We have already mentioned the most important consequence of a finite net Chern number: the presence of chiral edge states in the gap of a magnetic insulator. We have not yet discussed the consequences of this state of affairs, and we will do so next. The quantum states available in the bulk of trivial materials, i.e. Bloch states, are delocalized over the entire crystal, and as a result, when Bloch states are present at the Fermi level, electronic transport between any two points in the crystal can occur through the rapid local occupation and depletion of these quantum states. The edge states that appear in Chern magnets support a lower-dimensional analog of this property: they are delocalized quantum states restricted to the edge of a two dimensional crystal, and as a result they support electronic transport along the edge of the crystal through the rapid local occupation and depletion of these semi-localized quantum states. They do not support electronic transport through the bulk, and edge states that are not simply connected cannot transmit electrons through the bulk region separating them. As mentioned, the Chern number is a signed integer, and we have not yet discussed the physical meaning of the sign of the Chern number. The edge states in Chern magnets are chiral, meaning that electrons populating a particular edge state can only propagate in one direction around the edge of a two-dimensional crystal. The sign of the Chern number determines the direction or chirality with which propagation of the electronic wave function around the crystal occurs. Electronic bands with opposite Chern numbers produce edge states with opposite chiralities. So in summary, a two dimensional crystal that is a Chern magnet supports electronic transport through chiral edge states that live on its boundaries.
These systems remain bulk insulators, black plastic plant pots bulk and edge states separated by the bulk cannot exchange electrons with each other. The sign of the Chern number is determined by the spin state that is occupied, and thus the chirality of the available edge state is hysteretically switchable, just like the magnetization of the two dimensional magnet.It is important to remember that these quantum states are just as real as Bloch states, and apart from the short list of differences discussed above, they can be analyzed and understood using many of the same tools. For example, in a metallic system, the Fermi level can be raised by exposing a crystal to a large population of free electrons and using an electrostatic gate to draw electrons into the crystal. These Bloch states have a fixed set of allowed momenta associated with their energies, and experiments that probe the momenta of electrons in a crystal will subsequently detect the presence of electrons in newly populated momentum eigenstates. Similarly, attaching a Chern magnet to a reservoir of electrons and using an electrostatic gate to draw electrons into the magnet will populate additional chiral edge states. Properties that depend on the number of electrons occupying these special quantum states will change accordingly. In all of these systems, conductivity strongly depends on the number of quantum states available at the Fermi level. For metallic systems, the number of Bloch states available at any particular energy depends on details of the band structure. The total conductance between any two points within the crystal depends on the relative positions of the two points and the geometry of the crystal.
Thus conductivity is an intrinsic property of a metal, but conductance is an extrinsic property of a metal, and both are challenging to compute precisely from first principles.At finite temperature, electrons occupying Bloch states in metals can dissipate energy by scattering off of phonons, other electrons, or defects into different nearby Bloch states. This is possible because at every position in real space and momentum space there is a near-continuum of available quantum states available for an electron to scatter into with arbitrarily similar momentum and en-ergy. This is not the case for electrons in chiral edge states of Chern magnets, which do not have available quantum states in the bulk. As a result, electrons that enter chiral edge state wave functions do not dissipate energy. There is a dissipative cost for getting electrons into these wave functions this was discussed in the previous paragraph- but this energetic cost is independent of all details of the shape and environment of the chiral edge state, even at finite temperature. This is why the Hall resistance Rxy in a Chern magnet is so precisely quantized; it must take on a value of C 1 e h 2 , and processes that would modify the resistance in other materials are strictly forbidden in Chern magnets. All bands have finite degeneracy- that is, they can only accommodate a certain number of electrons per unit area or volume of crystal. If electrons are forced into a crystal after a particular band is full, they will end up in a different band, generally the band that is next lowest in energy. This degeneracy depends only on the properties of the crystal. Chern bands have electronic degeneracies that change in response to an applied magnetic field; that is to say, when Chern magnets are exposed to an external magnetic field, their electronic bands will change to accommodate more electrons.Simple theoretical models that produce quantized anomalous Hall effects have been known for decades.
