Contents

Preface

1 Chemical Bonding in Solids

1.1 Attractive and Repulsive Forces

1.2 Ionic Bonding

1.3 Covalent Bonding

1.4 Metallic Bonding

1.5 Hydrogen Bonding

1.6 van der Waals Bonding

1.7 Discussion and Problems

2 Crystal Structures

2.1. General Description of Crystal Structures

2.2. Some Important Crystal Structures

2.3. Crystal Structure Determination

2.4. Discussion and Problems

3 Mechanical Properties

3.1 Elastic Deformation

3.2 Plastic Deformation

3.3 Discussion and Problems

4 Thermal Properties of the Lattice

4.1 Lattice Vibrations

4.2 Heat Capacity of the Lattice

4.3 Thermal Conductivity

4.4 Thermal Expansion

4.5 Allotropic Phase Transitions and Melting

4.6 Discussion and Problems

5 Electronic Properties of Metals: Classical Approach

5.1 Basic Assumptions of the Drude Model

5.2 Results from the Drude Model

5.3 Shortcomings of the Drude Model

5.4 Discussion and Problems

6 Electronic Properties of Metals: Quantum Mechanical Approach

6.1 The Idea of Energy Bands

6.2 Free Electron Model

6.3 The General Form of the Electronic States

6.4 Nearly Free Electron Model

6.5 Energy Bands in Real Solids

6.6 Transport Properties

6.7 Brief Review of Some Key Ideas

6.8 Discussion and Problems

7 Semiconductors

7.1 Intrinsic Semiconductors

7.2 Doped Semiconductors

7.3 Conductivity of Semiconductors

7.4 Semiconductor Devices

7.5 Discussion and Problems

8 Magnetism

8.1 Macroscopic Description

8.2 Magnetic Effects in Atoms

8.3 Weak Magnetism in Solids

8.4 Magnetic Ordering

8.5 Discussion and Problems

9 Dielectrics

9.1 Macroscopic Description

9.2 Microscopic Polarization

9.3 The Local Field

9.4 Frequency Dependence of the Dielectric Constant

9.5 Other Effects

9.6 Discussion and Problems

10 Superconductivity

10.1 Basic Experimental Facts

10.2 Some Theoretical Aspects

10.3 Experimental Detection of the Gap

10.4 Coherence of the Superconducting State

10.5 Type I and Type II Superconductors

10.6 High-Temperature Superconductivity

10.7 Concluding Remarks

10.8 Discussion and Problems

11 Finite Solids and Nanostructures

11.1 Quantum Confinement

11.2 Surfaces and Interfaces

11.3 Magnetism on the Nanoscale

11.4 Discussion and Problems

Appendix

A.1 Explicit Forms of Vector Operations

A.2 Microscopic Form of the Maxwell Equations

A.3 The Maxwell Equations in Matter

References

Further Reading

Physical Constants and Energy Equivalents

Index

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The Author

Dr. Philip Hofmann

Institute for Storage Ring Facilities

Interdisciplinary Nanoscience Center

University of Aarhus

Denmark

Cover

Band structure of aluminum determined by angle–resolved photoemission. The color plot shows the photoemission intensity along the Γ–X direction. The dashed lines qualitatively track the maximum of the intensity and correspond to the band structure. Data taken from Physical Review B 66, 245422 (2002), see also Fig. 6.10 in this book.

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ISBN: 978-3-527-40861-0

Preface

This book emerged from a course on solid-state physics for third-year students of physics and nanoscience, but it should also be useful for students of related fields such as chemistry and engineering. The aim is to provide a bachelor-level survey over the whole field without going into too much detail. With this in mind, a lot of emphasis is put on a didactic presentation and little on stringent mathematical derivations or completeness. For a more in-depth treatment, the reader is referred to the many excellent advanced solid-state physics books. A few are listed in the Appendix.

To follow this text, a basic university-level physics course as well as some working knowledge of chemistry, quantum mechanics, and statistical physics is required. A course in classical electrodynamics is of advantage but not strictly necessary.

Some remarks on how to use this book: Every chapter is accompanied by a set of ‘‘discussion’’ questions and problems. The intention of the questions is to give the students a tool for testing their understanding of the subject. Some of the questions can only be answered with knowledge of later chapters. These are marked by an asterisk. Some of the problems are more of a challenge in that they are more difficult mathematically or conceptually or both. These problems are also marked by an asterisk. Not all the information necessary for solving the problems is given here. For standard data, for example, the density of gold or the atomic weight of copper, the reader is referred to the excellent resources available on the World Wide Web. Several useful resources for lectures, including equations, suggestions for lectures and the solutions to the problems in this book, can be found on the publisher’s web pages. More information, such as an updated list of relevant www pages, can be found on the author’s web pages.

