Chapter 1: Crystal Structure

Chapter 1: Crystal Structure

Chapter 1: Crystal Structure The Nobel “Booby” Prize!
See the “Ig Nobel” Prize discussed at:

The (Common) Phases of Matter
Gases Liquids & Liquid Crystals Solids This doesn’t include Plasmas, but these are the “common” phases!! “Condensed Matter” includes both of these. We’ll focus on Solids!

Gases Gases have atoms or molecules that do not bond to one another in a range of pressure, temperature & volume. Also, these molecules have no particular order & they move freely within a container.

Liquids & Liquid Crystals
Similar to gases, Liquids have no atomic or molecular order & they assume the shape of their containers. Applying low levels of thermal energy can easily break the existing weak bonds. Liquid Crystals have mobile molecules, but a type of long range order can exist; the molecules have a permanent dipole. Applying an electric field rotates the dipole & establishes order within the collection of molecules. 5

Solids Solids can be crystalline, polycrystalline, or amorphous.
Solids consist of atoms or molecules undergoing thermal motion about their equilibrium positions, which are at fixed points in space. Solids can be crystalline, polycrystalline, or amorphous. Solids (at a given temperature, pressure, volume) have stronger interatomic bonds than liquids. So, Solids require more energy to break the interatomic bonds than liquids.

Crystal Structure Topics 1. Periodic Arrays of Atoms
2. Fundamental Types of Lattices 3. Index System for Crystal Planes 4. Simple Crystal Structures 5. Direct Imaging of Crystal Structure 6. Non-ideal Crystal Structures 7. Crystal Structure Data

At the end of this Chapter, you should:
Objectives At the end of this Chapter, you should: 1. Be able to identify a unit cell in a symmetrical pattern. 2. Know that (in 3 dimensions) there are 7 (& ONLY 7!!) Possible unit cell shapes. 3. Be able to define cubic, tetragonal, orthorhombic & hexagonal unit cell shapes

Periodic Arrays of Atoms
Experimental Evidence of periodic structures. (See Kittel, Fig. 1.) The external appearance of crystals gives some clues to this. Fig. 1 shows that when a crystal is cleaved, we can see that it is built up of identical “building blocks”. Further, the early crystallographers noted that the index numbers that define plane orientations are exact integers. Cleaving a Crystal

Elementary Crystallography
Solid Material Types Crystalline Polycrystalline Amorphous Single Crystals

The Three General Types of Solids
Single Crystal Polycrystalline Amorphous Each type is characterized by the size of the ordered region within the material. An ordered region is a spatial volume in which atoms or molecules have a regular geometric arrangement or periodicity.

Crystalline Solids A Crystalline Solid is the solid form of a substance in which the atoms or molecules are arranged in a definite, repeating pattern in three dimensions. Single Crystals, ideally have a high degree of order, or regular geometric periodicity, throughout the entire volume of the material.

A Single Crystal has an atomic structure that repeats periodically across its whole volume. Even at infinite length scales, each atom is related to every other equivalent atom in the structure by translational symmetry. Single Pyrite Crystal Amorphous Solid Single Crystals

Polycrystalline Solids
A Polycrystalline Solid is made up of an aggregate of many small single crystals (crystallites or grains). Polycrystalline materials have a high degree of order over many atomic or molecular dimensions. These ordered regions, or single crystal regions, vary in size & orientation with respect to one another. These regions are called grains (or domains) & are separated from one another by grain boundaries. The atomic order can vary from one domain to the next. The grains are usually 100 nm microns in diameter. Polycrystals with grains that are < 10 nm in diameter are called nanocrystallites. Polycrystalline Pyrite Grain

Amorphous Solids Amorphous (Non-crystalline) Solids are composed of randomly orientated atoms, ions, or molecules that do not form defined patterns or lattice structures. Amorphous materials have order only within a few atomic or molecular dimensions. They do not have any long-range order, but they have varying degrees of short-range order. Examples of amorphous material include amorphous silicon, plastics, & glasses.

Departures From the “Perfect Crystal”
A “Perfect Crystal” is an idealization that does not exist in nature. In some ways, even a crystal surface is an imperfection, because the periodicity is interrupted there. Each atom undergoes thermal vibrations around their equilibrium positions for temperatures T > 0K. These can also be viewed as “imperfections”. Real Crystals always have foreign atoms (impurities), missing atoms (vacancies), & atoms in between lattice sites (interstitials) where they should not be. Each of these spoils the perfect crystal structure.

Crystallography Crystallography ≡ The branch of science that deals with the geometric description of crystals & their internal arrangements. It is the science of crystals & the math used to describe them. It is a VERY OLD field which pre-dates Solid State Physics by about a century! So (unfortunately, in some ways) much of the terminology (& theory notation) of Solid State Physics originated in crystallography. The purpose of Ch. 1 of Kittel’s book is mainly to introduce this terminology to you.

Solid State Physics Crystals Can Diffract X-rays Quantum Mechanics
Started in the early 20th Century when the fact that Crystals Can Diffract X-rays was discovered. Around that same time the new theory of Quantum Mechanics was being accepted & applied to various problems. Some of the early problems it was applied to were the explanation of observed X-ray diffraction patterns for various crystals & (later) the behavior of electrons in a crystalline solid.

A Basic Knowledge of Elementary Crystallography is Essential
for Solid State Physicists!!! A crystal’s symmetry has a profound influence on many of its properties. A crystal structure should be specified completely, concisely & unambiguously. Structures are classified into different types according to the symmetries they possess. In this course, we only consider solids with “simple” structures.

(Scanning Tunneling Microscope)
Crystal Lattice Crystallography focuses on the geometric properties of crystals. So, we imagine each atom replaced by a mathematical point at the equilibrium position of that atom. A Crystal Lattice (or a Crystal) ≡ An idealized description of the geometry of a crystalline material. A Crystal ≡ A 3-dimensional periodic array of atoms. Usually, we’ll only consider ideal crystals. “Ideal” means one with no defects, as already mentioned. That is, no missing atoms, no atoms off of the lattice sites where we expect them to be, no impurities,…Clearly, such an ideal crystal never occurs in nature. Yet, 85-90% of experimental observations on crystalline materials is accounted for by considering only ideal crystals! Platinum Surface (Scanning Tunneling Microscope) Crystal Lattice Structure of Platinum Platinum

A Lattice is Defined as an Infinite Array of Points in Space
Crystal Lattice Mathematically A Lattice is Defined as an Infinite Array of Points in Space in which each point has identical surroundings to all others. The points are arranged exactly in a periodic manner. 2 Dimensional Example  α a b C B E D O A y x 21

The simplest structural unit for a given solid is called the BASIS
Ideal Crystal ≡ An infinite periodic repetition of identical structural units in space. The simplest structural unit we can imagine is a Single Atom. This corresponds to a solid made up of only one kind of atom ≡ An Elemental Solid. However, this structural unit could also be a group of several atoms or even molecules. The simplest structural unit for a given solid is called the BASIS

Crystal Structure ≡ Lattice + Basis
The structure of an Ideal Crystal can be described in terms of a mathematical construction called a Lattice. A Lattice ≡ A 3-dimensional periodic array of points in space. For a particular solid, the smallest structural unit, which when repeated for every point in the lattice is called the Basis. The Crystal Structure is defined once both the lattice & the basis are specified. That is Crystal Structure ≡ Lattice + Basis

Crystalline Periodicity
In a crystalline material, the equilibrium positions of all the atoms form a crystal Crystal Structure ≡ Lattice + Basis For example, see Fig. 2. Lattice  2 Atom Basis  Crystal  Structure

Crystal Structure ≡ Lattice + Basis For another example, see the figure. Crystal Structure  Lattice  Basis 

Crystal Structure ≡ Lattice + Basis For another example, see the figure. Basis  Crystal Structure  Lattice 

A Two-Dimensional (Bravais) Lattice with Different Choices for the Basis

Lattice with atoms at the corners
2 Dimensional Lattice Lattice with atoms at the corners of regular hexagons α a b C B E D O A y x O A C B F b G D x y a E H 28

Crystal Structure = Lattice + Basis
The atoms do not necessarily lie at lattice points!! Crystal Structure = Lattice + Basis Basis   Crystal Structure 29 29

Chapter 2: Wave Diffraction & The Reciprocal Lattice

Chapter Topics 2. Bragg Law 3. Scattered Wave Amplitude
1. Wave Diffraction by Crystals 2. Bragg Law 3. Scattered Wave Amplitude 4. Reciprocal Lattice 5. Brillouin Zones 6. Fourier Analysis of the Basis

First, A Brief Optics Review X-Rays & Their Production
A Brief Review of the Optics needed to understand diffraction by crystalline solids. Discussion & Overview of: Diffraction X-Rays & Their Production

Optics Review: The Ray Model of Light
Specular Reflection ≡ Mirror-like Reflection Assume that light can be treated as a ray, with a single ray incident on the surface & a single ray reflected. This leads to: The Law of Reflection: Incident Angle = Reflected Angle.

Optics Review: Diffraction
LIGHT IS A WAVE, so it will diffract (bend) around a single slit or obstacle. Figure If light is a wave, a bright spot will appear at the center of the shadow of a solid disk illuminated by a point source of monochromatic light. Diffraction is not limited to visible light, but will happen with any wave, including other electromagnetic waves (X-Rays, ..) & including De Broglie waves associated with quantum mechanical particles (electrons, neutrons, ..)

Review of Diffraction Diffraction is a wave phenomenon. It is the apparent bending & spreading of waves when they meet an obstruction. Diffraction occurs with electromagnetic waves, such as light & radio waves, but also in sound waves & water waves. The most conceptually simple example of Diffraction is the double-slit diffraction of visible light. See the figure. Variable Slit Width ( nm) Constant Wavelength (600 nm) Distance d = Constant

Young’s Double Slit Experiment
Light Diffraction is caused by light bending around the edge of an object. The interference pattern of bright & dark lines from the diffraction experiment can only be explained by the additive nature of waves. Wave peaks can add together to make a bright line (or a peak in the intensity) or a dark line (a trough in the intensity from 2 waves cancelling each other out). Young’s Double Slit Experiment was the first to prove that light has wavelike properties.

