Measuring a magnetic field with a SQUID does not require optical access; many other magnetic field measurement techniques do. Together, these facts mean that scanning SQUIDs are often the best tools available for probing extremely low temperature phenomena. NanoSQUID sensors also have many advantages over planar SQUIDs. The most obvious, of course, has already been discussed, and that is their higher spatial resolution. A less obvious advantage- indeed, an advantage that became clear only after the first nanoSQUID sensors were fabricated and tested- is the geometry of the thin superconducting contacts, which under normal circumstances are aligned with the axis of the applied magnetic field. Large magnetic fields tend to destroy superconducting phases, so superconducting devices are all limited by the maximum magnetic fields at which they can operate. This so-called critical field HC is not an intensive property; there is a large-size limit that can be measured and tabulated for different materials, but the critical field of an individual piece of superconductor is a strong function of geometry. A thin superconducting film in the plane of an applied magnetic field can accommodate much higher magnetic field magnitudes than can be accomodated by a large piece of the same superconductor. The bulk limit for lead at low temperature is about 80 mT; we routinely make lead nanoSQUIDs that can survive magnetic fields of 1 T, plastic planters wholesale and we have on occasion made nanoSQUIDs that can survive magnetic fields above 2 T.
It turns out that many of the most useful magnetic imaging techniques are limited to low field operation. This thesis will focus primarily on low field phenomena, but there are also many magnetic phenomena that require high magnetic fields to appear, including the quantum Hall effect and a variety of magnetic phase transitions. The nanoSQUID technique is useful for studying these as well.Above about 0.2 T, superconductivity begins to rapidly degrade in the nanoSQUID sensor, destroying the sensitivity of the sensor and rendering it useless as a sensor of magnetic field. This limits this particular nanoSQUID to operation in the regime −0.2T < B < 0.2T. This is fairly general to indium nanoSQUIDs; their precise critical fields vary, but are generally considerably below those of lead nanoSQUID sensors. As with any sensor, measurements with nanoSQUIDs are contaminated with noise, and the dependence of that noise on SQUID bias and magnetic field can be characterized. A characterization of the noise spectrum of the indium nanoSQUID shown in Fig. 1.6A is shown in Fig. 1.6B. In nanoSQUID sensors, local maxima in critical current are often associated with high noise and thus low magnetic field sensitivity . This produces ‘blind spots’ in magnetic field for nanoSQUID sensors. These blind spots often in practice include B = 0, making true zero-field operation challenging for nanoSQUID sensors. Technologies exist for circumventing this issue[54], but in practice we mostly work around it.
The low magnetic field DC response of a lead nanoSQUID is shown in Fig. 1.6C. A higher magnetic field characterization is shown in Fig. 1.6D, illustrating the collapse of superconductivity in this nanoSQUID at a considerably higher magnetic field of about 0.75 T. The inventor of the technique has been active in developing ways to deposit other materials onto micropipettes for use as nanoSQUID sensors, and has succeeded in producing aluminum, niobium, tin, and alloyed molybdenum/rhenium nanoSQUIDs, in addition of course to indium and lead nanoSQUIDs. The MoRe nanoSQUIDs in particular are capable of operating in extremely large ambient magnetic fields, up to about 5T. The magnetic field noise floor of nanoSQUID sensors seems to vary for different materials as well. We do not have a strong model explaining why this is the case, but it is empirically true that indium and lead nanoSQUIDs have particularly low noise floors. Plots illustrating the dependence on magnetic field of the magnetic field sensitivity of a lead nanoSQUID sensor 80 nm in diamater are shown in Fig. 1.7. NanoSQUID sensors have some unique disadvantages as well. Like planar SQUIDs, nanoSQUIDs require superconductivity to function, which limits them to fairly low operating temperatures. In planar SQUIDs it is often possible to keep the SQUID itself cold while scanning over a much hotter sample, but nanoSQUID sensors are extremely poorly thermalized to their scan heads, which means that they generally are thermalized either to the surface over which they are scanning or to the black body spectrum of the vessel in which they are contained .
