Sources and References

These notes synthesize material from the following textbooks and standard references. They are not transcribed from any specific term’s lectures.

Primary textbook

• C. Kittel, Introduction to Solid State Physics, 8th ed. (Wiley, 2005)

Supplementary texts

• S. H. Simon, The Oxford Solid State Basics (Oxford University Press, 2013)
• N. W. Ashcroft and N. D. Mermin, Solid State Physics (Holt, Rinehart and Winston, 1976)
• M. P. Marder, Condensed Matter Physics, 2nd ed. (Wiley, 2010)

Online lecture notes

• D. Tong, Solid State Physics (University of Cambridge) — damtp.cam.ac.uk/user/tong/solidstate.html
• S. H. Simon, supplementary materials for The Oxford Solid State Basics — physics.ox.ac.uk/~sjc/

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Chapter 1: Crystal Structure

Introduction to Condensed Matter Physics

Condensed matter physics is the study of the physical properties of matter in its condensed phases — solids and liquids. It is the largest branch of physics, both in terms of the number of practitioners and the breadth of phenomena it encompasses. From the electrical conductivity of metals to the magnetic properties of iron, from the transparency of glass to the exotic superconductivity of certain materials cooled near absolute zero, condensed matter physics seeks to understand how the microscopic interactions between atoms give rise to the macroscopic properties we observe and exploit in technology.

The central challenge of the field is one of scale: a cubic centimeter of copper contains roughly \(10^{23}\) atoms, and yet we can predict its electrical resistance with remarkable accuracy from first principles. The reason this is possible — that such an overwhelmingly complex many-body system admits clean, tractable descriptions — is the existence of organizing principles. Chief among these is symmetry. When atoms arrange themselves into a crystal, they spontaneously break the continuous translational symmetry of free space down to a discrete symmetry. This discrete symmetry severely constrains the possible forms of physical quantities and enables powerful mathematical methods.

The course follows the logical progression laid out in Kittel’s Introduction to Solid State Physics, arguably the most influential textbook in the field. We begin with the geometry of crystalline order, then use that geometry to understand how X-rays scatter from crystals, which in turn tells us about the forces holding atoms together. From the forces we extract the vibrational modes of the lattice — phonons — and from phonons we compute thermal properties. We then turn to the electrons, first treating them as free particles and then incorporating the periodic potential of the crystal lattice, which leads to band theory and ultimately to the distinction between metals, insulators, and semiconductors. Each topic builds naturally on the last, and the mathematical structure throughout is one of remarkable elegance.

Translational Symmetry and the Bravais Lattice

The defining property of a crystalline solid is long-range order: if you know the position of one atom, you can predict the positions of all other atoms to arbitrary accuracy. More precisely, a crystal is periodic — there exists a set of translation vectors such that the crystal looks identical after translation by any one of them.

Definition (Bravais Lattice): A Bravais lattice is an infinite array of discrete points generated by the set of position vectors

\[\mathbf{R} = n_1 \mathbf{a}_1 + n_2 \mathbf{a}_2 + n_3 \mathbf{a}_3\]

where \(\mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3\) are the primitive lattice vectors (also called basis vectors) and \(n_1, n_2, n_3\) range over all integers. The vectors \(\mathbf{a}_i\) are not unique — there are many choices of primitive vectors that generate the same lattice — but they must be linearly independent and chosen such that no smaller set of translations generates the same set of points.

It is important to distinguish the Bravais lattice from the physical crystal. In a real crystal, there may be more than one atom per lattice point. The full crystal structure is specified by the Bravais lattice together with a basis — a set of atoms with their positions relative to each lattice point. If the basis consists of a single atom sitting exactly at each lattice point, we have a simple (monatomic) Bravais lattice. More commonly, the basis contains multiple atoms, giving rise to structures like sodium chloride (NaCl), where each lattice point has both a sodium and a chlorine atom associated with it.

The unit cell is the fundamental repeat unit of the crystal. It need not be a parallelepiped — any region of space that tiles all of space under the lattice translations, with no overlaps and no gaps, is a valid unit cell. The Wigner-Seitz cell is the particularly elegant choice: it is the region of space closer to a given lattice point than to any other lattice point. It is constructed by drawing perpendicular bisector planes between a chosen lattice point and all its neighbors and taking the smallest enclosed volume. The Wigner-Seitz cell has the full point symmetry of the Bravais lattice, which makes it particularly useful for visualization.

The Seven Crystal Systems and Fourteen Bravais Lattices

In three dimensions, there are exactly 14 distinct Bravais lattices, grouped into 7 crystal systems based on the symmetry of the unit cell. The crystal systems are distinguished by constraints on the lattice parameters: the three primitive vector lengths \(a, b, c\) (where \(a = |\mathbf{a}_1|\), etc.) and the three angles \(\alpha, \beta, \gamma\) between them.

The seven crystal systems are:

The cubic system, with \(a = b = c\) and \(\alpha = \beta = \gamma = 90°\), has the highest symmetry. It contains three Bravais lattices: simple cubic (SC), body-centered cubic (BCC), and face-centered cubic (FCC).

The tetragonal system has \(a = b \neq c\) and \(\alpha = \beta = \gamma = 90°\), giving two lattices (simple and body-centered tetragonal).

The orthorhombic system has \(a \neq b \neq c\) with all right angles, giving four lattices.

The hexagonal system has \(a = b \neq c\), with \(\alpha = \beta = 90°\) and \(\gamma = 120°\), giving one lattice.

The trigonal (rhombohedral) system has \(a = b = c\) with \(\alpha = \beta = \gamma \neq 90°\), giving one lattice.

The monoclinic system has \(a \neq b \neq c\), with one angle differing from \(90°\), giving two lattices.

The triclinic system has all sides and angles different, giving one lattice.

For this course, the most important systems are cubic and hexagonal, as they describe many technologically relevant materials.

Cubic Crystal Structures in Detail

Simple Cubic

The simple cubic (SC) lattice has primitive vectors along the three Cartesian axes:

\[\mathbf{a}_1 = a\hat{x}, \quad \mathbf{a}_2 = a\hat{y}, \quad \mathbf{a}_3 = a\hat{z}\]

with lattice constant \(a\). The conventional unit cell is a cube of side \(a\) with lattice points at the corners. Each corner is shared among 8 cubes, giving \(8 \times \frac{1}{8} = 1\) lattice point per conventional cell — equal to one per primitive cell, as expected. The coordination number (number of nearest neighbors) is 6. The packing fraction (fraction of space filled by touching hard spheres) is only \(\pi/6 \approx 52\%\). Simple cubic is rare in nature; polonium is the only elemental solid that adopts this structure.

Body-Centered Cubic

The body-centered cubic (BCC) lattice has a lattice point at each corner and one at the center of the cube. A convenient choice of primitive vectors is

\[\mathbf{a}_1 = \frac{a}{2}(-\hat{x} + \hat{y} + \hat{z}), \quad \mathbf{a}_2 = \frac{a}{2}(\hat{x} - \hat{y} + \hat{z}), \quad \mathbf{a}_3 = \frac{a}{2}(\hat{x} + \hat{y} - \hat{z})\]

The conventional cubic cell contains \(1 + 8 \times \frac{1}{8} = 2\) atoms. The nearest-neighbor distance is \(\frac{\sqrt{3}}{2}a\), and the coordination number is 8. The packing fraction is \(\frac{\pi\sqrt{3}}{8} \approx 68\%\). BCC is adopted by many metals, including iron (at room temperature), tungsten, chromium, and the alkali metals (Li, Na, K, Rb, Cs).

Face-Centered Cubic

The face-centered cubic (FCC) lattice has lattice points at all cube corners and at the center of each face. Primitive vectors can be written as

\[\mathbf{a}_1 = \frac{a}{2}(\hat{y} + \hat{z}), \quad \mathbf{a}_2 = \frac{a}{2}(\hat{x} + \hat{z}), \quad \mathbf{a}_3 = \frac{a}{2}(\hat{x} + \hat{y})\]

The conventional cell contains \(6 \times \frac{1}{2} + 8 \times \frac{1}{8} = 4\) atoms. The nearest-neighbor distance is \(\frac{a}{\sqrt{2}}\), and the coordination number is 12 — the highest possible for a monatomic lattice. The packing fraction is \(\frac{\pi}{3\sqrt{2}} \approx 74\%\), which is the maximum possible for identical spheres (Kepler’s conjecture, proved by Hales in 1998). FCC is adopted by many metals: aluminum, copper, gold, silver, nickel, and the noble gases in their solid phases.

The FCC structure is closely related to hexagonal close-packed (HCP). Both achieve 74% packing and coordination number 12; they differ only in the stacking sequence of close-packed planes. In FCC, stacking goes ABCABC…, while in HCP it goes ABABAB….

The Sodium Chloride and Zinc Blende Structures

Many compounds crystallize in structures that are best described as a Bravais lattice with a multi-atom basis.

Sodium Chloride (NaCl) Structure: The NaCl structure consists of two interpenetrating FCC lattices, one of Na\(^+\) and one of Cl\(^-\), displaced relative to each other by \(\frac{a}{2}\hat{x}\) (or equivalently by \(\frac{a}{2}\) along any cube edge). The lattice is FCC with a two-atom basis: Na at \((0,0,0)\) and Cl at \((\frac{a}{2},0,0)\). Every Na\(^+\) has 6 nearest-neighbor Cl\(^-\) ions, and vice versa. This structure is adopted by all alkali halides except CsCl, as well as MgO, FeO, and many other ionic compounds.