Constructive & Destructive Interference of Waves
Constructive Interference is the result of synchronized light waves that add together in phase to give regions (lines) of increased intensity. Destructive Interference results when two out-of phase light waves cancel each other out, resulting in regions (lines) of darkness.

Light Diffraction

Diffraction Pattern on a Screen
Photo of Diffraction Pattern

Light Diffraction & Interference

The resulting pattern of light & dark stripes is called a Diffraction Pattern.
Figure Diffraction pattern of (a) a circular disk (a coin), (b) razor, (c) a single slit, each illuminated by a coherent point source of monochromatic light, such as a laser.

Wavelets that interfere with each other
This occurs because (by Huygens’ Principle) different points along a slit create Wavelets that interfere with each other just like a double slit. Also, for certain angles θ the diffracted rays from a slit of width D destructively interfere in pairs. Angles for destructive interference are: Dsinθ = mλ (m = 1, 2, 3, 4..) Figure Analysis of diffraction pattern formed by light passing through a narrow slit of width D.

The minima of the single-slit diffraction pattern occur when when
Figure Intensity in the diffraction pattern of a single slit as a function of sin θ. Note that the central maximum is not only much higher than the maxima to each side, but it is also twice as wide (2λ/D wide) as any of the others (only λ/D wide each).

Diffraction From a Particle & From a Solid
Diffraction from a Single Particle To understand diffraction we also must consider what happens when a wave interacts with a particle. The result is that A particle scatters the incident beam uniformly in all directions. Diffraction from a Solid Material What happens if the beam is incident on solid material? If it is a crystalline material, the result is that The scattered beams may add together in some directions & reinforce each other to give diffracted beams.

X-Rays & Their Properties
Review & Overview of X-Rays & Their Properties X-Rays were discovered in 1895 by German physicist Wilhelm Conrad Röntgen. They were called “X-Rays” because their nature was unknown at the time. He was awarded the Physics Nobel Prize in 1901. The 1st X-Ray photograph taken was of Röntgen’s wife’s left hand. Wilhelm Conrad Röntgen ( ) Bertha Röntgen’s Hand 8 Nov, 1895

Review of X-Ray Propertıes
X-Rays are invisible, highly penetrating Electromagnetic Radiation of much shorter wavelength (higher frequency) than visible light. Wavelength (λ) & frequency (ν) ranges for X-Rays: 10-8 m ~ ≤ λ ~ ≤ m 3 × 1016 Hz ~ ≤ ν ~ ≤ 3 × 1019 Hz

X-Ray Energies ν = Frequency c = Speed of Light λx-ray ≈ 10-10 m ≈ 1 Ǻ
In Quantum Mechanics, Electromagnetic Radiation is described as being composed of packets of energy, called photons. The photon energy is related to its frequency by the Planck formula: We also know that, in vacuum, the frequency & the wavelength are related as: Combining these gives: λ = Wavelength ν = Frequency c = Speed of Light λx-ray ≈ m ≈ 1 Ǻ  E ~ 104 eV

are produced by the movement This released energy is in
X-Ray Production Visible light photons, X-Ray photons, & essentially all other photons are produced by the movement of electrons in atoms. We know from Quantum Mechanics that electrons occupy energy levels, or orbitals, around an atom's nucleus. If an electron drops to a lower orbital (spontaneously or due to some external perturbation) it releases some energy. This released energy is in the form of a photon The photon energy depends on how far in energy the electron drops between orbitals.

Schematic Diagram of Photon Emission
Incoming particles excite an atom by promoting an electron to a higher energy orbit. Later, the electron falls back to the lower orbit, releasing a photon with energy equal to the energy difference between the two states: hν = ΔE Remember that this figure is a schematic “cartoon” only, shown to crudely illustrate how atoms emit light when one of the electrons transitions from one level to another. It gives the impression that the electrons in an atom are in Bohr-like orbits around the nucleus. From Quantum Mechanics, we know that this picture is not valid, but the electron wavefunction is spread all over the atom. So, don’t take this figure literally!

X-Ray Tubes X-Rays can be produced in a highly evacuated glass bulb, called an X-Ray tube, that contains two electrodes: an anode made of platinum, tungsten, or another heavy metal of high melting point, & a cathode. When a high voltage is applied between the electrodes, streams of electrons (cathode rays) are accelerated from the cathode to the anode & produce X-Rays as they strike the anode. Evacuated Glass Bulb Cathode Anode

Monochromatic & Broad Spectrum X-rays
X-Rays can be created by bombarding a metal target with high energy (> 104 eV) electrons. Some of these electrons excite other electrons from core states in the metal, which then recombine, producing highly monochromatic X-Rays. These are referred to as characteristic X-Ray lines. Other electrons, which are decelerated by the periodic potential inside the metal, produce a broad spectrum of X-Ray frequencies. Depending on the diffraction experiment, either or both of these X-Ray spectra can be used.

X-Ray Absorption The atoms that make up our body’s tissue absorb visible light photons very well. The energy level of the photon fits with various energy differences between electron states. Radio waves don't have enough energy to move electrons between orbitals in larger atoms, so they pass through most materials. X-Ray photons also pass through most things, but for the opposite reason: They have too much energy. You will never see something like this with Visible Light!!  X-Rays

Generation of X-rays (K-Shell Knockout) not to be taken literally!
An electron in a higher orbital falls to the lower energy level, releasing its extra energy in the form of a photon. It's a large drop, so the photon has high energy; it is an X-Ray photon. Another schematic cartoon diagram, not to be taken literally! A “free” electron collides with a tungsten atom, knocking an electron out of a lower orbital. A higher orbital electron fills the empty position, releasing its excess energy as an X-Ray Photon

X-Ray Absorption A larger atom is more likely to absorb an X-Ray Photon in this way than a smaller one because larger atoms have greater energy differences between orbitals.  The energy level difference then more closely matches the energy of an X-Ray Photon. Smaller atoms, in which the electron orbitals are separated by relatively low energy differences, are less likely to absorb X-Ray Photons. The soft tissue in our bodies is composed of smaller atoms, & so does not absorb X-Ray Photons very well. The calcium atoms that make up our bones are large, so they are better at absorbing X-Ray photons.

Brief Quantum Mechanics, Quantum Chemistry & Molecular Physics Review
Note!! The following discussion uses excerpts (with changes) from a lecture posted by Engineering Prof. McCoy at Sweet Briar College, Sweet Briar, VA.

Issues & Ideas to Discuss
1. What causes chemical bonding between atoms? 2. What types of chemical bonds are there? In crystalline solids, 3. What properties of a material can be inferred from understanding the bonding between atoms? An understanding of many of the physical properties of materials is predicated on a knowledge of the interatomic forces that bind the atoms together. We’ll start with a quick quantum mechanics & atomic-molecular physics (& chemistry!) review.

Bohr Model of the Atom We know that this model is wrong in detail, but it gives a qualitative understanding of electron orbits in atoms. Nucleus

What is an Electron? Diffraction & Wave/Particle Duality
Electron Probability Density Explain top figure in terms of dropping pebbles in water to define diffraction Explain diffraction pattern in terms of light Explain two slit setup. When you do it with electrons ONE AT A TIME you get wave particle duality Before it is detected, the electron exists as a probability wave, it “exists” at ALL POSSIBLE POINTS, but different points have different probabilities. The probability distribution is described by the Schroedinger wave equation. Also happens with PROTONS A good way to think of the “real” situation is to think of the electron shells as “probability clouds”, where you might find the electron. Young’s Double Slit Experiment - Light Young’s Double Slit Experiment - Electrons Electron Orbit Typical Diffraction Pattern for Waves

Electron Orbitals: – Probability Waves
The Quantum Mechanics of the atom (Schrödinger’s Equation) describes electrons in terms of probability distributions that can have only discrete values of energy. Atomic Orbital Shapes While the two-slit experiment seemed to show that light is a wave, other experiments at roughly the same time demonstrated that it was also a particle (photoelectric effect). Thus, light also falls under wave/particle duality. Furthermore, the amount of energy contained in one particle of light (photon) represents the SMALLEST POSSIBLE CHUNK OF ENERGY THAT YOU CAN HAVE. That is, ENERGY IS QUANTIZED. The size of the energy chunks is E=h*nu where nu is frequency. Planck’s constant h = 6.626· J·s What does this mean to us? Electrons can actually take on discrete energy states. At each shell level, there are a series of subshells that actually have slightly different energies. The different energy levels within a shell are often referred to as “subshells” and are lettered s, p, d, f These different shells are actually due to quantization of angular momentum. Within a subshell you can also have different values of the magnetic quantum numbers (energies change in a magnetic field), For s shell there is only 1 substate. For p shell there are 3; for d there are 5 Also, each electron has “spin” that can be up or down. Thus, for each subshell you have at least 2 different possible states. Pauli exclusion principle: No two electrons can occupy the same state. The Shapes of the atomic s, p, & d orbitals are predicted by Quantum Mechanics (Schrödinger’s Equation)

An s orbital holds 2 electrons with opposite spins
Atomic Structure An s orbital holds 2 electrons with opposite spins

Each p orbital holds 2e- with opposite spins

Each d orbital holds 2e- with opposite spins

Orbitals & the Periodic Table
The s suborbitals fill

The p suborbitals fill

The d suborbitals fill

Discrete Electron Energy Levels
The Discrete Energies of the atomic s, p, & d electrons are well-predicted by Quantum Mechanics (Schrödinger’s Equation) Quantum Mechanics (Schrödinger’s Equation) Bohr Model While the two-slit experiment seemed to show that light is a wave, other experiments at roughly the same time demonstrated that it was also a particle (photoelectric effect). Thus, light also falls under wave/particle duality. Furthermore, the amount of energy contained in one particle of light (photon) represents the SMALLEST POSSIBLE CHUNK OF ENERGY THAT YOU CAN HAVE. That is, ENERGY IS QUANTIZED. The size of the energy chunks is E=h*nu where nu is frequency. Planck’s constant h = 6.626· J·s What does this mean to us? Electrons can actually take on discrete energy states. At each shell level, there are a series of subshells that actually have slightly different energies. The different energy levels within a shell are often referred to as “subshells” and are lettered s, p, d, f These different shells are actually due to quantization of angular momentum. Within a subshell you can also have different values of the magnetic quantum numbers (energies change in a magnetic field), For s shell there is only 1 substate. For p shell there are 3; for d there are 5 Also, each electron has “spin” that can be up or down. Thus, for each subshell you have at least 2 different possible states. Pauli exclusion principle: No two electrons can occupy the same state.

