For my journal club I've chosen a paper that came out of Eli Zeldov's group at the Weizmann Institute in Israel. For the past decade or so their group has specialized in the development of nanoscale scanning SQUIDs. We're probably all at least somewhat familiar with SQUIDs. Superconducting quantum interference devices exploit the quantization of magnetic flux through a superconducting ring and the current-phase relationship of a Josephson junction as a sensitive flux or magnetic field sensors - and indeed the nanoSQUIDs created by the Zeldov group have been used to perform sensitive scanning magnetometry in the same fashion as more standard micron scale scanning SQUIDs. However, in addition to being excellent magnetometers, it turns out that nanoSQUIDs are also exceptionally sensitive to temperature and, additionally, the specific style of nanoSQUID the Zeldov group has developed is ideally suited to thermal scanning. 

This talk will be focused on the use of nanoSQUIDs for scanning thermometry - not magnetometry (though I will discuss one example briefly). I don't plan to spend much time discussing quantum interference effects. In fact, for thermometry, the nanoSQUID does not utilize any interference effect. The same essential physics applies to a single Josephson junction and their group has used both single and double junction devices for thermometry.

The big takeaway I want to leave people with is an intuition about why a nanoSQUID is so well suited for the task of scanning thermometry. I will start by discussing the actual structure itself. The fabrication of nanoSQUIDs took many years to develop and is quite ingenious and difficult. Standard micron-scale scanning squids are typically made by lithographic techniques. The standard resolution of ebeam lithography is around 40 nm. This limits the total SQUID loop size to roughly 500 nm or larger. In order to create a nanoscale SQUID, different techniques must be utilized. The Zeldov group borrows a technique from biology. When biologists want to use extremely small pipettes (think on the scale of a red blood cell or smaller), they use a method of drawing a heated quartz tube to a very sharp point. They begin with a roughly 1 mm quartz pipette clamped on both ends. A CO2 laser heats up the center, and the two clamped ends are pulled rapidly to draw the quartz into an hour-glass shape. A second round of heating and an even more abrupt pull create the narrow apex and cleave the quartz, resulting in two mirror image pipettes with tips on the order of 10s of nm. Using this technique, the Zeldov group can get reproducible quartz pipettes down to about 20 nm inner diameter. (The manufacturer of the latest tool claims to be able to achieve 10 nm diameter.) A SQUID is constructed onto the apex of the quartz pipette by an angled evaporation of a superconducting material. The choice of superconductor turns out to be very important. Aluminum was among their first attempts. It's very easy to work with, but it's Tc is around 1K which is somewhat limiting. The have had success with both niobium and lead (Tc around 10K). Both are difficult to evaporate for difference reasons. Apparently niobium is exceptionally sensitive to degradation and contamination. It has to be evaporated in a UHV environment and the quartz tubes require a prior sputtering of alumina to reduce contamination from the heated quartz. Lead is also difficult to evaporate. It has a tendency to diffuse on the surface of materials and winds up growing in isolated islands that do not percolate and form a smooth coating. To get around this, the surface being evaporated onto must be cooled down - below even liquid nitrogen temperatures. They developed a thermal evaporator with a He-4 flow cryostat that allowed them to rapidly cool the quartz pipette down to about 8 K to avoid difficulties related to diffusion. At the end of the day lead winds up having much better properties - it's less noisy than niobium possibly due to less issues with strain and contamination. It also may be easier to make smaller devices because it does not require a coating of alumina. 

The deposition occurs in three steps. The first deposition occurs perpendicular to the apex of the pipette - typically about 15 nm of lead. The thickness of this deposition is limited by the angle of the pipette tip. A small thickness of Pb is deposited around the conical section - 15nm*(1-cos(theta)). This must be thin enough that a uniform film does not form around the sides. Afterwards, two depositions at angles of +/-105 degrees of 25nm of material are performed to create contacts along the sides of the pipette down to the apex with a gap between the two leads. Where the side leads overlap the ring of the apex, strong superconductivity exists. The nanobridges between these regions form the Josephson weak links. 