The challenge, then, lay in realizing real materials with all of the ingredients necessary to produce a Chern magnet. These are, in short: high Berry curvature, a two-dimensional or nearly two-dimensional crystal, and an interaction-driven gap coupled to magnetic order. It turns out that a variety of material systems with high Berry curvature are known in three dimensions; three dimensional topological insulators satisfy the first criterion, and are relatively straightforward to produce and deposit in thin film form using molecular beam epitaxy, satisfying the second. These systems do not, however, have magnetic order. Researchers attempted to induce magnetic order in these materials with the addition of magnetic dopants. It was hoped that by peppering the lattice with ions with large magnetic moments and strong exchange interactions that magnetic order could be induced in the band structure of the material, as illustrated in Fig. 3.11. This approach ultimately succeeded in producing the first material ever shown to support a quantized anomalous Hall effect. An image of a film of this material and associated electronic transport data are shown in Fig. 3.12.We have already discussed the notion of the Curie temperature and its origin. To reiterate, the Curie temperature is a temperature set by the lowest energy scale at which excitations that change the magnetic order can appear. It is worth emphasizing one point in particular: the set of excitations that change the magnetic order includes but is not limited to all those that promote an electron from the valence band to the conduction band, i.e. the excitations that support charge transport through the bulk of a magnetic insulator. For this reason, the energy scale of the Curie temperature is generally expected to be lower than the energy scale of thermal activation of electrons into the bulk conduction band of the magnetic insulator.There are many a priori reasons to suspect that magnetically doped topological insulators might have strong charge disorder. The strongest is the presence of the magnetic dopants- dopants always generate significant charge disorder; in a sense they are by definition a source of disorder. Because their distribution throughout the host crystal is not ordered, procona system dopants can reduce the effective band gap through the mechanism illustrated in Fig. 3.14. It turns out this concern about magnetically doped topological insulators has been born our in practice; the systems have been improved since their original discovery, but in all known samples the Curie temperatures dramatically exceed the charge gaps . This puts these systems deep in the kBTC > EGap limit. The resolution to this issue has always been clear, if not exactly easy. If a crystal could be realized that had bands with both finite Chern numbers and magnetic interactions strong enough to produce a magnetic insulator, then we could expect such a system to be a clean Chern magnet . Such a system would likely support a QAH effect at much higher temperature then the status quo, since it would not be limited by charge disorder.Other researchers predicted that breaking inversion symmetry in graphene would open a gap nearcharge neutrality with strong Berry curvature at the band edges. The graphene heterostructures we make in this field are almost always encapsulated in the two dimensional crystal hBN, which has a lattice constant quite close to that of graphene. The presence of this two dimensional crystal technically always does break inversion symmetry for graphene crystals, but this effect is averaged out over many graphene unit cells whenever the lattices of hBN and graphene are not aligned with each other.
Therefore the simplest way to break inversion symmetry in graphene systems is to align the graphene lattice with the lattice of one of its encapsulating hBN crystals. Experiments on such a device indeed realized a large valley hall effect, an analogue for the valley degree of freedom of the spin Hall effect discussed in the previous chapter, a tantalizing clue that the researchers had indeed produced high Berry curvature bands in graphene. Twisted bilayer graphene aligned to hBN thus has all of the ingredients necessary for realizing an intrinsic Chern magnet: it has flat bands for realizing a magnetic insulator, it has strong Berry curvature, and it is highly gate tunable so that we can easily reach the Fermi level at which an interaction-driven gap is realized. Magnetism with a strong anomalous Hall effect was first realized in hBN-aligned twisted bilayer graphene in 2019. Some basic properties of this phase are illustrated in Fig. 4.3. This system was clearly a magnet with strong Berry curvature; it was not gapped and thus did not realize a quantized anomalous Hall effect, but it was unknown whether this was because of disorder or because the system did not have strong enough interactions or small enough bandwidth to realize a gap. The stage was set for the discovery of a quantized anomalous Hall effect in an intrinsic Chern magnet in hBN-aligned twisted bilayer graphene.We return now to our discussion of twisted bilayer graphene; we will be discussing domain dynamics. To investigate the domain dynamics directly, we compare magnetic structure across different states stabilized in the midst of magnetic field driven reversal. Figure 5.13A shows a schematic depictionof our transport measurement, and Fig. 5.13B shows the resulting Rxy data for both a major hysteresis loop spanning the two fully polarized states at Rxy = ±h/e2 and a minor loop that terminates in a mixed polarization state at Rxy ≈ 0 . All three states represented by these hysteresis loops can be stabilized at B = 22 mT for T = 2.1 K, where our nanoSQUID has excellent sensitivity, allowing a direct comparison of their respective magnetic structures . Figures 5.13, F and G, show images obtained by subtracting one of the images at full positive or negative polarization from the mixed state, as indicated in the lower left corners of the panels. Applying the same magnetic inversion algorithm used in Fig. 5.1 produces maps of m corresponding to these differences , allowing us to visualize the domain structure generating the intermediate plateau Rxy ≈ 0 seen in the major hysteresis loop. The domains presented in Figs. 5.13, H and I, are difference images; the domain structures actually realized in experiment are illustrated schematically in Fig. 5.13, J-L. Evidently, the anomalous Hall resistance of the device in this state is dominated by the interplay of two large magnetic domains, each comprising about half of the active area. Armed with knowledge of the domain structure, it is straightforward to understand the behavior of the measured transport in the mixed state imaged in Fig. 5.13D. In particular, the state corresponds to the presence of a single domain wall that crosses the device, separating both the current and the Hall voltage contacts . In the limit in which the chiral edge states at the boundaries of each magnetic domain are in equilibrium, there will be no drop in chemical potential across the domain wall, leading to Rxy = 0. This is very close to the observed value of Rxy = 1.0 kΩ = 0.039 h/e 2 .