a shows a circuit diagram representation of an SED. The tunnel barriers define an isolated island whose charge can only be changed by an integer number of electrons. As shown in the figure, the SED is a three terminal device where the gate electrode primarily controls the electrostatic potential of the island. Figure 1
b shows an SEM image of one type of SED fabricated in silicon. The device consists of an etched and oxidized silicon nanowire with three finger gates wrapping around the device. These gates are used to form electrostatic tunnel barriers while an upper gate covering the wire from source to drain provides carriers (not shown). Dots are formed between the finger gates. The center finger gate can act as a barrier (to form two dots) or as a plunger gate when one dot is formed. Because the island is small, its total capacitance to the other electrodes is small. This implies a measurable electrostatic energy barrier for adding an additional electron to the island,
is known as the charging energy and produces a ladder of chemical potential levels on the island separated by
. Transport is only allowed when one or more of the dot levels falls within the source-drain bias window,
. At low source-drain bias only one charging level may fit in the bias window and one measures current resonances, between which the number of electrons on the island is fixed. This is shown schematically in Figure 1
c. The capacitance between the island and the plunger gate determines the spacing in voltage between the peaks. As the source-drain bias is increased, one reaches a point where more than one charging level is in the source-drain bias window and transport is no longer blockaded. This occurs when
passes the top or bottom of the diamonds shown in Figure 1
d. Therefore, the height of the diamonds allows measurement of the charging energy. At high bias, rather than measuring current resonances with zero current between them, the current oscillates sinusoidally without reaching zero current as shown in Figure 1
c. The period of the oscillations is again defined by the gate capacitance,
. The current oscillates, increasing as a new level enters the bias window and decreasing as a level leaves the bias window. Since the plunger gate capacitance is determined by the geometry, the period of oscillations as a function of
are as well. Thus, the period of oscillations is a consequence of the geometry of the device and can be simulated accurately through electrostatic solver tools such as Fastcap [28
]. Therefore, two identically fabricated devices should have very nearly the same period in the single electron curve.
While the period of the single electron curve should be very similar in different, but identically fabricated devices, the phase of the curve is not. Since the single electron oscillations are a function of gate voltage the phase corresponds to a voltage offset between successively measured curves. The difference in phase has its origin in the contribution from the local charge environment [29
]. This is known as the charge offset,
. Devices fabricated simultaneously will exhibit different values of
because the local charge environment is randomly determined by the different processing steps such as gate oxidation, etching, dopant distribution, anneals, etc. However, so long as the local charge environment remains stable, the voltage on the plunger gate can be used to compensate for this difference and each device can be brought into phase. If instead, the local charge environment changes with time, then the phase of the device will fluctuate randomly in time and such phase synchronization becomes essentially impossible. In this case, the device exhibits charge offset drift,
. In principle, the fluctuations in the local charge environment may occur at any frequency. Usually, the literature focuses on what is called the “long-term charge offset drift” which occurs at frequencies <1 Hz because the amplitude of the fluctuations is so large. This lack of synchronization, and concomitant difficulty in integration, drives our interest in studying
In the above,
corresponds to a current offset, while A
corresponds to the size of the oscillations, and
is the period of the oscillations as previously discussed. B
may be used to account for any slope in the sinusoidal curve. The phase,
in the fit is extracted and represents the charge offset drift. This is the most common measurement method. The second method is to measure the change in plunger gate voltage necessary to keep the current constant on a device (i.e., to keep a constant operating point in Figure 1
). Regardless of the method used, the data are usually presented such that the voltage variation in the phase of the curves is normalized by the period of the oscillation so that
ranges from 0 to 1e
. These two methods can detect a
As has already been mentioned, charge offset drift precludes device integration. Conventional Complementary Metal Oxide Semiconductor (CMOS) logic devices are built in state-of-the-art cleanrooms so that the devices operate as uniformly as possible. If the threshold voltage of a metal-oxide-semiconductor field-effect transistor (MOSFET) varies significantly from one device to another, the devices cannot be integrated together. This is because a threshold voltage variation can result in a particular device’s on-state occurring outside the range of voltages available to switch it. The same effect can occur in SEDs through temporal variation of the turn-on characteristics of a device. If, as alluded to above, a current standard is to be built out of parallelized SEDs, then each device must be made to operate identically to reduce the tunneling errors that limit the accuracy of the device. If a standard is built from 100 devices each tuned so that the phase of their curves are synchronized with a desired error rate of