I would like to thank the people who have helped me with many discussions, encouragement and suggestions. In particular, I would like to mention my colleagues Arne Nylandsted Larsen, Ivan Steensgaard, Maria Fuglsang Jensen, Justin Wells, Jørgen Bøttiger and many other instructors and students involved in teaching the course in Aarhus. Finally I thank Jørn Lyngesen for a careful proof-reading of the entire manuscript.

1

Chemical Bonding in Solids

In this chapter, we discuss different mechanisms that can lead to bonding between atoms so that they form solids. We will encounter different cases such as ionic, covalent, or metallic bonding. It has to be kept in mind that these are just idealized limiting cases. Often mixed bonding types are found, for example, a combination of metallic and covalent bonding in the transition metals.

As in conventional chemistry, only a restricted number of all the electrons participate in the bonding. These so-called valence electrons are the electrons in the outermost shell(s) of an atom. The electrons in the inner shells, or core electrons, are bound so tightly to the nucleus that they do not feel the presence of other atoms in their neighborhood.

1.1 Attractive and Repulsive Forces

Two different forces must be present to establish bonding in a solid or in a molecule. An attractive force is necessary for any bonding. Different types of attractive forces are discussed below. A repulsive force, on the other hand, is required in order to keep the atoms from getting too close to each other. An expression for an interatomic potential can be written as

(1.1)

where n > m, that is, the repulsive part has to prevail for short distances (sometimes this is achieved by assuming an exponential repulsion potential). Such a potential and the resulting force are shown in Figure 1.1. The reason for the strong repulsion at short distances is the Pauli exclusion principle. For a strong overlap of the electron clouds of two atoms, the wave functions have to change in order to become orthogonal to each other, because the Pauli principle forbids having more than two electrons in the same quantum state. The orthogonalization costs much energy, hence the strong repulsion.

1.2 Ionic Bonding

Ionic bonding involves the transfer of electrons from an electropositive atom to an electronegative atom. The bonding force is the Coulomb attraction between the two resulting ions. Ionizing both atoms usually costs some energy. In the case of NaCl, the ionization energy of Na is 5.1 eV but the electron affinity of Cl is only 3.6 eV. The net energy cost for creating a pair of ions is thus 5.1 – 3.6 = 1.5 eV. The energy gain is given by the Coulomb potential. For just one Na and one Cl ion separated by a distance a = 0.28 nm, this is –e²/4πε₀a, which amounts to 5.1 eV. Potential energies for more complicated structures are discussed below. It is important to distinguish between the different energy contributions involved: The ionization energy is the energy needed to turn atoms into ions, the lattice energy is the electrostatic energy gain by assembling a lattice, and the cohesive energy is the lattice energy minus the ionization energy, that is, it represents the total energy balance to form the ionic solid.

The fact that the total potential energy curve in Figure 1.1 contains a repulsive part in addition to the Coulomb potential means that the actual potential minimum for a given interatomic distance a is a bit shallower than expected from the pure Coulomb potential (10% or so). Therefore, the lattice energy can be calculated rather accurately using classical physics if the interatomic distance a is known. Predicting this distance, however, requires the inclusion of quantum effects. In any event, ionic bonding is very strong. The cohesive energy per atom is in the order of several electron volts.

1.3 Covalent Bonding

Covalent bonding is based on the true sharing of electrons between different atoms. The simplest case is that of the hydrogen molecule that we will discuss quantitatively below. In solids, covalent bonding is often found for elements with a roughly half- filled outer shell. A prominent example is carbon that forms solids as diamond or graphite as well as complex molecules such as Buckminster Fullerene C₆₀ or carbon nanotubes. The covalent bonds in diamond are constructed from a linear combination of the 2s orbital and three 2p orbitals. This results in four so-called sp³ orbitals that stick out in a tetrahedral configuration from the carbon atoms. In graphite, the 2s orbital is combined with only two 2p orbitals, giving three sp² orbitals, all in one plane and separated by an angle of 120°, and one p orbital oriented perpendicular to this plane. This linear combination of orbitals already reveals an important characteristic for the covalent bonding: it is highly directional. In addition to this, it is also very stable and the bonding energies are several electron volts.