Zinc Blende (ZnS) Structure: The zinc blende structure is also two interpenetrating FCC lattices, displaced by \(\frac{a}{4}(\hat{x}+\hat{y}+\hat{z})\) — a quarter of the body diagonal. This places the atoms in a tetrahedral coordination: each Zn has 4 nearest-neighbor S atoms at the corners of a tetrahedron, and vice versa. The tetrahedral coordination is characteristic of \(sp^3\) covalent bonding. Many important semiconductors adopt this structure: GaAs, InP, ZnS, CdTe. The diamond structure is a special case where both sublattices are occupied by the same element (C, Si, Ge).

Lattice Planes and Miller Indices

To describe X-ray diffraction, we need a systematic way to label planes within a crystal. A lattice plane is any plane that contains at least three non-collinear lattice points (and therefore contains an infinite 2D array of lattice points). Families of parallel, equally-spaced lattice planes tile all space.

Miller indices provide a compact notation for specifying lattice planes. The procedure is:

1. Identify the intercepts of the plane with the three crystallographic axes in units of the lattice constants. For example, a plane that intercepts the axes at \(2a\), \(3a\), \(1a\) has intercepts \((2, 3, 1)\).

2. Take the reciprocals: \((1/2, 1/3, 1)\).

3. Multiply through by the smallest integer that clears all fractions: \((3, 2, 6)\).

4. Enclose in parentheses: \((326)\).

The result \((hkl)\) are the Miller indices of the plane. Several conventions are worth noting. A plane parallel to an axis has Miller index 0 for that axis (infinite intercept, reciprocal zero). Negative intercepts are denoted with an overbar: \((\bar{1}10)\). Curly braces \(\{hkl\}\) denote the family of all symmetry-equivalent planes. Square brackets \([hkl]\) denote a direction in the crystal, and angle brackets \(\langle hkl \rangle\) denote all symmetry-equivalent directions.

The spacing \(d_{hkl}\) between adjacent planes of type \((hkl)\) in a cubic crystal is

\[d_{hkl} = \frac{a}{\sqrt{h^2 + k^2 + l^2}}\]

This formula will become central when we discuss Bragg diffraction.

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Chapter 2: The Reciprocal Lattice

Motivation and Definition

The direct lattice describes the periodic arrangement of atoms in real space. The reciprocal lattice is the Fourier transform analog of the direct lattice, living in wavevector (or momentum) space. It is not a mathematical abstraction — it is the natural space in which to discuss wave phenomena in crystals, including X-ray diffraction, electron states, and phonon dispersion. Understanding the reciprocal lattice deeply is perhaps the single most important conceptual step in solid state physics.

Consider any function \(f(\mathbf{r})\) that has the periodicity of the Bravais lattice:

\[f(\mathbf{r} + \mathbf{R}) = f(\mathbf{r}) \quad \text{for all lattice vectors } \mathbf{R}\]

We can expand this in a Fourier series. Which wavevectors \(\mathbf{G}\) can appear? We need \(e^{i\mathbf{G} \cdot (\mathbf{r}+\mathbf{R})} = e^{i\mathbf{G}\cdot\mathbf{r}}\) for all \(\mathbf{R}\), which requires

\[e^{i\mathbf{G}\cdot\mathbf{R}} = 1 \quad \text{for all lattice vectors } \mathbf{R} = n_1\mathbf{a}_1 + n_2\mathbf{a}_2 + n_3\mathbf{a}_3\]

This means \(\mathbf{G} \cdot \mathbf{R} = 2\pi \times \text{integer}\) for all integers \(n_1, n_2, n_3\), which is equivalent to \(\mathbf{G} \cdot \mathbf{a}_i = 2\pi \times \text{integer}\) for \(i = 1, 2, 3\).

The denominator \(\mathbf{a}_1 \cdot (\mathbf{a}_2 \times \mathbf{a}_3) = V_c\) is the volume of the primitive cell. The volume of the reciprocal primitive cell is \((2\pi)^3/V_c\).

Reciprocal Lattices of FCC and BCC

A fundamental and beautiful result is that the FCC and BCC lattices are reciprocals of each other.

For the FCC direct lattice with conventional cubic cell of side \(a\), the primitive vectors are \(\mathbf{a}_1 = \frac{a}{2}(\hat{y}+\hat{z})\), \(\mathbf{a}_2 = \frac{a}{2}(\hat{x}+\hat{z})\), \(\mathbf{a}_3 = \frac{a}{2}(\hat{x}+\hat{y})\). A straightforward calculation gives

\[\mathbf{b}_1 = \frac{2\pi}{a}(-\hat{x}+\hat{y}+\hat{z}), \quad \mathbf{b}_2 = \frac{2\pi}{a}(\hat{x}-\hat{y}+\hat{z}), \quad \mathbf{b}_3 = \frac{2\pi}{a}(\hat{x}+\hat{y}-\hat{z})\]

These are the primitive vectors of a BCC lattice with conventional cube side \(4\pi/a\). Conversely, the reciprocal of the BCC lattice is FCC.

For a simple cubic lattice with lattice constant \(a\), the reciprocal lattice is also simple cubic with lattice constant \(2\pi/a\). This makes sense dimensionally: real-space distances go as \(a\), so reciprocal-space distances go as \(1/a\) (up to the conventional factor of \(2\pi\)).

The Brillouin Zone

The Wigner-Seitz cell of the reciprocal lattice is called the first Brillouin zone (BZ). It is the set of wavevectors closer to the reciprocal lattice origin than to any other reciprocal lattice point. Because the reciprocal lattice has the same point group symmetry as the direct lattice, the Brillouin zone has that symmetry as well.

The Brillouin zone is fundamental to nearly every topic in solid state physics. Electron bands, phonon dispersions, optical selection rules — all are described as functions of wavevectors \(\mathbf{k}\) within the first Brillouin zone. The reason is that any wavevector \(\mathbf{k}\) outside the first BZ is equivalent (for the purposes of describing a periodic crystal) to one inside it, differing only by a reciprocal lattice vector \(\mathbf{G}\): states with wavevector \(\mathbf{k}\) and \(\mathbf{k} + \mathbf{G}\) are physically equivalent in a crystal.

The first Brillouin zone of the simple cubic lattice is a cube of side \(2\pi/a\) centered at the origin.

The first Brillouin zone of the BCC lattice (whose reciprocal is FCC) is a truncated octahedron — a cube with all eight corners cut off. It has 14 faces: 8 hexagons and 6 squares.

The first Brillouin zone of the FCC lattice (whose reciprocal is BCC) is a regular truncated octahedron. It has 14 faces: 8 hexagons and 6 squares (same shape as BCC’s BZ, by duality).

High-symmetry points in the Brillouin zone have conventional labels. For the cubic system, \(\Gamma = (0,0,0)\) is the zone center. For FCC, important points include \(L = \frac{\pi}{a}(1,1,1)\) (center of a hexagonal face), \(X = \frac{2\pi}{a}(1,0,0)\) (center of a square face), and \(K = \frac{3\pi}{2a}(1,1,0)\) (midpoint of an edge between hexagonal faces). These will appear repeatedly when we discuss band structures.

Fourier Analysis in a Crystal and the Structure Factor

Any function with the periodicity of the lattice can be expanded in reciprocal lattice vectors:

\[f(\mathbf{r}) = \sum_{\mathbf{G}} f_{\mathbf{G}} e^{i\mathbf{G}\cdot\mathbf{r}}\]

where the Fourier coefficients are

\[f_{\mathbf{G}} = \frac{1}{V_c} \int_{\text{cell}} f(\mathbf{r}) e^{-i\mathbf{G}\cdot\mathbf{r}} \, d^3r\]

This Fourier decomposition is the mathematical backbone of Bloch’s theorem (Chapter 7) and the theory of X-ray diffraction (Chapter 3). The electron density \(n(\mathbf{r})\) in a crystal is periodic and can be expanded this way; the Fourier coefficients \(n_{\mathbf{G}}\) are directly measurable through X-ray diffraction.

For a crystal with a basis (multiple atoms per unit cell), the Fourier coefficient of the electron density factors as

\[n_{\mathbf{G}} = S_{\mathbf{G}} \cdot (\text{atomic form factor})\]

where the geometric structure factor is

\[S_{\mathbf{G}} = \sum_j f_j e^{i\mathbf{G}\cdot\mathbf{d}_j}\]

The sum runs over atoms \(j\) in the basis, \(\mathbf{d}_j\) is the position of atom \(j\) within the unit cell, and \(f_j\) is the atomic scattering factor (related to the number of electrons on atom \(j\)). Crucially, if \(S_{\mathbf{G}} = 0\) for a particular reciprocal lattice vector \(\mathbf{G}\), then the corresponding diffraction peak is absent — this is a systematic absence. Structure factors thus provide a powerful tool for determining crystal structure.

Example — BCC structure factor: The BCC structure can be viewed as a simple cubic lattice with a two-atom basis: atom 1 at \((0,0,0)\) and atom 2 at \((\frac{a}{2}, \frac{a}{2}, \frac{a}{2})\). For identical atoms, \(f_1 = f_2 = f\), and the simple cubic reciprocal lattice vectors are \(\mathbf{G} = \frac{2\pi}{a}(h\hat{x}+k\hat{y}+l\hat{z})\). Then

\[S_{hkl} = f\left(1 + e^{i\pi(h+k+l)}\right) = f\left(1 + (-1)^{h+k+l}\right)\]

This is \(2f\) when \(h+k+l\) is even, and 0 when \(h+k+l\) is odd. Thus BCC crystals show systematic absences for all reflections with \(h+k+l\) odd — a powerful diagnostic.