, then each device must be stable for at least
s or 12 days. Thus, it is clear that charge offset drift prohibits integration of SEDs in a way analogous to the way threshold voltage differences prohibit integration of conventional transistor devices [9
]. It is worth mentioning that the
noise that typically occurs at
Hz in these devices does not limit the ability to parallelize devices such as for a current standard; only the large
Hz precludes integration [31
2.1. Metal-Based Devices
In general, metal-based devices suffer from the worst charge offset drift behavior. Here, we are referring to devices fabricated from Al/AlOx
/Al, where the AlOx
provides the tunnel barriers shown in Figure 1
a. These devices are typically fabricated through a technique known as double-angle evaporation [36
]. A pattern is generated in electron beam resist through electron beam lithography and Al is evaporated at an angle. This Al is oxidized in a controlled environment within the deposition chamber, and then the deposition angle is changed. With a carefully designed pattern and the right angles, an SED is formed with AlOx
tunnel barriers. Of course, devices have been fabricated from other metallic materials [37
]. These devices can exhibit “clean” Coulomb diamonds without large jumps in the data characteristic of large charge offset drift while still exhibiting some low frequency time instability [39
]. Unfortunately, we are unable to find rigorous measurements of charge offset drift, as outlined above, in metallic devices made from anything other than Al/AlOx
. Therefore, we focus on Al/AlOx
devices in this section.
A typical measurement of the charge offset drift for a metallic device is shown in Figure 3
]. As is plainly apparent from the data, the charge offset value wanders between 0 and
over the course of the measurement (and, in fact, probably wanders through several e
). The time scale for the drift is on the order of minutes to hours. In addition, the data show periods of rapid evolution, some periods of relative stability, and sudden jumps. The evolution from a very rapidly changing charge offset value to one which subsequently changes more gradually or only with sudden steps, is known as “transient relaxation”. We will discuss this relaxation in more detail below. Figure 3
b shows data that appear in Figure 3
a between days 19 and 22. These data mainly consist of two distinct levels of
between 0.2 and 0.3. This indicates that the drift is driven by some switching behavior and will be discussed in more detail later. Finally, vertical dashed lines in the figure denote either mechanical disturbances or deliberate temperature excursions to investigate the temperature dependence, the effects of thermal cycling, or the effects of exposure to air. Mechanical changes such as refilling the dewar with liquid helium have no discernible effect. The same is true for temperature excursions up to 40 K. However, it is possible that thermally cycling the device up to room temperature and/or exposing it to air does change the qualitative nature of the charge offset drift measured.
a shows source-drain current measured as a function of time through a metal-based SET for different values of the plunger gate voltage [40
]. As the voltage on the plunger gate,
, is increased, the fluctuations in the current become more rapid, while the amplitude and duty cycle remain constant. Looking closely at the data, one can see that it consists of discrete jumps, some of which appear as a quick succession of small steps and others which appear as a single large step. This is qualitatively similar to the data in Figure 3
b where only two levels are seen. If each jump has its origin in the switching event of a single TLS, we can interpret jumps of different size as being due to defects at different distances from the dot with smaller jumps being due to defects further away from the dot.
It is instructive to examine not only the time dependence but also the frequency dependence of the current noise in these devices. Examination of the power spectral density,
, allows us to separate equilibrium charge noise from non-equilibrium noise.
for the traces in Figure 4
a is shown in Figure 4
b along with a simulated set of data and Lorentzian fits. Consistent with the fits, the data show an approximately
dependence over some frequency range (different for each curve). This indicates that the device is dominated by a single or few TLS. The power spectral density does not always follow
at higher frequencies, however. In devices where the charge offset drift wanders continuously (without reproducible discrete jumps), the devices show a
dependence in the 1–100 Hz frequency range as shown in Figure 5
. This is indicative of a large number of TLSs contributing to the charge noise in the device as a
spectrum results from the addition of a large number of
spectra with different knee frequencies. This distinction separates devices into those dominated by a few defects or a single defect (
), and those dominated by many (
The devices in reference [41
] also show the transient relaxation present in Figure 3
. The inset of Figure 5
shows that the characteristic frequency of the charge offset drift follows a
dependence, where t
is the experimental run time. The
time dependence results in a long-tail evolution, which is characteristic of glassy relaxation, a non-equilibrium relaxation process. At frequencies above about 0.1 Hz, a
dependence is observed which does not change as a function of the experimental run time (not shown), indicating it is an equilibrium process.