Discrete Electron Energy Levels
The Discrete Energies of the atomic s, p, & d electrons are well-predicted by Quantum Mechanics (Schrödinger’s Equation) Quantum Mechanics (Schrödinger’s Equation) Later, we’ll see that in solids, the energy levels merge & broaden into bands & also shift, Leaving some gaps of forbidden energy. Bohr Model While the two-slit experiment seemed to show that light is a wave, other experiments at roughly the same time demonstrated that it was also a particle (photoelectric effect). Thus, light also falls under wave/particle duality. Furthermore, the amount of energy contained in one particle of light (photon) represents the SMALLEST POSSIBLE CHUNK OF ENERGY THAT YOU CAN HAVE. That is, ENERGY IS QUANTIZED. The size of the energy chunks is E=h*nu where nu is frequency. Planck’s constant h = 6.626· J·s What does this mean to us? Electrons can actually take on discrete energy states. At each shell level, there are a series of subshells that actually have slightly different energies. The different energy levels within a shell are often referred to as “subshells” and are lettered s, p, d, f These different shells are actually due to quantization of angular momentum. Within a subshell you can also have different values of the magnetic quantum numbers (energies change in a magnetic field), For s shell there is only 1 substate. For p shell there are 3; for d there are 5 Also, each electron has “spin” that can be up or down. Thus, for each subshell you have at least 2 different possible states. Pauli exclusion principle: No two electrons can occupy the same state.

Electronic Energy States
Note!!! The 4s state has a lower energy than the 3d state, so the N shell begins to fill before the M shell is filled. Electrons have discrete energy states Electrons will try to go to the lowest energy state (which means it takes more energy to remove it from the electron) When all electrons are in the lowest possible energy state the atom is in its ground state. Electrons in the outermost shell are called valence electrons. They are the primary driver in interactions with other atoms. Atoms are in stable configurations when the outermost shells are filled. Atoms in these states tend to be unreactive.

Atomic Electron Configurations for Some Elements
In most elements, the electron configuration is not stable. N Adapted from Table 2.2, Callister 6e. Why? Their Valence (outer) shell is usually not filled completely.

The Periodic Table Atoms in each column have similar Valence Electron Structure N Adapted from Fig. 2.6, Callister 6e. Electropositive Elements: Readily give up electrons to become + ions. Electronegative Elements: Readily acquire electrons to become - ions.

Smaller Electronegativity Larger Electronegativity
A measure of reactivity. It ranges from 0.7 to 4.0 Large Values  A larger tendency to acquire electrons. electronegativity increases going to the right and up Smaller Electronegativity Larger Electronegativity Adapted from Fig. 2.7, Callister 6e. (adapted from Linus Pauling, The Nature of the Chemical Bond, 3rd ed, Copyright 1939 & 1940, 3rd edition. Copyright 1960, Cornell U.).

The type and strength of the atomic bonds
Why have we been discussing all of this about electrons? Because the electron configurations determine how (& if) various atoms will bond. The type and strength of the atomic bonds determines material properties. A goal of this discussion: Understand why materials have the properties that they have. So, we’ll now look at different ways that atoms can form interatomic bonds, & consider the implications for material properties.

Bonding Force Bonding Energy
Discuss negative potential energy. Bonding Energy

Examples: Ionic Bonding
One or two (or more) electrons are transferred from an atom that has extra valence electrons to one that lacks them. Acquire electrons Give up electrons For 2 atoms to form an ionic bond, A large electronegativity difference is required.

Ionic Bonding Example: NaCl
This electron transfer leaves both atoms ionized, with opposite charges. They are then strongly attracted to each other through the Coulomb attraction: Example: NaCl

Ionic Bonding – Large Scale Structure (Solid)
Since a number of oppositely-charged ions are together, each ion of one charge will try to surround itself with atoms of the opposite charge. This causes ionic solids to form a 3-dimensional structure with alternating positive and negative ions, held together by relatively strong bonds: hard, brittle, insulating Do demo with strip magnets that repel when slid past each other. We would like to understand the strength, hardness, ductility, electrical conductivity, & other properties that result from this structure (as well as similar structures).

This is the primary bonding mechanism
Covalent Bonding Requires shared electrons Example: CH4 – C: has 4 valence e−, needs 4 more H: has 1 valence e−, needs 1 more Note! This depiction is very misleading! Electrons are NOT in Bohr like orbits but are waves spread out over the whole molecule!!! Their electronegativities are comparable. In a covalent bond, atoms share electrons so that both can assume states with a stable number of valence electrons. This is the primary bonding mechanism in many materials.

Examples: Covalent Bonding
3.5 Note that there can be a blend of ionic and covalent bonding. An atom with N valence electrons can bond with at most 8−N other atoms. Why?

Electrical conductivity
Metallic Bonding N Metal atoms have 1,2,or 3 valence electrons that are loosely bound. Each metal atom will “give up” these electrons to all of the other atoms to form a “sea of free electrons” in the metal The structure thus becomes a a group of ionized metal atom cores embedded in the “electron sea”. The “free” electrons bind the cores together and shield them from each other. We’re interested in how this structure affects: Electrical conductivity Thermal conductivity Ductility Strength Any other measurable properties. Do demo with strip magnets but slide them along the poles.

“Van der Waals” Bonding.
Secondary Bonding When atoms are covalently bonded, the resulting molecule will often have a positive end & a negative end (i.e. it forms a dipole). Molecules that are dipoles will be attracted to other dipoles through “Van der Waals” Bonding. These secondary, Van der Waals bonds tend to be weak compared to the primary bonds, but they can play an important role in determining the properties of polymeric materials & other molecular solids. General Case: Example: Liquid HCl Adapted from Fig. 2.14, Callister 6e. Example: Polymer

Aside: Win bets (. ), impress your friends (. ) & astound strangers (
Aside: Win bets (?), impress your friends (?) & astound strangers (?) by knowing the answer to this question: N Why does water expand when it freezes (unlike most materials, which contract when they freeze)? A water molecule is a strong dipole. Water molecules attract each other by “hydrogen bonding” - a special case of Van der Waals bonding. Discuss how this affects arctic sea ice and ice skating. When water freezes, the lattice structure that minimizes the potential energy due to hydrogen bonding has more space between atoms than liquid water (where the atoms are all jumbled together).

Summary: Bonding Types
Bond Type Bond Energy Material Type Ionic Large Ceramics, Salts Covalent Variable Polymers Large in Diamond Semiconductors Small in Bismuth Metallic Variable Metals Large in Tungsten Small in Mercury Secondary Smallest Solid Inert Gases (Van der Waals, Polymers Hydrogen)

Properties From Bonding:
Melting Temperature, Tm Look at the data & observe how the melting temperature varies with bonding energy. Why does it behave this way? Melting Temperature, Tm Tm is larger if |Eo| is larger.

Properties From Bonding:
Elastic Modulus, E How would you expect Young’s modulus to vary with the shape of the potential energy curve?

Properties From Bonding: Thermal Expansion Coefficient, a
Discuss anharmonicity of potential energy curve and how that results in thermal expansion. How would you expect Coefficient of thermal expansion to vary with the shape of the potential energy curve?

Summary: Primary Bond Categories
Ceramics Large bond energy large Tm large E small a (Ionic & covalent bonding): Metals Variable bond energy variable Tm variable E moderate a (Metallic bonding): Ceramics - Large bond energy large Tm large E small a Metals Variable bond energy moderate Tm moderate E moderate a Polymers Directional Properties Secondary bonding dominates small T small E large a Polymers Directional Properties Secondary bonding dominates small Tm small E large a (Covalent & Secondary Bonding):

Summary: Primary Bonding Categories
Material Type Bond Type Properties Ceramics Ionic & Large Bond Energy Covalent Large E Large Tm, Small α Metals Metallic Variable Bond Energy Variable E Variable Tm Moderate α Polymers Covalent & Directional Bonds Van der Waals Small E, Small Tm, (Van der Waals Large a dominates)

Chapter 4: Phonons I – Crystal Vibrations
A Cartoon About Solid State Chemistry!

Chapter 4: Phonons I – Crystal Vibrations
Change “Good-Chemist/Bad-Chemist” to “Good-Physicist/Bad-Physicist” This then becomes: A Cartoon About Solid State Physics!

Lattice Dynamics or “Crystal Dynamics”
Lattice Dynamics is a VERY LARGE subfield of solid state physics!

Lattice Dynamics is a VERY LARGE subfield of solid state physics! It is also a VERY OLD subfield!

Lattice Dynamics is a VERY LARGE subfield of solid state physics! It is also a VERY OLD subfield! It is also a “Dead” subfield!! That is, it is no longer an area of active research!

Most of our discussion will be very general & will apply to any crystalline solid.
We start the discussion more generally than Ch. 4 does: For all of this discussion, we will use a large amount of material from many sources outside of Ch. 4! At the beginning of this discussion, the material may seem abstract. But, don’t worry! Before this discussion is finished, it should hopefully be less abstract & it also should be a discussion that any upper level undergraduate in science or engineering should be able to understand.

