At the time I started my PhD, experiments in condensed matter physics had already begun a rapid expansion in their ability to conduct in situ modifications of the electronic structure of crystals, and in particular the electron density within crystals. The number of electrons per unit volume is an extremely important property of a crystal; it determines which quantum states within the crystal are filled and which are empty, and thus whether the system is metallic, insulating, or even something more exotic. Almost all of the properties of a crystal are impacted by electron density. Historically the number of electrons in a crystalline system has primarily been modified by adding dopants, i.e., additional atoms with more or fewer conduction electrons than the rest of the crystal .One can achieve dramatic changes in charge density using this technique, but that comes with a heavy cost- the crystal is no longer uniform, as every dopant contributes to disorder, and at high doping levels the band structure itself can be modified by the dopant atoms. More important than all of this for the purposes of experimental physics, however, is that under most circumstances the dopant concentration within a crystal can only be modified through laborious chemical treatments of a particular sample. Materials scientists working under these constraints who wish to explore electron density as an independent variable must either find ingenious material-specific techniques for modifying the dopant concentration in situ , or else they must make a separate sample for each data point they would like to present in their experiment. This is an incredibly labor-intensive process, hydroponic indoor growing system and it also comes with another significant downside: comparing the properties of two different samples with different doping densities exposes results to systematic differences in sample geometry and imperfections in protocol repeatability, and it is difficult to deconvolute these from the effects of changes in electron density.
For these reasons electron density has generally been an awkward and labor intensive independent variable to manipulate. I have found that there are a few ideas that occur naturally to newcomers and outsiders to the field that insiders know enough to immediately discount, and I’d like to discuss one of those ideas here. Chemical doping to manipulate electron density is an ingenious and important technique, but suppose we tried something much sillier- suppose we simply forcefully deposit electrons onto a crystal using some mechanical or electrical process. Would this not achieve our goal? In fact this does indeed work, we have machines that can do this- van de Graff generators can deposit charge onto a piece of metal mechanically, and a variety of other machines can mimic this behavior electronically. So why aren’t condensed matter physicists going around gluing interesting crystals to van de Graff generators so that we can controllably charge them up and measure their responses to changes in charge density? There are a few reasons, but the most important one is that there is a fundamental issue with manipulating charge density this way in three dimensions: this process does not produce a uniform distribution of electron density within the crystal we’d like to study. In three dimensional systems subjected to this treatment, as illustrated in Fig. 1.1B, excess charge accumulates on the surfaces of the crystal, and although we can force additional electrons into acrystal this way we do not ultimately get a system with a modified but still uniform electron density for us to study. This is not the case for two dimensional systems.
Those readers with any exposure to introductory physics have likely encountered parallel plate capacitors; these are highly idealized systems composed of a pair of infinitely thin conducting sheets separated by a small insulating space of consistent thickness. When a voltage is applied to one of these sheets with the other connected to a reservoir of mobile electrons, a uniform charge density per unit area appears on both sheets . Of course, in real metallic capacitors the charge density per unit volume is often still not microscopically uniform because the sheets are not actually infinitely thin, so electrons can redistribute themselves in the out-of-plane direction. To achieve true uniformity one of the plates of the capacitor must be atomically thin, so that electrons simply cannot redistribute themselves in the out-of-plane direction in response to the local electric field. An efficient technique for preparing atomically thin pieces of crystalline graphite was discovered in 2004 by Dr. Andre Geim and Dr. Konstantin Novoselov, an achievement for which they shared the Nobel prize in physics in 2010. The technique involves encapsulating a crystal within a piece of scotch tape and repeatedly ripping the tape apart; it works because the out-of-plane bonds in graphite are much weaker than the in-plane bonds. Graphite represents something of an extreme example of this condition, but it is satisfied to varying extents by a large class of other materials, and as a result the technique was rapidly generalized to produce a variety of other two-dimensional crystals. By constructing a capacitor with one gate replaced with one of these two dimensional crystals, as shown in Fig. 1.1D, researchers can easily access electron density as an independent variable in a condensed matter system.
