For this reason, orbital magnetism does not need spin-orbit coupling to support hysteresis, and it can couple to a much wider variety of physical phenomena than spin magnetism can- indeed, anything that affects the electronic band structure or real space wave function is fair game. For this reason we can expect to encounter many of the phenomena we normally associate with spin-orbit coupling in orbital magnets that do not possess it. I would also like to talk briefly about magnetic moments. It has already been said that magnetic moments in orbital magnets come from center-of-mass angular momentum of electrons, which makes them in some ways simpler and less mysterious than magnetic moments derived from electron spin. However, I didn’t tell you how to compute the angular momentum of an electronic band, only that it can be done. It is a somewhat more involved process to do at any level of generality than I’m willing to attempt here- it is described briefly in a later chapter- but suffice to say that it depends on details of band structure and interaction effects, which themselves depend on electron density and, drainage gutter in two dimensional materials, ambient conditions like displacement field. For this reason we can expect the magnitude of the magnetic moment of the valley degree of freedom to be much more sensitive to variables we can control than the magnetic moment of the electron spin, which is almost always close to 1 µB.
In particular, the magnetization of an orbital magnet can be vanishingly small, or it can increase far above the maximum possible magnetization of a spin ferromagnet of 1 µB per electron. Under a very limited and specific set of conditions we can precisely calculate the contribution of the orbital magnetic moment to the magnetization, and that will be discussed in detail later as well. Finally, I want to talk briefly about coercive fields. The more perceptive readers may have already noticed that we have broken the argument we used to understand magnetic inversion in spin magnets. The valley degree of freedom is a pair of electronic bands, and is thus bound to the two dimensional crystalline lattice- there is no sense in which we can continuously cant it into the plane while performing magnetic inversion. But of course, we have to expect that it is possible to apply a large magnetic field, couple to the magnetic moment of the valley µ, and eventually reach an energy µ · BC = EI at which magnetic inversion occurs. But what can we use for the Ising anisotropy energy EI ? It turns out that this model survives in the sense that we can make up a constant for EI and use it to understand some basic features of the coercive fields of orbital magnets, but where EI comes from in these systems remains somewhat mysterious. It is likely that it represents the difference in energy between the valley polarized ground state and some minimal-energy path through the spin and valley degenerate subspace, involving hybridized or intervalley coherent states in the intermediate regime. But we don’t need to understand this aspect of the model to draw some useful insights from it, as we will see later.
Real magnets are composed of constituent magnetic moments that can be modelled as infinitesimal circulating currents, or charges with finite angular momentum. It can be shown that the magnetic fields generated by the sum total of a uniform two dimensional distribution of these circulating currents- i.e., by a region of uniform magnetization- is precisely equivalent to the magnetic field generated by the current travelling around the edge of that two dimensional uniformly magnetized region through the Biot-Savart law. It turns out that this analogy is complete; it is also the case that a two dimensional region of uniform magnetization also experiences the same forces and torques in a magnetic field as an equivalent circulating current. The converse is also true- circulating currents can be modelled as two dimensional regions of uniform magnetization. The two pictures in fact are precisely equivalent. This is illustrated in Fig. 2.9. It is possible to prove this rigorously, but I will not do so here. One can say that in general, every phenomenon that produces a chiral current can be equivalently understood as a magnetization. All of the physical phenomena are preserved, although they need to be relabeled: Chiral edge currents are uniform magnetizations, and bulk gradients in magnetization are variations in bulk current current density. The details of this situation aren’t important; the lesson that is important is quite simple. In classical physics, we know how charged particles respond to local magnetic and electric fields. These rules are enough to completely explain the phenomena. This is apparently not the case in quantum mechanical systems. We can certainly attempt to describe systems this way, but in a wide variety of situations our models would be wrong, as in this one.