From the theory viewpoint, a solid is a system with a VERY LARGE
Lets start the discussion more generally than Ch. 4 does. Some of what we discuss in the following will be useful to us later when we discuss electronic band structures. Specifically, we know that, From the theory viewpoint, a solid is a system with a VERY LARGE number of coupled atoms. The form of the coupling between the atoms depends on the type of bonding that holds the solid together. Many possible bonding mechanisms were discussed in Kittel’s Ch. 3.

Many Coupled Electrons & Nuclei. H = He + Hn + He-n
From the theory viewpoint, a solid is a system with a VERY LARGE number of coupled atoms. The form of the coupling between the atoms depends on the type of bonding that holds the solid together. Many possible bonding mechanisms were discussed in Kittel’s Ch. 3. A solid can be considered as a system of Many Coupled Electrons & Nuclei. So, lets start the discussion by looking at the many electron, many nuclei Hamiltonian H (total mechanical energy) for a solid. H = He + Hn + He-n He = Electron Kinetic Energy + Interactions with other Electrons Nuclear Kinetic Energy + Interactions with other Nuclei Electron-Nuclear Interaction Energy

H = He + Hn + He-n The Classical, Many-Body Hamiltonian
(Mechanical Energy) for the solid has the form: H = He + Hn + He-n Exactly solving the equations of motion resulting from this Hamiltonian is impractical & intractable, even with the most powerful computers of 2013! Some Approximations obviously must be made! By making some approximations, which are rigorously justified in many advanced texts, after a lot of work, the complexity of the problem is significantly reduced.

The Classical, Many-Body Hamiltonian
(Mechanical Energy) for the solid has the form: H = He + Hn + He-n Some Approximations obviously must be made! By making some approximations, which are rigorously justified in many advanced texts, after a lot of work, the complexity of the problem is significantly reduced. The usual starting point (without even acknowledgement that approximations are being made!) for MOST Solid State texts (including Kittel) is to discussions after such approximations have already been made. Later (Ch. 7) we’ll discuss that, for electronic properties calculations (electronic bands, etc.) these approximations reduce H a to a One Electron Hamiltonian! For the lattice vibrational problem of interest here, the two most important approximations will now be briefly discussed.

Nuclei  Ions  He-n  He-i Hn  Hi
Approximation #1: Separate the electrons into 2 types: Core Electrons & Valence Electrons Core Electrons ≡ Those in the filled, inner shells of the atoms. They play NO role in determining electronic properties of the solid! Example: The Si free atom electron configuration is: 1s22s22p63s23p2 Core Electrons = 1s22s22p6 (filled shells!) The core electrons are localized near the nuclei.  We lump the core shells & nuclei together. So, in the Hamiltonian, we make the replacements: Nuclei  Ions [Core Electron Shells + Nucleus  Ion Core]  He-n  He-i Hn  Hi

The Si free atom electron configuration is: 1s22s22p63s23p2
The Valence Electrons These are the electrons in the unfilled, outer shells of the free atoms. These determine the electronic properties of the solid & take part in the bonding! Example The Si free atom electron configuration is: 1s22s22p63s23p2 Valence Electrons = 3s23p2 (unfilled shell!) In the solid, these hybridize with electrons on neighboring atoms. This results in very strong covalent bonds with the 4 Si nearest-neighbors in the Si lattice

H = He + Hi + He-i One Electron Hamiltonian! The Ion Motion Part of H.
So, the Classical, Many-Body Hamiltonian (Mechanical Energy) for the solid is now: H = He + Hi + He-i Later, when we focus on electronic properties calculations (bandstructures, etc.), we will make some approximations, to reduce this many electron Hamiltonian to a One Electron Hamiltonian! Now, however, we will focus our attention on The Ion Motion Part of H. Electron-Ion Interaction Energy He = Electron Kinetic Energy + Interactions with other Electrons Ion Kinetic Energy + Interactions with other Ions

The Hamiltonian (Mechanical Energy) for a Perfect, Periodic Crystal: Ne electrons, Ni ions; Ne, Ni ~ 1023 (huge!) Notation: i = electron; j = ion The classical, many-body Hamiltonian is: (Gaussian units!) H = He + Hi + He-i He = Pure electronic energy = KE(e-) + PE(e- -e-) He= ∑i(pi)2/(2mi) + (½)∑i∑i´[e2/|ri - ri´|] (i  i´) Hi = Pure ion energy = KE(i) + PE(i-i) Hi = ∑j(Pj)2/(2Mj) + (½)∑j∑j´[ZjZj´ e2/|Rj - Rj´|] (j  j´) He-i = Electron-ion interaction energy = PE(e--i) He-i= - ∑i∑j[Zje2/|ri - Rj|] Lower case r, p, m: Electron position, momentum, mass Upper case R, P, M: Ion position, momentum, mass

much slower (me/Mi) ~ 10-3 (<< 1) (or smaller)
Approximation # 2: The Born-Oppenheimer (Adiabatic) Approximation This approximation allows the separation of the electron & ion motions. A rigorous proof of it requires detailed, many body Quantum Mechanics. Qualitative (semiquantitative) justification: The very small ratio of the electron & ion masses!! (me/Mi) ~ 10-3 (<< 1) (or smaller)  Classically, the massive ions move much slower than the very small mass electrons!

 As far as the electrons are concerned, the ions are ~ stationary!
Typical ionic vibrational frequencies: υi ~ 1013 s-1  The time scale of the ion motion is: ti ~ s Electronic motion occurs at energies of about a bandgap: Eg= hυe = ħω ~ 1 eV  υe ~ 1015 s-1  te ~ s So, classically, the Electrons Respond to the Ion Motion ~ Instantaneously!  As far as the electrons are concerned, the ions are ~ stationary!  In the electron Hamiltonian, He the ions can be treated as  stationary!

 In the ion Hamiltonian, Hi, the electrons can be treated
Born-Oppenheimer (Adiabatic) Approximation Now, lets look at the Ions: The massive ions cannot follow the rapid, detailed electron motion.  The Ions ~ see an Average Electron Potential.  In the ion Hamiltonian, Hi, the electrons can be treated in an average way!

Implementation: Born-Oppenheimer (Adiabatic) Approximation
Write the coordinates for the vibrating ions as Rj = Rjo + δRj, Rjo = equilibrium ion position δRj = (small) deviation from equilibrium position The many body electron-ion Hamiltonian is (schematic!): He-i ~ = He-i(ri,Rjo) + He-i(ri,δRj) The New many body Hamiltonian in this approximation is: H = He(ri) + He-i(ri,Rjo) + Hi(Rj) + He-i (ri,δRj) (1)

We will now focus on HI only.
The Many body Hamiltonian in this approximation: H = He(ri) + He-i(ri,Rjo) + Hi(Rj) + He-i (ri,δRj) (1) For electronic energy band structure calculations (Ch. 7 of Kittel), we will neglect the last 2 terms. However, we will Keep ONLY THEM for the vibrational properties calculations. Now, rewrite the Hamiltonian H (Eq. (1)) in the form: H  HE [1st 2 terms of (1)] + HI [2nd 2 terms of (1)] HE  Electronic Part (Gives energy bands. Ch. 7 of Kittel) HI  Ionic Part (Gives the lattice vibrations). We will now focus on HI only.

HI  j[(Pj)2/(2Mj)] + Ee(R1,R2,R3,…RN)
Before making the Born-Oppenheimer Approximation, the Ion Hamiltonian was: HI = Hi + He-i = j[(Pj)2/(2Mj)] + (½)jj´[ZjZj´e2/|Rj-Rj´|] - ij[Zje2/|ri-Rj|] Because the ion ions are moving (vibrations), the ion positions Rj are obviously time dependent. After a tedious implementation of the Born Oppenheimer Approximation, the Ion Hamiltonian becomes: HI  j[(Pj)2/(2Mj)] + Ee(R1,R2,R3,…RN) Ee  Average electronic total energy for all ions at positions Rj. Equivalently, Ee  Average of the ion-ion interaction + electron-ion interaction.

HI  j[(Pj)2/(2Mj)] + Ee(R1,R2,R3,…RN)
So, we will use the Ion Hamiltonian in the form: HI  j[(Pj)2/(2Mj)] + Ee(R1,R2,R3,…RN) Ee  Total (average) electronic ground state total energy of the many electron problem as a function of all ion positions Rj

Ee acts as an effective Potential for the ion motion
So, we will use the Ion Hamiltonian in the form: HI  j[(Pj)2/(2Mj)] + Ee(R1,R2,R3,…RN) Ee  Total (average) electronic ground state total energy of the many electron problem as a function of all ion positions Rj This results in the fact that Ee acts as an effective Potential for the ion motion

the many electron problem
So, the total average electronic ground state energy Ee acts as an effective potential energy for the ion motion. Note that Ee depends on the electronic states of all e- AND the positions of all ions! To calculate it from first principles, the many electron problem must first be solved!

With modern computational techniques, it is possible to:
1. Calculate Ee to a high degree of accuracy (as a function of the Rj). 2. Then, Use the calculated electronic structure of the solid to compute & predict it’s vibrational properties. This is a HUGE computational problem! With modern computers, this can be done & often is done. But, historically, this was very difficult or even impossible to do. Therefore, people used many different empirical models instead.

teach us something about
Lattice Dynamics Most work in this area was done long before the existence of modern computers! This is an OLD field. It is also essentially DEAD in the sense that little, if any, new research is being done. The work that was done in this field relied on phenomenological (empirical), non-first principles, methods. However, it is still useful to briefly look at some of these empirical models because doing so will (hopefully) teach us something about the physics of lattice vibrations.

