resultWilhelm et al. (2001) of Wilhelm, et al., and is generally appropriate for dealing with the high-frequency part of Eq. (15) due to the relatively small values of and . However, this form may also be used for a sufficiently short and narrow section of transmission line.

In the opposite limit, for which , we find that the approximate analytic result

(23) |

holds for times . For typical temperature scales available in a dilution refrigerator (–), the logarithmic term is very small when (23) is applicable. We can therefore neglect it, and findLu et al. that

(24) |

This result is typically most useful for dealing with finite lines, for which is typically on the order of .

#### ii.2.2 Calculation of

Having obtained analytic forms for , a straightforward application of Eq. (3) allows one to calculate . Letting , we have from Eq. (22) for the large limit

(25) |

where is the beta function, in agreement with Wilhelm, et al.Wilhelm et al. (2001) For we have from (15) that and . While(25) is only valid for , in terms of this condition becomes , so that the result remains valid for quite large when is small.

In the small limit, we use (II.2.1) to obtainLu et al.

(26) | |||||

where is the incomplete gamma function, and and . Typically, this will be applied to a finite line, for which and . In some cases, for a short line, it may be more correct to use , and substitute the appropriate forms for and in Eq. (25) instead. ,

We show in Fig. 2(b) for the same transmission line parameters as used in Fig. 2(a). For comparison, we also show the form for an infinite transmission line given by where . Even at , is significantly broader than . While it is possible to obtain an analytic form for by expanding Eq. (2) in the high temperature limit, the resulting expression is of limited use for , since it involves only even powers of and therefore cannot satisfy detailed balance.Ingold and Nazarov (1992) As a result, such an expression cannot be used to calculate - characteristics, for instance, whereas in (26) can.

Ultimately, we are interested in calculating for the total impedance seen by the tunneling electrons. If we were to calculate the total kernel for the decomposition in (15), it would in general include all the terms in (21). We were unable to find an analytic form for under those circumstances. However, given the decomposition (15), it is possible to write where and correspond to the low- and high-frequency parts of , with corresponding and . The total is then given by the convolution , which can be performed numerically.

### ii.3 Multisection Transmission Lines

In our particular case, the sample leads do not have a single width. Instead, they broaden in sections from (section 1) to (section 4) as detailed in Table 2 below. As a result, we must generalize (9) to allow for the possibility of multiple sections. In general, we use Eq. (7) for a loaded transmission line, beginning closest to the SET with section 1. For this section, is taken to be the impedance of the second section, which is in turn terminated by the following sections. This cascading process is taken to end at our macroscopic contact pads, which are so broad as to provide very little impedance, and we therefore take for the last section, so that its impedance is given by (8). We also ignore a short () section with since it contributes only to , and its associated is very sharply peaked around .

If we were to use the exact form for given by the above cascading procedure, it would be too complex to be of use. Fortunately, a simple approximation gives fairly accurate results. We take

(27) |

where and are the resistance and capacitance per unit length of section , and is its length. We use the width and length of each section along with the 2DEG sheet resistance and depth to calculateCollin (1992) and , where is the dielectric constant of GaAs. To find an approximate form for we use the result (27) for in the decomposition (15). For comparison, we plot both this approximate result as well as the exact one obtained from the full form for in (4)–(6) and repeated applications of (7) versus frequency in Fig. 3 for different values of . The agreement is very good, especially considering the number of approximations required to develop a tractable approximate form for . The approximate version tracks the exact result very well except between the various corner frequencies of the transmission line sections, where its slope is generally too small. Agreement is better overall for larger values of , but even for the smallest values is still acceptable.

section | ||||||
---|---|---|---|---|---|---|

1 | 260 | 37 | ||||

2 | 884 | 0.76 | ||||

3 | 491 | |||||

4 | 375 |

We can calculate for tunneling through junction for the four section transmission line by choosing either or for a given section based on its value of in Table 2, and numerically convolving the four functions through

(28) |

While somewhat time consuming, this procedure needs to be performed only once for a given temperature since the 2DEG beneath the transmission lines is not affected by the Au gates, and so the transmission line parameters do not change with .

Finally, we then calculate the total for tunneling through junction by convolving with calculated from for the appropriate value of . This procedure typically must be performed many times, but can be done relatively quickly. Results for for tunneling through junction 1 of S2 are shown in Fig. 4 for a series of different values of . For , we take , which is already relatively broad, with a width several microvolts. In contrast, for small , is very sharply peaked around and approximates a delta function, as can be seen in the insets (a) and (b) in Fig. 4. As a result, is not strongly affected by until . Finally, for sufficiently large , begins to dominate and becomes very broad, indicating the high probability of inelastic transitions. Overall, the trend is for the transmission line to dominate energy exchange for small , while the 2DEG dominates energy exchange for large .

### ii.4 Calculation of - Curves

To calculate the - characteristics for the S-SET, we use a master equation approachSchön (1998) in which we assume that only two charge states, and where is the number of Cooper pairs, are important. This should be a valid approach for temperatures and biases small compared to the charging energy of the S-SET. We begin by calculating the free energy change for changing the island charge from to (or vice versa) due to tunneling through junction . We find

(29) | |||||

where is the gate charge and is the fraction of the bias voltage appearing across junction . We then use Eq. 1 to find the tunneling rates in terms of the sample parameters.

