Following Ashcroft and Mermin, the conduction electrons (“electrons” or “holes”) are assumed to move as wave packets. Dirac’s theorem states that the quantum wave packets representing massive particles always move, following the classical mechanical laws of motion. It is shown here that the conduction electron in an orthorhombic crystal moves classical mechanically if the primitive rectangular-box unit cell is chosen as the wave packet, the condition requiring that the particle density is constant within the cell. All crystal systems except the triclinic system have k-vectors and energy bands. Materials are conducting if the Fermi energy falls on the energy bands. Energy bands and gaps are calculated by using the Kronig-Penny model and its 3D extension. The metal-insulator transition in VO2 is a transition between conductors having three-dimensional and one-dimensional k-vectors.
Following Ashcroft and Mermin [
In simple cubic (SC), tetragonal (TET) and orthorhombic (ORC) crystals, the lattices have natural orthogonal axes. Their unit cells are different only in having one (1), two (2) and three (3) different sides. Elements Po (Pa) form SC (TET) crystals.
In Section 2 an electron in electric and magnetic fields, a conduction electron in solids, and crystal lattice structures are introduced and summarized. In Section 3 a theory of electron dynamics for an ORC lattice is developed. Choosing the rectangular-box unit cell as the wave packet, we establish that there are 3D k-vectors. The results are summarized, using Bloch theorem [
Let us take a classical electron in a free space moving in the electric field
where
is the electron velocity. We introduce a vector potential
where
Note that the Cartesian coordinates
In Hamiltonian formulation a Hamiltonian
The equations of motion are derived from
We may quantize the dynamics by introducing the fundamental commutation rules:
where j and k indicate components:
Wigner and Seitz used a primitive unit cell and lattice periodicity to obtain the ground-state energy of a metal [
where the Bravais vector
is specified by integers-set
The cubic cell may be chosen as the wave packet for the conduction electron. The center of mass of the wave packet is expected to move, following Hamilton’s equation of motion. The rigorous proof will be given later. Following Ashcroft and Mermin, we may set up a model of electron dynamics in solids. It is necessary to introduce
where
where
is the electron velocity, where
If an electron is in a continuous energy range (energy band), then it will be accelerated by the electric force, following Equation (13), and the material is a conductor. If the electron’s energy is discrete and is in a forbidden band (energy gap), it does not move under a small electric force, and the material is a insulator. If the acceleration occurs only for a mean free time
where n is the electron density and
For some crystals such as simple cubic (SC), face-centered-cubic (FCC), body-centered-cubic (BCC), tetragonal (TET) and ORC crystals, the choice of the orthogonal
We assume that a wave packet is composed of superposable plane waves characterized by k-vectors. The superposability is the basic property of the Schrödinger wave function in free space. A monoclinic (MCL) crystal can be generated from a TET crystal by distorting the rectangular faces perpendicular to the c-axis into parallelograms. Material plane waves proceeding along the c-axis exist since the x-y planes containing atoms are periodic in the z-direction in equilibrium. It has then one-dimensional (1D) k-vectors. In the x-y plane there is an oblique net whose corners are occuried by V’s for MCL (VO2). The Bravais vector may be defined by
where
where
To see this clearly, we first consider an electron in a simple square (SQ) lattice. The Schrödinger equation is
where the potential energy is periodic:
where
a key step for the separation-of-variable method. If we choose a periodic square boundary with the side length
We now go back to the original rhombic system. If we choose the x-axis along either
In SC, TET, and ORC crystals the lattices have natural orthogonal axes. We first take a rectangular 2D lattice. If the potential energy
then the energy-eigenvalue Schrödinger equation:
where E is the energy, is separable:
The 1D Schrödinger equation in x is
where
Clearly the wave function
where k is a real number, see below. Equation (26) represents a form of the Bloch theorem [
Let us discuss a few physical properties of the Bloch wave function
The following three main properties are observed.
The probability distribution function
The exponential function of a complex number
where m is an integer. We may choose the real number k in Equation (26), called the k-number (
the two end points are called the Brillouin boundary (points).
There are a number of energy gaps (forbidden regions of energy) in which no solutions of Equation (24) exist. The energy eigenvalues E are characterized by the k-number and the zone number (band index) j, which enumerates the energy bands:
This property (c) is not obvious, and it will be illustrated by examples later.