HI  ∑j[(Pj)2/(2Mj)] + Eo(Rjo) + E'(δRj)
Consider the coordinates of each vibrating ion: Rj  Rjo + δRj Rjo  equilibrium ion position δRj  vibrational displacement amplitude As long as the solid is far from it’s melting point, it is always true that |δRj| << a, where a  lattice constant. If this were not true, the solid would melt or “fall apart”! Lets use this fact to expand Ee in a Taylor’s series about the equilibrium ion positions Rjo. In this approximation, the ion Hamiltonian becomes: HI  ∑j[(Pj)2/(2Mj)] + Eo(Rjo) + E'(δRj) (Note that this is schematic; the last 2 terms are functions of all j) Eo(Rjo) = a constant & irrelevant to the motion E'(δRj) = an effective potential for the ion motion

 “The Harmonic Approximation”
Now, expand E'(δRj) in a Taylor’s series for small δRj The expansion is about equilibrium, so the first-order terms in δRj = Rj - Rjo are ZERO. That is, (Ee/Rj)o = 0. Stated another way, at equilibrium, the total force on each ion j is zero by the definition of equilibrium! The lowest order terms are quadratic in the quantities ujk = (δRj - δRk) (j,k, neighbors) If the expansion is stopped at the quadratic terms, the Hamiltonian can be rewritten as the energy for a set of coupled 3-dimensional simple harmonic oscillators.  “The Harmonic Approximation”

The Harmonic Approximation
A change of notation! Replace the Ion Hamiltonian HI with the Vibrational Hamiltonian Hv. Hv = ∑j[(Pj)2/(2Mj)] + E'(δRj) E'(δRj) is a function of all δRj & is quadratic in the displacements δRj.

Thermal Conductivity Κ
For the rest of the discussion, we will only be discussing The Vibrational Hamiltonian Hv in the Harmonic Approximation: Hv = ∑j[(Pj)2/(2Mj)] + E'(δRj) E'(δRj) is a function of all δRj & is quadratic in the displacements δRj. Caution!! There are limitations to the harmonic approximation! Some phenomena are not explained by it. For these, higher order (Anharmonic) terms in the expansion of Ee must be used. Anharmonic terms are necessary to explain the observed Linear Expansion Coefficient α. In the harmonic approximation, α  0!! Thermal Conductivity Κ In the harmonic approximation, Κ  !!

Ukℓ  Displacement of Ion k in Cell ℓ Vibrational Hamiltonian
Now, hopefully, a notation simplification: Ukℓ  Displacement of Ion k in Cell ℓ Pkℓ  Momentum of Ion k in Cell ℓ & of course Pkℓ = Mk(dUkℓ/dt) (p = mv) With this change, the Vibrational Hamiltonian in the Harmonic Approximation is: Hv = (½)∑kℓMk(dUkℓ/dt) (½)∑kℓ∑kℓUkℓΦ(kℓ,kℓ)Ukℓ Φ(kℓ,kℓ)  “Force Constant Matrix” (or tensor) Hv = The standard classical Hamiltonian for a system of coupled simple harmonic oscillators!

Φ(kℓ,kℓ)  (∂2E′/∂Ukℓ∂Ukℓ)
Look at the details & find that the matrix elements of the force constant matrix Φ are proportional to 2nd derivatives of the total electronic energy function E′ Φ(kℓ,kℓ)  (∂2E′/∂Ukℓ∂Ukℓ) E′ = Ion displacement-dependent portion of the electronic total energy. So, in principle, Φ(kℓ,kℓ) could be calculated using results from the electronic structure calculation. This was impossible before the existence of modern computers. Even with computers it can be computationally intense!

Φ(kℓ,kℓ)  (∂2E′/∂Ukℓ∂Ukℓ) still useful to look at SOME
E′ = Ion displacement-dependent portion of the electronic total energy. Before computers, Φ(kℓ,kℓ) was usually determined empirically within various models. That is, it’s matrix elements were expressed in terms of parameters which were fit to experimental data. Even though we now can, in principle, calculate them exactly, it is still useful to look at SOME of these empirical models because doing so will (hopefully) TEACH US something about the physics of lattice vibrations in crystalline solids.

To illustrate the procedure for treating the interatomic potential in the harmonic approximation, consider just two neighboring atoms. Assume that they interact with a known potential V(r). See Figure. Expand V(r) in a Taylor’s series in displacements about the equilibrium separation, keeping only up through quadratic terms in the displacements: r2 r1 V(r) a Repulsive Attractive This potential energy is the same as that associated with a spring with spring constant K:

The Vibrational Hamiltonian
in the Harmonic Approximation has the form: Hv = (½)∑kℓMk(dUkℓ/dt) (½)∑kℓ∑kℓUkℓΦ(kℓ,kℓ)Ukℓ This is a Classical Hamiltonian! So, when we use it, we are obviously treating the motion classically. So we can describe lattice motion using Hamilton’s Equations of Motion or, equivalently, Newton’s 2nd Law!

Fkl = Mk(d2Ukℓ/dt2) = - ∑kℓΦ(kℓ,kℓ)Ukℓ
The Classical Newton’s 2nd Law Equations of Motion for a system of coupled harmonic oscillators are all of the form: (Analogous to F = ma = -kx for a single mass & spring): Fkl = Mk(d2Ukℓ/dt2) = - ∑kℓΦ(kℓ,kℓ)Ukℓ (These are “Hooke’s Law” type forces!) The Force Constant matrix Φ(kℓ,kℓ) has two physical contributions: 1. A direct, ion-ion, Coulomb repulsion 2. An Indirect interaction The 2nd one is mediated by the valence electrons. The motion of one ion causes a change in its electronic charge distribution & this causes a force on it’s ion neighbors.

GOAL of the Following Discussion:
We will use the Newton’s 2nd Law equations of motion to find the allowed vibrational frequencies in various materials. In classical mechanics (see Goldstein’s graduate text or any undergraduate mechanics text) this means  Finding the normal mode vibrational frequencies of the system. Here, only a brief outline or summary of the procedure will be given. So, this will be an outline of how “Phonon Dispersion Curves” ω(q) are calculated (q is a wavevector).

Finding the normal mode vibrational frequencies of the system.
So, for various models of the vibrating solid, we will be Finding the normal mode vibrational frequencies of the system. Here, only a brief outline or summary of the procedure will be given. So, this will be an outline of how “Phonon Dispersion Curves” ω(q) are calculated (q is a wavevector). I again emphasize that this is a Classical Treatment! That is, this treatment makes no direct reference to PHONONS. This is because Phonons are Quantum Mechanical Quasiparticles. Here, first we’ll outline the method to find the classical normal modes. Once those are found, then we can quantize & start talking about Phonons. Shortly (Ch. 5) we’ll briefly summarize phonons also.

 The NORMAL MODE FREQUENCIES
The Classical Treatment of the Vibrational Hamiltonian Hv As already mentioned, Hv  Energy of a collection of N coupled simple harmonic oscillators (SHO) The classical mechanics procedure to solve such a problem is: 1. Find a coordinate transformation to re-express Hv written in terms of N coupled SHO’s to Hv written in terms of N uncoupled (1d) (independent) SHO’s. 2. The frequencies of the new, uncoupled (1d) SHO’s are  The NORMAL MODE FREQUENCIES  The allowed vibrational frequencies for the solid. 3. The amplitudes of the uncoupled SHO’s are  The NORMAL MODE Coordinates  The amplitudes of the allowed vibrations for the solid.

Chapter 5: Phonons II – Thermal Properties

What is a Phonon? It is necessary to QUANTIZE these normal modes.
We’ve seen that the physics of lattice vibrations in a crystalline solid Reduces to a CLASSICAL normal mode problem. The goal of the entire discussion so far has been to find the normal mode vibrational frequencies of the crystalline solid. In the harmonic approximation, this is achieved by first writing the solid’s vibrational energy as a system of coupled simple harmonic oscillators & then finding the classical normal mode frequencies & ion displacements for that system. Given the results of the classical normal mode calculation for the lattice vibrations, in order to treat some properties of the solid, It is necessary to QUANTIZE these normal modes.

These quantized normal modes of vibration are called
PHONONS PHONONS are massless quantum mechanical “particles” which have no classical analogue. They behave like particles in momentum space or k space.

PHONONS “Quasiparticles”
These quantized normal modes of vibration are called PHONONS PHONONS are massless quantum mechanical “particles” which have no classical analogue. They behave like particles in momentum space or k space. Phonons are one example of many like this in many areas of physics. Such quantum mechanical particles are often called “Quasiparticles”

Examples of other Quasiparticles:
These quantized normal modes of vibration are called PHONONS PHONONS are massless quantum mechanical “particles” which have no classical analogue. They behave like particles in momentum space or k space. Phonons are one example of many like this in many areas of physics. Such quantum mechanical particles are often called “Quasiparticles” Examples of other Quasiparticles: Photons: Quantized Normal Modes of electromagnetic waves.

PHONONS “Quasiparticles”
These quantized normal modes of vibration are called PHONONS PHONONS are massless quantum mechanical “particles” which have no classical analogue. They behave like particles in momentum space or k space. Phonons are one example of many like this in many areas of physics. Such quantum mechanical particles are often called “Quasiparticles” Examples of other Quasiparticles: Photons: Quantized Normal Modes of electromagnetic waves. Rotons: Quantized Normal Modes of molecular rotational excitations. Magnons: Quantized Normal Modes of magnetic excitations in magnetic solids

“Quasiparticles” + Many Others!!!
These quantized normal modes of vibration are called PHONONS PHONONS are massless quantum mechanical “particles” which have no classical analogue. They behave like particles in momentum space or k space. Phonons are one example of many like this in many areas of physics. Such quantum mechanical particles are often called “Quasiparticles” Examples of other Quasiparticles: Photons: Quantized Normal Modes of electromagnetic waves. Rotons: Quantized Normal Modes of molecular rotational excitations. Magnons: Quantized Normal Modes of magnetic excitations in magnetic solids Excitons: Quantized Normal Modes of electron-hole pairs Polaritons: Quantized Normal Modes of electric polarization excitations in solids + Many Others!!!