A very instructive example for covalent bonding is the hydrogen molecule H₂ for which we will sketch a solution here. We go into some detail, as much of this will be useful in later chapters. As a starting point, take two hydrogen atoms with their nuclei at R_A and R_B and we call |R_B — R_A| = R. We do, of course, know the solution of the Schrödinger equation for each of the atoms. Let these ground-state wave functions be Ψ_A and Ψ_B, respectively. The Hamilton operator for the hydrogen molecule can be written as

(1.2)

where r₁ and r₂ are the coordinates of the electrons belonging to the A and the B nucleus, respectively. The first two terms refer to the kinetic energy of the two electrons. The operators and act only on the coordinates r₁ and r₂, respectively. The electrostatic term contains the repulsion between the two nuclei and the repulsion between the two electrons as well as the attraction of each electron to each nucleus.

The solution of this problem is not simple. It would be greatly simplified by removing the electrostatic interaction between the two electrons because then the Hamiltonian could be written as the sum of two parts, one for each electron. This could then be solved by a product of the two wave functions that are solutions to the two individual Hamiltonians, that is, the two-particle wave function would look like Ψ (r₁, r₂) = Ψ_A (r₁)Ψ_B (r₂). Actually, this is not quite right because such a wave function is not in accordance with the Pauli principle. Since the electrons are fermions, the total wave function must be antisymmetric with respect to particle exchange and the simple product wave function does not fulfill this requirement.

The total wave function consists of a spatial part and a spin part and therefore there are two possibilities for forming an antisymmetric wave function. We can either choose a symmetric spatial part and an antisymmetric spin part or vice versa. This is achieved by constructing the spacial wave function of the form

(1.3)

(1.4)

The plus sign in (1.3) returns a symmetric spatial wave function that we can take for an antisymmetric spin wave function with the total spin equal to zero (the so-called singlet state); the minus in (1.4) results in an antisymmetric spatial wave function for a symmetric spin wave function with the total spin equal to 1 (the so-called triplet state).

The antisymmetric wave function (1.4) vanishes if r₁ = r₂, that is, the two electrons cannot be at the same place simultaneously. This leads to a depletion of the electron density between the nuclei and hence to an antibonding state. For the symmetric case, on the other hand, the electrons have opposite spins and can be at the same place, which leads to a charge accumulation between the nuclei and hence to a bonding state (see Figure 1.2).

An approximate way to calculate the eigenvalues of (1.2) was suggested by W. Heitler and F. London in 1927. The idea is to use the known single-particle 1s wave functions for atomic hydrogen for Ψ_A and Ψ_B to form a two-electron wave function Ψ (r₁, r₂), which is given by either (1.3) or (1.4). An upper limit for the ground-state energy eigenvalues can then be calculated by

(1.5)

The calculation is quite lengthy and shall not be given here. The resulting ground- state energies for the singlet and triplet states can be written as

(1.6)

(1.7)

E₀ is the ground-state energy for one hydrogen atom that appears here twice because we start with two atoms. The energies ΔE_↑↑ and ΔE_↑↓ are also shown in Figure 1.2. ΔE_↑↑ is always larger than zero and does not lead to any chemical bonding. ΔE_↑↓, however, shows a minimum below zero at approximately 1.5 times the Bohr radius. This is the binding state.

For long distances between the nuclei, (1.6) and (1.7) can further be simplified to give

(1.8)

that is, the energy change upon bonding has two parts, one that does depend on the relative spin orientations of the electrons (±X) and one that does not (C). The energy difference between the two states is then given by 2X, where X is called the exchange energy. In the case of the hydrogen molecule, the exchange energy is always negative.

We will encounter similar concepts in the chapter about magnetism where the underlying principle for magnetic ordering is very similar to what we have here. The total energy of a system of electrons depends on their relative spin directions through the exchange energy, and therefore a particular ordered spin configuration is favored. For two electrons, the “magnetic” character is purely given by the sign of X. For a negative X, the coupling with two opposite spins is favorable (the antiferromagnetic case) whereas a positive X would lead to a situation where two parallel spins give the lowest energy (the ferromagnetic case).

1.4 Metallic Bonding

In metals, the outer valence electrons are removed from the ion cores, but in contrast to ionic solids, there are no electronegative ions to bind them. Therefore, they are free to migrate between the remaining ion cores. These delocalized valence electrons are involved in the conduction of electricity and are therefore often called conduction electrons. One can expect metals to form from elements for which the energy cost of removing outer electrons is not too big. Nevertheless, this removal always costs some energy that has to be more than compensated by the bonding. Explaining the energy gain from the bonding in an intuitive picture is difficult but we can at least try to make it plausible. The ultimate reason must be some sort of energy lowering.

