Systems like chromium iodide have properties that are easy to understand in the context of the models we have so far discussed: strong on site interactions and exchange interactions produce full spin polarization, an interaction-driven band gap, and aligned magnetic moments within a single layer. As a result, these systems are electrical insulators. They support magnetic domain dynamics, and there is a temperature TC above which they cease to be magnetized , although they remain insulators far above that temperature. Extremely weak out-of-plane bonds produce highly anisotropic cleavage planes and make it relatively easy to prepare atomically thin crystals mechanically. As in other systems, this does not mean we will always be studying monolayers of the material. Bilayers, trilayers, four-layer crystals, and even thicker flakes can all have properties that differ significantly from those of a monolayer, often for reasons that we can understand, and CrI3 is no exception. Although it isn’t particularly relevant to the physics of magnetism, it’s worth mentioning that all of the chromium halides are highly unstable compounds, and decompose in a matter of seconds when exposed to air or moisture. These materials are difficult to study under normal circumstances, but two dimensional crystalline samples can be prepared inside of an inert-atmosphere glovebox. They can also be sandwiched, plant pot with drainage or ‘encapsulated,’ between other two dimensional crystals. Two dimensional crystals are so flat that this process produces an air- and water-proof barrier and protects the encapsulated crystal from degradation in atmosphere, facilitating easy measurements with tools like the nanoSQUID.
The crystalline structure of CrI3, projected onto a two-dimensional crystal, is visible in Fig. 2.6A. Unlike graphene, CrI3 has two different kinds of atoms in its unit cell; the chromium atoms are responsible for the magnetic moments producing magnetism. CrI3 has fairly strong spin-orbit coupling, and thus strong Ising anisotropy, with magnetic moments pointing out-of-plane . Most of the other chromium halides also support magnetic order, although the precise nature of each of their ground states differs somewhat. Both CrI3 and CrBr3 have ferromagnetic in-plane interactions and strong Ising anisotropy, but CrI3 has antiferromagnetic out-of-plane interactions, meaning that in the magnetic ground state of the crystal adjacent layers have their spins antialigned . Interestingly, CrCl3 also seems to have ferromagnetic in-plane interactions, but it is likely that it is not an Ising or easy-axis magnet, and instead has its spins pointed in the in-plane direction and thus free to rotate. It is evidently the case that although these systems are structurally very similar and all have strong spin-orbit coupling, their magnetic interactions and magnetocrystalline anisotropies vary wildly in response to modest differences in their electronic structure. As a result of all of the arguments discussed previously in this chapter, a CrI3 monolayer has finite magnetization even in the absence of an applied magnetic field, and its magnetic order experiences hysteresis in response to variations in the applied magnetic field, as illustrated in Fig. 2.6D . Antiferromagnetic interactions between adjacent layers in CrI3 mix in an interesting factor that can be easily understood: flakes with an even number of layers have no net magnetization in the absence of an applied magnetic field, but develop finite magnetization at higher magnetic fields as the applied magnetic field overwhelms interlayer interactions and realigns each layer in turn with the ambient magnetic field .
Layer realignments are close analogues of magnetic phase transitions we have already discussed, and they support magnetic hysteresis as well. We will be studying the magnetic phase transition highlighted in yellow with the nanoSQUID microscope. An optical microscope image of a large four-layer CrI3 sample and a much smaller CrI3 monolayer is shown in Fig. 2.6F; this sample has been encapsulated in hBN for protection in atmosphere. We will use this system to get a taste of what the nanoSQUID microscope is capable of. We will use the nanoSQUID to image the region outlined with a white box in Fig. 2.6F. We will be imaging magnetic order across the magnetic phase transition highlighted in yellow in Fig. 2.6E, starting at B = 720 mT and thus in a state in which the four-layer CrI3 sample has finite magnetization and ending at B = 540 mT, in a state in which the four-layer CrI3 sample has no net magnetization. We thus expect the four-layer CrI3 flake to have a finite net magnetization in the first image, and we expect an antiferromagnetic domain with no net magnetization to consume the magnetized region by the final image. The nanoSQUID sensor used to generate these images was about 80 nm in diameter, and was about 100 nm from the surface of the device, producing an imaging resolution of about 100 nm. A characterization of the SQUID used in this imaging campaign is available in Fig. 1.7. The fully magnetized state can be seen in Fig. 2.7A. The magnetic fields generated by the four-layer crystal are comparable to those emitted by the smaller monolayer, at right. In both flakes, the magnetic order is riven with linear defects, which we attribute to wrinkles or cracks in the two dimensional crystals. We can see in Fig. 2.7B that as we decrease the magnetic field, antiferromagnetic domains spread in from the edges of the flake, destroying the magnetization nearthe edges of the two dimensional crystal.
