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, drainage for plants in 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, growing raspberries in pots 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, 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. The fact that spins in ferromagnetic condensed matter systems are also aligned with each other does not affect this argument, and indeed there exist systems in both theory and experiment wherein electron spins align both with each other and with an applied magnetic field, smoothly and collectively following the direction of an applied magnetic field even as it varies. This is of course incompatible with ferromagnetic hysteresis, so we will need to mix in additional physics to explain this phenomenon. We have already discussed the fact the electron spins are orthogonal degrees of freedom from electronic wave functions, and do not couple to electric fields. This was something of an oversimplification. It is true in electrostatics problems, but in the relativistic limit- when electrons are moving at a non-negligible fraction of the speed of light- in their rest frames they experience static electric fields as large magnetic fields, as illustrated in Fig. 2.4.
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. When the magnitude of the magnetic field is increased past BC , all of the spins will suddenly rotate into alignment with the applied magnetic field. The simplest way is through polycrystallinity; in magnets composed of many microscopic domains, the magnetocrystalline anisotropy axes vary locally within the crystal, producing local variations in BC . In large,highly magnetized magnets, magnetic fields generated by the crystal itself can couple to its own magnetic domains . The resulting dispersion in individual domains’ coercive fields makes magnetization hysteresis loops of macroscopic samples rather smooth, instead of instantaneous at a well defined coercive field BC . However, the qualitative properties of the model apply rather well to individual domains, which do in many systems flip all together and rather suddenly at a well-defined, albeit local, BC . For this reason careful study of the detailed structure of magnetization curves of macroscopic samples often reveals a multitude of sharp steps in magnetization, corresponding to instantaneous repolarization of tiny, monocrystalline domains. This phenomenon is known as Barkhausen noise. In summary, the model we have built is very simple, and it requires both very clean samples and a lot of information about microscopic crystalline properties to provide insights into the behaviors of real spin ferromagnets. That said, there will be many situations in which it will have some utility in understanding the phenomena we encounter. We are now ready to discuss a real magnetic system. Chromium iodide is a two dimensional magnetic insulator.