Abstract In the early days of solid state theory, electrons were considered to be classical particles satisfying Newtonian mechanics. With the gradual establishment of quantum mechanics in the first half of the 20th century, the wave nature of microscopic particles became the key to an accurate understanding of the microscopic world. This has also profoundly affected people's understanding of the electrons in crystals: electrons exist in crystals in the form of Bloch waves, while their particle properties exist in the form of wave packets on spatial scales larger than the lattice spacing. Since 1980, the geometric phase of Bloch waves has been found to be indispensable in solid theory, which further refines the particle view of electrons in crystals. The purpose of this paper is to follow the progress of the whole solid-state physics research framework to outline the general outline of the evolution of the electron particle view, and to explain the connotation of the electron particle view with some key physical problems as examples, and to show its value in the study of solid-state physics.
Key words: crystal, electron, wave, particle view, classical dynamics, Bloch wave, wave packet, geometric phase, semi-classical dynamics
Particles and waves are the two basic forms that the material world is known to us. The two are two sides of the same coin, both opposing and harmonious. Historically, with the maturity of electrodynamics and the rise of quantum mechanics, our understanding of light has evolved from particles to waves and then to particles. For electrons, our knowledge began with cathode ray particles, but the understanding of the structure of the atomic shell forced people to accept its wave properties, allowing the physics world to experience another quantum revolution that touched the soul. Compared with the tiny size of atoms, electrons have a wider space to move in solids, allowing them to appear in the form of particles, throughout our thinking and discourse.
The narrative of solid state physics has three main sections [1]: structure, particles, and responses. "Structure" includes the order of atoms and certain physical quantities (e.g., spins) in space, and "particles" includes electrons and other meta-excitations (e.g., phonons); "Response" refers to the response of a solid to an external excitation in terms of force, heat, sound, light, electricity, magnetism, etc. As elementary particles, electrons interact with the nucleus and other electrons in quantum many, which not only supports the shell structure of the atom, but also provides a bond for the solid structure, thus becoming the key to connecting the microscopic unit and the macroscopic order. Electrons also often play a major role in the response of solids, and the most active part of the reaction often takes the form of a new particle.
The concept of particles in classical mechanics stems from an abstraction of a macroscopic object: emphasizing the position and momentum of its center of mass while ignoring its internal structure. It is also true that the electrons in the cathode rays behave like a charged classical particle under the electromagnetic field, but the electron state at the atomic scale needs to be described by quantum wave functions. This paper focuses on electrons in materials with periodic structures, reviewing how it underwent quantum mechanical transformations to emerge as a new type of particle at a larger than atomic scale. The new electrons follow a new set of equations of motion of particles, which are independent of each other but satisfy the Fermi-Dirac quantum statistics, and bridge the microstructure and macroscopic properties of materials concisely and effectively.
The understanding of electrons in solid-state physics has been going on for a hundred years [2—4]. In the early days, Arnold Sommerfeld, the pioneer of the old quantum theory, introduced quantum statistics into the theory of free electron gas, which solved the difficulties in the specific heat and heat transport of electrons. Then, Felix Bloch combined the quantum wave equation with the periodic potential field in solids to propose the concept of the Bloch particle. Based on this, people have sketched basic images of metals, semiconductors, and insulators. After World War II, after decades of efforts, people meticulously sorted out the many-body interaction of electrons, and carried out first-principles calculations of the Bloch state. At the same time, the quantum response of the Bloch band in the external field has been systematically studied, and the wave interpretation of the Bloch particle has been rediscovered. In the 80s of the last century, the study of the quantum Hall effect by David Thouless and others opened up the attention to the topological properties of Bloch electrons. Closely related to this, the influence of Berry phase and Berry curvature on Bloch electrons has gradually become clear, reshaping people's understanding of electrons in solids.
01 Classic electronic image
In 1897, Joseph Thomson discovered that the rays emitted from the cathode of a low-pressure gas discharge tube were charged particles, and the charge-to-mass ratio was determined by its acceleration and deflection in the electromagnetic field [5,6]. People later referred to it as electrons. Its discovery opened a chapter in the rigorous study of atoms, molecules, and solid-state physics at the theoretical level. Just three years later, Paul Drude established a microscopic image of a metal conducting electricity [7]: a group of free electrons is accelerated by an electromagnetic field, and the friction caused by collisional scattering stabilizes the motion to form an electric current proportional to the electric field (see Box 1). This image visually relates conductivity to charge density and the distance between collisions (mean free path) or time (relaxation time). His prediction of Hall resistivity is peculiar: the ratio of the transverse electric field to the current and magnetic fields depends only on the charge density and not on the relaxation time. Based on this, a standard method for measuring the current-carrying charge density in materials has been developed.