In figure one of the handout you can see an SEM of the lead SOT (SQUID on tip). The effective diameter is around 40 nm. 

Additionally, gold contacts are included above the narrow portion of the quartz tube and a cold shunting resistor is placed in close proximity on the quartz tube. The purpose of the shunting resistor will become clear later on. 

Now let's discuss the principle of operation. In the circuit diagram we have our SQUID being current biased by a voltage source and large bias resistor (several kOhm). Josephson weak links have a maximum Cooper pair supercurrent that they can sustain. The critical current is a function of the junction geometry and the specific properties of the superconducting material. If we imagine current biasing the SQUID by placing a large bias resistance Rb in front and applying a voltage V, the bias current Ib = V/Rb will be induced in the SQUID. As long as this Ib is less than the Josephson critical current (or more precisely twice the individual critical current), there is no voltage drop across the device due to the fact that Cooper pairs carry the current in a dissipationless fashion. Thus, all of the current chooses to go through the low resistance path (SOT) as opposed to the shunt. Now, if we consider what happens when Ib>Ic, the SQUID takes on a resistance that is of order it's "normal" state resistance Rn determined by the junction geometry. Thus, some of the bias current flows through the shunt resistor. To make things simple imagine there is a constant turn on resistance Rn of the SOT as soon as it exceeds its critical current. The the current flowing through the SOT will compete with the current flowing through the shunt: Ibias = Ishunt + Isot. Treating the SOT as an approximately ohmic device, we find Isot = Ibias*Rshunt/(Rshunt + Rn). (In reality the situation is a bit more complicated. There is a separate shunting resistor that is not on the tip directly. Additionally, there is the gold on-tip shunt discussed earlier. This is mainly used to prevent the I(V) curve from being nonlinear by forcing the load line of the SQUID to be monotonic and well-behaved. This results in a smooth turn-over from 0 resistance to finite resistance. The details are not crucial to understand the basic mechanism.) Thus, as Ibias is increased, we find that the slope of the Isot vs Ibias curve reduces from roughly 1 to Rshunt/(Rshunt+Rn) == alpha. If you look at Figure 1, you'll see just that - a series of ISOT vs Ibias traces at different temperatures. At a given temperature, you'll see the current in the SOT track the Ibias with a unit slope, then roll over at the critical current to a lower slope alpha. Importantly, this slope is the same for all temperatures. Also importantly, you'll notice that the kink in each curve has a strong temperature dependence ranging from 120 uA to 0 from 4.2 to 7 K. 

Imagine setting a bias current above the highest critical current. If we assume that the critical current changes with temperature as dI/dT, then the displacement in the Isot current at the given biasing current will be given by dI/dT*(1-alpha) where alpha is the lever arm for our shunt vs normal resistance bridge. Note that if alpha=1, then our sensitivity is ruined. If alpha = 0, this is ideal, but this would not be possible because we would need a vanishing shunt resistor. (This is why the on-tip resistor is used - very low parasitic contributions for a low alpha value.) 

Now we will consider the sensitivity of the critical current to changes in temperature. Some very smart people with hard to pronounce names worked out the critical current temperature dependence based on BCS theory after Josephson made his seminal prediction. The result of their detailed calculations is:

Ic ~ gap(T)/(eRn)tanh(gap(T)/2kT)

How can we understand this intuitively? Well gap/e looks like a voltage. Given that we're asking for a dissipationless supercurrent in the absence of an applied voltage across the junction, dimensional analysis practically begs us to consider the gap as the relevant energy scale. It turns out that Ic(T=0)Rn = material dependent constant due to the fact that both the resistance of the junction and the number of Cooper pair modes that a junction can support depend on the same geometrical/material properties. It makes sense that the critical current would be inversely proportional to this parameter. In the low temperature limit where the gap is strong, the tanh function saturates to 1, giving us Ic ~ gap(T)/(eRn) which we might have guessed by dimensional analysis alone. Now, if we start to turn up the temperature we might start to disturb our Josephson current and we will begin to wash out the Cooper pairs. In the high temperature limit (relative to Tc) we might expect the strength of the remaining supercurrent to depend on gap/kT (which goes to 0 as T->Tc) which is what you get if you expand the tanh function. 