The master equation can be solved exactly when only two charge states are considered.Schön (1998) Doing so, and using the detailed balance relation , we find

(30) |

where is the change in free energy for tunneling in the electrostatically favorable direction ( for junction 1 and junction 2), , , , and is the resistance quantum.

## Iii Experimental Results

### iii.1 Measurements

We have performed electrical measurements on the two samples described in Table 1 in a dilution refrigerator at mixing chamber temperatures ranging from to . Measurements were performed in a four-probe voltage biased configuration, with the bias applied symmetrically with respect to ground, in a shielded room using battery powered amplifiers. High frequency noise was excluded with -filters at room temperature and microwave filters at the mixing chamber.

Because the S-SET and 2DEG are electrically isolated from each other at dc, we were able to measure their conductances and separately. In both cases the conductance was measured by applying an ac bias voltage at and measuring the resulting current using standard lock-in techniques. The bias voltage used was 3 and respectively for the S-SET and 2DEG. We also performed measurements of dc - characteristics of the S-SET. For the 2DEG, the current contacts were positioned on opposite sides of the two QPCs, so that measured the series combination of their conductances. Since the S-SET sees the QPCs in parallel, we take for the pool geometry. For the stripe geometry we cannot measure the stripe resistance or capacitance directly, although we can estimate them from the sample design. We expect that in this geometry at low frequencies and at high frequencies, as discussed above. While the - measurements of the S-SET were usually performed at a fixed gate voltage , the conductance measurements were typically performed versus for a variety of temperatures. In the pool geometry all six Au gates were tied together and swept from 0 to , while in the stripe configuration only the four outermost gates were swept.

Results of these measurements are shown for S2 in Fig. 5 for mixing chamber temperatures ranging from 50 to . As is made more negative, the S-SET conductance oscillates due to the effects of the Coulomb blockade; the oscillations are clearly visible in Fig. 5(a)–(e). We show results for the pool geometry for various temperatures in Fig. 5(a)–(d), and results for the stripe geometry in Fig. 5(e). In both geometries, the envelope of the oscillations is relatively flat until , at which point the amplitude of oscillations begins to increase. For the pool geometry, the envelope continues to rise until , at which point it begins to fall again. For the lower temperatures (), the drop with more negative is quite steep, whereas for the higher temperatures () the drop is more gradual. For less negative , the envelope generally rises as decreases, and tends to saturate below . For more negative (beyond the maximum in the envelope at ) the envelope rises as decreases until , below which it decreases.

Since decreases as becomes more negative, the above behavior indicates that varies non-monotonically as decreases, first rising and then falling. A decrease in as decreases is expected. Physically, a larger tends to damp phase fluctuations, promoting superconducting behavior and therefore a higher . Alternatively, we can say that for low energies, the probability of exchanging energy with the environment increases as increases, as can be seen in Fig. 4. A higher therefore implies a higher probability of elastic (or nearly elastic) transitions, and a higher at low bias. In contrast, the decrease in for does not fit in with this general picture of energy exchange with the environment. While nonmonotonic behavior with can be expected,Wilhelm et al. (2001) it is generally associated with a crossover from the non-linear to linear portions of the - characteristic. We find that this is true in our simulations as well.

In contrast, for our experiments was always measured in the linear part of the - characteristic. We show in the inset to Fig. 5(d) the - characteristic for S2 at for , i. e., when the 2DEG is unconfined. The - characteristic is clearly linear to , so that our rms bias should be firmly in the linear regime. The non-monotonic behavior we observe cannot then be associated with changes in , and must arise from other physics.

We can find a clue as to the source of this behavior by examining vs. for the stripe geometry as shown in Fig. 5(e). In this geometry the envelope also begins to rise at . However, the rise is weaker, with the envelope increasing by only roughly half what is observed in the pool geometry. Furthermore, there is no decline in as is made more negative. This last observation is in agreement with our model for in this geometry, which predicts that approaches at the higher frequencies which dominate our measurement of . As a result, in this case never broadens for more negative as it does in the pool geometry, so that no decrease in is observed. Once again, the environmental theory cannot explain the reduction in for less negative .

One possible explanation for this decrease is that there are changes in the offset charge of the S-SET island that occur on a time scale short in comparison to the time constant of the lock-in, but long in comparison to the time scales associated with environmental fluctuations. It is well known that charge fluctuations in the substrate give rise to noise in SET-based electrometers.Visscher et al. (1995); Krupenin et al. (1998) Such charge noise typically has a magnitude of – at and a cutoff frequency (above which the intrinsic SET noise dominates) of about 100–. Let us assume that in our case the noise is somewhat larger than is typical, say at , due to the presence of the 2DEG. If we write , then the expected mean square charge varianceKogan (1996) between frequencies and is given by . Taking and we find , so that a typical variance of a few hundredths of an electronic charge is not unreasonable.

In a voltage biased configuration such as ours, these fluctuations would have the effect of averaging the measured current over an ensemble of charge states centered around the gate charge . Similar effects have been seen in measurements of other S-SET systems.Eiles and Martinis (1994) Since the S-SET current is sharply peaked around the charge degeneracy points, we expect that any such charge averaging would tend to reduce the measured peak current, and therefore the conductance . It is therefore possible that the reduction in for less negative arises due to increased charge averaging as the electrons in the 2DEG become less confined. We examine this possibility in more detail in the following section.

### iii.2 Comparison with Theory

To compare our measurements with theory, we must plot the measured