1st Edition
Understanding Solid State Physics Problems and Solutions
The correlation between the microscopic composition of solids and their macroscopic (electrical, optical, thermal) properties is the goal of solid state physics. This book is the deeply revised version of the French book Initiation à la physique du solide: exercices comméntes avec rappels de cours, written more than 20 years ago. It has five sections that start with a brief textbook introduction followed by exercises, problems with solutions, and comments and that are concluded with questions. It presents a quasi-systematic investigation of the influence of dimensionality changes, from 1D to 3D, via surfaces and 2D quantum wells, on the physical properties of solids. The aim of this book is to teach solid state physics through the use of problems and solutions giving orders of magnitude and answers to simple questions of this field. The numerous comments and problems in the book are inspired from some Nobel Prize–winning research in physics, such as neutron diffraction (1994), quantum Hall effect (1985), semiconducting heterostructures (1973), and tunnel microscope (1986), superconductivity (1987). The book will be helpful for undergraduate- and graduate-level students of solid state physics and chemistry and researchers in physics, chemistry, and materials science.
Preface
Tables
1.Crystal Structure and Crystal Diffraction
Course Summary
A. Crystal Structure
1. Definitions
2. Simple and Multiple Lattices
3. Lattice Rows and Miller Indices
4. Point Symmetry
5. The 7 Crystallographic Systems and the 14 Bravais Lattices
6. Space Symmetry
B. Diffraction and the Reciprocal Lattice
1. Bragg’s Law
2. X-Rays
3. Reciprocal Lattice (Exs. 12, 13, and 19)
4. More Detailed Analysis of Diffraction
Exercises
Exercise 1: Description of some crystal structures
Exercise 2: Mass per unit volume of crystals
Exercise 3: Construction of various crystal structures
Exercise 4: Lattice rows
Exercise 5a: Lattice rows and reticular planes
Exercise 5b: Lattice rows and reticular planes (continued)
Exercise 6: Intersection of two reticular planes
Exercise 7: Lattice points, rows and planes
Exercise 8: Atomic planes and Miller indices—Application to lithium
Exercise 9: Packing
Exercise 10a: Properties of the reciprocal lattice
Exercise 10b: Distances between reticular planes
Exercise 11: Angles between the reticular planes
Exercise 12: Volume of reciprocal space
Exercise 13: Reciprocal lattice of a face-centered cubic structure
Exercise 14: Reciprocal lattice of body-centered and face-centered cubic structures
Exercise 15: X-ray diffraction by a row of identical atoms
Exercise 16: X-ray diffraction by a row of atoms with a finite length
Exercise 17: Bravais lattices in 2D: Application to a graphite layer (graphene)
Exercise 18a: Ewald construction and structure factor of a diatomic row
Exercise 18b: Structure factor for a tri-atomic basis; Ewald construction at oblique incidence (variation of Ex. 18a)
Exercise 19: Reciprocal lattice, BZs, and Ewald construction of a twodimensional crystal
Exercise 20: X-ray diffraction patterns and the Ewald construction
Exercise 21a: Resolution sphere
Exercise 21b: Crystal diffraction with diverging beams (electron backscattered diffraction: EBSD)
Exercise 22: Atomic form factor
Exercise 23: X-ray diffusion by an electron (Thomson)
Problems
Problem 1: X-ray diffraction by cubic crystals
Problem 2: Analysis of an X-ray diffraction diagram
Problem 3: Low energy electron diffraction (LEED) by a crystalline surface: absorption of oxygen
Problem 4: Reflection high energy electron diffraction (RHEED) applied to epitaxy and to surface reconstruction
Problem 5: Identification of ordered and disordered alloys
Problem 6: X-ray diffraction study of a AuCu alloy
Problem 7: Neutron diffraction of diamond
Problem 8: Diffraction of modulated structures: application to charge density waves