BOX 1
Drood free electron model
In 1900, Drud developed a theoretical model of conductor conduction, which is known as Drudd's free electron model. Its interpretation of electrical and thermal conductivity shows that even a naïve image of a particle can be a puzzle piece for understanding the macroscopic properties of solids. Druder believed that there are equal amounts of positive and negative particles in a conductor, so the whole is electrically neutral. However, positive and negative particles can move freely, so they can generate electric current. For the sake of simplification, we will only consider the negatively charged particles, i.e., electrons. The laws of motion follow the classical Newtonian equations:
where r, p, e, m, and τ are the position, momentum, charge, mass, and relaxation time of the electrons, respectively. Eq. (2) gives the force on the particle: the first term is the usual electric field force, which causes a uniform change in momentum, and the second term is the frictional force, which causes the particle's momentum to be zero. As a result of the confrontation, the particle cannot be accelerated by the electric field all the time, but will reach a stable value, i.e., pf=-eτE. According to Eq. (1), this momentum corresponds to a steady velocity, resulting in a net current:
where n is the electron density. The corresponding conductivity σ is σ=ne2τ/m. The reciprocal of conductivity is resistivity, which is the part of the resistance that we know as Ohm's law that is independent of the size of the material.
Thermal conductivity can also be obtained within the framework of a free electron model, but since the temperature gradient is a force with statistical properties, the heat flow driven by it cannot be simply given by the equations of motion. Here we briefly explain the idea of calculating thermal conductivity, and omit the specific process. According to the energy equalization theorem, the statistical mean of a particle's kinetic energy is proportional to its temperature, i.e
。 Electrons moving in opposite directions come from different places, and the difference in their position also brings about a difference in temperature, thus carrying heat fluxes in opposite directions and of different magnitudes. The specific calculations show that the net heat flux is proportional to the temperature gradient, i.e., Jq=-κ∇T, while the thermal conductivity is determined by
Given.
When we divide the thermal conductivity by the electrical conductivity, the relaxation information is precisely eliminated:
。 The conclusion that the ratio of thermal conductivity to electrical conductivity given by the Drud free electron model is independent of the relaxation process is in good agreement with the experiment. Due to the lack of rigor of the theory at that time, the numerical coefficients given by Druder were also very close to the predicted value of quantum theory (π2/3). This was once considered an example of the success of the free electron model.
Druder's microscopic images can also be used to discuss the problem of heat conduction. A primary consideration is that electrons that are flying freely somewhere have come from different collision points, and that the rate of electrons coming from high temperatures is relatively large, based on the energy equalization theorem of classical statistics. This means that electrons tend to leave the hot region and flow towards the low region, resulting in a heat flow (Fourier's law) and an electric current (Zealbeck's law) that are proportional to the temperature gradient. He further discovered that if the thermal conductivity was divided by the electrical conductivity, the relaxation time could be eliminated. This result is proportional to temperature, and the proportionality factor is a universal constant consisting of the electron charge and the Boltzmann constant. His calculations were in good agreement with the experimental results of the time (Wiedmann-Franz law) and were once attributed to the success of his theory. However, if we divide the thermal conductivity (Zebeck coefficient) by the conductivity, the relaxation time can be eliminated, but the result is two orders of magnitude larger than the experimental value.
Of course, conductivity, etc., itself is proportional to the relaxation time. How to explain its magnitude and temperature dependence is also nerve-wracking. Hendrik Lorentz has devoted great efforts to this problem, systematically studying it with advanced transport theories established since the 19th century [8]. Starting from the classical statistical distribution, he basically repeats Druder's results under the approximation of relaxation time. He also envisaged microscopic mechanisms to specify the relaxation time, such as the solid scattering of positive ions by periodically arranged electrons. But the results of his predictions are always very different from the experiments.
Despite all its flaws, Druder's simple and intuitive images of transport have survived to this day and are used to describe the results of various experiments. This is because after Lorentz, it has undergone several major transformations in quantum mechanics. The first of these modifications was done by Wolfgang Pauli and Sommerfeld and others, who blessed quantum statistics for free electron gases. Sommerfeld was involved in the construction of a Bohr quantum model of atoms and introduced two quantum numbers of orbital angular momentum [9]. His disciple Pauli introduced the principles of spin and incompatibility [10], which eventually led to the establishment of an electronic shell model that could explain the periods of the elements. The principle of incompatibility leads to the Fermi-Dirac statistical distribution (see Box 2). Pauli applied it to free electron gases to explain the phenomenon of weak paramagnetism based on electron spin [11]. Sommerfeld calculated the specific heat of the electron and found that it was much smaller than predicted by the law of equal division of energy in classical statistics.
BOX 2
Maxwell-Boltzmann distribution and Fermi-Dirac distribution
The sign of the transition of conductor theory from classical to semiclassical is the substitution of the Fermi-Dirac distribution for the classical Maxwell-Boltzmann distribution. In the classical mechanics framework, when we consider that a large number of particles interact to reach an equilibrium state, the proportion of particles with a specific energy is given by the following Maxwell-Boltzmann distribution:
where E is the particle energy, kBT is the Boltzmann constant, T is the system temperature, and λ0 is the coefficient. In the distribution function originally obtained by Maxwell, E contains only kinetic energy, while Boltzmann generalized it to more general energy. With energy as the horizontal axis and occupancy as the vertical axis, we can plot the corresponding probability distribution (Fig. 1(a)). It can be seen that when the temperature is lower, the Maxwell-Boltzmann distribution has a narrow and sharp peak near the zero point. For a particular state, there can be more than one particle occupied.
