A third consideration is that the nuts or berries under study may simply have sufficient bio-activity to produce favorable outcomes, independent of potential researcher bias.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, snap clamps for greenhouse 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, 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 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. Electrostatic gating of graphene can produce crystals with an extra electron per hundred unit cells at most. 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 afore-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. Many magnetometers are sensitive to magnetic flux, not field, so very high magnetic field sensitivities are achievable by simply sampling a large region, but of course that is not a useful option when imaging microscopic magnetic systems. Suffice to say that nanoSQUID sensors, which had been invented in 2010 and integrated into a scanning probe microscope by their inventors by 2012, combine high spatial resolution with very high magnetic field sensitivity. This combination of performance metrics was and remains unique in its ability to probe the minute magnetic fields associated with gate-tunable electronic phenomena at the length scales demanded by the size of the devices. Gate-tunable phenomena in exfoliated heterostructures and nanoSQUID microscopy were uniquely well-matched to each other, greenhouse snap on clamp and although at the time I started my graduate research only a small handful of gate-tunable magnetic phenomena had so far been discovered in exfoliated two dimensional crystals, nanoSQUID microscopy seemed like the perfect tool for investigating them.So what exactly is nanoSQUID microscopy? We can start by discussing Superconducting Quantum Interference Devices, or SQUIDs. In summary, SQUIDs are electronic devices with properties that strongly depend on the magnetic field to which they are exposed, which makes them useful as magnetometers. I won’t delve into the details of how and why SQUIDs work here, but I will explain briefly how SQUIDs are made, since that will be necessary for understanding how nanoSQUID imaging differs from other SQUID-based imaging technologies. A SQUID is a pair of superconducting wires in parallel, each with a thin barrier in series . The electronic transport properties of this device depend strongly on the magnetic flux through the region between the wires, i.e. inside the hole in the center of the device in Fig. 1.3. To be a little bit more precise, superconductors transport current without dissipation, so long as the current density stays below a sharp threshold. When this threshold is exceeded, the superconductor reverts to dissipative transport, like a normal metal. Above this critical current, in the so-called ‘voltage state,’ electronic transport is dissipative and highly sensitive to B. Any non-superconductor can function as a barrier, including insulators, metals, and superconducting regions thinner than the coherence length. This makes SQUIDs excellent magnetic field sensors, with the caveat that they do not sample the magnetic field at a point, but averaged over a region A.
Making a SQUID is as easy as depositing a superconducting material onto a surface in the correct shape, and it can be done using many of the same techniques used to produce other microscopic electronic devices, like photolithography and thermal evaporation. This is sufficient for many applications, but it presents some issues for producing sensors for scanning probe microscopy. Scanning probe microscopy is a technique through which any sensor can be used to generate images; we simply move the sensor to every point in a grid, perform a measurement, and use those measurements to populate the pixels of a two dimensional array . This can of course be done with a SQUID, and many researchers have used SQUIDs fabricated this way to great effect. But the spatial resolution of a scanning SQUID magnetometry microscope is set by the size of the SQUID, and there are limits to how small SQUIDs can be fabricated using photolithography. It is also challenging to fashion these SQUIDs into probes that can be safely brought close to a surface for scanning; photolithography produces SQUIDs on large, flat silicon substrates, and these must subsequently be cut out and ground down into a sharp cantilever with the SQUID on the apex in order to get the SQUID close enough to a surface for microscopy. In summary, the ideal SQUID sensor for microscopy would be one that was smaller than could be achieved using traditional photolithography and located precisely on the apex of a sharp needle to facilitate scanning. As is so often the case when developing new technologies, we have to make the best of the tools other clever people have already developed. In the case of nanoSQUID microscopy, the inventors of the technique took advantage of a lot of legwork done by biologists. Long ago, glass blowers found that hollow glass tubes could be heated close to their melting point and drawn out into long cones without crushing their hollow interiors. Chemists used this fact to make pipettes for manipulating small volumes of liquid, and biologists later used the techniques they developed to fashion microscopic hypodermic needles that could be used to inject chemicals into and monitor the chemical environment inside individual cells in a process called patch-clamping. A rich array of tools exist for producing these structures, called micropipettes, for chemists and biologists. Eli Zeldov noticed that these structures already had the perfect geometry to serve as substrates for tiny SQUIDs. By depositing superconducting materials onto these substrates from a few different directions, one can produce superconducting contacts and a tiny torus of superconductor on the apex of the micropipette. The same group of researchers successfully integrated these sensors into a scanning probe microscope at cryogenic temperatures. The sizes of these SQUIDs are limited only by how small a micropipette can be made, and since the invention of the technique SQUIDs as small as 30 nm have been realized. We call these sensors nanoSQUIDs, or nanoSQUID-on-tip sensors. A few representative examples of nanoSQUID sensors are shown in Fig. 1.4. A characterization of the electronic transport properties of such a sensor, and in particular the sensor’s response to an applied magnetic field, is shown in Fig. 1.5. NanoSQUID microscopes share many of the core competencies of more traditional, planar scan- ning SQUID microscopes. They dissipate little power, and the measurements they generate are quantitative and can be easily calibrated by measuring the period of the SQUID’s electronic response to an applied magnetic field. 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.