To start with, if we attempt to proceed as we normally would- by assigning atomicorbitals to all of the atoms in the unit cell, computing overlap integrals, and then diagonalizing the resulting matrix to extract the hybridized eigenstates of the system- we would immediately run into problems, because the unit cell has far too many atoms for this calculation to be feasible. Some moir´e superlattices that have been studied in experiment have thousands of atoms per unit cell. There exist clever approximations that allow us to sidestep this issue, and these have been developed into very powerful tools over the past few years, but they are mostly beyond the scope of this document. I’d like to instead focus on conclusions we can draw about these systems using much simpler arguments. The physical arguments justifying the existence of electronic bands apply wherever and whenever an electron is exposed to an electric potential that is periodic, and thus has a set of discrete translation symmetries. For this reason, even though the moir´e super lattice is not an atomic crystal, we can always expect it to support electronic band structure for the same reason that we can always expect atomic crystals to support band structure. Two crystals with identical crystal symmetries will always produce moir´e superlattices with the same crystal symmetry, hydroponic growing supplies so we don’t need to worry about putting two triangular lattices together and ending up with something else.
Another property we can immediately notice is that the electron density required to fill a moir´e superlattice band is not very large. This can be made clear by simply comparing the original atomic lattice to a moir´e superlattice in real space . Full depletion of a band in an atomic crystal requires removing an electron for every unit cell , and full filling of the band occurs when we have added an electron for every unit cell. We have already discussed how this is not possible for the vast majority of materials using only electrostatic gating, because the resulting charge densities are immense. Full depletion of the moir´e band, on the other hand, requires removing one electron per moir´e unit cell, and the moir´e unit cell contains many atoms . So the difference in charge density between full filling and full depletion of an electronic band in a moir´e superlattice is actually not so great , and indeed this is easily achievable with available technology. Before we go on, I want to make a few of the limitations of this argument clear. There are two things this argument does not necessarily imply: the moir´e bands we produce might not be near the Fermi level of the system at charge neutrality, and the bandwidth of the moir´e superlattice need not be small. In the first case, we won’t be apply to modify the electron density enough to reach the moir´e band, and in the latter, we won’t be able to fill the moir´e band’s highest energy levels using our electrostatic gate. We know of examples of real systems with moir´e superlattice bands that fail each of those criteria.
But if these moir´e superlattice bands are near charge neutrality, and if their bandwidths are small, then we should be able to easily fill and deplete them with an electrostic gate.A variety of scanning probe microscopy techniques have been developed for examining condensed matter systems. It’s easy to justify why magnetic imaging might be interesting in gate-tuned two dimensional crystals, but magnetic properties of materials form only a small subset of the properties in which we are interested. Scanning tunneling microscopy is capable of probing the atomic-scale topography of a crystal as well as its local density of states, and a variety of scanning probe electrometry techniques exist as well, mostly based on single electron transistors. It’s worth pointing out that if you’re interested specifically in performing a scanning probe microscopy experiment on a dual-gated device, then these techniques both struggle, because the top gate both blocks tunnel current and screens out the electric fields to which a single electron transistor would be sensitive. Magnetic fields have an important advantage over electric fields: most materials have very low magnetic susceptibility, and thus magnetic fields pass unmodified through the vast majority of materials . This means that magnetic imaging is more than just one of many interesting things one can do with a dual-gated device; in these systems, magnetic imaging is a member of a very short list of usable scanning probe microscopy techniques. The simplest way in which we can use our nanoSQUID magnetometry microscope is as a DC magnetometer, probing the static magnetic field at a particular position in space .