In contrast, the Fermi-Dirac distribution is a type of quantum statistical distribution. It applies to particles with a semi-integer spin (e.g. an electron with a spin of 1/2). The expression is as follows:
It is used to describe the possession probability of a particular energy state with energy E. The Fermi-Dirac distribution (Fig. 1(b)) is significantly different from the Maxwell-Boltzmann distribution: the probability of possession of a particular quantum state is no more than 1, and when the temperature becomes zero (absolute zero), it has a share of 1 for states with a small energy ratio of μ and 0 for states with a large energy ratio of μ. Therefore, the energy μ represents the highest energy that a particle can fill at zero temperature, which is called Fermi energy. At finite temperatures, the energy state occupancy of very low or high energy is similar to that of the zero-temperature case, but in the energy interval of the size of kBT near the Fermi energy, the occupancy of the quantum state is significantly different from that of the zero-temperature case: there is a transition from the occupancy from 0 to 1.
We can visualize the difference between the two by Pauli's explanation of paramagnetic susceptibility. The magnetic susceptibility χFD obtained from the Fermi-Dirac distribution and the magnetic susceptibility χMB obtained from the Maxwell-Boltzmann distribution have the following ratio: χFD/χMB∝kBT/E0. It can be seen that at low temperatures, the values given by quantum statistics are greatly reduced compared with those of classical statistics. According to Fermi-Dirac statistics, at low temperatures, energy states with energies lower than Fermi energy are almost completely occupied. In a weak magnetic field, the distribution of most of the quantum states is almost constant—the only significant change is the apparently unoccupied region near the Fermi energy (Fig. 1(b)), i.e., the kBT-sized energy range. The proportion of this interval to the total energy is kBT/μ. This gives a rough indication of the reduced susceptibility of quantum statistics relative to classical statistics.
Fig.1 Maxwell-Boltzmann distribution (a) and Fermi-Dirac distribution (b) at different temperatures
We now know that the electron gas in a conductor is highly degenerate. At zero temperature, electrons are filled up from the lowest energy like water poured into a basin until somewhere called Fermi energy – the quantum energy level above it is not occupied. Depending on the density of electrons in the metal, this Fermi energy is two or three orders of magnitude higher than the thermal fluctuation energy corresponding to room temperature. As a result, the electrons in the metal are still very cold at room temperature: a large number of electrons buried deep under the Fermi energy are not affected, and only a few electrons in the immediate vicinity of the Fermi energy can be excited.
Quantum statistics requires the counting of quantum states, but the energy levels of electronic states in the macroscopic volume are very dense, so Pauli and Sommerfeld did continuous processing in the calculation process. According to quantum wave dynamics, free electrons can be represented by plane waves, and their energy and momentum are proportional to frequency and wave vector, respectively, and the scale coefficient is Planck's constant. Since the wave function is limited by boundary conditions, the wave vector can only take some discrete values, similar to standing waves on a string fixed at both ends. On average, the number of quantum states per unit space and momentum volume is inversely proportional to the cubic of Planck's constant. This corresponds to the number of microscopic states measured in classical statistical mechanics by phase space volume. Quantum wave dynamics has made the once obscure classical statistical measure clear.
In the case of transport, Sommerfeld et al. still used the classical particle image of electrons, and on the other hand, they used the relaxation time approximation to obtain a non-equilibrium distribution from the Fermi-Dirac distribution [12,13]. This free electron model, modified by quantum statistics, clarifies that the conduction electrons come from near the Fermi energy, which can better describe the thermal conduction and thermoelectric effects of metals. However, the particle image of the electrons still faces a great difficulty: the mean free path inferred from the resistance of the metal conductor can be much larger than the atomic spacing, as if the periodically arranged positive ion-pairing electrons no longer work after the electrons have been contributed. This is completely inconsistent with the origin of friction that was previously assumed.
02 Bloch electronic model
A new breakthrough awaited Sommerfeld's protégé, a young man from Werner Heisenberg, Bloch. In his doctoral dissertation, he applied the quantum wave equation he had just learned from Schrödinger to electrons in the periodic potential field of crystalline materials [14]. He was pleasantly surprised to find that the energy eigenwave function is still similar to a plane wave in the free state, except that the amplitude acquires a periodic modulation in space. This finding clears the way for solving the problem of excessive mean free path, because periodically arranged ions cannot block the movement of electrons with this wave function in the crystal. This quasi-plane wave came to be known as a Bloch wave. Its wave vector corresponds to a new type of momentum (for simplicity, it is still referred to as momentum) and is defined in a finite, unbounded, cyclic region (Brillouin zone). The spectrum is thus presented as bands in this region (see Box 3).
