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 the process of preparing a thin piece of such a crystal through mechanical means. In some of these materials, growing strawberries hydroponically 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, 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 flakes reach 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 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.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, growing tomatoes hydroponically 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, 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.