To further explore the nature of the Bloch wave function
and substitute it into Equation (26). If the function
then Equation (26) is satisfied. Equation (31) represents a second form of the Bloch theorem. The Bloch wave function
where
For a free particle, the k-number can range from
with no gaps. These features are different from the properties (b) and (c).
An important similarity arises when we write the time-dependent wave function
where the frequency
and the amplitude
Equation (35) shows that the Bloch wave function
The group velocity
By applying the (quantum) principle of wave-particle duality, we say that the Bloch electron moves with the dispersion (energy-momentum) relation:
The velocity
The lattice force
where the last line is obtained by partial integration. The first term on the right-hand side (r.h.s.) vanishes because both potential
Using this, we obtain from Equation (40)
This property also holds for the y-motion. Extending these results to a 3D motion, we obtain the desired result:
indicating that the conduction electron moves free from the lattice force. Only the external forces such as the electric and magnetic forces act on the conduction electrons.
Consider an infinite ORC lattice of lattice constants
where the Bravais lattice vector
The Schródinger equation is
The Bloch wave function
where
The three principal properties of the Bloch wave function are:
The probability distribution
The k-vector
the end points, which form a rectangular box, are called the Brillouin boundary.
The energy eigenvalues E have energy gaps, and the allowed energies E are characterized by the zone number j and the k-vectors:
Using Equation (47), we can express the Bloch wave function
The Bloch energy-eigenvalues in general have bands and gaps. We show this by taking the Kronig-Penny (K-P) model [
The Schrödinger equation can be written as in Equation (24). Since this is a linear homogeneous differential equation with constant coefficients, the wave function
According to the Bloch theorem, this function
The condition that the function
By solving Equation (56) with Equation (57), we obtain the eigenvalue E as a function of k. The band edges are obtained from
which corresponds to the limits of
At the lowest band edge
Near this edge the dispersion (energy-k) relation calculated from Equation (56) is
This one-dimensional K-P model can be used to study a simple 3D system. Let us take an ORC lattice of unit lengths
Let us now construct a model potential
Here the n are integers. A similar two-dimensional model is shown in
can now be reduced to three 1D K-P equations. We can then write an expression for the energy of our model system near the lowest band edge as
where
Equation (65) is what is intuitively expected of the energy-k relation for the electron in the ORC lattice. It is stressed that we derived it from first principles, assuming a 3D model Hamiltonian
The Coulomb potential
where
In classical statistical mechanics, a one-body distribution function
is introduced. Integrating over the phase space we obtain
An electron gas system is characterized by the Hamiltonian
The evolution equation for f is
where
If the system Hamiltonian
then the evolution equation for the field operator
In quantum field theory the basic dynamical variables are particle-field operators
Let us consider small oscillations for a system of atoms forming a SC lattice. Assume a longitudinal traveling wave along the x-axis. Imagine first hypothetical planes perpendicular to the x-axis containing atoms forming a square lattice. This plane has a mass per unit square of side length a (the lattice constant), equal to the atomic mass m. The plane is subjected to a restoring force per cm2 equal to Young modulus Y. The dynamics of a set of the parallel planes is similar to that of coupled harmonic oscillators.
Assume next a transverse wave traveling along the x-axis. The hypothetical planes containing many atoms are subjected to a restoring stress equal to the shear modulus S. The dynamics is also similar to the coupled harmonic oscillators in 1D.
Low-frequency phonons are those to which Debye’s continuum solid model [
where
where
The waves are superposable. Hence, phonons’ travels are not restricted to the crystal’s cubic directions. In short, there is a 3D k-vector,
Consider now the case of an ORC crystal. We may choose Cartesian coordinates
Phonons are quanta corresponding to the running plane-wave modes of lattice vibrations. Phonons are bosons, and the energies are distributed, following the Planck distribution function:
There is no activation energy unlike the case of the “electrons”. This arises from the boson nature of phonons. The temperature T alone determines the average number and energy.
Phonons and conduction electrons are generated based on the same lattice and k-space. This is important when describing the electron-phonon interaction.
