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. When the Fermi level is in the gap of a Chern magnet, there exists a number of quantum states at the Fermi level exactly equal to the Chern number. The conductance through a small number of delocalized quantum states is quite generally equal to e 2 h per quantum channel, and so the conductance between two electronic reservoirs in contact with the same edge of a Chern magnet is equal to C e 2 h . These facts together make the conductance and not just the conductivity an intrinsic property of a Chern magnet. This is a remarkable fact; indeed, as long as a researcher has access to a tool for measuring electrical resistance and a Chern magnet, they can directly measure a combination of the fundamental constants e and h through a simple electronic transport experiment- they don’t need to know anything about the geometry or band structure of the Chern magnet. This phenomenon is known as the quantized anomalous Hall effect. Chern magnets often support magnetic hysteresis, just like trivial magnetic insulators. Because the chirality of the edge state is determined by the sign of the Chern number, greenhouse vertical farming and the sign of the Chern number is determined by the bulk magnetization, quantized anomalous Hall effects usually exhibit magnetic hysteresis.
The quantized anomalous Hall effect is so unique to Chern mangets that that it is sometimes used as to define the entire class of systems; i.e., researchers historically have described these magnets as ‘QAH materials,’ or ‘QAH magnets.’ 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 functionsthis 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 1 C 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. 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.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, 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, 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. This is an interesting situation for a variety of reasons we have already covered. 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, nft vertical farming 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 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. An optical microscope image of the tBLG device discussed here is shown in Fig. 4.4A. The device is made using the “tear-and-stack” technique, in which one half of a graphene monolayer is torn off, rotated by a precise relative alignment angle , and then placed on top of the other half of the monolayer. The tBLG layer is sandwiched between two hBN flakes with thickness 40 and 70 nm, as shown in Fig. 4.4B. A few-layer-thick graphite flake is used as the bottom gate of the device, which has been shown to produce devices with low charge disorder . The stack rests on a Si/SiO2 wafer, which is also used to gate the contact regions of the device. The stack was assembled at 60◦ C using a dry-transfer technique with a poly film on top of a polydimethylsiloxane stamp. 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.