Thus far, we have established that (i) the charge offset drift in metal devices is large; (ii) that the noise spectrum in some devices shows that nearby two-level systems (TLSs) participate; (iii) that charge offset drift events occur on the tens of minutes to hours time scale; and iv) that the drift at low frequencies can be a “long-tail” non-equilibrium process. Table 1
, organized by device type, summarizes these results and the results for other types of devices discussed below. We can now discuss some attempts to mitigate
by establishing some empirical correlations between the level of
and some fabrication parameters or measurement protocols. Figure 5
shows the power spectral density for two different devices. The device plotted in red had extra oxide deposited on it to encapsulate the device while the blue device did not. This was done to investigate if the noise was associated with the presence of an amorphous insulator. Clearly, the blue curve’s low frequency noise is much lower than that of the red curve, indicating that this is indeed the case. Other tests have been performed, including: (i) metal deposition method (thermal vs e-beam); (ii) annealing in forming gas or inert gas; (iii) the use of ozone when forming tunnel junctions; (iv) thermal cycling to room temperature; (v) a nano-Faraday cage; (vi) cooling the device with or without electrical protection; and (vii) measuring in the normal vs. superconducting state. Unfortunately, none of these other tests substantially reduced the charge offset drift characteristics in metal devices [42
The above are the basic characteristics of charge offset drift in metal devices. The features shown in the data of Figure 3
, particularly that data of Figure 3
b, indicate a two-level system as the dominant noise source contributing to
. This conclusion is reinforced with the power spectral density data of Figure 4
and Figure 5
. The spectrum shown in Figure 4
, indicating that these devices are dominated by a single or few TLSs. However, the short-term noise in Figure 5
It has already been shown that the defects can be associated with the presence of amorphous insulators [41
]. Since metal devices are usually fabricated on insulating materials but also use AlOx
to form the tunnel barriers, it is natural to ask in what part of the device these defects are typically located. Presumably, TLSs located anywhere near the device could change their charge by e
and affect the charge offset noise. An appealingly simple argument by Song et al. [43
] concludes that the dominant TLSs must be located in the tunnel barriers of the device. Since the electric field is largest in the tunnel barriers, the defects residing there would induce the largest change in the electrostatic potential of the island. Defects located elsewhere would experience far smaller fields and, therefore, be incapable of producing such large (
) excursions. Within this argument, monotonic or continuous drift would come from capacitively coupled defects located elsewhere but near the device and contribute to higher frequency
Several experiments attempt to locate the defects. The work of Zorin et al. [44
] used two separate SEDs located 100 nm apart from each other and measured the low-frequency (1–10 Hz) correlations in the charge fluctuations. The expectation is that the experiment would find uncorrelated noise for defects entirely in the barrier and correlated noise for defects residing in the substrate (sputter deposited Al2
). They found an appreciable level of correlation between the devices, as much as 20%. Thus, this experiment shows that the level of noise coming from the substrate material is comparable to that coming from the barriers and renders the argument of Song et al. incomplete.
The experiment depicted in Figure 4
]) also located the defects outside of the tunnel barriers by comparing the switching rates as a function of the plunger gate voltage to that expected for TLSs located in the barriers. As can be seen in the inset of Figure 4
b, the switching rate is a monotonic and roughly exponential function of the gate voltage. This is in contrast to the voltage across a tunnel junction in the device, which should have a periodic dependence on gate voltage with period
mV for this device. Whatever the identity and the functional form of the defects’ switching rate, the dependence on gate voltage should have some periodic character and not a large-scale dependence as shown. Thus, many of the defects contributing to
are outside the tunnel barriers. Additionally, the data were found to be consistent with a cluster of TLSs which switched coherently when triggered by a particular TLS switch. The relative influence of a TLS on neighboring TLSs will play a crucial role in distinguishing between the behavior of metal and silicon-based devices.
