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.
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.
BRIEF REVIEW OF CLASSICAL THEORIES Harmonic Oscillator Boltzmann Transport Equation Maxwell's Equations MILESTONE DISCOVERIES AND OLD QUANTUM THEORY Blackbody Radiation and Quantum of Energy Specific Heat of Solids Photoelectric Effects Compton Scattering Duality of Matter Bohr?s H-Atom Theory SCHRODINGER EQUATION AND OPERATOR ALGEBRA Schrodinger Equation Momentum Eigenfunction and Fourier Series Hermitian Operator and Bra Ket Notations Orthgonality and Completeness of Eigenfunctions Basic Postulates of Quantum Mechanics Commutation Relations Conjugate Variables and Uncertainty Relation Operator Equation of Motion and Ehrenfest Theorem PARTICLE IN POTENTIAL WELL Infinite Square Well Potential Particle in 3-D Box Density of States: 1-D, 2-D and 3-D Particle in Quantum Well Particle in Delta Function Potential Well Quantum Well and Wire SCATTERING OF A PARTICLE AT 1-D POTENTIALS Scattering at Step Potential Scattering at Quantum Well TUNNELING AND ITS APPLICATIONS Tunneling across Square Potential Barrier Fowler-Nordheim and Direct Tunneling Resonant Tunneling The Applications of Tunneling PERIODIC POTENTIALS AND ENERGY BANDS 1-D Crystal and Kronig-Penny Model E-k Dispersion and Energy Bands Energy Bands and Resonant Tunneling THE HARMONIC OSCILLATOR Energy Eigenequation The Properties of Energy Eigenfunction The Operator Treatment THE ANGULAR MOMENTUM Angualr Momentum Operators Eigenfunctions and Spherical Harmonics THE HYDROGEN ATOM: THE SCHRODINGER TREATMENT Two-Body Central Force Problem The Hydrogen Atom The Atomic Orbital Virial Theorm and Doppler Shift SYSTEM OF IDENTICAL PARTICLES AND MANY-ELECTRON ATOMS Two Electron System Two Spin 1/2 System The Helium Atom The Periodic Table and the Structure of Atoms MOLECULES AND CHEMICAL BONDS The Ionized Hydrogen Molecule The Hydrogen Molecule Ionic Bond and Van der Waals Attraction THE PERTURBATION THEORY Time Independent Perturbation in Non-Degenerate Systems Time Dependent Perturbation Theory ATOM - FIELD INTERACTION Field Quantization Atom - Field Interaction Driven, Damped Two Level Atom INTERACTION BETWEEN EM WAVES AND OPTICAL MEDIA Attenuation and Dispersion of Waves Density Matrix and Ensemble Averaging Laser Devices QUANTUM STATISTICS General Background and Three Kinds of Particles Statistics for Distinguishable particles Statistics for Fermions and Fermi Distribution Function Statistics for Boson and Bose Einstein Distribution SEMICONDUCTOR STATISTICS Carrier Densities in Intrinsic Semiconductors Carrier Densities in Extrinsic Semiconductors Fermi Level in Extrinsic Semiconductors CHARGE TRANSPORT IN SEMICONDUCTORS Drift and Diffusion Currents Transport Coefficients Equilibrium and Non-Equilibrium Recombination and Generation Currents P-N JUNCTION DIODE The Junction Interface in Equilibrium The Junction Interface Under Bias The Diode I-V Applications of P-N Junction Diodes THE BIPOLAR JUNCTION TRANSISTOR: DEVICE PHYSICS AND TECHNOLOGY Bipolar Junction Transistor: Overview The Physics of Transistor Action Ebers Moll Equations Base Transit Time and Charge Control Model METAL OXIDE SILICON FIELD EFFECT TRANSISTORS (MOSFET) I: OVERVIEW OF DEVICE BEHAVIOR AND APPLICATIONS MOSFET: Overview Charge Control and Metal-Oxide-Silicon System NMOS I-V Applications of MOSFET METAL OXIDE SILICON FIELD EFFECT TRANSISTORS (MOSFET) II: DEVICE SCALING AND SCHOTTKY CONTACT Device Scaling: Physical issues and Limitations Metal - Semiconductor Contacts The I-V Behavior in Metal - Semiconductor Contacts
Chapter 4 deals with Quantum Mechanical (QM) phenomena. The first section describes the first attempts at quantization of energy, momentum, angular momentum, etc. We start with Plank's solution to the black-body problem, introducing the concept of photons: discrete packets of electromagnetic energy. We show how Plank's description of black-body radiation is perfectly fit by the cosmic microwave radiation, providing the strongest evidence for the Big-Bang at the origin of the Universe. We then describe Bohr's solution to the paradoxical stability of atoms, introducing quantization of angular momentum (or action) and correctly predicting the energy levels of the hydrogen atom. We show how Einstein used Plank's idea to explain the photo-electric effect (an effect exploited in CCD cameras and solar cells) and how he used Bohr's understanding of energy levels in atoms to provide an intuitive explanation of Plank's law. Einstein insight opened the way to the field of spectroscopy and the development of the laser. In the second section, following the lead of Dirac, we show that QM is equivalent to Linear Algebra in the space of complex functions (which describe the state of the system). We show that Schrödinger's equation is a natural consequence of that description, once the relation between energy and frequency established by Plank (for photons) and de Broglie (for particles) is taken into account. In the context of the chiral ammonia molecule we discuss the queerness of QM which describes physical systems as superposition of states (such as left and right chirality state of a molecule, dead and alive cat). We show that this queerness is nonetheless supported by experiments and is behind much of our technology: masers and lasers, atomic clocks, semiconductors (diodes, transistors, etc.). In the following section we introduce the representation of operators in QM (the equivalent of matrices in linear algebra): momentum, position, energy, etc. which correspond to measurements on the system. We show that the non-commutability of some operators (common with matrix operations) leads to Heisenberg uncertainty principle: the position and momentum of a particle cannot be determined with absolute precision. We discuss the solution of Schrödinger's equation for free particles demonstrating their equivalence