This procedure will also produce strong magnetic signals if magnetism couples strongly to current in the two dimensional crystal we’re studying, and this will be the case in several of the systems we will discuss later. As I mentioned either, the nanoSQUID is well-thermalized to its environment, and its properties are quite sensitive to temperature, so we can use the nanoSQUID microscope as a sensitive nanoscale thermometer. I will not present any scientific conclusions based on scans performed using this technique, but it is a powerful capability and it is also useful more practically for nanoSQUID navigation, which is discuseed further in the appendix. 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, stackable flower pots 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, 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 presence of these effects can produce a variety of phenomena. 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. In principle one must perform calculations to determine whether magnetism will occur in any specific system. In practice there exist good rules of thumb for making qualitative predictions. Same-site interactions and exchange interactions minimize energy by minimizing the number of minority spin species present in a crystal, and putting the electrons that would otherwise have occupied minority spin states into majority spin states. Of course, this process always requires that the system pay an additional energetic cost in kinetic energy, because those previously unoccupied majority spin states started out above the Fermi level. The competition between these energy scales determines whether magnetism will occur in any particular material. It follows that systems with a multitude of quantum states with very similar energies in their band structure will be more likely to form magnets; to put it more precisely, we are looking for situations in which, near the Ferm level at least, E = C, where C is some constant. We can say that under these circumstances, the energies of electrons in the crystal are independent of their momenta. We can also say that we have encountered a large local maximum or even a singularity in the density of states. We sometimes call this the ‘flat-bottomed band condition,’ or just the ‘flat band condition’ , and it can be made quantitative in the form of the Stoner criterion. Magnetism is perhaps the simplest phenomenon that can be understood in this context, but it turns out that this argument applies very generally, tower garden and physicists expect to find a variety of interesting phenomena dependent on electron interactions whenever we encounter these situations. It is important to be specific about what we mean by a flat band here: we expect to encounter magnetism whenever an electronic band is locally flat- it is fine for the band to have very high bandwidth as long as it has a region with E ≈ C. These systems will tend to produce magnetic metals. When we encounter bands that are truly flat- i.e., they have both weak dispersion and small bandwidths- we are more likely to encounter magnetic insulators, as illustrated in Fig. 2.3. 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 anti-parrallel 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.’We are now ready to discuss a real magnetic system. Chromium iodide is a two dimensional magnetic insulator. Systems like chromium iodide have properties that are easy to understand in the context of the models we have so far discussed: strong on site interactions and exchange interactions produce full spin polarization, an interaction-driven band gap, and aligned magnetic moments within a single layer. As a result, these systems are electrical insulators. They support magnetic domain dynamics, and there is a temperature TC above which they cease to be magnetized , although they remain insulators far above that temperature. Extremely weak out-of-plane bonds produce highly anisotropic cleavage planes and make it relatively easy to prepare atomically thin crystals mechanically. As in other systems, this does not mean we will always be studying monolayers of the material. Bilayers, trilayers, four-layer crystals, and even thicker flakes can all have properties that differ significantly from those of a monolayer, often for reasons that we can understand, and CrI3 is no exception. Although it isn’t particularly relevant to the physics of magnetism, it’s worth mentioning that all of the chromium halides are highly unstable compounds, and decompose in a matter of seconds when exposed to air or moisture. These materials are difficult to study under normal circumstances, but two dimensional crystalline samples can be prepared inside of an inert-atmosphere glovebox. They can also be sandwiched, or ‘encapsulated,’ between other two dimensional crystals. Two dimensional crystals are so flat that this process produces an air- and water-proof barrier and protects the encapsulated crystal from degradation in atmosphere, facilitating easy measurements with tools like the nanoSQUID. The crystalline structure of CrI3, projected onto a two-dimensional crystal, is visible in Fig. 2.6A. Unlike graphene, CrI3 has two different kinds of atoms in its unit cell; the chromium atoms are responsible for the magnetic moments producing magnetism. CrI3 has fairly strong spin-orbitcoupling, and thus strong Ising anisotropy, with magnetic moments pointing out-of-plane . Most of the other chromium halides also support magnetic order, although the precise nature of each of their ground states differs somewhat. Both CrI3 and CrBr3 have ferromagnetic in-plane interactions and strong Ising anisotropy, but CrI3 has antiferromagnetic out-of-plane interactions, meaning that in the magnetic ground state of the crystal adjacent layers have their spins antialigned . Interestingly, CrCl3 also seems to have ferromagnetic in-plane interactions, but it is likely that it is not an Ising or easy-axis magnet, and instead has its spins pointed in the in-plane direction and thus free to rotate. It is evidently the case that although these systems are structurally very similar and all have strong spin-orbit coupling, their magnetic interactions and magnetocrystalline anisotropies vary wildly in response to modest differences in their electronic structure. As a result of all of the arguments discussed previously in this chapter, a CrI3 monolayer has finite magnetization even in the absence of an applied magnetic field, and its magnetic order experiences hysteresis in response to variations in the applied magnetic field, as illustrated in Fig. 2.6D . Antiferromagnetic interactions between adjacent layers in CrI3 mix in an interesting factor that can be easily understood: flakes with an even number of layers have no net magnetization in the absence of an applied magnetic field, but develop finite magnetization at higher magnetic fields as the applied magnetic field overwhelms interlayer interactions and realigns each layer in turn with the ambient magnetic field .