Herein, relations between particle and wave properties for charge carriers in periodic potentials of crystalline metals and semiconductors are derived. The particle aspects of electrons in periodic potentials are considered using properties of quasimomentum (QM), while the wave aspects are described using wave packets of Bloch waves. The two aspects are combined in the derivation of QM–wavelength relations for energy bands of arbitrary nonparabolicity and nonsphericity. An effective mass relating electron QM to its average velocity for spherical energy bands is defined and used to calculate energy dependences of wavelengths for electrons in narrow‐gap semiconductors, graphene, and surface states of topological insulators. An uncertainty relation between electron QM and its spatial coordinate in periodic potentials is derived. It is emphasized that the described properties apply to the average (not instantaneous) electron behavior. Analogies and differences between the wave and uncertainty properties of electrons in crystalline solids and those in vacuum are traced. An interference experiment with semiconductor nanostructures is proposed to verify the theoretical predictions and demonstrate to what extent the wave properties of electrons in solids are distinct from those in vacuum.
Herein, relations between particle and wave properties for charge carriers in periodic potentials of crystalline metals and semiconductors are derived. The particle aspects of electrons in periodic potentials are considered using properties of quasimomentum (QM), while the wave aspects are described using wave packets of Bloch waves. The two aspects are combined in the derivation of QM–wavelength relations for energy bands of arbitrary nonparabolicity and nonsphericity. An effective mass relating electron QM to its average velocity for spherical energy bands is defined and used to calculate energy dependences of wavelengths for electrons in narrow‐gap semiconductors, graphene, and surface states of topological insulators. An uncertainty relation between electron QM and its spatial coordinate in periodic potentials is derived. It is emphasized that the described properties apply to the average (not instantaneous) electron behavior. Analogies and differences between the wave and uncertainty properties of electrons in crystalline solids and those in vacuum are traced. An interference experiment with semiconductor nanostructures is proposed to verify the theoretical predictions and demonstrate to what extent the wave properties of electrons in solids are distinct from those in vacuum.
In this work we are formulating a new theory for describing the waving nature of a microscopic electric particle. Based on the predictions of the Quantum Oscillatory Modulated Potential—QOMP, for describing the interaction between two microscopic electric particles, electron-electron, for instance, we are postulating that the waving behavior of these particles may be an attribute of the charges of the particles and not their masses as currently accepted. For a microscopic electric charge, we are presenting new arguments showing that the electric field in the vicinity of a microscopic charge is spatially waving and can be determined as the gradient per unit of charge of this new quantum interaction potential, with use of an appropriated phase factor to account for the behavior of an unbound electron. Differently of what is predicted by the classical Coulomb electric field, when a charged particle is moving under the action of a potential of V volts, the new electric field existing around the charge has the form of a wave packet. For typical values of the potential V, the wavelength of the waving electric field is in very good agreement with those experimentally observed with diffraction of electrons in crystalline solids.
The electron theory of metals revived by Sommerfeld assumes that an electron moves in a metal as though this were an equipotential medium. Considering the nuclei fixed and regularly spaced we obtain a potential periodic in space coordinates. To study the effect of such fields we may simplify the problem so as to contain only one periodic term for each coordinate in its expression for potential. This problem can be reduced further to a one-dimensional one, of which the simplest example is the motion of an electron in a field with potential cos x or sin x. Darwin has shown that a suitable combination or packet of elementary de Broglie waves is capable of moving coherently in several instances. The motion of such a packet is found to be equivalent to that of a particle in classical dynamics with the Heissenberg uncertainty relation. The wave packet is used here for the motion of the electron in a periodic field. The result obtained is equivalent to that of classical dynamics. The wave packet again moves as a particle with an uncertainty relation.
This chapter concerns the basic understanding of quantum mechanics from an engineering perspective. Understanding the topological behavior of elastic and acoustic waves demands physical knowledge and the mathematical treatments developed for quantum mechanics. This chapter thus presents quantum mechanics and its relation to the understanding of acoustic and elastic wave propagation for quantum analog applications. It is assumed in this chapter that the readers are familiar with the fundamentals of modern physics and the construct of an atom with nucleus and cloud of electrons. Assuming this familiarities, first the Schrödinger equation is derived. The derivation employed the concepts of relativistic particles and their properties of wave-particle-duality. Schrödinger wave equation and an elastic wave equation are solved in a one-dimensional system, simultaneously, to show the respective equivalency. Specific features such as the uncertainty principle of angular momentum and the relation to the Pauli matrices are derived for the eigenspinors. Next the Schrödinger equation is solved in a periodic crystal with the Bloch solution, and sample energy bands are presented., It is explained how, in these energy bands at a specific energy level, the Dirac equation for relativistic particle can help find the Hamiltonian operator. It is explained how using the Hamiltonian for spinors could help understand the band structure with their respective spin behavior at Dirac cones. These features and understanding will be useful for quantum analog applications of acoustic and elastic waves.