One energy contribution that is lowered is the kinetic energy of the conduction electrons. Consider the kinetic energy contribution in a Hamiltonian, T = –²∇²/2m. A matrix element (Ψ|T |Ψ) measures the kinetic energy of a particle. TΨ is proportional to the second spatial derivative of the wave function, that is, the curvature. For an electron that is localized to an atom, the curvature of its wave function is much higher than that for a nearly free electron in a metal and this is where the energy gain comes from.

The other contribution to the electron energy is the potential energy. One should think that the average electrostatic potential of any single electron in a solid is almost zero because there are (almost) as many other electrons as there are ions with the same amount of charge. But this turns out to be wrong. In fact, the electrons see an attractive potential. The reason is again partly due to the Pauli principle that, loosely speaking, does not allow two electrons with the same spin direction to be at the same place and therefore the electrons go “out of each others way.” In addition to this, there is also a direct Coulomb interaction between the electrons, which makes them avoid each other. We will discuss this in more detail when dealing with magnetism.

Typically, metallic bonding is not as strong as covalent or ionic bonding but it amounts to a few electron volts per atom. Stronger bonding is found in transition metals, that is, metals with both s and p conduction electrons and a partially filled d shell. The explanation for this is that we have a mixed bonding. The s and p electrons turn into delocalized metallic conduction electrons, whereas the d electrons create much more localized, covalent-type bonds.

1.5 Hydrogen Bonding

Hydrogen atoms have only one electron and can form one covalent bond. If the bond is to a very electronegative atom (like F or O), the electron is mostly located close to that atom and the hydrogen nucleus represents an isolated positive (partial) charge. This can lead to a considerable charge density because of the small size and it can therefore attract negative (partial) charges in other molecules to form an electrostatic bond. This type of bonding is called hydrogen bonding. It is usually quite weak but in some cases the cohesive energy can be up to several hundred meV per atom. It is responsible for the intermolecular attraction in water ice and for the bonding of the double helix in DNA.

1.6 van der Waals Bonding

The term van der Waals bonding refers to a weak, purely quantum mechanical effect. The electron cloud around an atom or a molecule has no static charge distribution but one governed by quantum mechanical fluctuations. For a simple atom with a closed shell, this can be viewed as a fluctuating dipole moment. This dipole moment can polarize other atoms nearby and the interaction of the two neighboring dipoles reduces the total energy, that is, it can lead to bonding. This type of interaction is present in any solid but it is much weaker than ionic, covalent, or metallic bonding. Typical binding energies are in the meV range and therefore van der Waals bonding is only observable for solids that do not show other bonding behavior, for example, noble gases. Pure van der Waals crystals can only exist at very low temperatures.

1.7 Discussion and Problems

1.7.1 Discussion

1. Why is a typical interatomic potential, such as in Figure 1.1, so asymmetric?

2. Which elements are likely to form crystals through ionic bonding?

3. What kind of forces are important for ionic bonding?

4. How does the lattice energy in an ionic crystal depend on the interatomic distance?

5. Explain the difference between cohesive energy and lattice energy.

6. Which elements are likely to form metals?

7. Where does the energy gain in metallic bonding come from?

8. What is the difference between a simple metal and a transition metal (definition and typical physical properties)?

9. Why is van der Waals bonding much weaker than most other bonding types?

1.7.2 Problems

1. Metallic bonding: One can use some simple-minded arguments to explain why metallic bonding is stable. One argument is that allowing the wave functions to be smeared out in space reduces the momentum uncertainty Δp and thereby the highest momentum (and energy) through ΔpΔx = /2. Estimate how much Δp would be reduced by allowing the wave function to spread from typical atomic dimensions (Bohr radius) to the typical dimensions of the lattice unit cell (a few Angstrom or 10 times the Bohr radius). If you assume that the highest p would correspond to a highest kinetic energy E=p²/2m_e, how much would this reduce the kinetic energy?

2. van der Waals force: Show that the bonding energy due to the van der Waals force between two atoms depends on their distance r as r^–6. Hint: The van der Waals interaction is due to the mutual interaction of fluctuating dipoles. Suppose that one atom forms a dipole moment, at some time. This can be modeled as two point charges, separated by a distance d. What is the electric field due to this dipole at some distance r that is much greater than d? Assume that a second atom at the distance r is polarized in the field such that its induced dipole moment is proportional to the field (see (9.4) in Chapter 9) and calculate how the potential energy of the system depends on r.