2.2. Silicon-Based Devices
While metal-based devices show a large amount of drift and the transient relaxation phenomenon, all-silicon devices show neither effect. Silicon-based devices are usually fabricated through gate oxidation and etching of one or more layers of doped polySi gates to both form tunnel barriers in the silicon and provide carriers. A typical charge offset drift curve for an all-silicon device is shown in Figure 6
]. Here, the device exhibits
over the course of 15 days when the temperature is stable (middle panel and left side of lower panel). These data were measured in a device with electrostatically tunable tunnel barriers defined by a set of gates and their voltages (see Figure 1
b). The same level of drift is seen in devices with fixed tunnel barriers prepared through a pattern dependent oxidation technique (PADOX) [31
]. This remarkable difference between silicon-only devices and metallic devices will be explained below.
does show some non-idealities in the device performance when all the experimental factors are not held fixed. Vertical red dashed lines represent temperature changes and black dashed lines represent liquid helium transfers and, therefore, a mechanical disturbance to the device. There is very little, if any, change in
as a result of the mechanical disturbances for this device. However, abrupt changes in temperature can produce a measurable change in
depicts the drift measured in another device from [45
]. Here, the temperature was held between 20 to 40 mK during the entirety of the measurement. Black lines here also represent the mechanical disturbance of helium transfers. These data are plotted on a similar y
-scale to that of the middle panel of Figure 6
and do exhibit some sensitivity to helium transfers.
Though these devices are very stable, from the above data, it seems clear that all-silicon devices with electrostatic barriers are somewhat sensitive to temperature excursions and/or mechanical disturbances. In fact, these devices are also more sensitive than fixed-barrier PADOX devices [31
]. This is probably a result of the increased number of gate voltages which are required to remain stable through mechanical or electrical disturbances in the tunable devices. However, this may depend on the details of fabrication since remarkable reproducibility has also been observed even after thermal cycling to room temperature [46
] (see Figure 8
inset) or with months between measurements [30
]. Thus, measurements of charge offset drift in the time domain reveal all-silicon devices (PADOX or electrostatic barrier devices) to be much more stable than metal devices. These results are summarized in Table 1
To further the discussion of the stability of all-silicon devices, and to compare more completely to metal based devices, we can examine the power spectral density of the charge noise in these devices as well. Figure 9
shows data measured on various days and at two different gate voltages [30
]. First, it is clear from the data at each gate voltage that the time dependence of the noise is quite small as the different color curves lie on top of each other for a particular gate voltage. Both the knee frequency and the absolute value of the noise remain unchanged. Secondly, one notices a difference in the character of the noise at
V (upper cluster of curves) and
V (lower cluster of curves).
shows a Lorentzian lineshape at
V as evinced by the
dependence above 10 Hz and the flat value of the noise below 1 Hz. The black line in the figure denotes a fit to this form. Therefore, in this range of gate voltage, there is a single TLS present in the device which dominates the noise. At
noise and has orders of magnitude smaller noise below 1 Hz. While the
noise form denotes that there are several TLSs active at this gate voltage, their combined effect is much smaller than the single TLS, which is active at the higher gate voltage. Finally, these data include multiple changes in gate voltage, and thermal cycling to room temperature. The implication of this data is that an individual TLS present in the Si/SiO2
system was stable as a function of temperature or slow changes in the gate voltage. Thus, the remarkable stability of all-silicon SEDs is due to a lack of interactions between individual TLSs and not to the complete absence of TLSs.