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Chapter 3: X-Ray Diffraction

Bragg’s Law

The wavelength of X-rays is comparable to the interatomic spacing in solids (\(\sim 0.1\)–\(1\) nm), making X-ray diffraction an ideal probe of crystal structure. The physical picture was first described by W.L. Bragg in 1913.

Consider a family of parallel lattice planes with Miller indices \((hkl)\), separated by spacing \(d_{hkl}\). When a collimated X-ray beam of wavelength \(\lambda\) strikes these planes at a glancing angle \(\theta\) (measured from the plane surface, not the normal), waves scattered from successive planes have a path length difference of \(2d\sin\theta\). Constructive interference occurs when this path difference is an integer multiple of the wavelength:

While Bragg’s law is elegantly simple, it is essentially a geometric statement about path-length differences and does not fully account for the intensity of diffraction peaks or their systematic absences. The more complete theory is the Laue formulation.

The Laue Condition

The Laue approach treats diffraction as the scattering of a plane wave by the periodic electron density of the crystal. Let \(\mathbf{k}\) be the wavevector of the incident X-ray beam and \(\mathbf{k}'\) the wavevector of the diffracted beam. For elastic scattering, \(|\mathbf{k}| = |\mathbf{k}'| = 2\pi/\lambda\). Define the scattering vector (or momentum transfer)

\[\Delta\mathbf{k} = \mathbf{k}' - \mathbf{k}\]

The amplitude of the scattered wave is proportional to the Fourier transform of the electron density, evaluated at \(\Delta\mathbf{k}\):

\[F(\Delta\mathbf{k}) = \int n(\mathbf{r}) e^{-i\Delta\mathbf{k}\cdot\mathbf{r}} \, d^3r\]

Because \(n(\mathbf{r})\) is periodic, its Fourier transform is nonzero only when \(\Delta\mathbf{k}\) equals a reciprocal lattice vector. This is the Laue condition:

The Laue condition and Bragg’s law are equivalent. To see this, note that for elastic scattering \(|\mathbf{k}'| = |\mathbf{k}|\), squaring \(\mathbf{k}' = \mathbf{k} + \mathbf{G}\) gives \(k^2 = k^2 + 2\mathbf{k}\cdot\mathbf{G} + G^2\), so \(2\mathbf{k}\cdot\mathbf{G} = -G^2\), or equivalently \(2k\sin\theta = G\). Since \(k = 2\pi/\lambda\) and \(G = 2\pi n/d_{hkl}\), this reduces to Bragg’s law.

The Ewald Sphere Construction

The Laue condition has a beautiful geometric interpretation due to Ewald. Draw the reciprocal lattice. Place the tip of the incident wavevector \(\mathbf{k}\) at a reciprocal lattice point. Draw a sphere (the Ewald sphere) of radius \(|\mathbf{k}| = 2\pi/\lambda\) centered at the tail of \(\mathbf{k}\). Diffraction occurs in the direction \(\mathbf{k}'\) whenever the Ewald sphere intersects another reciprocal lattice point, because then \(\mathbf{k}' - \mathbf{k} = \mathbf{G}\).

For a fixed wavelength (monochromatic X-rays), the Ewald sphere is fixed, and it will generally intersect few or no reciprocal lattice points — diffraction peaks are rare. This motivates the various experimental methods:

Laue method: Use white X-rays (a continuous range of wavelengths). This effectively thickens the Ewald sphere into a shell, making it more likely to intersect reciprocal lattice points. Useful for determining crystal orientation.

Rotating crystal method: Use monochromatic X-rays but rotate the crystal, sweeping the reciprocal lattice through the Ewald sphere.

Powder diffraction (Debye-Scherrer method): Use a powdered sample containing crystallites in all orientations. This is equivalent to rotating the entire reciprocal lattice about the origin, producing rings (Debye-Scherrer rings). Each ring corresponds to a different \((hkl)\) family.

Scattering Amplitude and Atomic Form Factors

The intensity of a diffraction peak is proportional to the square of the scattering amplitude. For a crystal with \(N\) unit cells each containing a basis of atoms at positions \(\mathbf{d}_s\) (relative to the lattice point), the scattering amplitude at the Laue condition \(\Delta\mathbf{k} = \mathbf{G}\) is

\[F(\mathbf{G}) = N \sum_s f_s(\mathbf{G}) e^{i\mathbf{G}\cdot\mathbf{d}_s} = N \cdot S(\mathbf{G})\]

where the atomic form factor is

\[f_s(\mathbf{G}) = \int n_s(\mathbf{r}) e^{i\mathbf{G}\cdot\mathbf{r}} \, d^3r\]

with \(n_s(\mathbf{r})\) the electron density of atom \(s\). For \(\mathbf{G} = 0\), \(f_s = Z_s\) (the atomic number). For \(\mathbf{G} \neq 0\), \(f_s < Z_s\) because the electron distribution is spread out over a volume comparable to the interatomic spacing.

The structure factor \(S(\mathbf{G}) = \sum_s f_s e^{i\mathbf{G}\cdot\mathbf{d}_s}\) determines which reflections are allowed and their relative intensities. By measuring the intensities of many diffraction peaks and solving the phase problem (the measured intensities give \(|S|^2\) but not the phase of \(S\)), crystallographers can reconstruct the electron density and determine atomic positions to sub-angstrom precision. This is the basis of X-ray crystallography, which has determined the structures of millions of molecules including DNA and thousands of proteins.

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Chapter 4: Crystal Bonding

Types of Chemical Bonds in Solids

The properties of a solid — its mechanical strength, melting point, electrical conductivity, optical response — are largely determined by the nature of the chemical bonds holding it together. There are four main types of bonding in solids, each with characteristic energy scales, geometries, and physical consequences.

Ionic Bonding

Ionic bonding occurs when electrons are transferred from electropositive atoms (metals) to electronegative atoms (non-metals), creating oppositely charged ions that attract via the Coulomb interaction. The classic example is NaCl: the Na\(^+\) and Cl\(^-\) ions arrange on the rock salt structure to minimize the total electrostatic energy.

The cohesive energy of an ionic crystal is dominated by the long-range Coulomb interactions. For a crystal with \(N\) ion pairs, the Coulomb contribution to the total energy per ion pair is

\[U_{\text{Coulomb}} = -\alpha \frac{e^2}{4\pi\epsilon_0 R}\]

where \(R\) is the nearest-neighbor distance and \(\alpha\) is the Madelung constant, a purely geometric quantity that depends only on the crystal structure. For the NaCl structure, \(\alpha_{\text{NaCl}} \approx 1.7476\). For the CsCl structure, \(\alpha_{\text{CsCl}} \approx 1.7627\).

The calculation of the Madelung constant for NaCl illustrates the slowly convergent nature of the Coulomb sum. Starting from a reference Na\(^+\) ion, the nearest neighbors are 6 Cl\(^-\) at distance \(R\), then 12 Na\(^+\) at \(R\sqrt{2}\), then 8 Cl\(^-\) at \(R\sqrt{3}\), and so on:

\[\alpha = 6 - \frac{12}{\sqrt{2}} + \frac{8}{\sqrt{3}} - \frac{6}{2} + \cdots = 1.7476...\]

This series converges only conditionally, requiring careful treatment (Ewald summation) in practice.

At short range, ion cores overlap and the Pauli exclusion principle creates a repulsive interaction. This is often modeled as a power-law repulsion \(B/R^n\) (Born-Mayer model), giving a total energy per ion pair

\[U(R) = -\frac{\alpha e^2}{4\pi\epsilon_0 R} + \frac{B}{R^n}\]

Minimizing with respect to \(R\) at equilibrium (where \(dU/dR = 0\)) gives \(B = \frac{\alpha e^2 R_0^{n-1}}{4\pi\epsilon_0 n}\), and the cohesive energy is

\[U(R_0) = -\frac{\alpha e^2}{4\pi\epsilon_0 R_0}\left(1 - \frac{1}{n}\right)\]

For NaCl, \(n \approx 9\), and this formula gives \(U \approx -8\) eV per ion pair, in good agreement with experiment. The factor \((1-1/n)\) is close to 1 for large \(n\), confirming that most of the cohesion comes from the Coulomb attraction.

van der Waals Bonding

Van der Waals bonding (also called dispersive bonding or London bonding) arises from quantum mechanical fluctuations in the electron distribution of neutral atoms. Even a neutral atom has a fluctuating electric dipole moment; this induces a dipole on a neighboring atom, and the correlated fluctuations lead to a net attractive interaction. The van der Waals interaction between two atoms decays as \(-C/R^6\) at large separation \(R\).

This is the dominant bonding in noble gas solids (He, Ne, Ar, Kr, Xe in the solid phase) and molecular crystals. The cohesive energies are small (of order 0.01–0.1 eV per atom), and the melting points are correspondingly low. The Lennard-Jones potential is the standard model:

\[U(R) = 4\epsilon\left[\left(\frac{\sigma}{R}\right)^{12} - \left(\frac{\sigma}{R}\right)^6\right]\]

The \(R^{-6}\) term is the van der Waals attraction; the \(R^{-12}\) term is a steeply repulsive model of the Pauli repulsion. The potential has minimum \(-\epsilon\) at \(R = 2^{1/6}\sigma\). For argon, \(\epsilon/k_B \approx 120\) K and \(\sigma \approx 3.4\) Å.

Covalent Bonding

Covalent bonding occurs when atoms share electrons, with the shared electrons occupying bonding orbitals that have enhanced density between the nuclei. The hydrogen molecule is the simplest example: the bonding orbital is the symmetric combination \(\psi_b \propto (\phi_A + \phi_B)\) and has lower energy than the isolated atomic orbitals due to the reduction in kinetic energy (a subtlety — the covalent bond is primarily a kinetic energy effect, not a potential energy effect, as first pointed out by Hellmann and Feynman).

