These results have consequently been interpreted as evidence for purely orbital ferromagnetism

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.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. 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, vertical towers for strawberries 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 superlattice 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[13, 19], 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. To date, quantum anomalous Hall effects have been observed at band fillings ν = 1 and ν = 3 in various graphene heterostructures, where ν = An corresponds to the number of electrons per unit cell area A with n the carrier density. Although orbital magnetism is generally expected theoretically in twisted bilayer graphene, no direct experimental probes of magnetism have been reported because of the relative scarcity of magnetic samples, their small size, and the low expected magnetization density they are predicted to have.In the absence of significant magnetic disorder ferromagnetic domain walls minimize surface tension. In two dimensions, domain walls are pinned geometrically in devices of finite size with convex internal geometry. As discussed in Fig. 5.15, we observe pinning of domain walls at positions that do not correspond to minimal length internal chords of our device geometry–suggesting that magnetic order couples to structural disorder directly.

This is corroborated by the fact that the observed domain reversals associated with the Barkhausen jumps are consistent over repeated thermal cycles between cryogenic and room temperature. Together, these findings suggest a close analogy to polycrystalline spin ferromagnets, which host ferromagnetic domain walls that are strongly pinned to crystalline grain boundaries ; indeed, these crystalline grains are responsible for Barkhausen noise as it was originally described. Although crystalline defects on the atomic scale are unlikely in tBLG thanks to the high quality of the constituent graphene and hBN layers, the thermodynamic instability of magic angle twisted bilayer graphene makes it highly susceptible to inhomogeneity at scales larger than the moir´e period, as shown in prior spatially resolved studies. For example, the twist angle between the layers as well as their registry to the underlying hBN substrate may all vary spatially, providing potential pinning sites. Moir´e disorder may thus be analogous to crystalline disorder in conventional ferromagnets, which gives rise to Barkhausen noise as it was originally described. A subtler issue raised by our data is the density dependence of magnetic pinning; as shown in Fig. 5.3, Bc does not simply track 1/m across the entire density range, in particular failing to collapse with the rise in m in the Chern magnet gap. This suggests nontrivial dependence of either the pinning potential or the magnetocrystalline anisotropy energy on the realized many body state. Understanding the pinning dynamics is critical for stabilizing magnetism in tBLG and the growing class of related orbital magnets, which includes both moir´e systems as well as more traditional crystalline systems such as rhombohedral graphite. In order to understand the microscopic mechanism behind magnetic grain boundaries in the Chern magnet phase in tBLG/hBN, we used nanoSQUID magnetometry to map the local moir´e superlattice unit cell area, and thus the local twist angle, in this device, using techniques discussed in the literature. This technique involves applying a large magnetic field to the tBLG/hBN device and then using the chiral edge state magnetization of the Landau levels produced by the gap between the moir´e band and the dispersive bands to extract the electron density at which full filling of the moir´e superlattice band occurs .