The “electrons” and “holes” have the same orthogonal unit cell size. The average phonon size is much greater than the electron size. The low-energy phonons have small k and great wavelengths. The average energy of a fermionic electron is greater than a bosonic phonon by two or more orders of magnitude. This establishes a usual physical picture that a point-like electron runs, and is occasionally scattered by a cloud-like phonon in the crystal.
We saw earlier that a MCL crystal has 1D k-vectors pointing along the c-axis for the electrons. There are similar 1D k-vectors for phonons. Besides, there are two other sets of 1D k-vectors. Plane waves running in the z-direction can be visualized by imagining the parallel plates, each containing a great number of atoms executing longitudinal and transverse small oscillations. Consider an oblique net of points (atoms) viewed from
the top, shown in
We next consider a TCL crystal, which has no k-vectors for the electrons. There are, however, a set of 1D k-vectors for phonons. Take a primitive TCL unit cell. The opposing faces are parallel to each other. There are restoring forces characterized by Young modulus Y and shear modulus S. Then, there are 1D k-vectors perpendicular to the faces. The set of 1D phonons can stabilize the lattice. These phonons in TCL are highly directional. There is no spherical wave formed.
We used the lattice property that the facing planes are parallel. This parallel-plane configuration is common to all seven crystal systems [
In 1959 Morin reported his discovery of a metal-insulator transition (MIT) in vanadium dioxide (VO2) [
A simpler view on the MIT we propose is as follows [
Whittaker et al. [
where
The MIT designation is a misnomer. The semiconductor-microconductor transition correctly describes the phenomenon since the transition is between the 3D-k semiconductor with an activation energy and the 1D-k semiconductor called here the microconductor. In the low temperature phase the resistance R decreases with the temperature T, indicating the semiconductor character. In the normal metal the resistance R increases with T, arising from the phonon population change. In the high temperature phase the resistance R is finite, and therefore the material is not insulator. There are sharp drop and rise in the resistance, and the phase change depends on the heating and cooling directions, arising from the domain-by-domain transitions.
Graphene forms a 2D honeycomb lattice. The WS unit cell is a rhombus (darkened area) shown in
The prevalent theory based on the WS rhombus unit cell model predicts a gapless semiconductor with an “electron”-“hole” symmetry. In our earlier work [
Thus, “electrons” are the majority carriers in graphene. The thermally activated electron densities are then given by
where
Graphite is composed of graphene layers stacked in the manner ABAB∙∙∙ along the c-axis. We may choose an orthogonal (Cartesian) unit cell shown in
The unit cell has three side-lengths:
Clearly, the system is periodic along the orthogonal directions with the three periods
Equation (82). Both “electron” and “hole” have the same unit cell size. The system is orthorhombic with the sides
The negatively charged “electron” (with the charge
It is sometimes said [
The construction of the orthogonal unit cell developed here can be followed in other materials forming HEX crystals: Zinc (Zn) and Beryllium (Be) form HEX crystals. The closed orbits on the coronet-like Fermi surface generate cyclotron resonance, which may be discussed using the orthogonal unit cells.
In summary, we established that
The conduction electron (“electron”, “hole”) in an ORC crystal moves when a primitive orthogonal unit cell is chosen as the quantum wave packet.
CUB, TET, ORC, RHL, HEX crystal systems have 3D k-spaces for electrons. The MCL system has a 1D k-space. The TCL has no k-vectors.
The MCL and TCL have 1D phonons, which are highly directional. No spherical phonon distributions are generated.
For RHL and HEX crystals the orthogonal unit cells different from the WS unit cells must be chosen for electron and phonon dynamics.
“Electrons” and “holes” have the same unit cell size, and they move with different effective masses. “Electrons” and “holes” in semiconductors are excited with different activation energies. Phonons are excited with no activation energies.
Both phonons and electrons are generated based on the same orthogonal unit cells. This fact is important when dealing with the electron-phonon interaction.
The electron size is the primitive unit cell size. The average phonon size is greater by two or more orders of magnitude at the room temperature.
Instantaneous interparticle Coulomb potential
The MIT in VO2 is in reality a transition between two semiconductors having 1D and 3D k-vectors. The 1D semiconductor (low temperature phase) is highly anisotropic.
The majority carriers in graphene and graphite are “electrons”.
Graphite has “electrons” and “holes”. Phonon exchange may generate Cooper pairs. Then graphite can be a superconductor.