Waves and particles in the crystal

This chapter provides the groundwork necessary to mathematically describe the crystalline solid that is to be the semiconductor used to explore the variety of interactions and cause and chance behaviors that physics builds insights into. The crystalline environment can be portrayed as a space-filling periodic arrangement consisting of a lattice with an atomic basis. The periodic arrangement leads to real space and reciprocal space descriptions with unit cells—the Wigner-Seitz cell and the Brillouin and Jones zones—where a variety of characteristics can be represented. Bloch’s theorem with its modulation function of the plane wave of a quantized wavevector, momentum versus crystal momentum, together with the consequences of symmetries and periodic perturbation in the appearance of bandgaps, is discussed for electron states. Phonons as particles for periodic oscillations of atoms, their modes and various branches, and consequences of ionicity leading to frequency-dependent permittivity are discussed.

Quantum to macroscale and linear response

This chapter focuses on the properties associated with linear response. Reversibility holds in linear transformations. Schrödinger and Maxwell equations are linear, yet the world is irreversible, with time marching forward and dissipation quite ubiquitous. The connections between the quantum and microscopic scale, which are reversible and non-deterministic, to the macroscale, where irreversibility and determinism abounds, arise through interactions where both linear and nonlinear responses can appear. Causality’s implication in linear response is illustrated through a toy example and a quantum-statistical view of response. Linear response theory—using Green’s functions—is applied to develop dispersion relationships and dielectric function. The tie-in between real and imaginary parts is illustrated as one example of the Kramers-Kronig relationship, and the linear response of a damped oscillator and the Lorentz model, together with the oscillating electron model, employed to illustrate the dielectric function implications.

Causality and Green’s functions

The chapter elucidates causality to prevent its erroneous use. It then develops Green’s functions to analyze and predict systems under stimulation in the presence of cause and chance. Causality is the tracing of an observed phenomenon entirely to an identified cause. Cause and chance both lead to observations in the natural world. Chance is tackled through probabilities. A toy quantum model emphasizing uncertainty is used to illustrate cause and chance. Several examples are discussed to illustrate the development of Green’s functions—examining both advancing Green’s functions and retarding Green’s functions—for classical and quantum evolution. The S-matrix is also discussed.

Particle generation and recombination

This chapter introduces a semi-classical interpretation of particle generation and recombination using the bimolecular recombination coefficient and radiative lifetime. Particles—electrons and holes in the semiconductor—can be generated and recombine because of the multitude of energetic interactions. Radiative recombination and generation arise in the interaction with photons and can be spontaneous or stimulated. Important non-radiative processes such as the Hall-Shockley-Read process and the Auger process, which arise in multiparticle interactions, are discussed. Auger recombination is common at small bandgaps and high concentrations but also appears in large bandgap materials under high injection conditions. Impact ionization is an example of Auger generation arising from high fields. The Auger process is analyzed quantum-mechanically to show how energy and momentum conservation equations and quantum restrictions lead to the observed behavior. The chapter also discusses recombination at surfaces, which is inevitably present because of the defects and confined states arising from symmetry breaking.

Scattering-constrained dynamics

This chapter discusses the statics and dynamics of particle ensemble evolution under multiple stimuli—electrical, magnetic and thermal, particularly (thermoelectromagnetic interaction)—by developing the evolution of the distribution function in a generalized form from its thermal equilibrium form. In the presence of electrical and magnetic fields, this shows the Hall effect, magnetoresistance, et cetera. Add thermal gradients, and one can elaborate additional consequences that can be calculated in terms of momentum relaxation times and the nature of impulse interaction, since momentum and energies carried by the ensemble are accounted for. So, parameters such as thermal conductivity due to the carriers can be determined, thermoelectric, thermomagnetic and thermoelectromagnetic interactions can be quantified and the Ettinghausen effect, the Nernst effect, the Righi-Leduc effect, the Ettinghausen-Nernst effect, the Seebeck effect, the Peltier effect and the Thompson coefficient understood. The dynamics also makes it possible to determine the frequency dependence of the phenomena.

