Because their distribution throughout the host crystal is not ordered, 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. We now have all of the tools we need to begin discussing real examples of Chern magnets on moir´e superlattices. Our discussion will begin with twisted bilayer graphene. We have already discussed the notion that moir´e superlattices can support electronic bands, and that we can expect these bands to accommodate far fewer electrons per unit area than bands in atomic lattices . This was pointed out in 2011 by Rafi Bistritzer and Allan MacDonald, but they also made another interesting observation: the band structure of the moir´e superlattice is highly sensitive to the relative twist angle of the two lattices, square pots and the bandwidth of the resulting moir´e bands can be finely tuned using the relative twist angle as a variational parameter.

It turns out twisted bilayer graphene moir´e superlattice bands can be made to have vanishingly small bandwidth by tuning the twist angle to the so-called ‘magic angle.’ The magic angle is around 1.10- 1.15◦ and a schematic of magic angle twisted bilayer graphene is shown in Fig. 4.2A. The computed band structure of twisted bilayer graphene is illustrated in Fig. 4.2B for a few different twist angles, including the magic angle. The other bands are grayed out at the magic angle to illustrate the low bandwidth of the moir´e superlattice bands. The low bandwidth of the moir´e superlattice bands combined with the low electronic density required to fill them makes them especially appealing targets for electrostatic gating experiments. The system is relatively easy to prepare; twisted bilayer graphene devices are produced by ripping a monolayer of graphene in half, rotating one crystal relative to the other using a mechanical goniometer, and then overlaying it on the other. The ‘flatness’ of the band also makes this system especially likely to support interaction-driven electronic phases like magnetism or superconductivity .It is also worth mentioning that it is extremely easy to identify situations in which interactions produce gaps in these systems. Because gaps appear when the moir´e superlattice bands are completely filled with electrons or with holes, and we already know that the moir´e superlattice bands are fourfold degenerate, we can expect any interaction-driven insulating phase to appear as an insulating phase at precisely 1/4, 1/2, or 3/4 of the electron density required to reach full filling of the moir´e superlattice band. These are sometimes called ‘filling factors’ of 1, 2, and 3, respectively, referencing the number of electrons per moir´e unit cell. This argument is presented in schematic form in Fig. 4.2 in the context of experimental data.

Interaction-driven gaps were first discovered in 2018, and this discovery was quickly followed by the dramatic discovery of superconductivity in twisted bilayer graphene. Other researchers predicted that breaking inversion symmetry in graphene would open a gap near charge 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. In an exfoliated heterostructure, the orientation of the crystal lattice relative to the edges of the flake can often be determined by investigating the natural cleavage planes of the flake. Graphene and hBN, being hexagonal lattices, have two easy cleavage planes – zigzag and armchair, each with six-fold symmetry, that together produce cleavage planes for every 30◦ relative rotation of the lattice. We tentatively identify crystallographic directions by finding edges of the flakes with relative angles of 30◦ . From the optical image we find that the cleavage planes of the tBLG layer and the top hBN are aligned.Four-terminal resistance measurements were carried out in a liquid helium cryostat with a 1 K pot and a base temperature of 1.6 K. The measurement was done using AC current excitations of 0.1 – 20 nA at 0.5 – 5.55 Hz using a DL 1211 current preamplifier, SR560 voltage preamplifier, and SR830 and SR860 lock-in amplifiers. Gate voltages and DC currents are applied, and amplified voltages recorded, with a home built data acquisition system based on AD5760 and AD7734 chips. Careful examination of this system produced some surprises as well- or at least, some phenomena that we did not naively expect to encounter in a generic intrinsic Chern magnet. Ferromagnetic domains in tBLG interact strongly with applied current. In our device, this allows deterministic electrical control over domain polarization using exceptionally small DC currents. Figure 4.7A shows Rxy at 6.5 K and B=0, measured using a small AC excitation of ∼ 100 pA to which we add a variable DC current bias. We find that the applied DC currents drive switching analogous to that observed in an applied magnetic field, producing hysteretic switching between magnetization states. DC currents of a few nanoamps are sufficient to completely reverse the magnetization, which is then indefinitely stable . Figure 4.7B shows deterministic writing of a magnetic bit using current pulses, large plastic pots and its nonvolatile readout using the large resulting change in the anomalous Hall resistance. High fidelity writing is accomplished with 20 nA current pulses while readout requires < 100 pA of applied AC current. Assuming a uniform current density in our micron-sized, two atom thick tBLG device results in an estimated current density J < 103 A·cm−2 . While current-induced switching at smaller DC current densities of J ≈ 102 A·cm−2 has been realized in MnSi, readout of the magnetization state in this material has so far only been demonstrated using neutron scattering[40]. Compared with other systems that allow in situ electrical readout, such as GaMnAs and Cr-2Te3 heterostructures, the applied current densities are at least one order of magnitude lower. More relevant to device applications, the absolute magnitude of the current required to switch the magnetization state of the system in our device is, to our knowledge, 3 orders of magnitude smaller than reported in any other system. Figure 4.7C shows the Hall resistance at T = 7 K measured as a function of magnetic field and current. DC currents compete directly with the magnetic field: opposite signs of current stabilize opposite magnetic polarizations, including states aligned opposite to the direction favored by the static magnetic field. Current breaks time reversal symmetry, it appears, and favors magnetic domains much as an applied magnetic field would. In the aftermath of the discovery of this phenomenon, a variety of theoretical models were proposed to explain it.