Problem 9: Structure factor of GaxAl1–xAs
Problem 10: Structure factor of superlattices
Problem 11: Diffraction of X-rays and neutrons from vanadium
Problem 12: X-ray diffraction of intercalated graphite
Questions
2. Crystal Binding and Elastic Constants
Course Summary
A. Crystal Binding
1. Statement of the Problem
2. Rare Gas Crystals
3. Ionic Crystals
4. Metallic Bonds
5. Covalent Bonds
B. Elastic Constants
1. Introduction
2. Stress
3. Strain
4. Hooke’s Law
5. Velocity of Elastic Waves
Exercises
Exercise 1: Compression of a ionic linear crystal
Exercise 2a: Madelung constant for a row of divalent ions
Exercise 2b: Madelung constant of a row of ions –2q and +q
Exercise 3: Cohesive energy of an aggregate of ions
Exercise 4: Madelung constant of a 2D ionic lattice
Exercise 5: Madelung constant of ions on a surface, an edge, and a corner
Exercise 6: Madelung constant of an ion on top of a crystal surface
Exercise 7: Madelung constant of parallel ionic layers
Exercise 8: Cohesive energy of a MgO crystal
Exercise 9: Ionic radii and the stability of crystals
Exercise 10: Lennard-Jones potential of rare gas crystals
Exercise 11: Chemisorption on a metallic surface
Exercise 12: Anisotropy of the thermal expansion of crystals
Exercise 13: Tension and compression in an isotropic medium. Relations between Sij, Cij, E (Young’s modulus) and s (Poisson coefficient), l and m (Lamé coefficients)
Exercise 14: Elastic anisotropy of hexagonal crystals
Exercise 15: Shear modulus and anisotropy factor
Exercise 16: Elastic waves in isotropic solids
Problems
Problem 1: Cohesion of sodium chloride
Problem 2: Cohesion and elastic constants of CsCl
Problem 3: Van der Waals–London interaction. Cohesive energy of rare gas crystals
Problem 4: Velocity of elastic waves in a cubic crystal: Application to aluminum and diamond
Problem 5: Strains in heteroepitaxy of semiconductors
Questions
3. Atomic Vibrations and Lattice Specific Heat
Course Summary
1. Vibrations in a Row of Identical Atoms
2. Lattices with More Than One Atom per Unit Cell
3. Boundary Conditions
4. Generalization to 3D
5. Phonons
6. Internal Energy and Specific Heat
7. Thermal Conductivity
Exercises
Exercise 1: Dispersion of longitudinal phonons in a row of atoms of type C=C–C=C–C=
Exercise 2a: Vibrations of a 1D crystal with two types of atoms m and M.
Exercise 2b: Vibrations of a 1D crystal with a tri-atomic basis
Exercise 3: Vibrations of a row of identical atoms. Influence of second nearest neighbors
Exercise 4: Vibrations of a row of identical atoms: Influence of the nth nearest neighbor
Exercise 5: Soft Modes
Exercise 6: Kohn Anomaly
Exercise 7: Localized phonons on an impurity
Exercise 8: Surface acoustic modes
Exercise 9: Atomic vibrations in a 2D lattice
Exercise 10: Optical absorption of ionic crystals in the infrared
Exercise 11: Specific heat of a linear lattice
Exercise 12a: Specific heat of a 1D ionic crystal
Exercise 12b: Debye and Einstein temperatures of graphene, 2D, and diamond, 3D
Exercise 13: Atomic vibrations in an alkaline metal: Einstein temperature of sodium
Exercise 14: Wave vectors and Debye temperature of mono-atomic lattices in 1-, 2-, and 3D.
Exercise 15: Specific heat at two different temperatures
Exercise 16: Debye temperature of germanium
Exercise 17: Density of states and specific heat of a monoatomic 1D lattice from the dispersion relation
Exercise 18: Specific heat of a 2D lattice plane
Exercise 19: Phonon density of states in 2D and 3D: evaluation from a general expression
Exercise 20a: Zero point energy and evolution of the phonon population with temperature
Exercise 20 b: Vibration energy at 0 K of 1, 2, and 3D lattices (variant of the previous exercise)
Exercise 21: Average quadratic displacement of atoms as a function of temperature
Problems
Problem 1: Absorption in the infrared: Lyddane–Sachs–Teller relation
Problem 2: Polaritons