Comparison of Phonons & Photons
Quantized normal modes of lattice vibrations. The energies & momenta of phonons are quantized PHOTONS Quantized normal modes of electromagnetic waves. The energies & momenta of photons are quantized Phonon Wavelength: λphonon ≈ a ≈ m (visible) Photon Wavelength: λphoton ≈ 10-6 m >> a

Quantum Mechanical Simple Harmonic Oscillator The energy is quantized.
Quantum mechanical results for a simple harmonic oscillator with classical frequency ω: The energy is quantized. n = 0,1,2,3,.. En E The energy levels are equally spaced!

(or “zero point”) Energy. The number of phonons is NOT conserved.
Often, we consider En as being constructed by adding n excitation quanta of energy to the ground state. Oscillator Ground State (or “zero point”) Energy. E0 = If the system makes a transition from a lower energy level to a higher energy level, it is always true that the change in energy is an integer multiple of Phonon Absorption or Emission ΔE = (n – n΄) n & n ΄ = integers In complicated processes, such as phonons interacting with electrons or photons, it is known that The number of phonons is NOT conserved. That is, phonons can be created & destroyed during such interactions.

Thermal Energy & Lattice Vibrations
As was already discussed in detail, the atoms in a crystal vibrate about their equilibrium positions. This motion produces vibrational waves. The amplitude of this vibrational motion increases as the temperature increases. In a solid, the energy associated with these vibrations is called the Thermal Energy 143

A knowledge of the thermal energy is fundamental to obtaining an understanding many of the basic properties (thermodynamic properties & others!) of solids. Examples Heat Capacity, Entropy, Helmholtz Free Energy, Equation of State, etc.... A relevant question is how is this thermal energy calculated? We would like to know how much thermal energy is available to scatter a conduction electron in a metal or a semiconductor. This is important because this scattering contributes to electrical resistance & other transport properties.

Thermal (Thermodynamic) Specific Heat or Heat Capacity
How is this thermal energy calculated? We would like to know how much thermal energy is available to scatter a conduction electron in a metal or a semiconductor. This is important because this scattering contributes to electrical resistance & other transport properties. Most important, the thermal energy plays a fundamental role in determining the Thermal (Thermodynamic) Properties of a Solid Knowledge of how the thermal energy changes wih temperature gives an understanding of heat energy necessary to raise the temperature of the material. An important, measureable property of a solid is its Specific Heat or Heat Capacity

Lattice Vibrational Contribution to
the Heat Capacity The Thermal Energy is the dominant contribution to the heat capacity in most solids. In non-magnetic insulators, it is the only contribution. Other contributions: Conduction Electrons in metals & semiconductors. Magnetic ordering in magnetic materials. The calculation of the vibrational contribution to the thermal energy & heat capacity of a solid has 2 parts: 1. Evaluation of the contribution of a single vibrational mode. 2. Summation over the frequency distribution of the modes.

Vibrational Specific Heat of Solids
cp Data at T = 298 K

Historical Background
In 1819, using room temperature data, Dulong and Petit empirically found that the molar heat capacity for solids is approximately Here, R is the gas constant. This relationship is now known as the Dulong-Petit “Law”

The Molar Heat Capacity
Assume that the heat supplied to a solid is transformed into the vibrational kinetic & potential energies of the lattice. To explain the Dulong-Petit “Law” theoretically, using Classical, Maxwell- Boltzmann Statistical Mechanics, a knowledge of how the heat is divided up among the degrees of freedom of the solid is needed.

The Molar Heat Capacity The Molar Energy of a Solid
The Dulong-Petit “Law” can be explained using Classical Maxwell-Boltzmann statistical physics. Specifically, The Equipartition Theorem can be used. This theorem states that, for a system in thermal equilibrium with a heat reservoir at temperature T, The thermal average energy per degree of freedom is (½)kT If each atom has 6 degrees of freedom: 3 translational & 3 vibrational, then R = NA k

The Molar Heat Capacity Heat Capacity at Constant Volume
By definition, the heat capacity of a substance at constant volume is Classical physics therefore predicts: A value independent of temperature

The Molar Heat Capacity
Experimentally, the Dulong-Petit Law, however, is found to be valid only at high temperatures.

Einstein Model of a Vibrating Solid
In 1907, Einstein extended Planck’s ideas to matter: he proposed that the energy values of atoms are quantized and proposed the following simple model of a vibrating solid: Each atom is independent Each vibrates in 3-dimensions Each vibrational normal mode has energy:

In effect, Einstein modeled one mole of a solid
as an assembly of 3NA distinguishable oscillators. He used the Canonical Ensemble to calculate the average energy of an oscillator in this model.

The Partition Function
To compute the average, note that it can be written as Z has the form: Here, b  [1/(kT)] Z is called The Partition Function

With b  [1/(kT)] and En = ne, , the partition
function Z for the Einstein Model is This follows from the geometric series result

Differentiating Z with
respect to b gives: Multiplying by –1/Z gives: This is the Einstein Model Result for the average thermal energy of an oscillator. The total vibrational energy of the solid is just 3NA times this result.

The Heat Capacity in the Einstein Model
is given by: Do the derivative & define TE  e/k. TE is called The Einstein Temperature Finally, in the Einstein Model, CV has the form:

The Einstein Model of a Vibrating Solid
Einstein, Annalen der Physik 22 (4), 180 (1907) CV for Diamond

Thermal Energy & Heat Capacity: The Einstein Model: Another Derivation
The following assumes that you know enough statistical physics to have seen the Cannonical Ensemble & the Boltzmann Distribution! The Quantized Energy of a Single Oscillator has the form: If the oscillator interacts with a heat reservoir at absolute temperature T, the probability Pn of it being in quantum level n is proportional to the Boltzmann Factor: Pn 

Quantized Energy of a Single Oscillator:
In the Cannonical Ensemble, a formal expression for the average energy of the harmonic oscillator & therefore of a lattice normal mode of angular frequency ω at temperature T is given by: The probability Pn of the oscillator being in quantum level n has the form: Pn  [exp (-β)/Z] where the partition function Z is given by:

Average Energy: Now, some straightforward math manipulation!
Putting in the explicit form gives: According to the Binomial expansion, for x << 1 where

The equation for ε can be rewritten:
Finally, the result is:

This is the Mean Phonon Energy. The first term in
(1) This is the Mean Phonon Energy. The first term in (1) is called the Zero-Point Energy. As mentioned before, even at 0ºK the atoms vibrate in the crystal & have a zero-point energy. This is the minimum energy of the system. The thermal average number of phonons n(ω) at Temperature T is given by The Bose-Einstein Distribution, & the denominator of the second term in (1) is often written:

<> = ћω[n() + ½] The number of phonons at temperature
(1) (2) By using (2) in (1), (1) can be rewritten: <> = ћω[n() + ½] In this form, the mean energy <> looks analogous to a quantum mechanical energy level for a simple harmonic oscillator. That is, it looks similar to: So the second term in the mean energy (1) is interpreted as The number of phonons at temperature T & frequency ω.

ħω << kBT High Temperature Limit: Temperature dependence of
the mean energy <> of a quantum harmonic oscillator. Taylor’s series expansion of ex  for x << 1 High Temperature Limit: ħω << kBT At high T, <> is independent of ω. This high T limit is equivalent to the classical limit, (the energy steps are small compared to the total energy). So, in this case, <> is the thermal energy of the classical 1D harmonic oscillator (given by the equipartition theorem).

ħω > > kBT “Zero Point Energy”
Temperature dependence of the mean energy <> of a quantum harmonic oscillator. Low Temperature Limit: ħω > > kBT “Zero Point Energy” At low T, the exponential in the denominator of the 2nd term gets larger as T gets smaller. At small enough T, neglect 1 in the denominator. Then, the 2nd term is e-x, x = (ħω/(kBT). At very small T, e-x  0. So, in this case, <> is independent of T: <>  (½)ħω

Einstein Heat Capacity CV
The heat capacity CV is found by differentiating the average phonon energy: Let

Einstein Heat Capacity CV
The specific heat CV in this approximation vanishes exponentially at low T & tends to the classical value at high T. These features are common to all quantum systems; the energy tends to the zero- point-energy at low T & to the classical value at high T. where Area =

The specific heat at constant volume Cv depends
qualitatively on temperature T as shown in the figure below. For high temperatures, Cv (per mole) is close to 3R (R = universal gas constant. R  2 cal/K- mole). So, at high temperatures Cv  6 cal/K-mole The figure shows that Cv = 3R at high temperatures for all substances. This is the classical Dulong-Petit law. This states that specific heat of a given number of atoms of any solid is independent of temperature & is the same for all materials!

Einstein Model for Lattice Vibrations in a Solid Cv vs T for Diamond
Einstein, Annalen der Physik 22 (4), 180 (1907) Points: Experiment Curve: Einstein Model Prediction

Einstein Model of Heat Capacity of Solids
The Einstein Model was the first quantum theory of lattice vibrations in solids. He made the assumption that all 3N vibrational modes of a 3D solid of N atoms had the same frequency, so that the whole solid had a heat capacity 3N times In this model, the atoms are treated as independent oscillators, but the energies of the oscillators are the quantum mechanical energies. This assumes that the atoms are each isolated oscillators, which is not at all realistic. In reality, they are a huge number of coupled oscillators. Even this crude model gives the correct limit at high temperatures, where it reproduces the Dulong-Petit law of 3R per mole.

At high temperatures, all crystalline solids have a vibrational specific heat of 6 cal/K per mole; they require 6 calories per mole to raise their temperature 1 K. This arrangement between observation and classical theory breaks down if the temperature is not high. Observations show that at room temperatures and below the specific heat of crystalline solids is not a universal constant. In each of these materials (Pb,Al, Si,and Diamond) specific heat approaches a constant value asymptotically at high T. But at low T, the specific heat decreases towards zero which is in a complete contradiction with the above classical result.