We will discuss one of them in more depth in the context of another system later in this thesis.As discussed in the previous chapter, electrons in moir´e flat band systems can spontaneously break time reversal symmetry, giving rise to a quantized anomalous Hall effect. In this chapter we use a superconducting quantum interference device to image stray magnetic fields in twisted bilayer graphene aligned to hexagonal boron nitride. We find a magnetization of several Bohr magnetons per charge carrier, demonstrating that the magnetism is primarily orbital in nature. Our measurements reveal a large change in the magnetization as the chemical potential is swept across the quantum anomalous Hall gap consistent with the expected contribution of chiral edge states to the magnetization of an orbital Chern insulator. Mapping the spatial evolution of field-driven magnetic reversal, we find a series of reproducible micron scale domains pinned to structural disorder whose boundaries host chiral edge states. In crystalline solids, orbital magnetization arises from the Berry curvature of the bands and intrinsic angular momentum of the Bloch electron wave packet. Although the orbital magnetization often contributes—at times substantially—to the net magnetization of ferromagnets, all known ferromagnetism involves partial or full polarization of the electron spin. Theoretically, however, ferromagnetism can also arise through the spontaneous polarization of orbital magnetization without involvement of the electron spin. Recently, hysteretic transport consistent with ferromagnetic order has been observed in heterostructures composed of graphene and hexagonal boron nitride, neither of which are intrinsically magnetic materials. Notably, spin-orbit coupling is thought to be vanishingly small in these systems, effectively precluding a spin-based mechanism. These results have consequently been interpreted as evidence for purely orbital ferromagnetism. To host purely orbital ferromagnetic order, a system must have a time reversal symmetric electronic degree of freedom separate from the electron spin as well as strong electron-electron interactions. Both are present in graphene heterostructures, where the valley degree of freedom provides degenerate electron species related by time reversal symmetry and a moir´e superlattice can be used to engineer strong interactions. In these materials, a long wavelength moir´e pattern, arising from interlayer coupling between mismatched lattices, modulates the underlying electronic structure and leads to the emergence of superlattice minibands within a reduced Brillouin zone. The small Brillouin zone means that low electron densities are sufficient to dope the 2D system to full filling or depletion of the super lattice bands, which can be achieved using experimentally realizable electric fields. For appropriately chosen constituent materials and interlayer rotational alignment, the lowest energy bands can have bandwidths considerably smaller than the native scale of electronelectron interactions, EC ≈ e 2/λM, where λM is the moir´e period and e is the electron’s charge. The dominance of interactions typically manifests experimentally through the appearance of ‘correlated insulators’ at integer electron or hole filling of the moir´e unit cell, consistent with interaction-induced breaking of one or more of the spin, valley, or lattice symmetries. Orbital magnets are thought to constitute a subset of these states, in which exchange interactions favor a particular order that breaks time-reversal symmetry by causing the system to polarize into one or more valley projected bands. Remarkably, the large Berry curvature endows the valley projected bands with a finite Chern number, so that valley polarization naturally leads to a quantized anomalous Hall effect at integer band filling.