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. Wethus 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, stacking flower pot tower 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/antiferromagnet 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, 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 magnetize, 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 equaland opposite angular momenta, danish trolley 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 in graphene 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. For this reason, orbital magnetism does not need spin-orbit coupling to support hysteresis, and it can couple to a much wider variety of physical phenomena than spin magnetism can- indeed, anything that affects the electronic band structure or real space wave function is fair game. For this reason we can expect to encounter many of the phenomena we normally associate with spin-orbit coupling in orbital magnets that do not possess it. I would also like to talk briefly about magnetic moments. It has already been said that magnetic moments in orbital magnets come from center-of-mass angular momentum of electrons, which makes them in some ways simpler and less mysterious than magnetic moments derived from electron spin. However, I didn’t tell you how to compute the angular momentum of an electronic band, only that it can be done.
It is a somewhat more involved process to do at any level of generality than I’m willing to attempt here- it is described briefly in a later chapter- but suffice to say that it depends on details of band structure and interaction effects, which themselves depend on electron density and, in two dimensional materials, ambient conditions like displacement field. For this reason we can expect the magnitude of the magnetic moment of the valley degree of freedom to be much more sensitive to variables we can control than the magnetic moment of the electron spin, which is almost always close to 1 µB. In particular, the magnetization of an orbital magnet can be vanishingly small, or it can increase far above the maximum possible magnetization of a spin ferromagnet of 1 µB per electron. Under a very limited and specific set of conditions we can precisely calculate the contribution of the orbital magnetic moment to the magnetization, and that will be discussed in detail later as well. Finally, I want to talk briefly about coercive fields. The more perceptive readers may have already noticed that we have broken the argument we used to understand magnetic inversion in spin magnets. The valley degree of freedom is a pair of electronic bands, and is thus bound to the two dimensional crystalline lattice- there is no sense in which we can continuously cant it into the plane while performing magnetic inversion. But of course, we have to expect that it is possible to apply a large magnetic field, couple to the magnetic moment of the valley µ, and eventually reach an energyµ · BC = EI at which magnetic inversion occurs. But what can we use for the Ising anisotropy energy EI ? It turns out that this model survives in the sense that we can make up a constant for EI and use it to understand some basic features of the coercive fields of orbital magnets, but where EI comes from in these systems remains somewhat mysterious. It is likely that it represents the difference in energy between the valley polarized ground state and some minimal-energy path through the spin and valley degenerate subspace, involving hybridized or intervalley coherent states in the intermediate regime. But we don’t need to understand this aspect of the model to draw some useful insights from it, as we will see later. Real magnets are composed of constituent magnetic moments that can be modelled as infinitesimal circulating currents, or charges with finite angular momentum. It can be shown that the magnetic fields generated by the sum total of a uniform two dimensional distribution of these circulating currents- i.e., by a region of uniform magnetization- is precisely equivalent to the magnetic field generated by the current travelling around the edge of that two dimensional uniformly magnetized region through the Biot-Savart law.