In solids, the directionality of covalent bonds determines the crystal structure. Carbon in diamond adopts \(sp^3\) hybridization, with four equivalent orbitals pointing toward the corners of a tetrahedron. Silicon and germanium adopt the same diamond cubic structure. The strong, directional covalent bonds give diamond its exceptional hardness (the highest of any natural material).

Covalent cohesive energies are large (several eV per bond), giving high melting points and large elastic moduli. The directionality of the bonds also produces the semiconducting behavior of Si, Ge, and related materials, as we will discuss extensively in Chapter 8.

Metallic Bonding

Metallic bonding arises when electrons are delocalized over the entire crystal rather than localized in bonds between specific pairs of atoms. The valence electrons form a “sea” of free electrons that simultaneously screens the ion-ion repulsion and provides the attractive glue holding the lattice together.

Metals are characterized by high electrical and thermal conductivity (due to the mobile electrons), opacity and high reflectivity at optical frequencies, and high ductility (the non-directional bonding allows planes of atoms to slide past each other without breaking bonds). Metallic cohesive energies are intermediate (1–10 eV per atom).

The free electron model, developed in Chapter 6, treats the metallic electrons as an ideal Fermi gas moving in a uniform positive background (the jellium model). Despite its simplicity, this model correctly predicts many properties of simple metals like Na, K, Al, and Cu.

Compressibility and Phonon Velocity

Regardless of bonding type, we can characterize the stiffness of the crystal by its bulk modulus \(B\), defined by

\[B = -V\frac{dP}{dV} = V\frac{d^2U}{dV^2}\bigg|_{V_0}\]

where \(U\) is the total energy and \(V_0\) is the equilibrium volume. The bulk modulus has dimensions of pressure (Pa) and is a measure of the crystal’s resistance to uniform compression.

The velocity of sound in the crystal is directly related to the bulk modulus:

\[v_s = \sqrt{\frac{B}{\rho}}\]

where \(\rho\) is the mass density. For NaCl, \(B \approx 25\) GPa and \(\rho \approx 2.16 \times 10^3\) kg/m\(^3\), giving \(v_s \approx 3400\) m/s. We will return to elastic waves and their quantum mechanical counterparts (phonons) in the next chapter.

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Chapter 5: Crystal Vibrations and Phonons

Vibrations of a Monatomic Lattice

The atoms in a crystal are not static — they vibrate about their equilibrium positions, driven by thermal energy. These vibrations carry thermal energy (heat capacity), scatter electrons (electrical resistance), and interact with light (Raman spectroscopy). Understanding them is therefore essential to virtually every macroscopic property of the solid.

Consider the simplest model: a one-dimensional chain of identical atoms of mass \(M\), separated by lattice constant \(a\), interacting with their nearest neighbors via a spring constant \(C\). Let \(u_n\) denote the displacement of the \(n\)-th atom from its equilibrium position. The equation of motion is

\[M\ddot{u}_n = C(u_{n+1} - u_n) + C(u_{n-1} - u_n) = C(u_{n+1} + u_{n-1} - 2u_n)\]

We seek normal mode solutions of the form \(u_n = u e^{i(kna - \omega t)}\), where \(k\) is the wavevector and \(\omega\) is the angular frequency. Substituting into the equation of motion:

\[-M\omega^2 u = C u(e^{ika} + e^{-ika} - 2) = 2Cu(\cos ka - 1) = -4Cu\sin^2\!\left(\frac{ka}{2}\right)\]

This gives the phonon dispersion relation for a monatomic chain:

\[\omega(k) = 2\sqrt{\frac{C}{M}}\left|\sin\!\left(\frac{ka}{2}\right)\right|\]

Several features of this dispersion deserve comment:

Periodicity in \(k\)-space: The dispersion is periodic with period \(2\pi/a\) in \(k\). This means that wavevectors \(k\) and \(k + 2\pi/a\) describe the same physical vibration — exactly as expected from the lattice periodicity. We therefore restrict attention to the first Brillouin zone \(-\pi/a < k \leq \pi/a\), which contains exactly \(N\) allowed wavevectors for a chain of \(N\) atoms (using periodic boundary conditions).

Long-wavelength limit: For small \(k\), \(\sin(ka/2) \approx ka/2\), and \(\omega \approx v_s k\) where \(v_s = a\sqrt{C/M}\) is the speed of sound. This linear acoustic dispersion is just what we expect for a continuous elastic medium.

Zone boundary: At the Brillouin zone boundary \(k = \pm\pi/a\), the dispersion has a maximum \(\omega = \omega_{\max}\) and a zero slope \(d\omega/dk = 0\). This zero slope reflects the standing-wave character of the mode: adjacent atoms move in opposite directions (\(k = \pi/a\) means a wave with wavelength \(2a\), the shortest possible). The group velocity \(v_g = d\omega/dk\) vanishes at the zone boundary.

Number of modes: With \(N\) atoms and periodic boundary conditions, the allowed values of \(k\) are \(k_j = 2\pi j / (Na)\) for \(j = 0, \pm 1, ..., \pm(N/2-1), N/2\) — exactly \(N\) modes, one for each degree of freedom. This is the phonon branch analogous to acoustic sound waves; since it starts at \(\omega = 0\) for \(k = 0\), it is called the acoustic branch.

Diatomic Lattice and Optical Phonons

When the unit cell contains more than one atom, the phonon dispersion develops additional branches. Consider a one-dimensional chain with two atoms per unit cell: atoms of mass \(M_1\) and \(M_2\), alternating, connected by spring constant \(C\). Let \(u_n\) and \(v_n\) be the displacements of the two atoms in the \(n\)-th unit cell.

The equations of motion are:

\[M_1 \ddot{u}_n = C(v_n + v_{n-1} - 2u_n)\]\[M_2 \ddot{v}_n = C(u_{n+1} + u_n - 2v_n)\]

Substituting \(u_n = u \, e^{i(kna - \omega t)}\) and \(v_n = v \, e^{i(kna - \omega t)}\) yields the eigenvalue problem:

\[\begin{pmatrix} \frac{2C}{M_1} - \omega^2 & -\frac{C}{M_1}(1 + e^{-ika}) \\ -\frac{C}{M_2}(1 + e^{ika}) & \frac{2C}{M_2} - \omega^2 \end{pmatrix} \begin{pmatrix} u \\ v \end{pmatrix} = 0\]

Setting the determinant to zero gives a quadratic equation in \(\omega^2\):

\[\omega^2 = C\left(\frac{1}{M_1} + \frac{1}{M_2}\right) \pm C\sqrt{\left(\frac{1}{M_1}+\frac{1}{M_2}\right)^2 - \frac{4\sin^2(ka/2)}{M_1 M_2}}\]

This gives two branches — the two signs yield two solutions at each wavevector \(k\):

The acoustic branch (minus sign) has \(\omega \to 0\) as \(k \to 0\). In this limit, \(u \approx v\), meaning both atoms in the cell move together in phase. This corresponds to long-wavelength sound waves.

The optical branch (plus sign) has \(\omega \to \omega_0 > 0\) as \(k \to 0\). At \(k = 0\):

\[\omega_{\text{opt}}(0) = \sqrt{2C\left(\frac{1}{M_1} + \frac{1}{M_2}\right)} = \sqrt{\frac{2C}{\mu}}\]

where \(\mu = M_1 M_2/(M_1+M_2)\) is the reduced mass. At \(k=0\) in the optical branch, the center of mass is stationary, but the two atoms oscillate against each other.

The name “optical” comes from the fact that in ionic crystals (like NaCl), this mode involves the two oppositely charged ions moving in opposite directions, creating an oscillating electric dipole that couples strongly to light. The transverse optical (TO) phonon frequency of NaCl at \(k=0\) is in the infrared range, \(\omega_{\text{TO}} \approx 3 \times 10^{13}\) rad/s. This leads to the phenomenon of reststrahlen (residual ray): strong absorption and reflection of infrared light near the optical phonon frequency.

At the zone boundary \(k = \pm\pi/(2a)\) (remembering the unit cell has period \(2a\)), the acoustic and optical branches split, with

\[\omega_{\text{ac}} = \sqrt{2C/M_1}, \quad \omega_{\text{opt}} = \sqrt{2C/M_2}\]

(assuming \(M_1 < M_2\), so \(\omega_{\text{ac}} > \omega_{\text{opt}}\) at the zone boundary — but not in the center!). There is a gap between the two branches in the range \(\sqrt{2C/M_1} < \omega < \sqrt{2C/M_2}\): no phonon modes exist at these frequencies.

Extension to Three Dimensions

In three dimensions, a crystal with \(p\) atoms per primitive cell has \(3p\) phonon branches: 3 acoustic branches and \(3p - 3\) optical branches. The 3 acoustic branches correspond to the three translational modes of the cell (2 transverse, 1 longitudinal). For NaCl with \(p=2\), there are 3 acoustic and 3 optical branches (TO, LO, and 2 degenerate transverse modes).

The acoustic branches split into one longitudinal acoustic (LA) branch, where atoms move parallel to \(\mathbf{k}\), and two transverse acoustic (TA) branches, where atoms move perpendicular to \(\mathbf{k}\). The LA branch generally has higher frequency than the TA branches because the restoring force for longitudinal compression (related to bulk modulus) is larger than for transverse shear (related to shear modulus) in many materials.

