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.We perform spatially resolved magnetometry to image the submicron magnetic structure of the same sample presented in the previous chapter and in Ref. , which consists of a twisted graphene bilayer aligned to one of the hexagonal boron nitride encapsulating layers. Figure 5.1B shows a schematic representation of our experimental setup. We use a superconducting quantum interference device fabricated on the tip of a quartz tube from cryogenically deposited indium with a magnetic field sensitivity of approximately 15 nT/Hz1/2 at select outof-plane magnetic fields of less than 50 mT. The SQUID is mounted to a quartz tuning fork and rastered in a 2D plane parallel to, and at a fixed height above, the tBLG heterostructure. Here n is the nominal density inferred from a parallel plate capacitor model, with the capacitance determined from the lowfield Hall density . Images are acquiredin the same background magnetic field B = 22 mT but on opposite branches of the hysteresis loop shown in Fig. 5.1A. As demonstrated in Fig. 8.16, and explained in more detail in the appendix, raspberry container the measured BT F contains contributions from magnetic signals as well as other effects arising from electric fields or thermal gradients. To isolate the magnetic structure that gives rise to the observed hysteretic transport, we subtract data from Figs. 5.1, C and D, from each other.

The result is shown in Fig. 5.1E, which depicts the gradient magnetometry signal associated with the fully polarized orbital ferromagnet. To reconstruct the static out-of-plane magnetic field, Bz, we integrate BT F along ˆa from the lower and left boundaries of the image . We infer the total magnetization density m from the Bz data using standard Fourier domain techniques as shown in Fig. 5.1G. The algorithm used for this process is covered in depth in the associatd publication. Figures 5.1, H and I, show a comparison of BT F and m plotted along the contours indicated in Figs. 5.1, E and G. The shaded regions in Fig. 5.1I denote confidence intervals, whose absolute error bounds form the dominant systematic uncertainty in |ˆa|. Our measurements are taken close to ν = 3, equivalent to a single hole per unit cell relative to the nonmagnetic state at ν = 4 that corresponds to full filling of the lowest energy bands. We find that the magnetization density is considerably larger than 1 µB per unit cell area A ≈ 130/nm2 , where we have taken g = 2 as appropriate for graphene, in which spin orbit coupling is negligible. Without any assumptions about the nature of the broken symmetries, this state has a maximum spin magnetization of 1 µB per moir´e unit cell. Our data reject this hypothesis, finding instead a maximum magnetization density of m in the range 2 − 4 µB per moir´e unit cell corresponding to an orbital magnetization of 1.8-3.6× 10−4µB/carbon atom. We conclude that the magnetic moment associated with the Chern magnet phase in tBLG is dominated by its orbital component.In an intrinsic orbital magnet in which all moments arise from conduction electrons, the magnetization depends strongly on the density. Additional density dependence arises from the fact that contributions to the orbital magnetization from both wave-packet angular momentum and Berry curvature need not be uniformly distributed within the Brillouin zone.

Transport observations of aquantum anomalous Hall effect measure only the total Berry curvature of a completely filled band. At partial band filling, however, extrinsic contributions from scattering complicate the relationship between transport and band properties. In contrast, measuring m provides direct information about the density-dependent occupation of the Bloch states in momentum space. 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, large plastic pots for plants 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, 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. 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. The closest imaginable analog of the tBLG/hBN Chern magnet in this system is one in which interactions favor the formation of a valley-polarized ferromagnet, at which point the finite Chern number of the valley subbands would produce a Chern magnet.