As mentioned earlier, all-silicon devices with fixed barriers (PADOX) are marginally more stable than those with electrostatic barriers. Given this slight dependence on architecture one might think that the overall stability of devices might be the product of some specific steps in the fabrication. This has been directly addressed in the literature, most directly in reference [47
], where devices were fabricated in different foundries. It was found that the stability of all-silicon devices is a robust property of the materials. Regardless of how the devices are fabricated, even devices with relatively poor performance (gate leakage and unintentional barriers) exhibit excellent charge offset stability 100 times better than that of metal-based devices [47
In summary, single electron devices based in the Si/SiO2 material system show much smaller charge offset drift () than metallic SEDs () and much smaller sensitivity to external perturbations such as temperature, mechanical disturbance, and voltage pulses. This stability is not associated with the lack of TLSs in SiO2 but rather with the degree of interaction between TLSs which are present in both AlOx and SiO2. Before we discuss the TLSs in these systems within the context of glassy relaxation, we first discuss work on SEDs fabricated in the combined system of silicon dots defined with Al gates to complete the experimental picture.
2.3. Mixed Devices
The observations of charge offset drift which have been enumerated so far paint a striking dichotomy between metal-based devices and silicon-based devices. This drives a measurement of charge offset drift on a device where the SED quantum dot is formed in silicon with Al gates [48
]. Devices based on this architecture have been extremely successful as quantum information devices and have recently been used to demonstrate the first two-qubit gate in silicon [24
]. Earlier, the same group fabricated SEDs within 100 nm of each other to use one as a sensor of charge transitions on the other [35
]. To be sensitive, the sensor SED has to remain on the steep portion of the SED curve at low bias shown schematically in Figure 1
. To combat the charge offset drift induced excursions in the operating point of the sensor, the authors implemented a computer-controlled feedback loop to return the sensor to the appropriate place on the sensor control curve.
A similar device which contained only one SED was examined in reference [48
]. The device is shown in an SEM micrograph in Figure 10
a along with the measurement circuit. It consists of two barrier gates labeled B1 and B2. Two “lead gates” (labeled L1 and L2) provide carriers out to the heavily-doped source and drain without inducing a large number of electrons on the island. The island potential is modulated with the plunger gate labeled as P. The corresponding charge offset drift data appears in Figure 11
. The size of the drift,
, is a value intermediate between that of silicon based devices and that of metal based devices. This is a powerful result. One might expect that since the dot is formed in the silicon away from the Al gates and the tunnel barriers are created electrostatically, that the charge offset drift would be similar to an all-silicon device. The fact that it is not similar lends strong support to the discussion of glassy relaxation. A hint as to why this is the case is shown in Figure 10
b. It shows a TEM micrograph taken from a slice of the device along the line from source to drain under gate P, the dot plunger gate. As is clear from the micrograph, the deposited Al and SiO2
layer react to form a non-deliberate AlOx
layer at the upper interface of the SiO2
gate dielectric. This is an important fabrication feature in determining the level of charge offset drift in the device.
2.4. Summary of Experimental Observations
Experimentally, it is clear that metal-based devices exhibit a large amount of charge offset drift ), that silicon-only devices show very little (), and that devices made in silicon with metallic gates show an intermediate level of drift ().
Metal-based devices also show the phenomenon of transient relaxation that is correlated with the time from fabrication. More drift is also associated with the amount of amorphous insulator in or near the device. The frequency dependence of the charge offset drift noise in metal devices appears to be either in some devices or in others. The former is correlated with observed steps or clusters of steps in the charge offset drift or current measured as a function of time, while the latter is correlated with a more continuous variation. This invites interpretation in terms of TLSs in the amorphous AlOx. These defects affect the device’s stability when they are found anywhere nearby as well as in the tunnel barriers of metal devices through their electrostatic coupling. Finally, while many experimental knobs have been turned to investigate correlations of charge offset drift with various fabrication and measurement parameters, they have revealed no correlations.
Silicon devices show very little charge offset drift even after thermal cycling, voltage pulses, and long periods of time between measurements. The relative lack of drift in silicon devices, combined with the observation of TLSs in certain energy ranges (temperature or gate voltage ranges), intimates that the crucial difference with metallic devices lies in the lack of interactions between TLSs in SiO2, not their absence.
Measurements on devices which combine the above material systems where the dot is formed in the silicon below a SiO2 gate dielectric with aluminum gates exhibit . That these results occupy a middle ground between metallic and all-silicon devices, in combination with the observation of a non-deliberate AlOx layer below the gates, is consistent with the idea that the effect of TLSs reduces with distance from the dot.