There are situations in which this is a valuable tool, and we will look at some DC magnetometry data shortly, but in practice our nanoSQUID sensors often suffer from 1/f noise, spoiling our sensitivity for signals at low or zero frequency. One of the primary advantages of the technique is its sensitivity, and to make the best of the sensor’s sensitivity we must measure magnetic fields at finite frequencies. We have already discussed how we can use electrostatic gates to change the electron density and band structure of two dimensional crystals. We will discuss shortly a variety of gate-tunable phenomena with magnetic signatures that appear in these systems. It follows, of course, that we can modulate the magnetic fields emitted by these electronic phases and phenomena by modulating the voltages applied to the electrostatic gates we use to stabilize these phases. This is illustrated in Fig. 1.15C: an AC voltage is applied to the bottom gate relative to the two dimensional crystal, and the local magnetic field is sampled at the same frequency by the SQUID. Electrons carry a degree of freedom that we have not yet extensively discussed: spin. Electron spin is a fundamentally quantum mechanical property; it can be more or less understood using analogies to classical physics, but it also has some properties that don’t have simple classical analogues. Spin can be understood as a quantized unit of angular momentum that an electron can never be rid of. Although an electron is, as far as we know, a point particle, this unit of angular momentum couples to charge and produces a quantized electron magnetic moment, which we call the Bohr magneton, µB. Electron spins both couple to and emit local magnetic fields, and they are orthogonal to the electronic wave function- changing an electron’s wave function will not under normal circumstances influence its spin, and vice versa. Electrons are fermions; they obey the Pauli exclusion principle, which states that no two electrons can be placed into the same quantum state. The simplest consequence the existence of electron spin has is the fact that electronic wave functions can fit two electrons instead of one, because an electron can have either an ‘up’ spin or a ‘down’ spin. We say that electron spin produces an energetic degeneracy, because each electronic wave function can thus support two electrons. Electron spin is not the only degree of freedom that can produce energetic degeneracies; we will discuss a different one later. All of the above arguments apply for electron spin in condensed matter systems as well, and we can expect every electronic band to support both spin ‘up’ and spin ‘down’ electrons. These arguments say nothing about interactions between electrons, and all of the physical laws we normally expect to encounter still apply. In particular, nft hydroponic electrons of opposite spin can occupy the same wave function, but a pair of electrons have like charges, so they repel each other. There is thus an energetic cost to putting two electrons with opposite spin into the same wave function, and this cost can be quite large. This consideration is outside the realm of the physical models we have so far discussed, because electronic bands in the simplest possible picture are independent of the extent to which they are filled.
We are introducing an effect that will violate this assumption; the energies of electronic bands may now change in response to the extent to which they are filled. In particular, when an electronic wave function is completely filled with one spin species , it will remain possible to add additional electrons with opposite spins, but there will be an additional energetic cost to doing so. It is important to be precise about the fact that the displacement of the unfavorable spin species upward in energy occurs after the wave function is filled with its first spin. As a result, which spin species gets displaced upward in energy is arbitrary, and is determined by the spin polarization of the first electron we loaded into our wave function. This is an example of a ‘spontaneously broken symmetry,’ because before the addition of that first electron, the two spin species were energetically degenerate, and after the band is completely filled with both electron species, they will again be energetically degenerate.All of the above arguments apply to localized electronic wave functions and do not say anything specific about condensed matter systems, which involve many separate atoms that each support their own wave functions. A similar but somewhat subtler argument applies to electronic wave functions on adjacent atoms in condensed matter systems. When electronic wave functions on two adjacent atoms overlap, the structure of the delocalized electronic band that will emerge from them when they hybridize depends strongly on their relative spin polarization. When electrons on adjacent atoms have the same spin, the Pauli exclusion principle will prevent them from overlapping, thus minimizing their Coulomb interaction energy. When electrons on adjacent atoms have opposite spins, the Pauli exclusion principle doesn’t apply, because the two electrons are already in different quantum states, and they can overlap. This produces a larger interaction energy for arrangements wherein electrons on adjacent atoms have antialigned spins . Like all qualitative rules there are exceptions wherein other energetic contributions are more important, but this argument applies to a wide variety of condensed matter systems. These systems are known as ‘ferromagnets.’ They have interaction-driven displacements of minority spin bands, are at least partially spin polarized, and have electron spins that are largely aligned with each other. Both of these energy scales, the ‘same-site interaction’ and the ‘exchange interaction’ respectively, can be quite large in real condensed matter systems. The displacement of a spin subband upward in energy can produce partially spin-polarized metals , fully spin-polarized metals which we call ‘half-metals’ , and spin-polarized insulators which we call ‘magnetic insulators’ . Examples of each of these kinds of systems are known in nature, and all of these phenomena represent manifestations of magnetism.Most electrons in condensed matter systems are not moving at relativistic velocities. However, in the outermost valence shells of very large atoms , electrons can end up in such high angular momentum states that their velocities become relativistic.We can thus expect electrons in bands formed from orbitals supported by heavy atoms to respond to local electric potential variations as if they provide a local magnetic field. This phenomenon is known as spin-orbit coupling, and it provides a mechanism through which the energy of an electron spin can couple to the electrostatic environment inside of an atomic lattice. Predicting the global minima in energy as a function of spin orientation is very challenging, but it is often true that a discrete set of minima exist, and of course they must obey the symmetries of the atomic lattice. For this reason in many magnetic materials there is a discrete set of magnetic ground states defined by axes along which the electron spin can point. It is very often the case that there exist two global minima in energy that are antiparrallel along an axis of high symmetry; when this is the case, we say that the system is an Ising ferromagnet. The axis along which the ground state spin orientation points is called the ‘easy axis.’This is the origin of magnetic hysteresis in ferromagnets. According to the model of ferromagnetism we have so far developed, all of the spins in a ferromagnetic crystal are always aligned. When we apply a small magnetic field antialigned with the magnetization of our ferromagnet, nothing will occur at first.