It is also not very useful for probing metastable states

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, hydroponic bucket 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 of the 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. Consider the following procedure: we obtain a pair of identical two dimensional atomic crystals. We slightly rotate one relative to the other, and then place the rotated crystal on top of the other . The resulting pattern brings the top layer atoms in alignment with the bottom layer atoms periodically, but with a lattice constant that is different from and in practice often much larger than the lattice constant of the original two atomic lattices. We call the resulting lattice a ‘moir´e superlattice.’ The idea to do this with two dimensional materials is relatively new, but the notion of a moir´e pattern is much older, and it applies to many situations outside of condensed matter physics. Pairs of incommensurate lattices will always produce moir´e patterns, and there are many situations in daily life in which we are exposed to pairs of incommensurate lattices, like when we look out a window through two slightly misaligned screens, or try to take pictures of televisions or computer screens with our camera phones. Of course these ‘crystals’ differ pretty significantly from the vast majority of crystals with which we have practical experience, so we’ll have to tread carefully while working to understand their properties. To start with, if we attempt to proceed as we normally would- by assigning atomicorbitals to all of the atoms in the unit cell, computing overlap integrals, and then diagonalizing the resulting matrix to extract the hybridized eigenstates of the system- we would immediately run into problems, because the unit cell has far too many atoms for this calculation to be feasible. Some moir´e superlattices that have been studied in experiment have thousands of atoms per unit cell. There exist clever approximations that allow us to sidestep this issue, and these have been developed into very powerful tools over the past few years, but they are mostly beyond the scope of this document. I’d like to instead focus on conclusions we can draw about these systems using much simpler arguments.

The physical arguments justifying the existence of electronic bands apply wherever and whenever an electron is exposed to an electric potential that is periodic, and thus has a set of discrete translation symmetries. For this reason, even though the moir´e superlattice is not an atomic crystal, we can always expect it to support electronic band structure for the same reason that we can always expect atomic crystals to support band structure. Two crystals with identical crystal symmetries will always produce moir´e superlattices with the same crystal symmetry, so we don’t need to worry about putting two triangular lattices together and ending up with something else.Another property we can immediately notice is that the electron density required to fill a moir´e superlattice band is not very large. This can be made clear by simply comparing the original atomic lattice to a moir´e superlattice in real space . Full depletion of a band in an atomic crystal requires removing an electron for every unit cell , and full filling of the band occurs when we have added an electron for every unit cell. We have already discussed how this is not possible for the vast majority of materials using only electrostatic gating, because the resulting charge densities are immense. Full depletion of the moir´e band, on the other hand, requires removing one electron per moir´e unit cell, and the moir´e unit cell contains many atoms . So the difference in charge density between full filling and full depletion of an electronic band in a moir´e superlattice is actually not so great , and indeed this is easily achievable with available technology. Before we go on, I want to make a few of the limitations of this argument clear. There are two things this argument does not necessarily imply: the moir´e bands we produce might not be near the Fermi level of the system at charge neutrality, and the bandwidth of the moir´e superlattice need not be small. In the first case, we won’t be apply to modify the electron density enough to reach the moir´e band, and in the latter, stackable planters we won’t be able to fill the moir´e band’s highest energy levels using our electrostatic gate. We know of examples of real systems with moir´e superlattice bands that fail each of those criteria. But if these moir´e superlattice bands are near charge neutrality, and if their bandwidths are small, then we should be able to easily fill and deplete them with an electrostic gate. A variety of scanning probe microscopy techniques have been developed for examining condensed matter systems. It’s easy to justify why magnetic imaging might be interesting in gate-tuned two dimensional crystals, but magnetic properties of materials form only a small subset of the properties in which we are interested. Scanning tunneling microscopy is capable of probing the atomic-scale topography of a crystal as well as its local density of states, and a variety of scanning probe electrometry techniques exist as well, mostly based on single electron transistors. It’s worth pointing out that if you’re interested specifically in performing a scanning probe microscopy experiment on a dual-gated device, then these techniques both struggle, because the top gate both blocks tunnel current and screens out the electric fields to which a single electron transistor would be sensitive. Magnetic fields have an important advantage over electric fields: most materials have very low magnetic susceptibility, and thus magnetic fields pass unmodified through the vast majority of materials . This means that magnetic imaging is more than just one of many interesting things one can do with a dual-gated device; in these systems, magnetic imaging is a member of a very short list of usable scanning probe microscopy techniques.

The simplest way in which we can use our nanoSQUID magnetometry microscope is as a DC magnetometer, probing the static magnetic field at a particular position in space . There are situations in which this is a valuable tool, and we will look at some DC magnetometry data shortly, but in practice our nanoSQUID sensors often suffer from 1/f noise, spoiling our sensitivity for signals at low or zero frequency. One of the primary advantages of the technique is its sensitivity, and to make the best of the sensor’s sensitivity we must measure magnetic fields at finite frequencies. We have already discussed how we can use electrostatic gates to change the electron density and band structure of two dimensional crystals. We will discuss shortly a variety of gate-tunable phenomena with magnetic signatures that appear in these systems. It follows, of course, that we can modulate the magnetic fields emitted by these electronic phases and phenomena by modulating the voltages applied to the electrostatic gates we use to stabilize these phases. This is illustrated in Fig. 1.15C: an AC voltage is applied to the bottom gate relative to the two dimensional crystal, and the local magnetic field is sampled at the same frequency by the SQUID. We can use this techniqueto extract δV δB at an array of positions above the two dimensional crystal. This technique is very simple and powerful, but it has a few important drawbacks. It can only produce a quantitative measurement of B if the same scan is performed for a large set of gate voltages, so that δV δB can be integrated. Many ferromagnets, for example, can be locked into quantum states that aren’t their ground states using a ferromagnetic hysteresis loop, and rapidly tuning the electron density tends to relax these phases to their ground states. So whenever we are interested in probing metastable magnetic states, we need to be careful about using this measurement method. Of course, we can also modulate the magnetic field through the nanoSQUID by modulating the position of the nanoSQUID. Since the magnetic field varies rapidly in space, we can often expect to get strong signals when we probe δB δx this way . The position of the nanoSQUID is rapidly modulated using a piezoelectric tuning fork pressed against the side of the nanoSQUID sensor; the details of the tuning fork hardware and measurement are discussed further in the appendix. This measurement method allows us to use the nanoSQUID to probe metastable or even non-gate-tunable magnetic phenomena at finite frequency. It has a few drawbacks of its own, though. The nanoSQUID sensors have parasitic sensitivities to local temperature and electric potential , and if these vary in space the resulting signals will contaminate our magnetic field data. As a result, whenever we use this contrast mechanism we must try to extract differences between two different magnetic states if we want quantitatively precise information about the magnetic field. We can also apply an AC current in the plane of the two dimensional crystal. Large currents will emit detectable magnetic fields through the Biot-Savart law, and under those conditions we can use this contrast mechanism to reconstruct the current density through our two dimensional crystal.