Phonons as Quantum Mechanical Excitations

The quantization of lattice vibrations follows from the standard treatment of the harmonic oscillator. Each normal mode \((\mathbf{k}, s)\) (where \(\mathbf{k}\) is the wavevector and \(s\) is the branch index) is an independent quantum harmonic oscillator. The energy of that mode is

\[\epsilon_{\mathbf{k}s} = \hbar\omega_{\mathbf{k}s}\left(n_{\mathbf{k}s} + \frac{1}{2}\right)\]

where \(n_{\mathbf{k}s} = 0, 1, 2, ...\) is the occupation number. A phonon is a single quantum of vibration — adding one phonon to mode \((\mathbf{k}, s)\) increases the energy by \(\hbar\omega_{\mathbf{k}s}\).

Phonons are bosons: their occupation numbers are governed by the Bose-Einstein distribution

\[\langle n_{\mathbf{k}s} \rangle = \frac{1}{e^{\hbar\omega_{\mathbf{k}s}/k_BT} - 1}\]

At temperature \(T\), the average energy of mode \((\mathbf{k},s)\) (excluding the zero-point energy) is \(\hbar\omega_{\mathbf{k}s}/\left(e^{\hbar\omega/k_BT}-1\right)\).

Phonons carry crystal momentum \(\hbar\mathbf{k}\). In phonon-phonon scattering or phonon-electron scattering, crystal momentum is conserved modulo a reciprocal lattice vector \(\hbar\mathbf{G}\). Processes that conserve crystal momentum exactly (\(\mathbf{G}=0\)) are called Normal processes (N-processes); those with \(\mathbf{G}\neq 0\) are called Umklapp processes (U-processes, from the German “to flip over”). Umklapp processes are responsible for thermal resistance in insulators at low temperatures.

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Chapter 6: Thermal Properties of Solids

Heat Capacity: Classical vs. Quantum

The heat capacity (or specific heat) of a solid is one of its most fundamental thermodynamic properties. Classically, the equipartition theorem predicts that each atom contributes \(3k_B\) to the heat capacity at constant volume (3 kinetic + 3 potential degrees of freedom), giving

\[C_V = 3Nk_B = 3R\]

for a mole of atoms. Here \(R = N_A k_B = 8.314\) J/mol·K is the gas constant. This is the Dulong-Petit law, empirically established in 1819. It works well for most solids at room temperature and above.

However, at low temperatures, the heat capacity drops dramatically — far below the Dulong-Petit value. Experimentally, \(C_V \propto T^3\) at low temperatures for insulators, and \(C_V \propto T\) for metals. The classical theory has no explanation for this. The resolution requires quantum mechanics.

Einstein Model

In 1907, Einstein proposed that each atom in a solid vibrates independently with the same frequency \(\omega_E\) (the Einstein frequency). Each oscillator is quantized, with energy levels \(\hbar\omega_E(n + 1/2)\). The average energy per oscillator is

\[\langle E \rangle = \hbar\omega_E\left(\frac{1}{e^{\hbar\omega_E/k_BT} - 1} + \frac{1}{2}\right)\]

The heat capacity per oscillator is

\[C_E = k_B\left(\frac{\hbar\omega_E}{k_BT}\right)^2 \frac{e^{\hbar\omega_E/k_BT}}{\left(e^{\hbar\omega_E/k_BT} - 1\right)^2}\]

Defining the Einstein temperature \(\Theta_E = \hbar\omega_E/k_B\), the total molar heat capacity is

\[C_V = 3R\left(\frac{\Theta_E}{T}\right)^2 \frac{e^{\Theta_E/T}}{(e^{\Theta_E/T}-1)^2}\]

Limiting behaviors: At high temperature \(T \gg \Theta_E\), we expand the exponentials: \(e^{\Theta_E/T} \approx 1 + \Theta_E/T\), giving \(C_V \approx 3R\) — the Dulong-Petit result. At low temperature \(T \ll \Theta_E\), \(e^{\Theta_E/T} \gg 1\), and

\[C_V \approx 3R\left(\frac{\Theta_E}{T}\right)^2 e^{-\Theta_E/T}\]

This is exponentially suppressed at low temperature — qualitatively right (the heat capacity drops to zero) but the exponential dependence does not match the observed \(T^3\) law for insulators.

The failure of the Einstein model at low temperatures arises from its treatment of all modes as having the same frequency. In reality, there are long-wavelength acoustic modes with very low frequencies, and these are thermally excited at temperatures much lower than \(\Theta_E\).

Debye Model

In 1912, Debye improved on Einstein’s model by using the actual phonon dispersion. The key insight is that at low temperatures, only long-wavelength (low-frequency) phonons are thermally excited. For these modes, the dispersion is approximately linear: \(\omega = v_s k\).

Debye treated the crystal as a continuous elastic medium with linear dispersion \(\omega = v_s k\). The density of phonon modes in 3D is

\[g(\omega) = \frac{3V\omega^2}{2\pi^2 v_s^3}\]

(the factor 3 counts the three acoustic polarizations). However, a crystal with \(N\) atoms has exactly \(3N\) phonon modes, so the density of states must be cut off at a maximum frequency — the Debye frequency \(\omega_D\), determined by

\[\int_0^{\omega_D} g(\omega) \, d\omega = 3N \implies \omega_D = v_s\left(\frac{6\pi^2 N}{V}\right)^{1/3}\]

The Debye temperature is defined as \(\Theta_D = \hbar\omega_D/k_B\).

The total internal energy (excluding zero-point contribution) is

\[U = \int_0^{\omega_D} \frac{\hbar\omega}{e^{\hbar\omega/k_BT}-1} g(\omega) \, d\omega = \frac{3V\hbar}{2\pi^2 v_s^3}\int_0^{\omega_D}\frac{\omega^3}{e^{\hbar\omega/k_BT}-1} \, d\omega\]

Substituting \(x = \hbar\omega/k_BT\) and \(x_D = \Theta_D/T\):

\[U = 9Nk_BT\left(\frac{T}{\Theta_D}\right)^3\int_0^{\Theta_D/T}\frac{x^3}{e^x-1} \, dx\]

and the heat capacity is \(C_V = dU/dT\).

The Debye \(T^3\) Law

At low temperatures \(T \ll \Theta_D\), the upper limit \(x_D = \Theta_D/T \to \infty\), and we can use

\[\int_0^{\infty}\frac{x^3}{e^x-1}\,dx = \frac{\pi^4}{15}\]

This gives

\[U = 9Nk_BT\left(\frac{T}{\Theta_D}\right)^3 \cdot \frac{\pi^4}{15} = \frac{3\pi^4}{5}Nk_B\frac{T^4}{\Theta_D^3}\]

and

\[C_V = \frac{12\pi^4}{5}Nk_B\left(\frac{T}{\Theta_D}\right)^3 \propto T^3\]

The physical origin of the \(T^3\) law is transparent: the number of thermally excited phonons is \(\sim (k_BT/\hbar v_s)^3\) (from the volume in \(k\)-space up to wavenumber \(k_BT/\hbar v_s\)), and each carries energy \(\sim k_BT\), giving \(U \propto T^4\) and \(C_V \propto T^3\).

At high temperature \(T \gg \Theta_D\), \(x = \hbar\omega/k_BT \ll 1\) for all phonons, and \(e^x - 1 \approx x\), so the integral becomes \(\int_0^{x_D} x^2 dx = x_D^3/3\), giving \(U = 3Nk_BT\) and \(C_V = 3Nk_B\) — the Dulong-Petit law again.

Typical Debye temperatures: Pb (\(\Theta_D \approx 105\) K), Al (\(\Theta_D \approx 428\) K), Fe (\(\Theta_D \approx 470\) K), C-diamond (\(\Theta_D \approx 2230\) K). Light, stiff materials have high Debye temperatures.

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Chapter 7: Free Electron Gas

The Sommerfeld Model

With the structure of the crystal and its vibrations understood, we turn to the electrons. The most important electrons for determining the electrical and thermal properties of metals are the conduction electrons — the valence electrons that, in a metal, are not bound to any particular atom but move freely throughout the crystal.

Sommerfeld’s model treats these electrons as a free Fermi gas: non-interacting electrons confined to a box of volume \(V = L^3\), subject only to the Pauli exclusion principle. The interactions between electrons are enormous (the Coulomb repulsion between nearby electrons in a metal is several eV), yet this model works extraordinarily well for many metals. The reason — justified by Landau’s Fermi liquid theory — is that the key effects of interactions can be absorbed into renormalized quasiparticle parameters.

The single-particle energy eigenstates are plane waves \(\psi_{\mathbf{k}}(\mathbf{r}) = \frac{1}{\sqrt{V}}e^{i\mathbf{k}\cdot\mathbf{r}}\), with energy

\[\epsilon_{\mathbf{k}} = \frac{\hbar^2 k^2}{2m}\]

Applying periodic boundary conditions \(\psi(x+L,y,z) = \psi(x,y,z)\) etc. gives allowed wavevectors \(k_i = 2\pi n_i/L\) for integer \(n_i\). The allowed values of \(\mathbf{k}\) form a cubic lattice in \(k\)-space with spacing \(2\pi/L\), and the volume per allowed \(\mathbf{k}\) point is \((2\pi/L)^3 = (2\pi)^3/V\).

Ground State Properties at \(T=0\)

At \(T = 0\), electrons fill the lowest available states, with at most two electrons (spin up and spin down) per wavevector state. The occupied states form a sphere in \(k\)-space — the Fermi sphere — and the surface of this sphere is the Fermi surface. The radius \(k_F\) is determined by the electron density \(n = N/V\):

The total number of electrons is

\[N = 2 \times \frac{\text{Volume of Fermi sphere}}{(2\pi/L)^3} = 2 \times \frac{\frac{4}{3}\pi k_F^3}{(2\pi)^3/V} = \frac{k_F^3 V}{3\pi^2}\]

Solving for \(k_F\):

\[k_F = \left(3\pi^2 \frac{N}{V}\right)^{1/3} = (3\pi^2 n)^{1/3}\]

The Fermi energy \(E_F\) is the energy of the highest occupied state at \(T=0\):

\[E_F = \frac{\hbar^2 k_F^2}{2m} = \frac{\hbar^2}{2m}(3\pi^2 n)^{2/3}\]

The Fermi temperature \(T_F = E_F/k_B\) and Fermi velocity \(v_F = \hbar k_F/m\) characterize the energy and velocity scales of electrons at the Fermi surface.