There is no point in this experiment at which an electron interacted with a magnetic field through the Lorentz force, and yet it turns out to be true that the magnetic field impacts the kinematics of electrons participating in the experiment. In a landmark result published in 1984, Michael Berry showed that our understanding of a variety of systems- including crystalline systems in condensed matter theory- suffered from a close analog of this misunderstanding. Researchers have since gone back to fix this oversight, plastic gutter and this led to the introduction of Berry curvature in condensed matter systems. Every crystal is defined by a periodic electric potential profile. In two dimensional crystals this is a scalar function of two dimensions over the lattice in real space. Let us switch our focus to momentum space. The periodic electric potential in real space produces a set of functions over momentum space E that define quantum states that electrons within the crystal can occupy. A correctly executed attempt to account for the effects of the Berry phase in crystalline systems produces a new vector-valued function over momentum space Ω that affects the kinematics of electrons in electronic bands. In two dimensional systems Ω is always oriented out-of-plane, but it can be positively or negatively oriented. We call this function the Berry curvature, and it must be accounted for to correctly explain a vast array of electronic phenomena, including electronic transport in metals, electronic transport in insulators, and angular momentum and magnetization in magnetic systems. In the same way that the Berry phase impacts the kinematics of free electrons moving through a two slit interferometer, Berry curvature impacts the kinematics of electrons moving through a crystal. You’ll often hear people describe Berry curvature as a ‘magnetic field in momentum space.’ You already know how electrons with finite velocity in an ambient magnetic field acquire momentum transverse to their current momentum vector. We call this the Lorentz force. Well, electronswith finite momentum in ‘ambient Berry curvature’ acquire momentum transverse to their current momentum vector. The difference is that magnetic fields vary in real space, and we like to look at maps of their real space distribution. Magnetic fields do not ‘vary in momentum space,’ at non-relativistic velocities they are strictly functions of position, not of momentum. Berry curvature does not vary in real space within a crystal. It does, however, vary in momentum space; it is strictly a function of momentum within a band. And of course Berry curvature impacts the kinematics of electrons in crystals. Condensed matter physicists love to say that particular phenomena are ‘quantum mechanical’ in nature. Of course this is a rather poorly-defined description of a phenomenon; all phenomena in condensed matter depend on quantum mechanics at some level. Sometimes this means that a phenomenon relies on the existence of a discrete spectrum of energy eigenstates. At other times it means that the phenomenon relies on the existence of the mysterious internal degree of freedom wave functions are known to have: the quantum phase. I hope it is clear that Berry curvature and all its associated phenomena are the latter kind of quantum mechanical effect. Berry curvature comes from the evolution of an electron’s quantum phase through the Brillouin zone of a crystal in momentum space. It impacts the kinematics of electrons for the same reason it impacts interferometry experiments on free electrons; the quantum phase has gauge freedom and is thus usually safely neglected, but relative quantum phase does not, so whenever coherent wave functions are being interfered with each other, scattered off each other, or made to match boundary conditions in a ‘standing wave,’ as in a crystal, we can expect the kinematics of electrons to be affected.
We will shortly encounter a variety of surprising and fascinating consequences of the presence of this new property of a crystal. Berry curvature is not present in every crystal- in some crystals there exist symmetries that prevent it from arising- but it is very common, and many materials with which the reader is likely familiar have substantial Berry curvature, including transition metal magnets, many III-V semiconductors, and many elemental heavy metals. It is a property of bands in every number of dimensions, although the consequences of finite Berry curvature vary dramatically for systems with different numbers of dimensions. A plot of the Berry curvature in face-centered cubic iron is presented in the following reference: We will not be discussing this material in any amount of detail, the only point I’d like you to take away from it is that Berry curvature is really quite common. For reasons that have already been extensively discussed, we will focus on Berry curvature in two dimensional systems. This equation is telling us that in systems with significant Berry curvature, applying an electric field will produce current density transverse to that electric field. It is illustrated in Fig. 3.1 for an isolated electron in a specific Bloch state. Berry curvature has a few general properties that are worth knowing. Kramers’ pairs- i.e., pairs of spin subbands related by time reversal symmetry- must have opposite Berry curvature. As a result, systems that don’t break time reversal symmetry cannot have any net current flow as a result of Eq. 3.3. The equation still applies, but each spin-polarized current density is precisely balanced by the other spin polarization’s contribution. This does not mean that Berry curvature has no consequences in such systems; in these systems, spin concentrates on opposite sides of the system, transverse to the applied electric field. This state of affairs is known as the spin Hall effect, and it is illustrated in Fig. 3.2A,B. In the presence of magnetism, electrons can occupy states with unbalanced Berry curvature, and as a result Eq. 3.3 produces a net current density. The resulting electron accumulation transverse to the applied electric field produces a transverse voltage called the Hall voltage. It is often useful to put contacts on the edges of devices in order to probe this voltage, as illustrated in Fig. 3.1. Of course, the sign of this voltage is a spontaneously broken symmetry, and it follows the magnetization of the magnetic order, as illustrated in Fig. 3.2C-F. It is possible for magnetic insulators to form in systems with bands that have finite net Berry curvature . This produces an extremely special situation, and the bulk of this thesis will be devoted to probing and understanding the properties of these kinds of systems. They are called Chern magnets.Several chapters of this thesis focus on the properties of a particular class of magnetic insulator that can exist in two dimensional crystals. These materials share many of the same properties with the magnetic insulators described in Chapter 2. They can have finite magnetization at zero field, and this property is often accompanied by magnetic hysteresis. The spectrum of quantum states available in the bulk of the crystal is gapped, and as a result they are bulk electrical and thermal insulators. They have magnetic domain walls that can move around in response to the application of an external magnetic field, or alternatively be pinned to structural disorder. And of course they emit magnetic fields which can be detected by magnetometers.