These systems also facilitate an additional degree of control, with no real analogue in three dimensional systems. By placing capacitor plates on both sides of the two dimensional crystal and applying opposite voltages to the opposing gates, researchers can apply out-of-plane electric fields to these systems . A semiclassical model- in which electrons within the system redistribute themselves in the out-of-plane direction to screen this electric field- does not apply; instead, the wave functions hosted by the two dimensional crystal are themselves deformed in response to the applied electric field . This changes the electronic band structure of the crystal directly, without affecting the electron density. So to summarize, when a two dimensional crystal is encapsulated with gates to produce a three-layer capacitor, researchers can tune both the electron density and the band structure of the crystal at their pleasure. In the first case, this represents a degree of control that would require the creation of many separate samples to replicate in a three dimensional system. The second effect cannot be replicated in three dimensional systems with any known technique.There is a temptation to focus on the exotic phenomena that these techniques for manipulating the electronic structure of two dimensional crystals have allowed us to discover, and there will be plenty of time for that. I’d first like to take a moment to impress upon the reader the remarkable degree of control and extent of theoretical understanding these technologies have allowed us to achieve over those condensed matter systems that are known not to host any new physics. I’ve included several figures from a publication produced by Andrea’s lab with which I was completely uninvolved. It contains precise calculations of the compressibility of a particular allotrope of trilayer graphene as a function of electron density and out-of-plane electric field based on the band structure of the system . It also contains a measurement of compressibility as a function of electron density and out-of-plane electric field, performed using the techniques discussed above . The details of the physics discussed in that publication aren’t important for my point here; the observation I’d like to focus on is the fact that, for this particular condensed matter system, quantitatively accurate agreement between the predictions of our models and the real behavior of the system has been achieved. I remember sitting in a group meeting early in my time working with Andrea’s lab, long before I understood much about Chern magnets or any of the other ideas that would come to define my graduate research work, and marvelling at that fact. Experimental condensed matter physics necessarily involves the study of systems with an enormous number of degrees of freedom and countless opportunities for disorder and complexity to contaminate results. Too often work in this field feels uncomfortably close to gluing wires to rocks and then arguing about how to interpret the results, grow hydroponic with no real hope of achieving full understanding, or closure, or even agreement about the conclusions we can extract from our experiments.
Within the field of exfoliated heterostructures, it is now clear that we really can hope to pursue true quantitative accuracy in calculations of the properties of condensed matter systems. Rich datasets like these, with a variety of impactful independent variables, produce extremely strong limits on theories. They allow us to be precise in our comparisons of theory to experiment, and as a result they have allowed us to bring models based on band structure theory to new heights of predictive power. But most importantly, under these conditions we can easily identify deviations from our expectations with interesting new phenomena- in particular, situations in which electronic interactions produce even subtle deviations from the predictions of single particle band structure theory.This is more or less how I would explain the explosion of interest in the physics of two dimensional crystalline systems within experimental condensed matter physics over the past decade. If you ask a theorist if two dimensional physical systems have any special properties, they will tell you that they do. They might say that the magnetic phase transitions in a Heisenberg model on a two dimensional lattice differ dramatically from those on a three dimensional one. They might say thathigh Tc superconductivity is apparently a two dimensional phenomenon. They might note that two dimensional electronic systems can support quantum Hall effects and even be Chern magnets , while three dimensional systems cannot. But it is easy to miss the forest for the trees here, and I would argue that interest in these particular physical phenomena is not behind the recent explosion in the popularity of the study of exfoliated two dimensional crystals in condensed matter physics. Instead, much more basic technical considerations are largely responsible- it is simply much easier for us to use charge density and band structure as independent variables in two dimensional crystals than in three dimensional crystals, and that capability has facilitated rapid progress in our understanding of these systems. The techniques described above still have some limitations, and chief among them is a limited range of electronic densities that they can reach. Of course, the gold standard of electron density modulation is the ability to completely fill or deplete an electronic band, which requires about one electron per unit cell in the lattice. Chemical doping can achieve enormous offsets in charge density, sometimes as high as one electron per unit cell. This limitation isn’t fundamental and there are some ideas in the community for ways to improve it, but for now it remains true that electrostatic gates can modify electron densities only slightly relative to the total electron densities of real two dimensional crystals. As it stands, electrostatic gating can only substantially modify the properties of a crystal if the crystal happens to have large variations in the number and nature of available quantum states near charge neutrality. For many crystals this is not the case; thankfully it is for graphene, and for a wide variety of synthetic crystals we will discuss shortly. Electrostatic gating of two dimensional crystals was rapidly becoming a mature technology by the time I started my PhD. So where does nanoSQUID magnetometry fit into all of this? A variety of other techniques exist for microscopic imaging of magnetic fields; the most capable of these other technologies recently developed the sensitivity and spatial resolution necessary to image stray magnetic fields from a fully polarized two dimensional magnet, with a magnetization of about one electron spin per crystalline unit cell, and this was widely viewed within the community as a remarkable achievement. We will shortly be discussing several ferromagnets composed entirely of electrons we have added to a two dimensional crystal using electrostatic gates. Because of the a fore-mentioned limitations of electrostatic gating as a technology, this necessarily means that these will be extremely low density magnets with vanishingly small magnetizations, at least 100 times smaller than those produced by a fully polarized two dimensional magnet like the one in the reference above. It is difficult to summarize performance metrics for magnetometers, especially those used for microscopy.