The early quantum theory of electrical conductivity in crystal lattices by Bloch and Zener [1,2] led to the striking prediction that a homogeneous static electric field induces an oscillatory rather than uniform motion of the electrons. These so-called Bloch oscillations have never been observed in natural crystals because the scattering time of the electrons by the lattice defects is much shorter than the Bloch period. In semiconductor superlattices the larger spatial period leads to a much shorter Bloch period (,600 fs) and Bloch oscillations have recently been observed through the emission of THz radiation by the electrons [3]. Here we present Bloch oscillations of atoms in the fundamental energy band of a periodic optical potential. We directly measure the atomic momentum distribution evolving in time under the influence of a constant inertial force for various potential depths. We experimentally observe oscillation periods in the millisecond range as well as positive and negative effective masses. Bloch oscillations are a pure quantum effect which can be explained in a simple one-dimensional model. The periodicity of the lattice (period d) leads to a band structure (Fig. 1) of the energy spectrum of the particle and the corresponding eigenenergies Ensqd and eigenstates jn, ql (Bloch states) are labeled by the discrete band index n and the continuous quasimomentum q; Ensqd and jn, ql are periodic functions of q with period 2pyd and q is conventionally restricted to the first Brillouin zone g2pyd, pydg [4]. Under the influence of a constant external force F, weak enough not to induce interband transitions, a given Bloch state jn, qs0dl evolves (up to a phase factor) into the state jn, qstdl according to qstd qs0d 1 Ftyh . (1) This evolution is periodic with a period tB hyjFjd corresponding to the time required for the quasimomentum to scan a full Brillouin zone. The mean velocity in jn, qstdl
The identity and size of atoms is explicable only in quantum physics. The double slit experiment illustrates the wave-particle duality of light and of matter. To describe quantum interference the concept of a complex probability is introduced, the squared amplitude of which is the probability of a particle being at a particular location. The uncertainty relation requires atomic motion in solids even at absolute zero. The symmetry of exchanging indistinguishable particles leads to the classification of particles as fermions or bosons. The exclusion principle applies to electrons and rationalises the Periodic Table and much more. Electrons in solids exist in bands of energy. Band theory explains why some materials are electrical conductors, others are insulators or semiconductors. Chemical bonding involves quantum tunnelling of electrons. Hydrogen may diffuse in solids by quantum tunnelling. The temperature dependence of the specific heat of a solid is explicable only in quantum physics.
Chapter 3 provides a brief review of a few concepts from classical Newtonian physics, and how this fails to explain phenomena at the nanoscopic scale. As is detailed within, the classical mechanical model is able to describe all observed physical phenomena; however, when it comes to the nanoscale this model breaks down, because the properties of the materials change dramatically. At this scale, the surface-to-volume ratio increases, gravitational forces become negligible and random molecular motion becomes more important. This changes the properties of the material and affects the devices that are fabricated at this scale. It is hence necessary to develop a new model that can explain such physical phenomena. This has led to a new branch of physics, i.e. quantum physics (QP) or quantum mechanics, which is able to explain the behavior observed at nanoscale. QP enables us to explain phenomena such as the photoelectric effect, Compton effect. etc. In this chapter, we introduce quantum mechanics and how it evolved in the late 19th and early 20th centuries. Further, light’s behaviour as a particle as well as a wave is described by the photoelectric effect and Young’s double-slit experiment, respectively. Further, the dual nature of the particle is explained using the de Broglie principle. Schrodinger’s equation is then explained, which gives the probability of finding a subatomic-scale particle at a certain position in a quantum well. In addition to the formulation of Schrodinger’s equation, its solution, the concept of a potential well, the characteristics of the wave function and its interpretation and significance, and, finally, the application of Schrodinger’s equation in the structure of an atom form subsequent parts of this chapter. The equation serves as a basis with which to understand band theory. Band theory explains the difference between conductors, insulators and semiconductors. Materials are distinguished according to the differences in their band gaps. Adding impurities or varying temperature changes the band gap in semiconductors, which consequently changes the properties of the semiconductor and its electrical conduction. One more important measure to understand and exploit the properties and behaviour of materials is Fermi–Dirac statistics. It is useful to acquire the probability of electrons at a given energy level at a specific temperature. These concepts can be utilised by physicists and chip makers to make more energy-efficient devices. Hence, this chapter details the fundamentals of quantum physics and how they relate to the electrical properties of nanodevices.