Problem 3: Longitudinal and transverse phonon dispersion in CsCl
Problem 4: Improvement of the Debye model: Determination of qD from elastic constants application to lithium
Problem 5: Specific heats at constant pressure Cp and constant volume Cv: (Cp – Cv) correction
Problem 6: Anharmonic oscillations: thermal expansion and specific heat for a row of atoms
Problem 7: Phonons in germanium and neutron diffusion
Problem 8: Phonon dispersion in a film of CuO2
Problem 9: Phonons dispersion in graphene
Questions
4. Free Electrons Theory: Simple Metals
Course Summary
1. Hypothesis
2. Dispersion Relation and the Quantization of the Wave Vector
3. Electron distribution and density of states at 0°K: Fermi energy and Fermi surface in 3D
4. Influence of Temperature on the Electron Distribution: Electron-Specific Heat
5. Electronic Conductivity
6. Wiedemann–Franz Law
7. Other Successful Models Obtained From the Free Electron Formalism
Exercises
Exercise 1: Free electrons in a 1D system. Going from an atom to a molecule and to a crystal
Exercise 2: 1D metal with periodic boundary conditions
Exercise 3: Free electrons in a rectangular box (FBC)
Exercise 4: Periodic boundary conditions, PBC, in a 3D metal
Exercise 5: Electronic states in a metallic cluster: Influence of the cluster size
Exercise 5b: Electronic states in metallic clusters: Influence of the shape
Exercise 6 (Variation of Ex. 5 and 5b): F center in alkali halide crystals and Jahn–Teller effect
Exercise 7: Fermi energy and Debye temperature from F and P boundary conditions for objects of reduced dimensions
Exercise 8: Fermion gas
Exercise 9: Fermi energy and thermal expansion
Exercise 10: Electronic specific heat of copper
Exercise 11: Density of electronic states in 1, 2, and 3D from a general formula
Exercise 12: Some properties of lithium
Exercise 13: Fermi energy, electronic specific heat, and conductivity of a 1D conductor
Exercise 14: Fermi energy and electronic specific heat of a 2D conductor
Exercise 14b: p-electrons in graphite (variation of Ex. 14 and simplified approach for graphene)
Exercise 14ter: Fermi vector and Fermi energy (at 0 K) of an electron gas in 1, 2, and 3D. Comparison with the residual vibration energy of atoms.
Exercise 15: Surface stress of metals
Exercise 16: Effect of impurities and temperature on the electrical resistivity of metals: Matthiessen rule
Ex. 17: Effect of the vacancy concentration on the resistivity of metals
Exercise 18: Effect of impurity concentration on the resistivity
Exercise 19: Another expression for the conductivity σ
Exercise 20: Size effects on the electrical conductivity of metallic films
Exercise 21: Anomalous skin effect
Exercise 22: Pauli paramagnetism of free electrons in 1, 2, and 3D.
Exercise 23: Quantum Hall Effect
Exercise 24: Simplified evaluation of the interatomic distance, compression modulus, B, and cohesive energy of alkali metals
Exercise 25: Pressure and compression modulus of an electron gas: Application to sodium
Exercise 26: Screening effect
Exercise 27: Thermionic emission: The Richardson–Dushman equation
Exercise 28: Thermal Field Emission: the energy width of the emitted beam
Exercise 28b: Thermionic emission in 2D
Exercise 29: UV Reflectivity of alkali metals (simplified variation of Pb 6).
Exercise 30: Refractive Index for X-rays and total reflection at grazing incidence
Exercise 31: Metal reflectivity in the IR: The Hagen–Rubens relation
Problem 1: Cohesive energy of free electron metals.
Problem 2: Dipole layer and work function at surfaces of free electron metals.
Problem 2b: Electronic density and Energy of metal surfaces: Breger–Zukovitski Model
Problem 3: X-ray photoelectron emission (XPS), X-ray absorption fine structure (EXAFS); Auger electron and X-ray photon emissions
Problem 4: Refraction of electrons at metal/vacuum interface and angle-resolved photo-electron spectroscopy (ARPES).