The total ground-state energy is

\[E_0 = \int_0^{E_F} \epsilon \, g(\epsilon) \, d\epsilon\]

where the density of states

\[g(\epsilon) = \frac{V}{2\pi^2}\left(\frac{2m}{\hbar^2}\right)^{3/2}\epsilon^{1/2} = \frac{3N}{2E_F}\left(\frac{\epsilon}{E_F}\right)^{1/2}\]

gives the number of electron states per unit energy interval. Evaluating:

\[E_0 = \frac{3}{5}NE_F\]

The average energy per electron is \(\frac{3}{5}E_F\), reflecting the large kinetic energies due to the Pauli principle. This is the quantum degeneracy pressure — even at \(T=0\), the Fermi gas exerts a pressure

\[P = -\frac{\partial E_0}{\partial V} = \frac{2}{3}\frac{E_0}{V} = \frac{2nE_F}{5}\]

For copper, \(P \approx 38\) GPa — comparable to the bulk modulus! This Fermi pressure plays an important role in stabilizing the crystal against collapse.

The Fermi-Dirac Distribution

At finite temperature, the occupation of single-particle states is governed by the Fermi-Dirac distribution:

\[f(\epsilon) = \frac{1}{e^{(\epsilon-\mu)/k_BT} + 1}\]

where \(\mu\) is the chemical potential (also called the Fermi level when used loosely). At \(T=0\), \(f(\epsilon) = 1\) for \(\epsilon < \mu(0) = E_F\) and 0 for \(\epsilon > E_F\) — a sharp step. At finite temperature, this step is smeared over an energy range of order \(k_BT\) around \(\mu\).

The chemical potential at temperature \(T\) is determined by the constraint that the total number of electrons is fixed:

\[N = \int_0^\infty g(\epsilon) f(\epsilon) \, d\epsilon\]

For \(T \ll T_F\) (the typical situation for metals at room temperature), the Sommerfeld expansion gives

\[\mu(T) = E_F\left[1 - \frac{\pi^2}{12}\left(\frac{k_BT}{E_F}\right)^2 + \cdots\right]\]

The chemical potential decreases slightly with temperature; the shift is of order \((k_BT)^2/E_F\), which for copper at room temperature is only about 0.01% of \(E_F\).

Electronic Heat Capacity

Classically, each free electron should contribute \(\frac{3}{2}k_B\) to the heat capacity, giving a total \(C_{\text{el}} = \frac{3}{2}Nk_B\). This grossly overestimates the electronic contribution to heat capacity — experiments show it is about 100 times smaller at room temperature. The resolution is quantum mechanical.

Only electrons within an energy range \(\sim k_BT\) of the Fermi level can be thermally excited (those deeper than \(k_BT\) below \(E_F\) cannot find empty states nearby). The fraction of “active” electrons is roughly \(T/T_F\), so the thermal energy is roughly

\[U_{\text{thermal}} \sim N \cdot \frac{T}{T_F} \cdot k_BT = Nk_B\frac{T^2}{T_F}\]

giving \(C_{\text{el}} \sim Nk_B T/T_F\) — linear in \(T\) and smaller than the classical value by \(T/T_F \ll 1\).

A careful Sommerfeld expansion gives the exact result:

\[C_{\text{el}} = \frac{\pi^2}{3}k_B^2 T \cdot g(E_F) = \frac{\pi^2}{2}Nk_B\frac{T}{T_F} \equiv \gamma T\]

The coefficient \(\gamma = \frac{\pi^2 k_B^2}{3} g(E_F)\) is the Sommerfeld coefficient (or electronic heat capacity coefficient). Experimentally, the total low-temperature heat capacity of a metal is \(C_V = \gamma T + AT^3\), where the \(T^3\) term is the Debye phonon contribution. Plotting \(C_V/T\) versus \(T^2\) gives a straight line with intercept \(\gamma\) and slope \(A\), enabling independent determination of the Fermi level and Debye temperature.

Electrical Conductivity and the Drude Model

Drude model (1900): Before Sommerfeld’s quantum treatment, Drude modeled the conduction electrons as classical particles undergoing random scattering from the lattice ions with a characteristic relaxation time \(\tau\). In an electric field \(\mathbf{E}\), the equation of motion for the average electron momentum \(\mathbf{p} = m\mathbf{v}\) is

\[\frac{d\mathbf{p}}{dt} = -e\mathbf{E} - \frac{\mathbf{p}}{\tau}\]

The collision term \(-\mathbf{p}/\tau\) captures the dissipative effect of scattering. In steady state \(d\mathbf{p}/dt = 0\):

\[\mathbf{v}_d = \frac{-e\tau}{m}\mathbf{E}\]

This gives current density \(\mathbf{J} = -ne\mathbf{v}_d = \frac{ne^2\tau}{m}\mathbf{E} = \sigma\mathbf{E}\), with the Drude conductivity

\[\sigma = \frac{ne^2\tau}{m}\]

or equivalently, the resistivity \(\rho = 1/\sigma = m/(ne^2\tau)\).

The Drude model gives a correct qualitative picture but the microscopic interpretation of \(\tau\) is wrong: Drude thought electrons scatter off ion cores, giving \(\tau \sim 10^{-14}\) s. Modern understanding (from Bloch’s theorem) is that electrons in a perfect periodic potential travel without scattering — all scattering is from deviations from periodicity: phonons, impurities, defects. This does not change the form of the Drude result, but changes the interpretation of \(\tau\).

Sommerfeld’s refinement: In the quantum treatment, only electrons near the Fermi surface participate in transport. The appropriate velocity scale is \(v_F\), not the classical thermal velocity. The mean free path is \(\ell = v_F \tau\). For copper, \(\tau \approx 2.5 \times 10^{-14}$ s at room temperature, giving \(\ell \approx v_F\tau \approx 40\) nm — much larger than the lattice spacing, confirming that Bloch states are largely undisturbed by the lattice.

Thermal Conductivity and the Wiedemann-Franz Law

In a metal, heat is conducted primarily by electrons. The thermal conductivity is

\[\kappa = \frac{1}{3}C_{\text{el}} v_F^2 \tau = \frac{\pi^2 k_B^2 T n \tau}{3m}\]

The ratio of thermal to electrical conductivity is the Wiedemann-Franz ratio:

\[\frac{\kappa}{\sigma T} = \frac{\pi^2 k_B^2}{3e^2} \equiv L_0\]

where \(L_0 = 2.44 \times 10^{-8}\) W·\(\Omega\)/K\(^2\) is the Lorenz number. This universal ratio, predicted by the Sommerfeld model and observed experimentally for many metals, is a triumph of the free electron theory. Its universality follows from the fact that both heat and charge are carried by the same electrons; the \(\tau\) and \(v_F\) cancel in the ratio.

The Hall Effect

The Hall effect (discovered by E.H. Hall in 1879) provides a direct measurement of the sign and density of charge carriers in a conductor. When a current flows in the \(x\)-direction and a magnetic field \(\mathbf{B}\) is applied in the \(z\)-direction, the Lorentz force deflects electrons toward the \(y\)-direction. Charge builds up on the sample surface until the resulting electric field \(E_y\) balances the magnetic force:

\[\mathbf{F} = -e(\mathbf{E} + \mathbf{v}\times\mathbf{B}) = 0\]

In steady state, \(E_y = v_x B_z\). Since \(J_x = -nev_x\), we have \(E_y = -J_x B_z/(ne)\). The Hall coefficient is defined as

\[R_H = \frac{E_y}{J_x B_z} = -\frac{1}{ne}\]

The sign of \(R_H\) reveals whether the carriers are electrons (\(R_H < 0\)) or holes (\(R_H > 0\)), and the magnitude gives the carrier density \(n\). For free electrons, the simple Drude result works well for many simple metals. For transition metals and semiconductors, the story is more complex, involving band structure effects (Chapter 8).

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Chapter 8: Electrons in a Periodic Potential

Bloch’s Theorem

The Drude and Sommerfeld models treat conduction electrons as completely free, ignoring the periodic potential \(V(\mathbf{r})\) of the ion cores. In reality, this potential is not small — the energy difference between an electron in the atom and in the solid is of order 10 eV, comparable to the bandwidth. Yet the free electron model works surprisingly well. Understanding why requires Bloch’s theorem.