As a further approximation we can take the BCS limiting form of the gap's temperature dependence when T near Tc: gap(T) ~ gap(0)sqrt(1-T/Tc). Expanding the tanh which is appropriate when the gap is small (near Tc) we find:

Ic ~ gap(0)sqrt(1-T/Tc)/(eRn)(1/2kT)gap(0)sqrt(1-T/Tc) ~ gap(0)/(eRn)(1-T/Tc) = I(0)(1-T/Tc).

This leads to a lever arm given by I(0)/Tc. Plugging in realistic values of I(0) ~ 100 uA and Tc ~ 10 we find a result on the order of dIc/dT ~ 10 uA/K. 

We can see that the strong temperature dependence of the superconducting gap near Tc is responsible for the strong change in current with temperature. This is the hallmark of a generic second order phase transition - the order parameter falls off as sqrt(1-T/Tc). In principle any proxy for an order parameter at a phase transition should be a good candidate for thermometry (perhaps susceptibility of a magnetic system). It is convenient that the Josephson supercurrent is directly proportional to the order parameter.

Far from Tc, this approximation fails and the critical current does not change much. This is a fundamental limitation of the technique, but luckily there are other superconductors with a wide range of Tc. The appropriate measurement range is something like Tc/2 < T < Tc.

Other considerations for good thermal sensitivity: You need to be able to measure your proxy for temperature with as minimal noise as possible. Fortunately, there are fantastic low noise amplifiers that operate at cryogenic temperatures. Martinis developed an amplifier known as a series SQUID array amplifier in which a single current-carrying line is inductively coupled to a series of about 100 SQUIDs. This array of SQUIDs helps to impedance match the ~1 Ohm output impedance of the nanoSQUID to a 50 Ohm line that can be amplified using conventional room temperature electronics. The noise figures for the SSAA are quite good. Zeldov's group quotes a noise floor of less than 10 pA/root Hz which is essentially the same order of magnitude as the Johnson and shot noise for a 100 Ohm resistor carrying 100 microamps at 4K. 

In total the thermal sensitivity is given by taking: dI/root Hz / dI/dT ~ 10^(-11)*10^5 K/rootHz = 1 uK/root Hz. The next best techniques in Figure 1 are approximately 10 mK/root Hz: an astonishing 4 orders of magnitude improvement. 

This sensitivity is good, but there are many existing low temperature thermometers (Coulomb blockade thermometer, resistive thermometers, etc) which offer very good resolution. Where this technique stands out are the ideal thermal properties of the entire SQUID apparatus that allow scanning to operate in a realistic sensitivity window. Let's consider some important inequalities:

In order for the SQUID sensor to come to equilibrium with the sample, it must be sufficiently thermally isolated from the support structure by a thermal resistance Rss which is much greater than the thermal resistance from the sensor to device Rsd. So Rss >> Rsd. Furthermore in order for the sample to be well heatsunk the sample must have much stronger thermal coupling to the bulk substrate Rdb. Thus Rsd >> Rdb so that Rss >> Rsd >> Rdb. 

The thermal resistance of a ~100 nm x 100 nm metal on top of a SiOx surface at 4 K is given by Rdb ~ 10^7 K/W. By introducing a variable pressure of He exchange gas into their cryostat, they can vary the thermal resistance between device and SOT between 10^10 at 1 mbar to 10^8 K/W at 30 mbar. Finally, the quartz pipette actually has a remarkably high Rss. Due to the fact that it's hollow, it has quenched radial phonon modes. The longitudinal modes are strongly constrained by the nanoscale cross section. One phonon channel has a resistance of ~10^11 K/W. They estimate that their smaller 40 nm devices possess only a few available phonon modes and have resistances of order 10^11 K/W. This would not be the case for a planar Josephson junction which would have much stronger coupling to its support structure.

Finally, before concluding our section on the performance of the nanoSQUID thermometer, let's consider some fundamental limits. We are used to thinking about quanta of flux, charge, noise limits, etc. What is a good figure of merit for heat dissipation? 

