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. The properties of crystals differ from the properties of atoms floating in free space because the atomic orbitals of the atoms in a crystal are close enough to those of adjacent atoms for electrons to hop between atoms. The resulting hybridization of atomic orbitals produces quantum states delocalized over the entire crystal with the capacity to carry momentum. This situation is shown in schematic form in Fig. 1.9A. For quantum states delocalized over the entire crystal, position ceases to be a useful basis. Instead, under these conditions we label electronic wave functions by their momenta, kx and ky. The atomic orbitals that prior to hybridization had discrete energy spectra now have energy spectra given by discrete functions of momentum, f. We call these functions electronic bands. Electrons loaded into the electronic bands of a two dimensional crystal will occupy the quantum states with the lowest available energies, so we can specify a maximum energy at which we expect to find electrons for any given electron density. We call that energy EF , the Fermi level. We can raise the Fermi level by adding additional electrons to the crystal, as shown in Fig. 1.9B. We have already discussed how two dimensional crystals naturally allow for manipulation of the electron density, square plant pots and thus the Fermi level. We have also already discussed how the application of an out-of-plane electric field to a two dimensional crystal will change the structure of the atomic orbitals supported by that crystal.

It naturally follows that atomic orbitals so modified will produce different electronic bands, as shown in Fig. 1.9C. It is relatively straightforward to compute how electronic bands will respond to the application of a displacement field. We will be using the momentum and energy basis for the rest of this document; this basis is known as momentum space. The simplest experiment we can perform to probe the electronic properties of a two dimensional crystal in this geometry is an electronic transport experiment, in which a voltage is applied to a region of the crystal with another region grounded, so that electrical current flows through the crystal. We can check if the crystal supports any electrical transport at all, and if it does we can measure the electrical resistance of the crystal this way, in close analogy to how this is done for three dimensional crystals. Crystals will only accept and thus conduct electrons if there are available quantum states at the Fermi level; we call these crystals metals , and they can be identified in band structure diagrams by the intersection of the Fermi level with an electronic band . Crystals without empty quantum states at the Fermi level will not accept and conduct electrons , and they can be identified in band structure diagrams with crystals for which the Fermi level does not intersect with an electronic band . There exists a variety of other experiments we can perform on two dimensional crystals in order to understand their properties. Two dimensional crystals can support electronic transport in the in-plane direction if they are metals, as shown in Fig. 1.10. Capacitors can also support electronic transport in the out-of-plane direction, as long as that electronic transport occurs at finite frequency. The same structure that we use to modify the electron density and ambient out-of-plane electric field of a two dimensional crystal can also be used as a capacitive AC conductor, as illustrated in Fig. 1.11A.

The conductance will depend only on the frequency at which an AC voltage is applied and the geometry of the parallel plate capacitor. However, if a two dimensional crystal is added in series, the capacitance of the top gate to the bottom gate may be substantially modified. If the two dimensional crystal is an insulator, electric fields will penetrate it and the capacitance between the two gates will not change. However, if the two dimensional crystal is a metal it will accept electrons and cancel the applied electric field, dramatically reducing the capacitance between the top and bottom gates and neutralizing the AC current through the capacitor. This technique can be used to measure the electronic properties of specifically the bulk of a two dimensional crystal; it is the property that was both calculated and measured in Fig. 1.2C and D. These two techniques are the bread and butter of the experimental study of two dimensional crystals, because they require only the ability to create stacks of two dimensional crystals and access to tools common to the study of all other microelectronic systems. We will discuss a considerable amount of electronic transport and capacitance data as well. However, the primary focus of this thesis will be on systems for which the nanoSQUID microscope can provide important information that is inaccessible to these techniques, and so we will discuss a few such systems next. 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.

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 canal ways 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 , plastic pots for planting 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, 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.Finally, moir´e superlattice bands inherit any electronic degeneracies- like, for example, electron spin- that came with the original lattice. We haven’t discussed electronic degeneracies yet, and we will shortly. So if a moir´e superlattice satisfies all of these criteria, then it will provide a set of electronic bands that can be completely filled or depleted with an electronic gate. I’m sure this seems to the reader like a pretty niche system, and that’s more or less because it is. There aren’t too many material systems that need their atomic bonds aligned with a mechanical goniometer, and it’s hard to imagine ever integrating such a procedure into an industrial fabrication line. However, it’s tough to adequately express how hard it would be to replicate the properties of a moir´e superlattice band in an atomic crystal. I made an attempt to do so in the introduction to this thesis; suffice to say the control we have over the properties of these systems is more or less unprecedented within experimental condensed matter physics, and this means that we can perform experiments on electronic phases in these systems that would be difficult or impossible in atomic crystals. 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 .