The strength of this Landau level’s magnetization can be mapped in real space , and the density at which maximum magnetization occurs can be processed into a local twist angle as a function of position . It was noted in that the moir´e superlattice twist angle distribution in tBLG is characterized by slow long length scale variations interspersed with thin wrinkles, across which the local twist angle changes rapidly. These are also present in the sample imaged here . The magnetic grain boundaries we extracted by observing the domain dynamics of the Chern magnet appear to correspond to a subset of these moir´e superlattice wrinkles. It may thus be the case that these wrinkles serve a function in moir´e superlattice magnetism analogous to that of crystalline grain boundaries in more traditional transition metal magnets, pinning magnetic domain walls to structural disorder and producing Barkhausen noise in measurements of macroscopic properties. In tBLG, a set of moir´e subbands is created through rotational misalignment of a pair of identical graphene monolayers. In twisted monolayer-bilayer graphene a set of moir´e subbands is created through rotational misalignment of a graphene monolayer and a graphene bilayer. These systems both support Chern magnets. Both systems are also members of a class of moir´e superlattices known as homobilayers; in these systems, the 2D crystals forming the moir´e superlattice share the same lattice constant, and the moir´e superlattice appears as a result of rotational misalignment, as illustrated in Fig. 5.17A. Homobilayers have many desirable properties; the most important one is that the twist angle can easily be used as a variational parameter for minimizing the bandwidth of the moir´e subbands, container vertical farming producing the so-called ‘magic angle’ tBLG and tMBG systems. Homobilayers do, however, have some undesirable properties. Although local variations in electron density are negligible in these devices, the local filling factor of the moir´e superlattice varies with the moir´e unit cell area, and thus with the relative twist angle. The tBLG moir´e superlattice is shown for two different twist angles in 5.17B-C across the magic angle regime; it is clear that the unit cell area couples strongly to twist angle in this regime, illustrating the sensitivity of these devices to twist angle disorder. The relative twist angle of the two crystals in moir´e superlattice devices is never uniform. Imaging studies have clearly shown that local twist angle variations provide the dominant source of disorder in tBLG . It is hard to exaggerate the significance of this problem to the study of moir´e superlattices. Phenomena discovered in tBLG devices are notoriously difficult to replicate. Orbital magnetism at B = 0 has only been realized in a handful of tBLG devices, and quantization of the anomalous Hall resistance has only been demonstrated in a single tBLG device, in spite of years of sustained effort by several research groups. A mixture of careful device design limiting the active area of devices and the use of local probes has allowed researchers to make many important discoveries while sidestepping the twist angle disorder issue- indeed, some exotic phases are known in tBLG only from a single device, or even from individual scanning probe experiments- but if the field is ever to realize sophisticated devices incorporating these exotic electronic ground states the problem needs to be addressed.There is another way to make a moir´e superlattice. Two different 2D crystals with different lattice constants will form a moir´e superlattice without a relative twist angle; these systems are known as heterobilayers . These systems do not have ‘magic angles’ in the same sense that tBLG and tMBG do, and as a result there is no meaningful sense in which they are flat band systems, but interactions are so strong that they form interaction-driven phases at commensurate filling of the moir´e superlattice anyway. Indeed, many of the interaction-driven insulators these systems support survive to temperatures well above 100 K.

The most important way in which heterobilayers differ from homobilayers, however, is in their insensitivity to twist angle disorder. In the small angle regime, the moir´e unit cell area of a heterobilayer is almost completely independent of twist angle, as illustrated in 5.17E-F. A new intrinsic Chern magnet was discovered in one of these systems, a heterobilayer moir´e superlattice formed through alignment of MoTe2 and WSe2 monolayers. The researchers who discovered this phase measured a well-quantized QAH effect in electronic transport in several devices, demonstrating much better repeatability than was observed in tBLG. The unit cell area as a function of twist angle is plotted for three moir´e superlattices that support Chern insulators in 5.17G, with the magic angle regime highlighted for the homobilayers, demonstrating greatly diminished sensitivity of unit cell area to local twist angle in the heterobilayer AB-MoTe2/WSe2. MoTe2/WSe2 does have its own sources of disorder, but it is now clear that the insensitivity of this system to twist angle disorder has solved the replication issue for Chern magnets in moir´e superlattices. Dozens of MoTe2/WSe2 devices showing well-quantized QAH effects have now been fabricated, and these devices are all considerably larger and more uniform than the singular tBLG device that was shown to support a QAH effect, and was discussed in the previous chapters. The existence of reliable, high-yield fabrication processes for repeatably realizing uniform intrinsic Chern magnets is an important development, and this has opened the door to a wide variety of devices and measurements that would not have been feasible in tBLG/hBN. The basic physics of this electronic phase differs markedly from the systems we have so far discussed, and we will start our discussion of MoTe2/WSe2 by comparing and contrasting it with graphene moir´e superlattices. In tBLG/hBN and its cousins, valley and spin degeneracy and the absence of significant spin-orbit coupling combine to make the moir´e subbands fourfold degenerate. When inversion symmetry is broken the resulting valley subbands can have finite Chern numbers, so that when the system forms a valley ferromagnet a Chern magnet naturally appears. Spin order may be present but is not necessary to realize the Chern magnet; it need not have any meaningful relationship with the valley order, since spin-orbit coupling is absent. MoTe2/WSe2 has strong spin-orbit coupling, and as a result, the spin order is locked to the valley degree of freedom. This manifests most obviously as a reduction of the degeneracy of the moir´e subbands; these are twofold degenerate in MoTe2/WSe2 and all other TMD-based moir´e superlattices.