The Einstein model also gives correctly a specific heat tending to zero at absolute zero, but the temperature dependence near T= 0 does not agree with experiment. Taking into account the actual distribution of vibration frequencies in a solid this discrepancy can be accounted using one dimensional model of monoatomic lattice

Thermal Energy & Heat Capacity
Debye Model Density of States According to Quantum Mechanics if a particle is constrained; the energy of particle can only have special discrete energy values. it cannot increase infinitely from one value to another. it has to go up in steps.

This is the case of classical mechanics.
These steps can be so small depending on the system that the energy can be considered as continuous. This is the case of classical mechanics. But on atomic scale the energy can only jump by a discrete amount from one value to another. Definite energy levels Steps get small Energy is continuous

In some cases, each particular energy level can be associated with more than one different state (or wavefunction ) This energy level is said to be degenerate. The density of states is the number of discrete states per unit energy interval, and so that the number of states between and will be

There are two sets of waves for solution; Running waves Standing waves
These allowed k wavenumbers correspond to the running waves; all positive and negative values of k are allowed. By means of periodic boundary condition an integer Length of the 1D chain These allowed wavenumbers are uniformly distibuted in k at a density of between k and k+dk. running waves

for running waves for standing waves Standing waves:
In some cases it is more suitable to use standing waves,i.e. chain with fixed ends. Therefore we will have an integral number of half wavelengths in the chain; These are the allowed wavenumbers for standing waves; only positive values are allowed. for running waves for standing waves

DOS of standing wave DOS of running wave
These allowed k’s are uniformly distributed between k and k+dk at a density of DOS of standing wave DOS of running wave The density of standing wave states is twice that of the running waves. However in the case of standing waves only positive values are allowed Then the total number of states for both running and standing waves will be the same in a range dk of the magnitude k The standing waves have the same dispersion relation as running waves, and for a chain containing N atoms there are exactly N distinct states with k values in the range 0 to

The density of states per unit frequency range g():
The number of modes with frequencies  &  + d will be g()d. g() can be written in terms of S(k) and R(k). modes with frequency from  to +d corresponds to modes with wavenumber from k to k+dk

; Choose standing waves to obtain
Let’s remember dispertion relation for 1D monoatomic lattice

Multibly and divide Let’s remember: True density of states

True DOS(density of states) tends to infinity at ,
True density of states by means of above equation constant density of states True DOS(density of states) tends to infinity at , since the group velocity goes to zero at this value of . Constant density of states can be obtained by ignoring the dispersion of sound at wavelengths comparable to atomic spacing.

The energy of lattice vibrations will then be found by integrating the energy of single oscillator over the distribution of vibration frequencies. Thus for 1D Mean energy of a harmonic oscillator One can obtain same expression of by means of using running waves. It should be better to find 3D DOS in order to compare the results with experiment.

y + - L - + + - x L 3D DOS Let’s do it first for 2D, then for 3D.
Consider a crystal in the shape of 2D box with crystal sides L. y + - L - + + - x L Standing wave pattern for a 2D box Configuration in k-space

Let’s calculate the number of modes within a range of wavevector k.
Standing waves are choosen but running waves will lead same expressions. Standing waves will be of the form Assuming the boundary conditions of Vibration amplitude should vanish at edges of Choosing positive integer

y + - L - + + - x L Standing wave pattern for a 2D box
x L Standing wave pattern for a 2D box Configuration in k-space The allowed k values lie on a square lattice of side in the positive quadrant of k-space. These values will so be distributed uniformly with a density of per unit area. This result can be extended to 3D.

L L L Octant of the crystal: kx,ky,kz(all have positive values)
The number of standing waves; L L

is a new density of states defined as the number of states per unit magnitude of in 3D.This eqn can be obtained by using running waves as well. (frequency) space can be related to k-space: Let’s find C at low and high temperature by means of using the expression of

High and Low Temperature Limits
Each of the 3N lattice modes of a crystal containing N atoms = This result is true only if at low T’s only lattice modes having low frequencies can be excited from their ground states; long  w Low frequency sound waves k

and at low T depends on the direction and there are two transverse, one longitudinal acoustic branch: Velocities of sound in longitudinal and transverse direction

Zero point energy = at low temperatures

Debye Model of Heat Capacity of Solids

How good is the Debye approximation at low T?
The lattice heat capacity of solids thus varies as T3 at low temperatures; this is referred to as the Debye T3 law. The figure illustrates the excellent agreement of this prediction with experiment for a non-magnetic insulator. The heat capacity vanishes more slowly than the exponential behaviour of a single harmonic oscillator because the vibration spectrum extends down to zero frequency.

The Debye interpolation scheme
The calculation of is a very heavy calculation for 3D, so it must be calculated numerically. Debye obtained a good approximation to the resulting heat capacity by neglecting the dispersion of the acoustic waves, i.e. assuming for arbitrary wavenumber. In a one dimensional crystal this is equivalent to taking as given by the broken line of density of states figure rather than full curve. Debye’s approximation gives the correct answer in either the high and low temperature limits, and the language associated with it is still widely used today.

The Debye approximation has two main steps:
1. Approximate the dispersion relation of any branch by a linear extrapolation of the small k behaviour: Debye approximation to the dispersion Einstein approximation to the dispersion

Debye cut-off frequency
2. E nsure the correct number of modes by imposing a cut-off frequency , above which there are no modes. The cut-off freqency is chosen to make the total number of lattice modes correct. Since there are 3N lattice vibration modes in a crystal having N atoms, we choose so that

The lattice vibration energy of
becomes and, First term is the estimate of the zero point energy, and all T dependence is in the second term. The heat capacity is obtained by differentiating above eqn wrt temperature.

The heat capacity is Let’s convert this complicated integral into an expression for the specific heat changing variables to and define the Debye temperature

The Debye prediction for lattice specific heat
where

How does limit at high and low temperatures?
High temperature  x is always small 

We obtain the Debye law in the form
How does limit at high and low temperatures? Low temperature For low temperature the upper limit of the integral is infinite; the integral is then a known integral of T < < We obtain the Debye law in the form

Lattice heat capacity due to Debye interpolation scheme
Figure shows the heat capacity between the two limits of high and low T as predicted by the Debye interpolation formula. 1 Because it is exact in both high and low T limits the Debye formula gives quite a good representation of the heat capacity of most solids, even though the actual phonon-density of states curve may differ appreciably from the Debye assumption. Lattice heat capacity of a solid as predicted by the Debye interpolation scheme 1 Debye frequency and Debye temperature scale with the velocity of sound in the solid. So solids with low densities and large elastic moduli have high Values of for various solids is given in table. Debye energy can be used to estimate the maximum phonon energy in a solid. Solid Ar Na Cs Fe Cu Pb C KCl

5335 Homepage: http://www.phys.ttu.edu/~cmyles/Phys5335/5335.html.
Electronic Bandstructures Information from Kittel’s book (Ch. 7) + many outside sources. Some lectures on energy bands will be based on those prepared for Physics 5335 Semiconductor Physics. That course was taught last in the Fall of It is scheduled to be taught next in the Fall of 2012!! As discussed at the start of the semester, Phys clearly has overlap with this Solid State course, but the 2 courses are complementary & are NOT the same. I encourage you to take Phys. 5335! More information (last update, Dec., 2010!!) about Phys. 5335 is found on the course webpage: 5335 Homepage: 5335 Lecture Page:

Bandstructure  E(k) We are interested in understanding the PHYSICS of the behavior of electronic energy levels in crystalline solids as a function of wavevector k or momentum p = ħk. Much of our discussion will be valid in general, for metals, insulators, & semiconductors. Group Theory: The math of symmetry that can be useful to simplify calculations of E(k). We won’t be doing this in detail. But we will introduce some of it’s results & notation. Bandstructure Theory: A Mathematical Subject! Detailed math coverage will be kept to a minimum. Results and the PHYSICS will be emphasized over math! Many Methods to Numerically Calculate E(k) Exist: They are highly sophisticated & computational! We’ll only have an overview of the most important methods.

Basic knowledge that I must assume that you know:
1. Electron energies are quantized (discrete). 2. Have at least seen the Schrödinger Equation The fundamental equation that governs (non-relativistic) quantum mechanics If you are weak on this or need a review, get & read an undergraduate quantum mechanics book! 3. Understand the basic Crystal Structures of some common crystals (as in Kittel’s Ch. 1). 4. In a crystal, electronic energy levels form into regions of allowed energy (bands) & forbidden energy (gaps).

Electronic Energy Bands 
Overview A qualitative & semi-quantitative treatment now. Later, a more detailed & quantitative treatment. Electronic Energy Bands  Bandstructure  E(k) Gives the dependence of the electronic energy levels in crystalline materials as a function of wavevector k or momentum p = ħk. For FREE electrons, E(k) = (p)2/2mo = (ħk)2/2mo, (mo= free electron mass) In Crystalline Solids, E(k) forms into regions of allowed energies (bands) & regions of forbidden energies (gaps).

The E(k) can be complicated!
As we’ll see later, Often in solids, for k in some high symmetry regions of the 1st Brillouin Zone, a good Approximation is: E(k) = (ħk)2/2m* m* is not mo! m* = m*(k) = “effective mass” The E(k) can be complicated! Calculation of E(k) requires sophisticated quantum mechanics AND computational methods to obtain them numerically!

Bands In “r-space” (functions of position in the solid)
One way to distinguish between solid types is by how the electrons fill the bands & by the band gaps! Note: Bandstructures E(k) are bands in “k-space” (functions of k or momentum in solids), which is a completely different picture than is shown here.

Electronic Energy Bands Qualitative Picture
Electrons occupying the quantized energy states around a nucleus must obey the Pauli Exclusion Principle. This prevents more than 2 electrons (of opposite spin) from occupying the same level. Allowed band Forbidden gap Allowed band Forbidden gap Allowed band …………N Number of Atoms

On solving the Schrödinger Equation, it is found that
In a solid, the energy differences between each of the discrete levels is so tiny that it is reasonable to consider each of these sets of energy-levels as being continuous BANDS of energy, rather than considering an enormous number of discrete levels. On solving the Schrödinger Equation, it is found that 1. There are regions of energy E for which a solution exists (that is, where E is real). These regions are called the Allowed Bands (or just the Bands!). and 2. There are regions of energy E for which no solution exists (for real E). These regions are called the Forbidden Gaps. Obviously, electrons can only be found in the Allowed Bands & they can’t be found in the Forbidden Gaps.

