Energy quantization in nanoscale materials is manifested in a “blue-shift” of the emission spectrum of nanoparticles of decreasing size. The phenomenon is known as the “quantum size effect,” namely, the increasing gaps between energy levels for a spatially confined particle. The effect is demonstrated by solving the one-dimensional Schrödinger equation for the “particle in a box” model. The confining potential translates into boundary conditions, which result in energy quantization, where the corresponding standing wave solutions demonstrate remarkable differences from the classical description. Different energy levels are obtained by changing boundary conditions to periodic, for a “particle on a ring,” where the phenomenon energy level degeneracy is introduced. Extending the discussion to multidimensional “boxes” enables one to analyze the energy spectrum and the density of states nanostructures including quantum dots, wires, and wells, with references to the devices based on a two-dimensional electron gas, and to quantized conductance through point contacts.

Born-von Karman boundary conditions; density of states in bulk materials, quantum wells, and quantum wires. Carrier statistics in semiconductors: Fermi-Dirac distribution function, electron and hole density in the conduction and valence bands. Nondegenerate semiconductors; effective density of states; intrinsic semiconductors. Mass-action law. Doped semiconductors: donors and acceptors; hydrogen-like model. Degenerate semiconductors. Quasi-Fermi levels in nonequilibrium systems. Charge transport in semiconductors. Diffusion current.

We explore the energy eigenvalues and eigenstates of a periodic series of potential energy wells with the purpose of creating a rudimentary model of a solid. Our model uses an approximate approach that emphasizes the interaction between neighboring atoms. We learn how the eigenstates of the periodic potential can be constructed from the eigenstates of the single elements of the periodic potential. We also learn that the eigenstates of a solid are characterized by a wavelength, and that the energies of those eigenstates form bands centered near the atomic energy eigenvalues. We model electron motion in solids with the use of a wave packet, a superposition of delocalized Bloch states.

Chapter 3 derives the gap equation and determines the critical transition temperature as well as the zero-temperature energy gap as a function of coupling constant for d-wave superconductors. The energy dependence of the density of states and its effects on the temperature dependence of the gap function, entropy and other thermodynamic quantities are also discussed. Low energy nodal excitations lead to characteristic power-law behaviors in the specific heat or other thermodynamic response functions of d-wave superconductors at low temperatures, in contrast to the activated behaviors in s-wave superconductors. The probability density current and charge density current operators of d-wave quasiparticles, together with the gap operators in the continuum limit, are derived and discussed with the BdG framework.

Chapter 8 studies the many-impurity scattering effects in d-wave superconductors, particularly in the unitary or Born scattering limit. The impurity corrections to self-energy, density of states, superconducting critical temperature, entropy and specific heat are derived and compared with measurement data for high-Tc superconductors.

Kinetic theory is a framework for calculating macroscopic physical properties of systems from their microscopic degrees of freedom.This idea is applied to an ideal gas to derive the Maxwell--Boltzmann velocity distribution, which is demonstrated to be compatible with the ideal gas law and is used to calculate the rate of effusion of an ideal gas.When molecular collisions are important, the mean free path and collision time are quantities that can characterize these collisions.Situations in which collisions are important, such as Brownian motion and diffusion, are presented, along with relevant equations: the Langevin equation and Fick's Law.

Unlike classical particles, quantum particles are indistinguishable.Fermions and bosons differ in their quantum statistics, and the consequences of this for their statistical mechanics are explored in the grand canonical ensemble.The Fermi--Dirac and Bose--Einstein distribution functions are derived, and utilized to write thermal averages using the density of states.

Following the discovery of the two laws of thermodynamics, theorists turned to the issue of finding an interpretation of the laws in terms of the atomic and molecular properties of the gases. Following Clausius's interpretation in terms of the kinetic theory of gases, Maxwell discovered the Maxwell velocity distribution and related it to Gaussian statistics. This marked the beginnings of statistical mechanics and the realisation that the law of increase of entropy has only a statistical validity. Out of these analysis Boltzmann and Gibbs created the discipline of statistical mechanics. The Gibbs entropy is also central to information theory.