The Hamiltonian of an electron in a crystal is

\[H = -\frac{\hbar^2}{2m}\nabla^2 + V(\mathbf{r})\]

where \(V(\mathbf{r} + \mathbf{R}) = V(\mathbf{r})\) for all Bravais lattice vectors \(\mathbf{R}\). The Schrödinger equation is \(H\psi = \epsilon\psi\). Bloch proved that the eigenstates of this Hamiltonian can always be written in a specific form:

Proof: Consider the translation operator \(T_{\mathbf{R}}\) defined by \(T_{\mathbf{R}}\psi(\mathbf{r}) = \psi(\mathbf{r}+\mathbf{R})\). Since \(V\) is periodic, \([H, T_{\mathbf{R}}] = 0\) for all \(\mathbf{R}\). Moreover, all translation operators commute: \(T_{\mathbf{R}}T_{\mathbf{R}'} = T_{\mathbf{R}+\mathbf{R}'} = T_{\mathbf{R}'}T_{\mathbf{R}}\). Therefore, we can simultaneously diagonalize \(H\) and all \(T_{\mathbf{R}}\). The eigenvalues of \(T_{\mathbf{R}}\) are \(e^{i\mathbf{k}\cdot\mathbf{R}}\) (since \(|T_\mathbf{R}| = 1\) and we need \(T_{\mathbf{R}}T_{\mathbf{R}'} = T_{\mathbf{R}+\mathbf{R}'}\)). A state with these translation eigenvalues must have the form \(\psi_{n\mathbf{k}}(\mathbf{r}) = e^{i\mathbf{k}\cdot\mathbf{r}}u_{n\mathbf{k}}(\mathbf{r})\), which is the Bloch form. \(\square\)

The Bloch state is a plane wave modulated by a periodic function. When the potential vanishes, \(u_{n\mathbf{k}} = \text{const}\) and we recover free electrons. The crystal momentum \(\hbar\mathbf{k}\) is conserved in scattering processes only modulo \(\hbar\mathbf{G}\) — it is not the true mechanical momentum \(m\mathbf{v}\).

Nearly Free Electron Model

The simplest way to see how the periodic potential opens up band gaps is the nearly free electron (NFE) model, treating \(V(\mathbf{r})\) as a weak perturbation of the free electron gas.

The Hamiltonian in the plane wave basis \(|\mathbf{k}\rangle = \frac{1}{\sqrt{V}}e^{i\mathbf{k}\cdot\mathbf{r}}\) is

\[H_{\mathbf{k},\mathbf{k}'} = \frac{\hbar^2k^2}{2m}\delta_{\mathbf{k}\mathbf{k}'} + V_{\mathbf{k}-\mathbf{k}'}\]

where \(V_{\mathbf{G}} = \frac{1}{V}\int V(\mathbf{r})e^{-i\mathbf{G}\cdot\mathbf{r}}d^3r\) are the Fourier components of the potential. Non-zero matrix elements connect states \(\mathbf{k}\) and \(\mathbf{k} + \mathbf{G}\) — states differing by a reciprocal lattice vector.

Far from any zone boundary, first-order perturbation theory applies, and the energy is simply

\[\epsilon_{\mathbf{k}} \approx \frac{\hbar^2k^2}{2m} + V_0\]

where \(V_0\) is the average potential. However, near a zone boundary where two free electron states become degenerate — \(\frac{\hbar^2k^2}{2m} = \frac{\hbar^2|\mathbf{k}+\mathbf{G}|^2}{2m}\) — we must use degenerate perturbation theory.

Consider the zone boundary at \(k = \pi/a\) in 1D. The states \(|k\rangle\) and \(|k-G\rangle = |k-2\pi/a\rangle\) are degenerate at exactly \(k = \pi/a\). The \(2\times2\) secular problem gives

\[\epsilon_{\pm} = \frac{\hbar^2}{2m}\left(\frac{\pi}{a}\right)^2 \pm |V_G|\]

where \(V_G = V_{2\pi/a}\) is the relevant Fourier component of the potential. The two eigenstates are

\[\psi_{\pm} \propto e^{ikx} \pm e^{i(k-G)x} \propto \begin{cases}\cos(\pi x/a) \\ \sin(\pi x/a)\end{cases}\]

The cosine state concentrates charge density near the ion positions (high potential, lower kinetic energy), while the sine state places charge between ions (lower potential energy). The energy splitting \(2|V_G|\) between them is the band gap.

This is the central result of band theory. Even an arbitrarily weak periodic potential creates band gaps. The Bloch states just below and above the gap are standing waves — the result of Bragg reflection at the zone boundary.

Energy Bands and the Band Structure

The full solution of the Bloch Hamiltonian for all \(\mathbf{k}\) in the first Brillouin zone gives a set of energy bands \(\epsilon_n(\mathbf{k})\), \(n = 1, 2, 3, ...\). Several general features:

Periodicity: \(\epsilon_n(\mathbf{k}+\mathbf{G}) = \epsilon_n(\mathbf{k})\) — the bands are periodic in reciprocal space.

Time reversal symmetry: In the absence of a magnetic field, \(\epsilon_n(\mathbf{k}) = \epsilon_n(-\mathbf{k})\). Combined with crystal symmetry, this often leads to additional degeneracies.

Band crossings and anticrossings: Bands can cross at high-symmetry points if they transform as different irreducible representations of the little group. Otherwise, the off-diagonal matrix element of \(H\) hybridizes the bands and they “anticross” — analogous to the avoided crossing of two coupled quantum levels.

Zone boundary behavior: Every band approaches the zone boundary with zero slope (\(d\epsilon/dk = 0\)), analogous to standing waves.

Density of states: The density of states \(g(\epsilon)\) can be quite different from the free-electron parabola, with peaks called van Hove singularities at energies where \(\nabla_{\mathbf{k}}\epsilon_n(\mathbf{k}) = 0\).

Metals, Insulators, and Semiconductors

The most profound consequence of band theory is that it explains the distinction between metals, insulators, and semiconductors purely from the electronic structure.

Metals occur when at least one band is partially filled. The Fermi level cuts through a band, leaving a Fermi surface. Electrons near the Fermi surface can be excited to slightly higher energies at negligible cost, explaining electrical conduction, large heat capacity contributions, and optical absorption at all frequencies.

Insulators occur when the Fermi level lies in a band gap: all bands below are completely full (the valence bands), all bands above are completely empty (the conduction bands). For an electron to be excited, it must overcome the gap \(E_g\). At \(T = 0\), there are no current-carrying states available, and the material is a perfect insulator. Typical insulators have \(E_g > 4\) eV (diamond: 5.5 eV, SiO\(_2\): 9 eV, NaCl: 9.4 eV).

Semiconductors are insulators with small band gaps, typically \(E_g \lesssim 3\) eV. At room temperature, a small fraction of electrons are thermally excited across the gap, creating both free electrons in the conduction band and holes in the valence band. Si has \(E_g = 1.12\) eV, Ge has \(E_g = 0.67\) eV, GaAs has \(E_g = 1.42\) eV.

The existence of insulators is the most startling prediction of band theory. Without it, we might expect all solids to be metallic (since all atoms contain electrons). Band theory explains why, for example, diamond (carbon with 4 valence electrons, filling 2 full bands in a zone containing 4 states per atom per spin) is a perfect insulator.

Tight-Binding Model

While the nearly free electron model starts from extended plane waves and adds periodicity, the tight-binding model starts from localized atomic orbitals and allows them to overlap. It is more appropriate for narrow bands (d-electrons in transition metals, for example).

Consider a crystal with one atom per unit cell, each with an atomic orbital \(\phi(\mathbf{r})\) of energy \(\epsilon_0\). The Bloch state constructed from these orbitals is

\[\psi_{\mathbf{k}}(\mathbf{r}) = \frac{1}{\sqrt{N}}\sum_{\mathbf{R}} e^{i\mathbf{k}\cdot\mathbf{R}} \phi(\mathbf{r}-\mathbf{R})\]

(a Bloch sum of atomic orbitals). Computing the expectation value of the crystal Hamiltonian \(H = H_{\text{at}} + \Delta U(\mathbf{r})\), where \(\Delta U\) is the difference between the crystal and atomic potentials, gives

\[\epsilon(\mathbf{k}) = \epsilon_0 - \beta - \gamma\sum_{\mathbf{R}_j \neq 0} e^{i\mathbf{k}\cdot\mathbf{R}_j}\]

where \(\beta = -\int \phi^*(\mathbf{r})\Delta U(\mathbf{r})\phi(\mathbf{r})\,d^3r\) is a shift in the site energy (usually positive, lowering the energy), and \(\gamma = -\int \phi^*(\mathbf{r})\Delta U(\mathbf{r})\phi(\mathbf{r}-\mathbf{R}_j)\,d^3r\) is the transfer integral (or hopping matrix element) between neighboring sites.

For a simple cubic lattice with lattice constant \(a\), including only nearest-neighbor hopping:

\[\epsilon(\mathbf{k}) = \epsilon_0 - \beta - 2\gamma(\cos k_x a + \cos k_y a + \cos k_z a)\]

The bandwidth (energy range of the band) is \(W = 2 \times 2 \times 3\gamma = 12\gamma\) for the simple cubic case. The bandwidth grows with the overlap integral \(\gamma\), which increases as atoms are brought closer together. This is why under pressure, bands broaden and insulators can become metallic.

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Chapter 9: Band Structure of Real Materials

Effective Mass

Near a band minimum (e.g., the bottom of the conduction band), the dispersion can be approximated as quadratic:

\[\epsilon_n(\mathbf{k}) \approx \epsilon_c + \frac{\hbar^2}{2m^*}k^2\]

where \(m^*\) is the effective mass. More generally, the effective mass is a tensor:

\[\left(\frac{1}{m^*}\right)_{ij} = \frac{1}{\hbar^2}\frac{\partial^2\epsilon}{\partial k_i \partial k_j}\]

The effective mass captures the effect of the band structure on the dynamics: an electron in a band responds to external forces as if it had mass \(m^*\), which can be much less than the bare electron mass (e.g., \(m^* \approx 0.067m\) in GaAs) or much larger (heavy fermion materials).

For a band maximum, the effective mass is negative (the dispersion curves downward). An electron near a band maximum with negative effective mass behaves like a positively charged particle with positive effective mass — this is the hole concept.