Problem 5: Scanning Tunneling Microscope (STM)
Problem 6: DC electrical conductivity. Influence of a magnetic field
Problem 7: Drude model applied to the reflectance of alkali metals in the ultraviolet and to characteristic electron energy losses
Problem 8: Dispersion of surface plasmons
Problem 9: Metallic superconductors, London equations, and the Meissner effect
Problem 10: Density of Cooper pairs in a metallic superconductor
Problem 11: Dispersion relation of electromagnetic waves in a two-fluid metallic superconductor
Solution:
Questions
5. Band Theory: Other Metals, Semiconductors, and Insulators
1. Introduction
2. Band Theory
3. Filling of Available States: The Fermi Surface
4. Density of states, effective mass, electrons, and holes
5. Success of Band Theory
6. Semiconductors (Generalities)
8. Different Types of Semiconductors
9. Allotropes of Carbon: Graphene, C-nanotubes, and Buckyballs
Exercises
Exercise 1: s-electrons bonded in a row of identical atoms: 1D
Exercise 2: Electrons bounded in a 2D lattice
Exercise 2b: Band structure of high Tcsuperconductors. Influence of 2D nearest neighbors (variation of Ex. 2)
Exercise 3: Tight binding in a simple cubic lattice (3D)
Exercise 3b: Tight bindings in the bcc and fcc lattices (variation of Ex. 3)
Exercise 4: Dimerization of a linear chain
Exercise 5: Conductors and Insulators
Exercise 5b: Nearly free electrons in a rectangular lattice
Exercise 6: Phase transition in the substitution alloys. Application to CuZn alloys
Exercise 7: Why nickel is ferromagnetic and copper is not
Exercise 8: Cohesive energy of transition metals
Exercise 9: Semi-metals
Exercise 10: Elementary study of an intrinsic semiconductor
Exercise 11: Density of states and bandgap
Exercise 12: Conductivity of semiconductors in the degenerate limit
Exercise 13: Carrier density of a degenerated semiconductor
Exercise 14: Semi-insulating gallium arsenide
Exercise 15: Intrinsic and extrinsic electrical conductivity of some semiconductors
Exercise 16: Impurity orbitals
Exercise 17: Donor ionization
Exercise 18: Hall effect in a semiconductor with two types of carriers
Exercise 19: Transverse magnetoresistance in a semiconductor with two types of carriers
Exercise 20: Excitons
Exercise 21: III–V compounds with a direct bandgap: Light and heavy holes
Exercise 22: Electronic specific heat of intrinsic semiconductors
Exercise 23: Specific heat and the bandgap in metallic superconductors
Exercise 24: The Burntein–Moss effect
Exercise 25: Bandgap, transparency, and dielectric constant of ionic crystals
Exercise 26: Dispersion of light: Sellmeier formula
Exercise 27: Back to the optical index and absorption coefficient of X-rays
Exercise 28: Optical absorption and colors of semiconductors and insulators
Exercise 29: Optoelectronic properties of III–V compounds
Exercise 30: The Gunn diode
Problems
Problem 1: Krönig–Penney Model. Periodic potential in 1D
Problem 2: Nearly free electrons in a 1D lattice
Problem 3: 1D semiconductor: electronic specific heat
Problem 4: DC conductivity of intrinsic and doped Ge and Si
Problem 5: Degenerated and nondegenerated semiconductors
Problem 6: Electron transitions: Optical properties of semiconductors and insulators
Problem 7: The p-n junction
Problem 8: The transistor
Problem 9: Electronic states in semiconductor quantum wells and superlattices
Problem 9b: Electronic states in 2D quantum wells (variation of Problem 9)
Problem 10: Band structure and optical properties of graphite in the ultraviolet
Problem 11: p-p* band structure of graphene
Problem 12: Single-wall-carbon nanotubes (SWCNTs)
Questions
Index
Biography
Jacques Cazaux (1934–2014) was emeritus professor at the University of Reims, France. He did his undergraduate work in physics at the University of Sorbonne, Paris, and obtained his PhD in 1970 from the College of France, Paris, by submitting the thesis titled Anisotropy of Plasmons in Graphite. He then joined the University of Reims as a professor of solid state physics, and there he initiated a research laboratory on surface analysis (XPS and Auger) and material characterization (electron probe microanalysis, electron and X-ray microscopies). His research focused on the physics of secondary electron emission, and he authored more than 150 articles published in scientific journals. In recognition of his special contribution to scientific knowledge, Prof. Cazaux was invited as a speaker at more than 50 international meetings and was on the board of several scientific committees.
"This book is an English translation of the late author’s French book Initiation à la physique du solide: exercices comméntes avec rappels de cours, and is a largely extended and up-to-date edition based on the rapid development in materials science, nanochemistry, and solid state physics. Beyond the standard discussion on crystal structures and diffraction, the author has included neutron diffraction, quantum Hall effects, and novel material like graphene. This highly recommendable book could benefit a broad readership, including graduate students and scientists involved with solid state matter, as it provides an integrated approach by posing numerous problems on each topic and then providing their solutions along with extended commentaries. It presents treatment and problems focused on the specific heat of atomic vibrations and lattices in particular and metals in relation to advancement in band theory, specifically including semiconductors and insulators."
—Axel Mainzer Koenig, CEO, 21st Century Data Analysis, Portland, USA
"This book is a compendium of questions and problems from essential topics in solid state physics. It is a translated version of an older French text and can serve as a strong supplement to a course on this subject. Overall, the author's main purpose is to assist students studying the subject. The author engages students through insightful questions and then, in most cases, provides detailed answers. Each chapter begins with a background summary of the relevant theory and equations for problem-solving, followed by numerous solved examples. At the end of every chapter, qualitative and quantitative questions are presented; brief answers are supplied at the end of the book. The work contains helpful diagrams and graphs."
— CHOICE