with EM phenomena: particles undergo refraction and diffraction, exemplifying the particle/wave duality inherent to QM systems. We discuss QM tunneling and its use in the Scanning Tunneling Microscope. In the following section we study Dirac's equation: the successful attempt by Dirac to unify QM with Einstein's relativity, with the resulting prediction of anti-particles and spin. We then solve for the energy levels of electrons in the hydrogen atom (from Schrödinger's equation) and show that the existence of spin profoundly affects the behavior of electrons in an atom: they cannot share the same energy level, a rule known as Pauli's exclusion principle. Pauli's principle is central to the understanding of the chemical properties of atoms and fully explains Mendeleev's periodic table of the elements. In the next section we discuss along with two exactly soluble systems (diatomic rotational and vibrational levels), various approximations used to study systems (e.g. molecules) for which there are no analytical solutions. In particular, we introduce Pauling's variational approach, Hückel molecular orbital theory and time independent perturbation theory. We apply those approaches to determine the energy levels in alkenes, in multi-electron atoms and in systems under constant electric or magnetic field (such as in Magnetic Resonance Imaging). In the last section we discuss time dependent perturbation theory. We use it to establish the rules allowing for transitions between energy levels and revisit Einstein description of absorption/emission phenomena.
Abstract We have already encountered several aspects of quantum theory: the presence of discrete energy levels in atoms; the exclusion principle which tells us how these levels are occupied; and the formation of energy bands in solids. We did not make a big deal of it at the time since, once we accepted the premise of the band gap in semiconductors, we were able to understand the operation of the transistor without worrying about whether we were using results of quantum theory. However, if we wish to probe further—to understand the devices of the future, those under development, and some which are already commercially available—then there is no escape. We have to confront quantum theory head on. We will not be concerned with any of the deep philosophical questions which arise from quantum theory. These are addressed in many other books, and do not concern us here. Instead, we will concentrate on the practical effects—how quantum theory affects the behaviour of semiconductor devices. From this point of view, there are two principal consequences, and we will devote a chapter to each. This chapter deals with wave-particle duality. If you have not encountered this phrase before, do not worry—all will be revealed in due course. The following chapter describes a phenomenon called tunnelling. It is only after exploring these concepts that it is possible to understand the operation of the quantum transistor, a subject which we turn to in Chapter 9.
An easy-to-understand guide to the complex subject of quantum physics. Quantum physics is how scientists describe the world of the very small. For other people, however, the rules of quantum physics seem to violate all logic: How can a particle be in more than one place at the same time? How can it tunnel through an impenetrable barrier? How can a cat in a box be both alive and dead? This book explains the complexities of quantum physics in bite-sized "lessons" that make it clear and accessible to all readers. The sections and chapters are: 1. Atoms -- quantum; quantum physics in everyday life; the periodic table; atoms and nuclei; isotopes; hydrogen atom (energy levels and spectra) 2. Photons -- photoelectric effect; thermal emission and the Planck distribution; wave particle duality (Young's slit experiment) 3. Quantum devices -- superconductors; transistor, diode; light-emitting diode; laser 4. Spin -- spin; fermions; exclusion principle; Fermi Dirac distribution; Bose-Einstein statistics 5. Wave Mechanics -- Heisenberg uncertainty principle; wave functions; standing waves; wave particle duality 6. Subatomic Particles -- radioactivity; atomic particles; protons; neutrons; electrons 7. Particle Physics -- elementary particles; quarks; leptons; electron and electron neutrino; muon and muon neutrino; tauon and tau neutrino; bosons; photon; W and Z boson; gluon; Higgs boson; composite particles; hadrons; matter-anti-matter 8. Quantum Physics in Medicine and industry -- magnetic resonance imaging; superfluidity; cancer treatment by radiation 9. Into the Future -- quantum physics of space; quantum gravity; graviton; Hawking radiation (grand unified theory); dark matter; strings; dark energy. Throughout the book, there are timelines and profiles about the scientists who developed the theories and made important discoveries in quantum physics, including Dmitri Rutherford, Albert Einstein, Heike Kamerlingh Onnes, William Shockley, Isama Akasaki, Stephen Hawking, Enrico Fermi, and Max Born.
Because atoms and molecules are small, their behavior can only be understood using quantum mechanics, rather than classical mechanics. Quantum mechanics amounts to studying the behavior of electrons and nuclei and their interactions. We will limit ourselves to slowly moving electrons and nuclei, and thus will not need to discuss relativistic quantum mechanics. Large particles also obey quantum mechanics, but the relevant wavelengths are so small that we don't notice them. Small particles absolutely require the use of quantum mechanics for any meaningful understanding. This treatment is not merely an inconvenience: it leads to useful and interesting phenomena such as the Pauli exclusion principle, the Aufbau principle, Hund's rules, the Heisenberg uncertainty principle, wave-particle duality, the photoelectric effect, Ehrenfest's theorem, and the correspondence principle.