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Understanding Solid State Physics

Problems and Solutions

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## Book Description

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.

## Table of Contents

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 Ga*x*Al1–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 –2*q *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 *n*th 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 *q*D from elastic constants application to lithium

Problem 5: Specific heats at constant pressure *C*p and constant volume *C*v: (*C*p – *C*v) 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

## Author(s)

### 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.

## Reviews

"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