Semiconductors, Insulators, Conductors
Full Band Empty Band All energy levels are empty All energy levels are occupied (contain electrons) It can be shown that Neither full nor empty bands participate in electrical conduction.

Calculated Si Bandstructure
GOALS After this chapter, you should: 1. Understand the underlyingPhysics behind the existence of bands & gaps. 2. Understand how to interpret this figure. 3. Have a rough, general idea about how realistic bands are calculated. 4. Be able to calculate energy bands for some simple models of a solid. Note Si has an indirect band gap!  Eg

Brief Quantum Mechanics (QM) Review
QM results that I must assume that you know! The Schrödinger Equation (time independent; see next slide) describes electrons. Solutions to the Schrödinger Equation result in quantized (discrete) energy levels for electrons.

Quantum Mechanics (QM)
The Schrödinger Equation: (time independent) Hψ = Eψ (this is a differential eigenvalue equation) H  Hamiltonian operator for the system (energy operator) E  Energy eigenvalue, ψ  wavefunction Particles are QM waves! |ψ|2  probability density ψ is a function of ALL coordinates of ALL particles in the problem!

The Physics Behind E(k)
E(k)  Solutions to the Schrödinger Equation for an electron in a solid. QUESTIONS Why (qualitatively) are there bands? Why (qualitatively) are there gaps?

These can be understood from 2 very different qualitative pictures!!
Bands & Gaps These can be understood from very different qualitative pictures!! The 2 pictures are models & are the Opposite Limiting Cases of the true situation! Consider an electron in a perfectly periodic crystalline solid: The potential seen by this electron is perfectly periodic The existence of this periodic potential is  the cause of the bands & the gaps!

Almost Free Electrons (done in detail in Kittel’s Ch. 7!)
Qualitative Picture #1 “A Physicist’s viewpoint”- The solid is looked at “collectively” Almost Free Electrons (done in detail in Kittel’s Ch. 7!) For free electrons, E(k) = (p)2/2mo = (ħk)2/2mo Almost Free Electrons: Start with the free electron E(k), add small (weakly perturbing) periodic potential V. This breaks up E(k) into bands (allowed energies) & gaps (forbidden energy regions). Gaps: Occur at the k where the electron waves (incident on atoms & scattered from atoms) undergo constructive interference (Bragg reflections!)

The Pseudopotential Method (the modern method of choice!)
Qualitative Picture #1 Forms the basis for REALISTIC bandstructure computational methods! Starting from the almost free electron viewpoint & adding a high degree of sophistication & theoretical + computational rigor:  Results in a method that works VERY WELL for calculating E(k) for metals & semiconductors! An “alphabet soup” of computational techniques: OPW: Orthogonalized Plane Wave method APW: Augmented Plane Wave method ASW: Antisymmetric Spherical Wave method Many, many others The Pseudopotential Method (the modern method of choice!)

Atomic / Molecular Electrons
Qualitative Picture #2 “A Chemist’s viewpoint”- The solid is looked at as a collection of atoms & molecules. Atomic / Molecular Electrons Atoms (with discrete energy levels) come together to form the solid Interactions between the electrons on neighboring atoms cause the atomic energy levels to split, hybridize, & broaden. (Quantum Chemistry!) First approximation: Small interaction V! Occurs in a periodic fashion (the interaction V is periodic). Groups of levels come together to form bands (& also gaps). The bands E(k) retain much of the character of their “parent” atomic levels (s-like and p-like bands, etc.) Gaps: Also occur at the k where the electron waves (incident on atoms & scattered from atoms) undergo constructive interference (Bragg reflections!)

The Pseudopotential Method (the modern method of choice!)
Qualitative Picture #2 Forms the basis for REALISTIC bandstructure computational methods! Starting from the atomic / molecular electron viewpoint & adding a high degree of sophistication & theoretical & computational rigor  Results in a method that works VERY WELL for calculating E(k) (mainly the valence bands) for insulators & semiconductors! (Materials with covalent bonding!) An “alphabet soup” of computational techniques: LCAO: Linear Combination of Atomic Orbitals method LCMO: Linear Combination of Molecular Orbitals method The “Tightbinding” method & many others. The Pseudopotential Method (the modern method of choice!)

Theories of Bandstructures in Crystalline Solids
 Pseudopotential Method   Tightbinding (LCAO) Method   Electronic Interaction                              Semiconductors, Insulators Metals Almost Free Electrons Molecular Electrons Isolated Atom, Atomic Electrons Free Electrons

5335 Homepage: http://www.phys.ttu.edu/~cmyles/Phys5335/5335.html.
Introduction to Semiconductors Information from Kittel’s book (Ch. 8) + many outside sources. Most lectures on semiconductors will be based on those prepared for Physics 5335 Semiconductor Physics. That course was taught last in the Fall of It is scheduled to be taught next in the Fall of 2012!! As discussed at the start of the semester, Phys clearly has overlap with this Solid State course, but the 2 courses are complementary & are NOT the same. I encourage you to take Phys. 5335! More information (last update, Dec., 2010!!) about Phys is on the course webpage: 5335 Homepage: 5335 Lecture Page:

An Alternate Semiconductor Definition!

What is a Semiconductor? Kittel Ch. 8 & many other sources
Classification of Solids by their Conductivity/Resistivity (σ = conductivity, ρ = resistivity) Metals: Good Conductors! 103 ≤ σ ≤ 108 (Ω-cm)-1 & ≤ ρ ≤ Ω-cm Insulators: Poor Conductors! σ ≤ 10-8 (Ω-cm)-1; ρ ≥ Ω-cm Semiconductors/Semimetals: 10-8 ≤ σ ≤ 103 (Ω-cm)-1; ≤ ρ ≤ Ω-cm Note the HUGE range!! Note also that there are no rigid boundaries!

Semiconductors - Conductivity/Resistivity Definition
 Metals   Semimetals 

Semiconductors - Bandgap Definition
Semiconductors are ~ Small Bandgap Insulators (we defined bandgap Eg earlier). Strictly speaking, a semicondcutor must also be capable of being doped (we’ll define doping later). Typical Bandgaps Semiconductors: 0 ~ ≤ Eg ≤ ~ 3 eV Metals & Semimetals: Eg = 0 eV Insulators: Eg ≥ 3 eV An Exception is Diamond: Eg = ~ 6 eV. Diamond is usually an insulator, but it can be doped & used as a semiconductor! Also, sometimes there is confusing terminology like: GaAs: Eg = 1.5 eV is sometimes called semi-insulating!

More Semiconductor Characteristics
In pure materials (very rare): The electrical conductivity σ  exp(cT) T = Kelvin Temperature, c = constant Impure materials (most): The electrical conductivity σ depends strongly on impurity concentrations. “Doping” means to add impurities to change σ The electrical conductivity σ can be changed by light or electron radiation & by injection of electrons at contacts Transport of charge can occur by the motion of electrons or holes (defined later).

The Best Known Semiconductor is Silicon (Si)
However, there are HUNDREDS (maybe THOUSANDS) of others! Elemental: Si, Ge, C (diamond) Binary Compounds: GaAs, InP, . Organic Compounds: (CH)n (polyacetyline) Magnetic Semiconductors: CdxMn1-xTe, … Ferroelectric Semiconductors: SbI, … Superconducting Compounds: GeTe, SrTiO3, (“High Tc materials”)

The Periodic Table The Relevant Parts for Elemental & Binary Semiconductors
III IV V VI II II Group IV Materials & III-V & II-VI Compounds

The Periodic Table Cloth!

Group IV Elements & III-V & II-VI Compounds

Group IV Elements, III-V & II-VI Compounds
Diamond Lattice  Zincblende or Wurtzite Lattices Diamond→     (α-Sn or gray tin)  Band gap (mostly) decreases, n-n distance (mostly) increases within a row going from IV elements to III-V compounds to II-VI compounds. Band gap (mostly) decreases, n-n distance (mostly) increases going from IV elements to III-V compounds to II-VI compounds. Band gap (mostly) decreases, n-n distance (mostly) increases going down a column.

Many Materials of Interest:
Have crystal lattice structures = Diamond or Zincblende (discussed in detail earlier!): Each atom is tetrahedrally coordinated with four (4) nearest-neighbors. The bonding is (mostly) sp3 hybrid bonding (strongly covalent). Two atoms/unit cell (repeated to form an infinite solid).

Zincblende (ZnS) Lattice
Zincblende Lattice The Cubic Unit Cell. If all atoms are the same, it becomes the Diamond Lattice! Zincblende Lattice A Tetrahedral Bonding Configuration

Zincblende/Diamond Lattices
The Cubic Unit Cell Zincblende Lattice The Cubic Unit Cell Semiconductor Physicists & Engineers need to know these structures!

Semiconductor Physicists & Engineers need to know these structures!
Diamond Lattice Diamond Lattice The Cubic Unit Cell Semiconductor Physicists & Engineers need to know these structures!

Zincblende (ZnS) Lattice
Zincblende Lattice The Cubic Unit Cell.

Some Materials of Interest:
Have crystal lattice structures = Wurtzite Structure (discussed in detail earlier!): Similar to Zincblende, but has hexagonal symmetry instead of a cubic. Each atom is tetrahedrally coordinated with four (4) nearest-neighbors. The bonding is (mostly) sp3 hybrid bonding (strongly covalent). Two atoms/unit cell (repeated to form an infinite solid).

Semiconductor Physicists & Engineers need to know these structures!
Wurtzite Lattice Semiconductor Physicists & Engineers need to know these structures!

Chapter 1: Crystal Structure

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Chapter 1: Crystal Structure

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