Landauer calculated a limit on the minimum energy dissipated in a bit operation: E = kTlog(2). If we imagine such an irreversible process occurring repeatedly at 4K and 1 GHz (typical qubit conditions), this dissipates about 40 fW of power. Looking in figure 1, panel e shows proof that their nanoSQUID can beat this limit by considering the power dissipated in a small 120 x 120 nm device made from copper. As the power is reduced, the ultimate noise floor (with the given amount of signal averaging) is found to be about 6 fW - below the 40 fW threshold demonstrating good thermal sensitivity on a scale relevant for quantum dissipative processes.

Without further adieu let's look at some data. 

In figure 2 you can see a demonstration of the multiple functionalities of the scanning nanoSQUID device. We've focused on its role as a thermometer, but of course it can also detect DC and AC magnetic fields. To demonstrate the three capabilities, the authors have fabricated a zigzag array of copper (nonmagnetic) and permalloy (ferromagnetic). A magnetic field is applies both out of plane as well as parallel to the permalloy easy axis. The fringing fields at the end of the wires indicate that they are monodomain and aligned along their length. The z-axis field provides evidence of some tilting as well. Additionally, they pass an alternating current at a frequency f through the copper-permalloy structure. This alternating current creates a Biot-Savart magnetic field oscillating at the excitation frequency f. By reading the output current on a lockin amplifier they can sense the ac magnetic field. It clearly only emanates from the portion of the metals which pass a current. Finally, heating occurs at the second harmonic of the excitation 2f due to the fact that both polarities of the current swing cause Joule heating. The permalloy is much more resistive than the copper and so dissipates much more heat. Some dissipation is visible in the copper and heat diffusion into the protrusions is also evident. The great feature of their technique is that they can read out the DC, first harmonic, and second harmonic simultaneously. One beautiful feature of the SSAA amplifier is that because the SQUID array is sensitive to not only the standard AC mutual inductance with the SOT current carrying wire but also the DC Biot-Savart field carried by the line, they are able to recover DC and low frequency response easily. This would not be possible with a standard inductive coupler. 

Moving on to figure 3 we can see in the lower panels SEM images of two different single wall carbon nanotubes. The nanotubes are coiled into loops and it is clear from the dissipation that the second CNT is electrically shorted at its crossover. Additionally, you'll notice a fine set of rings tracking along the length of the nanotubes. These are believed to be enhanced dissipation from a series of quantum dots. When the CNTs are placed on the rough substrate, surface disorder creates sections of CNT that form small islands. These islands exhibit Coulomb blockade - a phenomenon in which the classical electrostatic energy to add a single electron to the island is significant. The SOT has a slight local gating affect - it acts as a scanning gate. When the SOT is brought within a certain distance from one of these dots, its potential modulates the discrete electron levels within the quantum dot and brings an unoccupied state into resonance with the Fermi level of the CNT. This allows electrons flowing in the CNT to hop on and off the island and increases the amount of dissipation. This perspective is reinforced by observing changes in the ring radii as a function of scanning height. Panel c is a zoom in of one such ring with a lower scan height than in panels a,b. When the tip is within about 20 nm of the quantum dot it allows enhanced conduction and increases the local temperature by about 50 uK.

In figure 4 we can see a similar dramatic demonstration of quantum dissipation. We can see on the left panel a washer shaped encapsulated boron nitride and graphene stack. It has been etched into this particular shape by a reactive ion etching process. Current is flown between the top and bottom right lead. Surprisingly, one can see an array of dissipation points along the etched portion of the graphene. The current causes a baseline local heating of the graphene, but there is enhanced dissipation around the etched boundaries. This is a bit of a different situation from the CNT. It is proposed that these defects are coming from adatoms and vacancies in the van der Waals materials. Instead of forming isolated quantum dots with multiple charging states they form single resonant states pinned to the Dirac point as recently seen in STM measurements. Only one ring is seen around each defect and they act as dissipation hot spots where the electron-lattice coupling is strong.