BOX 3
Bloch's Theory and Wave Packets
In 1928, Bloch published his famous article "Quantum Mechanics of Electrons in a Lattice", which pushed the study of solid state theory into the quantum age. In the one-dimensional case, he considers the electrons in the lattice to have the following Hamiltonian:
。 Unlike free electrons, the potential field U(x) perceived by electrons is the sum of the potential fields produced by the individual ions (Figure 2). Due to the periodic nature of the ion configuration, U(x) inherits this property: U(x+a)=U(x). where a is the interval between adjacent ions. Bloch found that in the periodic potential field, the electron eigenwave function is a modulated plane wave, i.e., the Bloch wave:
where the amplitude u(x) has the same periodicity as the potential field: uk(x+a)=uk(x). It is important to note here that, unlike the plane wave function of free electrons, the indicator k of the wave function here is not exactly the momentum of the electron, but a lattice momentum. Its value is no longer negative infinity to positive infinity, but is in a finite-sized Brillouin zone, as discovered by Brillouin. For a one-dimensional lattice, this Brillouin region is [-π/a,π/a), and its left and right endpoints are equivalent.
图2 单层二硫化钨(WS2)晶格与二维能带 (a)二硫化钨晶格的单元格构型(上)及俯视图(下)； (b)二维能带以及晶格动量所在的布里渊区(图(a)摘自Ye Y et al. Nature Photonics,2015,9:733;图(b)摘自Bussolotti F et al. Nano Futures,2018,2:032001)
The eigenvalues corresponding to each eigenstate can form the structure of the bands. The formation of energy bands is very similar to the energy levels in a molecule (Figure 3). When adjacent ions in the lattice are very far apart, the lattice is actually a collection of atoms that do not interact with each other. At this time, the eigenvalue of the energy of the electron is the energy level of the electron in the atom. As the distance between adjacent ions gradually closes, the electron cloud overlap of the electron wave function in the adjacent atoms gradually becomes significant, resulting in the covalent coupling of the atomic energy levels, forming a bonded state (energy level decrease) and an antibonding state (energy level increase). In other words, in a crystal, atomic energy levels are diffused from a single value into an energy range, and the further arrangement of these energies in the momentum space forms bands (Fig. 2, Fig. 3).
Fig.3. Periodic potential field and energy bands perceived by electrons in solids. The yellow line shows the electron energy levels in each atom, the initial value of which is the same for atoms at different locations. The purple region is the energy band formed by the dispersion of atomic energy levels due to the overlapping electron clouds of different atoms
Bloch also considered the acceleration behavior of electrons under an electric field. In a general three-dimensional case, he constructs the wave packet in a linear combination of Bloch states:
The wavepacket form can be further constrained so that it is localized in momentum space near a central point kc. According to the uncertainty principle, the wave envelope state can be greatly broadened in real space (Figure 4): although the wave packet has the central point rc of the real space, its width can be much larger than the ionic spacing (this is the period felt by the Bloch states that make up the wave packet). When considering the responsive properties of electrons, we often need to add certain external forces, and at the same time require these forces to be small enough that they can detect rather than change the intrinsic properties of electrons. In this case, the external potential corresponding to the external force changes in space. In order to rationally use wave packets to describe the motion of electrons, we need to further require that the spatial spread is much smaller than the characteristic change scale of the external potential (Fig. 4). Under the action of the electric field E, Bloch discovered the following force equation under the framework of quantum mechanics:
The equation for the velocity of the electron becomes
Except that the momentum here is a finite lattice momentum, this set of dynamical equations is exactly the same as in the classical case (Equations (1) and (2)). Later, Jones and Zener found that the magnetic field could be added in a similar way, and the corresponding force equation remained consistent with the classical case:
Fig.4. Distribution of wave packets in momentum space (a) and real space (b) (b) The black sphere represents the atomic position in the figure)
What does the particle of a Bloch wave in a band look like? We can linearly superimpose the Bloch waves near a certain momentum to obtain a wave packet with a relatively definite momentum (relative to the size of the Brillouin zone) (see Box 3). And if we broaden the field of view to a scale much larger than the atomic interval, we will find that it is also relatively localized in space, and thus has a relatively definite position. This wave packet with a definite position and momentum exhibits the particle properties of Bloch waves. The group velocity is the velocity expectation of the Bloch state at the center of the momentum, which is equal to the partial derivative of its energy to the momentum. This velocity increases with momentum in the lower part of the band, but it reverses when it reaches the upper part, as if the mass changes from positive to negative.
Under the electromagnetic field of general intensity, the perturbation of the Schrödinger equation by the gauge potential field changes very slowly in space, and can change little in the spatial width of the Bloch wave packet. Using this condition, Bloch found that the momentum of the wave packet changes at a uniform velocity under the action of an electric field, like a free electron. Later, this result was popularized by Herbert Jones and Clarence Zener, adding the Lorentz force of the magnetic field to the electric field force [15]. These discoveries give a new lease of life to the image of particles with electrons in the bands: we simply replace the kinetic relationship of the free electrons with the band function on the Brillouin region, and then hide the existence of the lattice potential field. The resulting Bloch electrons follow the original dynamics of the electromagnetic field.