This gives nanoSQUID sensors some interesting capabilities, namely that under the right conditions they can function as extremely sensitive scanning probe thermometers, but it also comes with some drawbacks. NanoSQUIDs composed of superconductors with critical temperatures below 4.2 K, the boiling point of helium-4 at atmospheric pressure, must thus have actively cooled thermal radiation shields to operate in very high vacuum, and of course imaging of hot samples is completely out of the question for these sensors. A variety of exciting opportunities exist for the application of sensitive magnetic imaging techniques to biological systems, and this is not a realistic option for nanoSQUID sensors. NanoSQUIDs are quite fragile and can be easily destroyed by vibrations, necessitating vibration isolation systems, and the superconducting film on the apex of the micropipette is quite thin, typically between 15 and 20 nm, so superconducting materials that oxidize in air will be quickly degraded. Thankfully indium and lead do not oxidize rapidly, but they do oxidize at a finite rate, so nanoSQUIDs composed of these materials only last for a few days when left in air. Storage in high vacuum can improve their lifespan, but generally not indefinitely. In summary, scanning probe microscopes fitted with nanoSQUID sensors can function as magnetometry microscopes with 30-250 nm resolution. They are capable of operating at very low temperatures and magnetic fields of up to several Tesla. Their high sensitivities allow them to detect the minute magnetic fields emitted by electronic phases composed entirely of electrons forced into a two dimensional heterostructure with an electrostatic gate. We will discuss some of the properties of two dimensional heterostructures next.Many crystalline compounds have cleavage planes; that is, planes along which cracks propagate most readily. When such compounds are stressed beyond their yield strength, they tend to break up into pieces with characteristic shapes that inherit the anisotropy of the chemical bonds forming the crystal out of which they are composed. Indeed, this observation was a compelling piece of early evidence for the existence of crystallinity, and even atoms themselves. There exists a class of materials withcovalent bonds between unit cells in a two dimensional plane and much weaker van der Waals bonds in the out-of-plane direction, producing extraordinarily strong chemical bond anisotropy. In these materials, known as ‘van der Waals’ or ‘two dimensional’ materials, this anisotropy produces cleavage planes that tend to break bulk crystals up into two dimensional planar pieces. Exfoliation is theprocess of preparing a thin piece of such a crystal through mechanical means. In some of these materials, the chemical bond anistropy is so strong that it is possible to prepare large flakes that are atomically thin . These two dimensional crystals have properties quite different from their bulk counterparts. They do have a set of discrete translation symmetries, plastic plant pot which makes them crystals, but they only have these symmetries along two axes- there is no sense in which a one-atom-thick crystal has any out-of-plane translation symmetries. For this reason they have band structures that differ markedly from their three dimensional counterparts. A variety of techniques have been developed for preparing atomically thin flakes from van der Waals materials, but by far the most successful has been scotch tape exfoliation. In this process, a chunk of a van der Waals crystal is placed on a piece of scotch tape. This separates the chunk of van der Waals crystal along its cleavage planes into two pieces of comparable size on opposite sides of the piece of tape. This process is repeated several times, further dividing the number of atomic layers in each chunk with each successive repetition. If we assume we are dividing the number of atomic layers in each piece roughly in half with each round, after N repetitions the resulting crystals should have thicknesses reduced by a factor of 2 1 N .
This is enough to reduce each flake to atomic dimensions after a small number of repetitions of the process. This process is, of course, self-limiting; once the flakesreach atomic dimensions, they cannot be further subdivided. Graphite can be exfoliated through this process into flakes one or a few atoms thick and dozens of microns in width, a width-to-thickness aspect ratio of 105 . As previously discussed, this process cannot be executed on every material. It depends critically on scotch tape bonding more strongly to a layer of the crystal than that layer bonds to other layers within the crystal. It also depends on very strong in-plane bonds within the material, which must support the large stresses associated with reaching such high aspect ratios; materials with weaker in-plane bonds will rip or crumble. In practice these materials are almost always processed further after they have been mechanically exfoliated, and the preparation process typically begins when they are pressed onto a silicon wafer to facilitate easy handling. Samples prepared in this way are called ‘exfoliated heterostructures.’ It is of course interesting that this process allows us to prepare atomically thin crystals, but another important advantage it provides is a way to produce monocrystalline samples without investing much effort in cleanly crystallizing the material; mechanical separation functions in these materials as a way to separate the domains of polycrystalline materials. Graphene was the first material to be more or less mastered in the context of mechanical exfoliation, but a variety of other van der Waals materials followed, adding substantial diversity to the kinds of material properties that can be integrated into devices composed of exfoliated heterostructures. Monolayer graphene is metallic at all available electron densities and displacement fields, but hexagonal boron nitride, or hBN, is a large bandgap insulator, making it useful as a dielectric in electronic devices. Exfoliatable semiconductors exist as well, in the form of a large class of materials known as transition metal dichalcogenides, or TMDs, including WSe2, WS2, WTe2, MoSe2, MoS2, and MoTe2. Exfoliatable superconductors, magnets, and other exotic phases are all now known, and the preparation and mechanical exfoliation of new classes of van der Waals materials remains an area of active research. Once two dimensional crystals have been placed onto a silicon substrate, they can be picked up and manipulated by soft, sticky plastic stamps under an optical microscope. This allows researchers to prepare entire electronic devices composed only of two dimensional crystals; these are known as ‘stacks.’ These structures have projections onto the silicon surface that are reasonably large, but remain atomically thin- capacitors have been demonstrated with gates a single atom thick, and dielectrics a few atoms thick. Researchers have developed fabrication recipes for executing many ofthe operations with which an electrical engineer working with silicon integrated circuits would be familiar, including photolithography, etching, and metallization. I think it is important to be clear about what the process of exfoliation is and what it isn’t. It is true that mechanical exfoliation makes it possible to fabricate devices that are smaller than the current state of the art of silicon lithography in the out-of-plane direction. However, these techniques hold few advantages for reducing the planar footprint of electronic devices, so there is no meaningful sense in which they themselves represent an important technological breakthrough in the process of miniaturization of commercial electronic devices. Furthermore, and perhaps more importantly, it has not yet been demonstrated that these techniques can be scaled to produce large numbers of devices, and there are plenty of reasons to believe that this will be uniquely challenging. What they do provide is a convenient way for us to produce two dimensional monocrystalline devices with exceptionally low disorder for which electron density and band structure can be conveniently accessed as independent variables. That is valuable for furthering our understanding of condensed matter phenomena, independent of whether the fabrication procedures for making these material systems can ever be scaled up enough to be viable for use in technologies.