This process continues in Fig. 2.7C, but it is clear that the linear defects present in the magnetic order stop and redirect ferromagnet/anti-ferromagnet domain walls. These defects protect a small patch of magnetization at the center of the flake as the magnetic field continues to decrease in Fig. 2.7D. In Fig. 2.7E, even this internal patch of magnetized material is overwhelmed, and the entire four-layer flake has completed its phase transition to antiferromagnetic order. The monolayer remains fully magnetized. Chromium iodide is a very simple magnetic system, at least at the level of its macroscopic magnetic properties, but there are still a few conclusions we can draw from our nanoSQUID imaging campaign. First, although it is true that CrI3 supports magnetic hysteresis, it does not behave as a single macrospin, instead supporting rich domain dynamics dominated by internal structural disorder. For this reason we cannot expect to learn anything about the energy scale of magnetoelectric anisotropy from the coercive field. This puts CrI3 in a very large class of magnets that includes almost all large polycrystalline samples of transition metal magnets. We will later on discuss several magnets for which the macrospin approximation is more or less valid. Second, it is apparently the case that magnetic domains cost the least energy to nucleate near the edges of the sample. This isn’t surprising, since this region of the crystal experiences the weakest exchange interactions because the nucleated domain is not completely surrounded by the metastable magnetization state, but it is a nice sanity check for our understanding of these systems. Finally, growing blueberries in pots the fact that regions of high disorder remain highly magnetized even in the antiferromagnetic ground state may provide a hint towards the nature of internal disorder in this system. Other experiments have shown that regions of high strain in CrI3 become highly magnetized, so it could be that these one dimensional defects are wrinkles in the two dimensional crystal. We have thus used the nanoSQUID microscope to image magnetic domain dynamics in a two dimensional chromium iodide crystal with approximately 100 nm resolution at magnetic fields as high as 720 mT. This system is a relatively simple one, an uncomplicated magnetic insulator, without the physical phenomena that will form the scientific focus of this thesis. I think it’s useful to illustrate the capabilites of the nanoSQUID microscopy technique under ideal circumstances, i.e. in a system with high magnetization and strong internal disorder, but also to distinguish the physics of magnetism from the physics of Berry curvature, orbital magnetism, and Chern numbers. These phenomenatogether will form the main focus of this thesis, and we will discuss all of them in the next chapter. Let us take a closer look at the crystalline structure of graphene, armed with the knowledge we have gained about spontaneously broken symmetries and magnetism. We have already discussed how each atomic orbital of each atom contributes a band to the crystalline band structure, although of course the orbitals hybridize to produce new, decocalized quantum states. And of course we know that because either spin species can occupy a band, in reality each band can accommodate two electrons. It contains two different atoms, but those atoms have almost precisely the same environment- in fact, the only difference between them is the fact that the distribution of atoms with which they are surrounded is inverted. We can say that graphene has inversion symmetry, and furthermore that there exists a pair of different atoms that are swapped by inversion in real space and time reversal symmetry in momentum space.
Because these two atoms have almost exactly the same environment, they produce bands that are also strikingly similar. In particular, they produce pairs of bands that are related both by inversion symmetry and time reversal symmetry. These are not the same bands, but they do have precisely the same density of states at every energy. As a result, these bands are in practice energetically degenerate. This means that all of the phenomena associated with spontaneously broken symmetry can apply to this pair of bands, which together form a new degree of freedom. For reasons having to do with the shape of graphene’s band structure, we often call this new degree of freedom the ‘valley degeneracy.’ We have already seen how spin degeneracy produced magnetism. This is now joined by the valley degeneracy, so in graphene we can expect to encounter both of these twofold degeneracies, together producing a fourfold degeneracy. Every graphene band can thus accommodate four electrons. There is one more important point to make about the valley degeneracy. I mentioned in passing that these two states can be related to each other by time reversal symmetry in momentum space. In practice, this means that if the function describing one valley’s band structure is E, then we can immediately say that the other valley’s band structure is E. Suppose we found a set of conditions under which one of the bands in a graphene allotrope had finite angular momentum in its ground state. This is actually not so uncommon, so far as physical phenomena go- many atomic orbitals have finite angular momentum, and in condensed matter systems they can hybridize to form delocalized bands with finite angular momentum. The above condition tells us that we can then expect to find another band with equal energies and equal and opposite angular momenta, and thus magnetic moments. These are precisely the conditions satisfied by the electron spin degree of freedom that allowed it to produce magnetism! So, under these circumstances- i.e., assuming we can find conditions under which a band in graphene has finite anguluar momentum in its ground state, strong electronic interactions, and a flat-bottomed or flat band- we can expect to find a new form of magnetism, dubbed by theorists ‘orbital magnetism,’ wherein center of mass angular momentum coupling to the electron charge is responsible for the magnetic moment, instead of electron spin. There are many important corollaries of the arguments we’ve just discussed, and many more of them will appear later, but there are a few I’d like to focus some special attention on. We discussed earlier how the orientation of electron spin generally does not interact with electronic band structure unless we invoke relativistic effects in the form of spin-orbit coupling. Carbon atoms are extremely light, and as a result the energy scale of spin-orbit coupling in graphene is quite low. For this reason condensed matter researchers in the distant past na¨ıvely expected not to find magnetic hysteresis ingraphene systems. The type of magnet proposed here does not invoke spin-orbit coupling; in fact, it does not even invoke spin. Instead, the two symmetry-broken states are themselves electronic bands that live on the crystal, and they differ from each other in both momentum space and real space.