Based on the particle image and relaxation time approximation of Bloch electrons, the electron theory of Sommerfeld et al. can be applied to actual metal materials with a slight modification. Bloch's theory suggests that the cause of electron scattering can only be caused by periodic damage to crystals, such as impurity defects and atomic vibrations. In his doctoral dissertation, Bloch also specifically considered the latter's scattering of electrons, successfully explaining the tendency of resistivity to rise with temperature. In the high-temperature region, the lattice vibration satisfies the classical law of energy equalization, and the intensity increases linearly with temperature, and its scattering of electrons also increases, which ultimately leads to an increase in resistivity. In the cryogenic region, using the phonon quantization that Peter Debye had just obtained at the time, Bloch also predicted the resistance behavior of the famous temperature to the fifth power. Because the scattering of impurities leads to a finite electrical resistance at zero temperatures, his prediction was later confirmed in very pure crystals.
Without scattering, the constant growth of the electron's momentum under the electric field causes the Bloch electrons to move in a circular motion in a band, which is known as the Bloch oscillation. This ideal state of DC to AC is difficult to achieve in solid materials, as the relaxation time is often much smaller than the period of this oscillation. This phenomenon was observed for the first time when Leo Esaki invented a superlattice system that greatly reduced the Bloch oscillation period [16]. After many more years, the Bloch oscillation was finally perfectly realized in the cold atom system in the optical lattice, which became an accurate method for measuring gravitational acceleration [17].
Bloch's theory can also depict insulators and semiconductors. Alan Wilson noted that if a band is filled in its entirety by electrons, the total current is zero, and that the momentum circulates in the band under the action of an electric field, without changing the band filling, the total current remains zero [18,19]. In this way, if all the energy bands are either empty or completely filled, the material is an insulator. This is like the inertness of a shell filled with a quantum model of an atom. However, as we will see later, the above considerations ignore the topological geometry effects of Bloch's electrons, and that the cycle of momentum can actually lead to a quantized anomalous Hall current perpendicular to the direction of the electric field.
Pure and perfect semiconductors are also insulators at zero temperatures. However, the energy gap between the full band and the empty band is relatively small, and it is easy to make the quantum state at the bottom of the empty band occupy or the quantum state at the top of the full band vacant under the condition of room temperature and doping. The latter of these is called a hole, which is equivalent to the top of the full band being occupied by particles with a positive charge. These particles behave like free electrons, but their effective mass is determined by the curvature of the bands. Interestingly, since the holes are positively charged, the corresponding Hall effect has the opposite sign of the electron, which perfectly explains a major doubt left by Sommerfeld's theory.
03Quantum many-body action
The particle view of electrons also undergoes a tempering of quantum many-body action. To put it simply, electrons are subjected to Coulomb repulsion between electrons in addition to feeling the Coulomb gravitational force of the nucleus. The periodic structure of the lattice and the order of other degrees of freedom are the result of how these coulombs manifest themselves in quantum mechanics. Strictly speaking, the Bloch electron is only a presence in some kind of mean field sense.
In extreme cases, such as superconductors and Mott insulators, electrons take on a completely new look. In the former, the electrons pair into bosons due to some attraction and further condense to a state of superconductivity and complete diamagnetism. In the latter, due to relatively strong Coulomb repulsion, the charge of electrons is confined to individual cells, and the spins of adjacent cells are arranged in opposite directions, so that the system exhibits an insulating antiferromagnetic order. There is also a class of systems that exhibit topological order, including the recently observed anomalous fractional quantum Hall state, in which meta-excitations occur in the form of fractional charges and exhibit quantum statistics that differ from fermions and bosons.
This paper does not intend to discuss such strongly correlated systems, but rather focuses on the commonly encountered solid phases, where the many-body quantum ground state of electrons can be seen as bottom-up filling of the respective Bloch bands. A key figure in early research in many-body physics was John Slater, who traveled from the United States to Europe for postdoctoral research. He popularized Douglas Hartree's self-consistent mean-field method, which incorporated spin and orbital states into a many-body determinant wave function that he later named, quantitatively explaining the fine structure of atoms [20]. After meeting Heisenberg and Bloch, he set about comparing the many-body wave functions based on the bound and malleable states they used, and found that the latter was more applicable to metals in general [21].
Further research on the many-body wave function has led to the birth of first-principles calculations of modern solid-state electronic structures. Based on the image of the Bloch state, the Cohen-Shen-Lujiu equation in density functional theory also has the form of the Schrödinger equation, except that the potential field perceived by each electron is derived from a set of universal self-consistent equations. This calculation can accurately give the binding energy of solids, predict the microgeometry of crystals and the macroelasticity of materials, etc.
For metals, supported by Landau's Fermi liquid theory, the first-principles calculation of the Fermi surface structure can be rigorously tested by the de Haas-Van Alfing effect and applied to various linear response problems. For band insulators, first-principles calculations can also give good ground-state properties. However, when it comes to the description of the excited states of semiconductors, such as energy gap and exciton binding energy, the first-principles method still needs some many-body perturbation corrections.
Regarding ferromagnetism, there were two opposing views on ferromagnetism in the early days: Heisenberg's local magnetic moment and Bloch's cruising magnetic moment [22,23]. Modern spin density functional theories are based on the latter, which can accurately predict the ground-state magnetization of transition metals such as iron, cobalt, and nickel. Heisenberg's spin coupling model is still popular in textbooks and scientific research because it is more convenient to describe spin wave excitation and phase transitions, but the coupling coefficients are still quantified in combination with linear response theory and first-principles calculations.