Transport and evolution of classical and quantum ensembles

This chapter explores the evolution of an ensemble of electrons under stimulus, classically and quantum-mechanically. The classical Liouville description is derived, and then reformed to the quantum Liouville equation. The differences between the classical and the quantum-mechanical description are discussed, emphasizing the uncertainty-induced fuzziness in the quantum description. The Fokker-Planck equation is introduced to describe the evolution of ensembles and fluctuations in it that comprise the noise. The Liouville description makes it possible to write the Boltzmann transport equation with scattering. Limits of validity of the relaxation time approximation are discussed for the various scattering possibilities. From this description, conservation equations are derived, and drift and diffusion discussed as an approximation. Brownian motion arising in fast-and-slow events and response are related to the drift and diffusion and to the Langevin and Fokker-Planck equations as probabilistic evolution. This leads to a discussion of Markov processes and the Kolmogorov equation.

Introduction

Semiconductors, as crystalline, polycrystalline or amorphous inorganic solids, as ordered or disordered organic solids or even in glassy and liquid forms, form a large set of materials useful in active and passive devices. The control of their properties arising in an interaction of particles—atoms, electrons, photons, their elementary one- and many-body excitations, transport and the exchange between different energy forms—has been a fruitful human endeavor since the birth of the transistor, where they found their first large-scale use. Integrated electronics, through its social and commercial informational ubiquity; optoelectronics, through lasers and photovoltaics; and thermoelectronics and magnetoelectronics, with their use in energy transformation and signal detection, are but a few of these gainful uses. Nanoscale, within this milieu, opens up a variety of perturbative and significantly more substantial and sensitive effects. Some are very useful, and some can be quite a bother....

Remote processes

This chapter discusses remote processes that influence electron transport and manifest themselves in a variety of properties of interest. Coulomb and phonon-based interactions have appeared in many discussions in the text. Coulomb interactions can be short range or long range, but phonons have been treated as a local effect. At the nanoscale, the remote aspects of these interactions can become significant. An off-equilibrium distribution of phonons, in the limit of low scattering, will lead to the breakdown of the local description of phonon-electron coupling. Phonons can drag electrons, and electrons can drag phonons. Soft phonons—high permittivity—can cause stronger electron-electron interactions. So, plasmon scattering can become significant. Remote phonon scattering too becomes important. These and other such changes are discussed, together with phonon drag’s consequences for the Seebeck effect, as illustrated through the coupled Boltzmann transport equation. The importance of the zT coefficient for characterizing thermoelectric capabilities is stressed.

Onsager relationships

This chapter discusses Onsager relationships. These relationships result from the linear response at the macroscale off-equilibrium from the reversibility of the microscale and represent an example of cause and chance at work. Flux-flow formalism—flux densities tied to thermodynamic forces—is developed to build the generalized linear relationships for heat, electric, chemical composition and free energy. The relationships are then applied to examples from previous chapters—thermoelectric and others—to show how results of interest can be derived more easily through exploiting Onsager relationships’ linearity and reciprocity relationships. The chapter discusses Onsager relationships with respect to Ohm’s law, Fourier’s law, Fick’s law, Darcy’s law, Gibbs free energy, thermoelectric effects and fluctuation-dissipation.

Major scattering processes

This chapter discusses major scattering processes found in semiconductors, including phonon scattering (deformation scattering, piezoelectric scattering, polar scattering and non-polar scattering), scattering arising from impurities (charged, so a Coulomb scattering, and charge neutral) and scattering arising in compositional randomness, from carrier-carrier events and due to coupled-particle interactions. The discussion starts by making connections between the classical scattering cross-section and its quantum-mechanical origins through the matrix elements for scattering. The ability to write the matrix element is employed for describing scattering by phonons in its various forms, for impurities and their various levels of accuracy of the description. Umklapp processes are described. When multiple scattering processes are present, the resulting transport manifests the processes’ independence and dependence. With an understanding of the scattering, observed behavior in semiconductors of interest is summarized to show their relative importance. The chapter concludes by discussing frequency and high field behavior manifested by electron ensembles.