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. We can use this technique to extract δV δB at an array of positions above the two dimensional crystal. This technique is very simple and powerful, but it has a few important drawbacks. It can only produce a quantitative measurement of B if the same scan is performed for a large set of gate voltages, so that δV δB can be integrated. It is also not very useful for probing metastable states. Many ferromagnets, for example, can be locked into quantum states that aren’t their ground states using a ferromagnetic hysteresis loop, and rapidly tuning the electron density tends to relax these phases to their ground states. So whenever we are interested in probing metastable magnetic states, we need to be careful about using this measurement method. Of course, 25 liter pot plastic we can also modulate the magnetic field through the nanoSQUID by modulating the position of the nanoSQUID. Since the magnetic field varies rapidly in space, we can often expect to get strong signals when we probe δB δx this way .
The position of the nanoSQUID is rapidly modulated using a piezoelectric tuning fork pressed against the side of the nanoSQUID sensor; the details of the tuning fork hardware and measurement are discussed further in the appendix. This measurement method allows us to use the nanoSQUID to probe metastable or even non-gate-tunable magnetic phenomena at finite frequency. It has a few drawbacks of its own, though. The nanoSQUID sensors have parasitic sensitivities to local temperature and electric potential , and if these vary in space the resulting signals will contaminate our magnetic field data. As a result, whenever we use this contrast mechanism we must try to extract differences between two different magnetic states if we want quantitatively precise information about the magnetic field. We can also apply an AC current in the plane of the two dimensional crystal. Large currents will emit detectable magnetic fields through the Biot-Savart law, and under those conditions we can use this contrast mechanism to reconstruct the current density through our two dimensional crystal. 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.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, 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, 25 litre plant pot 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.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 ‘flatband 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.Hysteresis in response to variations in applied magnetic field is perhaps the most well-known property of ferromagnetic systems, and it is not yet clear from the explanation so far provided why it occurs. Electron spins couple to magnetic fields through their magnetic moments, with energy E = µB · B,so spins will minimize their energies by aligning themselves with an applied magnetic field. 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. 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.