Spin wave excitation is a typical boson that describes the deviation mode of a system from a fixed magnetic sequence, such as a ferromagnetic sequence. Their thermal excitation causes the long-range ferromagnetic order to decrease in the three-dimensional case to the third power of temperature and completely lose it in the low-dimensional case. Similarly, the thermal excitation of phonons (the bose quanta that vibrates in the lattice) can have a similar effect on the translationally symmetrical order of the lattice. On scales much larger than atoms, these emerging bosons tend to behave in classical particle form, and their response to external conditions can also be treated in the case of electrons.
04 Single-band quantum theory
The magnetic properties of materials have always been an important topic to promote the development of solid theory. When considering the concept of spin in quantum physics, Pauli has shown that free electron gases exhibit paramagnetism. Later, Lev Landau considered the effect of the magnetic field on the orbital wave function, solved a series of discrete energy levels (Landau levels), and discovered the diamagnetism of free electron gas at the weak field limit [24]. This came as a surprise, as around 1911 Niels Bohr and others had demonstrated that classical orbital motion did not give any magnetic properties in an equilibrium distribution [25]. Their argument is also true for the Fermi-Dirac statistics. Landau's work has aroused Rudolf Peierls' curiosity: how to recover the orbital quantum effect of a Bloch electron in an energy band?
Bloch solves his wave function with a tight-bound approximation: the wave functions on each atom are linearly superimposed, and the atomic energy levels are broadened into energy bands through quantum tunneling. Pyles notes that even if the weak magnetic field is ignored for the correction of the atomic wave function, the magnetic vector potential still assigns different phases to the wave function on each atom, so that the tunneling coefficient between the atoms has a corresponding phase difference [26]. This causes the momentum operator of the tightly bound Hamiltonian to produce a movement proportional to the magnetic vector potential. This is known as the Pyles substitution, and the momentum of the substitution is no longer commuted in both directions perpendicular to the magnetic field. Years later, Solis et al. used a Hamiltonian model to study the quantum Hall effect.
Through the perturbation calculation of the free energy, Pyles obtained the diamagnetic coefficient of the Bloch electron gas, and also discovered the oscillatory behavior of the magnetization in the low temperature region, which opened the explanation for the de Haas-Van Alven effect [27]. In the case of free electrons, this oscillation corresponds to Landau levels crossing the Fermi surface. Many years later, Lars Onsager used the Bohr-Sommerfeld quantization method to reduce this oscillation phenomenon to the quantization of the area enclosed by the trajectory mapped by the Bloch electron cyclotron motion in momentum space, which became a guideline for studying the geometry of the Fermi surface [28].
In 1937, in the process of studying semiconductor excitons, Gregory Wannier quantized the fixed energy bands in a slowly changing external standard potential [29]. Slater summed up this method as Vanier's theorem: the effective Hamiltonian is the band function plus the external potential field, and the momentum in the former and the position in the latter become a pair of regular variables, satisfying the quantum reciprocity relation. The exciton spectroscopy and impurity energy levels similar to the hydrogen atomic energy levels mentioned in the textbook are derived from this effective Hamiltonian.
In the course of his research, Vanier also developed his famous Vanier function, which can be used to make a Fourier expansion of the Bloch wave function in the energy band. This is similar to the atomic wave function used in the tightly bound method, but instead of tail overlap, it is always orthogonally normalized. In 1951, Joaquin Luttinger used the Wattier function to generalize the Pyers substitution described above to the case beyond the tight binding[30]. Thus, with the addition of Varnier's theorem to the Pyles substitution, we get an effective quantum theory of general bands under electromagnetic fields.
As mentioned above, when not scattered, electrons can oscillate repeatedly in the band under the action of an electric field. Vanier found that the eigentheorized states corresponding to such oscillations are a set of steps (Stark ladders) with equally spaced energy levels and a localized wave function near each cell [31]. Interestingly, this stationary wave function is also a Vanier function with a minimum width. Years later, the Stark Ladder was clearly observed in the cold atom system of the optical lattice [32].
However, as we will see below, this single-band quantum theory based on the Vanier function does not always hold true due to the topological geometry effects of the Bloch bands. For example, Solis proves that when the Chen number of a band is non-zero, the local Vanier function of the band no longer exists. After the above-mentioned work of Rattinger, Walter Kohn et al. developed a set of methods based on regular transformations to obtain the effective Hamiltonian of electrons [33], which no longer relied on the local Wattier function, and this method can also be generalized to the higher order of electromagnetic fields, but their derivation process and results are quite complex and have not been widely used. The reason for this was that there was a lack of clear understanding of the topological geometry of the Bloch state.
05 Topological geometry effects
In 1980, Klaus von Klitzing discovered that the Hall resistance of two-dimensional electron gases under strong magnetic fields presents a series of quantized platforms determined by fundamental physical constants (Planck's constant and electron charge) [34]. To explain this quantum Hall effect, Solis et al. studied the subbands of two-dimensional Bloch electrons in a strong magnetic field and found that each full subband contributes a quantized transverse conductance [35]. Their results reveal a topological Chen in a physical context, which is universal to the Bloch band and can also be generalized to disordered and many-body interactions. This theory kicked off the prelude to the study of Bloch's electron topology and geometric properties in condensed matter physics, and the predicted quantum anomalous Hall effect was later confirmed by Xue Qikun's team [36], which also promoted the surging research of various topological materials. For this, Solis was awarded the 2016 Nobel Prize in Physics.
The topological Chan number is closely related to the Berry geometric phase of the Bloch state in the momentum space: it is equal to the integral of the density of the phase (i.e., the Berry curvature) in the Brillouin zone (see Box 4). A careful analysis of the Bloch wave packet shows that, in addition to the familiar group velocity, electrons have an anomalous velocity that is proportional to the Berry curvature under external forces [37—39]. When this term is retrieved, the equations of motion of the electrons appear more symmetrical: the anomalous velocities have a form similar to the Lorentz force, and the Berry curvature acts like a "magnetic field" in momentum space. This "magnetic field" causes the electron's trajectory to shift laterally perpendicular to the direction of the external force, resulting in an anomalous Hall effect. This corresponds to the topological quantization result described in Chan number mentioned above in the full-band case.
BOX 4
Berry curvature and semi-classical theory
From 1994 to 1999, Zhang Mingzhe, Niu Qian, and G. Sundaram constructed a complete semi-classical theoretical framework through variational methods. They considered the following Lagrangian quantities:
cause
Variation of the wave function gives the complete Schrödinger equation. In this sense, it is complete and precise. Semi-classical theory begins with the limitation of the form of the wave function. After limiting the form of the wave packet in momentum space and real space as mentioned above, Zhang Mingzhe et al. found that the wave function containing quantum behavior in the Lagrangian quantity disappeared and was replaced by the classical wave packet position and momentum, and its time derivative:
。 It is still important to note that although this Lagrangian quantity contains only classical mechanical quantities, it is still different from the Lagrangian quantities in classical mechanics: in classical mechanics, the Lagrangian quantities are only related to position, velocity, and time, and here there is a lattice momentum and its time derivative. The semi-classical kinetic equation derived from this Laslanis quantity is as follows:
Compared with the classical equations of electron motion (Equations (1) and (2), Equations (7) and (8)), this set of kinetic equations is markedly different: although the electron is still forced, the velocity of the electron acquires a new contribution
, which is an anomalous velocity. The Ω is called the Berry curvature, which reflects the geometric properties of the Bloch state itself.
The calculation of Berry curvature involves complex band structures in materials. The commonly used calculation expressions are as follows:
where En is the intrinsic energy of the electron and Vnm is the velocity operator
Matrix elements in the Bloch states of the nth and m-bar bands. Eq. (13) shows that the practical calculation of the Berry curvature on a particular band requires the participation of the Bloch states of other bands. Here, we need to emphasize that although the Berry curvature can also be given by the curl of the Berry contact in a single Bloch state, this relationship is not suitable for the numerical calculation of the Berry curvature because the Berry contact is related to the canonical choice.
Equation (11) is also different from the classical velocity equation: the energy in the first term is not a simple band energy, but contains the coupling of the orbital magnetic moment and the magnetic field originating from the orbital motion, i.e., ε=ε0-B⋅m, whereas in classical mechanics, the energy of the system does not change under the external magnetic field. If we look at it in the language of the wave packet, this orbital magnetic moment m actually represents the rotation of the wave wrap around its own center of mass. The Berry curvature mentioned above represents the revolution of the center of mass of the wave packet.
If we further compare the velocity equation with the force equation, we can see that the two have a similar structure: after replacing position with momentum and potential energy with kinetic energy, the force equation can be roughly changed to the form of the velocity equation. This similarity suggests that we can put the
Think of the "Lorentz force" as the "Lorentz force" of the momentum space, while Ω as the "magnetic field" of the momentum space. In fact, Ω not only have the same space-time transformation properties as magnetic fields, but can even have the same kinetic effects: like magnetic fields, the Berry curvature can deflect the trajectory of electrons laterally (Figure 5). This is known as the anomalous Hall effect. The current it produces is perpendicular to the electric field, i.e., J=E×σ, and the electrical conductivity can be written as
For two-dimensional semiconductors with Fermi energy in the energy band, the anomalous Hall conductance described above can only take the discrete quantized value (quantum Hall effect). This is because the integral of Berry curvature in a closed Brillouin zone is quantized:
where n is an integer.
Fig.5 Hall effect (a) and anomalous Hall effect (b). The current is driven by an electric field, while the lateral deflection of the electron trajectory is due to the magnetic field and the Berry curvature, respectively (Image taken from Chang C Z, Li M. J. Phys.:Condens. Matter，2016，28：123002)
For more general perturbations that change slowly in time and space, the response of the Bloch electrons involves more components of the Berry curvature in phase space and time [40]. For example, in addition to the components between the different momentum indicators mentioned above, there are also adiabatic velocity components between momentum and time, components between different indicators of four-dimensional space-time, and components similar to Planck's constant between momentum and position. These various aspects of Berry curvature have a wide range of applications in materials research.
The energy of the electron used in the equation of motion is equal to the energy band value of the local approximation plus some gradient terms [39]. The latter is proportional to the dipole moment of certain physical quantities, which can be calculated by the Bloch function and is an important part of the Bloch electronic intrinsic information. An example of this is the frequently used magnetic moment of the electron orbital, which depicts the linear movement of electron energy in a magnetic field. Both the g-factor of the magnetic moment of the electron in the solid and the magnetic moment of the valley electron are derived from this. Even the Bohr magneton with electron spin can be thought to come from the velocity dipole moment in the band of the Dirac equation [41].
The equations of motion of the above electrons are applicable to non-degenerate bands, and the external force is not sufficient to cause Zener transitions between different bands. For degenerate or near-degenerate bands, which often occur in spin-related problems, it is necessary to use the probability amplitude to characterize the transfer of electrons between different bands [42]. The physical quantities in question in the equation of motion of the electron become matrices, i.e., non-abelian forms, plus a Schrödinger equation satisfied by a simultaneous probability amplitude.
Due to the effect of Berry curvature, the equation of motion of the Bloch electron is no longer a regular Hamiltonian form. Liuville's theorem that the volume of phase space does not evolve with time no longer holds. In order to conserve the number of quantum states in phase space, the density of states previously used from Pauli and Sommerfeld to Paul [43]. By transmuting the actual equation of motion into a regular form, it is found that the physical momentum and position of the electron are translated relative to the momentum and position of the regular body. This led to a correction to the Pyles substitution, and also unexpectedly revealed the origin of the spin-orbit coupling in the degenerate energy band.
Like Bloch's bands, first-principles calculations can also give Berry curvature and other parameters in the equations of motion. This not only gives the topological number of the insulator, but also allows for the systematic study of various intrinsic properties of real metals and semiconductor materials, including orbital magnetization and anomalous Hall effect, in combination with the Fermi-Dirac distribution. Over the past two decades, remarkable achievements have been made in this area. In addition, the equations of motion of the Bloch electron mentioned above can also be incorporated into the framework of Boltzmann scattering theory, which can improve the description of the exogenous transport properties of materials.
06Summary and outlook
The classical Newtonian view of particles failed at the atomic scale, but through the transformation of quantum statistics and wave dynamics, as well as the tempering of many-body interactions, electrons appeared as new particle forms on a larger scale. Its momentum is confined to a finite region, and the energy is isolated into bands. When an energy band is filled with electrons, it exhibits inertness and no longer participates in electrical and thermal conductivity. In the highest energy band occupied by electrons in metals, there is a Fermi surface that represents the energy of the highest monomer of electrons. At room temperature, only a few electrons near the Fermi surface can be excited and participate in the response to the external field.
For a long time, Bloch's electron theory in solids was based mainly on their energies and could correctly give a basic picture of the binding energy, lattice structure, and macroscopic elasticity of materials, as well as their response to various external fields. Nearly 40 years of research have shown that the Berry phase of the Bloch state and the associated geometric topology also deeply affect the behavior of electrons. This paper briefly reviews how the Berry curvature modifies the equations of motion of electrons and their fundamental properties. We will provide a more detailed background on how this has changed the understanding and calculation of various aspects of the physical properties of solids.
Similar to electrons, other particles in crystals, such as the phonons and magnetons mentioned above [44], follow similar equations of motion. For quasiparticles in superconductors, the order parameters can not only determine their energy spectrum, but also affect the Berry curvature and the corresponding topological properties [45]. We can expect that the order parameters corresponding to phonons and magnetons will have a similar effect.
When the effects of high-order gradients need to be considered, we need to make appropriate adjustments to the equations of motion, but still retain the original form [46]. For example, by taking into account the first-order perturbations of the Bloch states in the local approximation, taking into account the first-order mixing of other quantum states caused by the external field gradient, the Berry curvature can be accurate to the first order, and the energy can be accurate to the second order. In this way, the resulting equations of motion can be used to calculate the second-order response problem [47]. This has also started to have a lot of applications in recent years.
When the periodicity defined by the Bloch state changes, such as the strain of a material slowly changes with space and time, the motion of the particles feels a space-time bending effect [48]. When the periodicity of space expands to the periodicity of space-time, the Bloch state is replaced by the Floquet-Bloch state, a plane wave whose amplitude is periodically modulated over space-time. The energy and momentum of the corresponding particles are confined to the four-dimensional torus, satisfying the equations of motion that are similar to the form of the Bloch particle, but more abundant [49]. When this space-time cycle is superimposed on a slow modulation similar to the spatiotemporal strain, the equations of motion are modified into a generalized covariant form, thus opening up a possible path for the study of artificial gravity in the laboratory [50].
Acknowledgments This article received valuable comments from Li Canguo and Sun Fengguo during the conception stage, and received professional comments from Jin Xiaofeng, Shen Shunqing, Yu Yue, Shi Yu, Zhou Jianhui, Wu Congjun and other colleagues after it was basically written.
Author: Niu Qian Gao Yang
(School of Physics, University of Science and Technology of China)
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