Holes

A full valence band carries no net current. If we remove one electron from state \(\mathbf{k}_e\) in an otherwise full band (by thermal excitation or optical absorption), the missing electron behaves as a particle called a hole with:

• wavevector \(\mathbf{k}_h = -\mathbf{k}_e\)
• energy \(\epsilon_h(\mathbf{k}_h) = -\epsilon_v(\mathbf{k}_e)\) (relative to the valence band maximum)
• charge \(+e\) (positive)
• effective mass \(m_h^* = -m_e^*\) (positive at a band maximum)

The hole concept is extremely useful in semiconductor physics. Instead of tracking \(N-1\) electrons in a nearly full valence band, we track one hole in an empty band. Holes move and respond to fields as if they were positive charge carriers.

The current in a semiconductor with electron concentration \(n\) in the conduction band and hole concentration \(p\) in the valence band is

\[\mathbf{J} = -ne\mathbf{v}_e + pe\mathbf{v}_h = \sigma\mathbf{E}\]

where the conductivity is \(\sigma = ne\mu_e + pe\mu_h\), with \(\mu_{e,h} = e\tau_{e,h}/m^*_{e,h}\) the mobilities.

Optical Properties and Band Gaps

The interaction of light with a semiconductor is governed by the band gap. A photon of energy \(\hbar\omega\) can be absorbed if its energy exceeds the band gap: \(\hbar\omega > E_g\). This determines the optical absorption edge and explains why different semiconductors absorb different colors of light.

A direct band gap semiconductor (like GaAs) has its conduction band minimum and valence band maximum at the same \(\mathbf{k}\) value (usually \(\Gamma\)). Optical transitions occur vertically in \(k\)-space with no change in \(\mathbf{k}\), giving strong absorption above the gap.

An indirect band gap semiconductor (like Si, Ge) has its conduction band minimum at a different \(\mathbf{k}\) than the valence band maximum. For optical absorption at the indirect gap, momentum conservation requires the simultaneous emission or absorption of a phonon. This makes indirect transitions weaker by the small electron-phonon coupling. Silicon therefore absorbs light much less efficiently than GaAs near its band edge — which is why silicon solar cells need to be much thicker than GaAs ones, and why GaAs (not Si) is used in LEDs and laser diodes.

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Chapter 10: Semiconductor Physics

Intrinsic Semiconductors

In a pure (intrinsic) semiconductor at temperature \(T > 0\), thermal fluctuations excite electrons from the valence band to the conduction band. For each electron excited, a hole is left behind. Charge neutrality requires \(n = p \equiv n_i\), the intrinsic carrier concentration.

The electron concentration in the conduction band is

\[n = \int_{E_c}^{\infty} g_c(\epsilon) f(\epsilon) \, d\epsilon\]

where \(g_c(\epsilon) \propto (\epsilon - E_c)^{1/2}\) is the conduction band density of states (assuming parabolic bands) and \(f(\epsilon)\) is the Fermi-Dirac distribution. Similarly, the hole concentration involves the density of states in the valence band and the hole occupation probability \(1 - f(\epsilon)\).

Assuming the Fermi level is well inside the gap (\(E_c - \mu \gg k_BT\) and \(\mu - E_v \gg k_BT\)), the Fermi-Dirac distribution can be approximated by the Boltzmann distribution:

\[n = N_c \, e^{-(E_c - \mu)/k_BT}, \qquad p = N_v \, e^{-(\mu - E_v)/k_BT}\]

where

\[N_c = 2\left(\frac{m_e^* k_BT}{2\pi\hbar^2}\right)^{3/2}, \qquad N_v = 2\left(\frac{m_h^* k_BT}{2\pi\hbar^2}\right)^{3/2}\]

are the effective density of states in the conduction and valence bands, respectively. These have the form of the classical ideal gas density of states with the effective mass replacing the free electron mass.

The product \(np\) is independent of the Fermi level:

\[np = N_c N_v \, e^{-(E_c - E_v)/k_BT} = N_c N_v \, e^{-E_g/k_BT}\]

This is the law of mass action for semiconductors. It is a powerful constraint: regardless of doping, the product \(np\) depends only on \(T\) and the fundamental material parameters. Setting \(n = p = n_i\):

\[n_i = \sqrt{N_c N_v} \, e^{-E_g/2k_BT}\]

The position of the Fermi level in an intrinsic semiconductor is found by setting \(n = p\):

\[\mu_i = \frac{E_c + E_v}{2} + \frac{k_BT}{2}\ln\!\left(\frac{N_v}{N_c}\right) = \frac{E_c + E_v}{2} + \frac{3k_BT}{4}\ln\!\left(\frac{m_h^*}{m_e^*}\right)\]

For equal effective masses (\(N_c = N_v\)), the intrinsic Fermi level lies exactly at midgap. In silicon, \(m_h^* > m_e^*\), so the intrinsic level is slightly above midgap.

Impurity Doping

The intrinsic carrier concentration in Si at room temperature (\(n_i \approx 10^{10}\) cm\(^{-3}\)) is tiny compared to the density of atoms (\(\sim 5 \times 10^{22}\) cm\(^{-3}\)). The enormous range of carrier concentration and type achievable through doping — adding controlled impurities — is what makes silicon the basis of modern electronics.

Donors

Donor impurities are atoms with one more valence electron than the host. In silicon (valence 4), phosphorus (valence 5) is a donor: four of its electrons participate in covalent bonds with neighboring Si atoms, and the fifth is bound to the P\(^+\) ion core by a Coulomb potential screened by the semiconductor dielectric constant \(\kappa \approx 11.7\) for Si.

This is analogous to a hydrogen atom in a dielectric medium, with electron mass replaced by the effective mass \(m_e^*\). The binding energy is

\[E_d = \frac{m_e^*}{m} \cdot \frac{1}{\kappa^2} \cdot 13.6 \text{ eV} \approx \frac{0.26}{(11.7)^2} \times 13.6 \approx 25 \text{ meV}\]

for Si. This is much smaller than the band gap (1.12 eV) and much less than \(k_BT\) at room temperature (26 meV). Therefore, at room temperature, essentially all donor electrons are thermally ionized (complete ionization or ionization saturation), contributing free electrons to the conduction band.

With donor concentration \(N_D\) (cm\(^{-3}\)) under complete ionization, the electron concentration satisfies

\[n \approx N_D, \qquad p = \frac{n_i^2}{N_D}\]

(from the law of mass action). The Fermi level shifts toward the conduction band: \(\mu = E_c - k_BT\ln(N_c/N_D)\). This is an n-type semiconductor (electrons are the majority carriers).

Acceptors

Acceptor impurities have one fewer valence electron than the host. Boron (valence 3) in silicon creates a hole: it can accept an electron from the valence band, leaving a mobile hole and a negatively charged B\(^-\) ion. The hole is bound to B\(^-\) with binding energy \(\sim 45\) meV in Si. At room temperature, all acceptors are ionized, giving hole concentration \(p \approx N_A\) and \(n = n_i^2/N_A\). The Fermi level shifts toward the valence band. This is a p-type semiconductor.

Compensation

When both donors and acceptors are present, they compensate each other. The charge neutrality condition (counting all charged species) is

\[n + N_A^- = p + N_D^+\]

where \(N_A^-, N_D^+\) are the ionized acceptor and donor concentrations. In the regime of complete ionization, this gives

\[n - p = N_D - N_A\]

Combined with \(np = n_i^2\), this yields

\[n = \frac{N_D - N_A}{2} + \sqrt{\left(\frac{N_D - N_A}{2}\right)^2 + n_i^2}\]

The p-n Junction

The p-n junction is formed by bringing a p-type and an n-type semiconductor into contact (in practice, by doping different regions of the same crystal). It is the fundamental building block of modern electronics: transistors, diodes, LEDs, solar cells, and CCDs all rely on p-n junctions.

Equilibrium and the Built-in Potential

When the two regions are joined, electrons diffuse from the n-side (high \(n\)) to the p-side, and holes diffuse from the p-side to the n-side. This leaves behind ionized donors on the n-side (positive charge) and ionized acceptors on the p-side (negative charge), creating a depletion region depleted of mobile carriers and a built-in electric field directed from n to p.

In equilibrium, the drift current due to the electric field exactly cancels the diffusion current. The condition for equilibrium is that the Fermi level is spatially constant throughout the device: \(\mu(x) = \text{const}\). This forces the band structure to bend as shown schematically, with a built-in potential

\[V_{bi} = \frac{k_BT}{e}\ln\!\left(\frac{N_A N_D}{n_i^2}\right)\]

For Si with \(N_A = N_D = 10^{17}\) cm\(^{-3}\), \(V_{bi} \approx 0.8\) V.

Depletion approximation: In the depletion region, the charge density is simply \(\rho = eN_D\) on the n-side and \(\rho = -eN_A\) on the p-side (all ionized, all mobile carriers swept away). Poisson’s equation \(\nabla^2\phi = -\rho/\epsilon\) then gives a linear electric field and parabolic potential profile. The depletion width is

\[W = \sqrt{\frac{2\epsilon V_{bi}}{e}\left(\frac{1}{N_A} + \frac{1}{N_D}\right)}\]

For the symmetric case \(N_A = N_D = 10^{17}\) cm\(^{-3}\) in Si (\(\epsilon \approx 11.7\epsilon_0\)), \(W \approx 100\) nm.

Current-Voltage Characteristics: the Diode Equation

Applying a forward bias voltage \(V > 0\) (p-side positive) reduces the barrier, allowing more carriers to diffuse across. The resulting current increases exponentially with voltage. Reverse bias (\(V < 0\)) increases the barrier, suppressing diffusion; only a small reverse saturation current due to minority carrier generation flows.

The result — derived by Shockley in 1949 — is the ideal diode equation:

\[I = I_0\left(e^{eV/k_BT} - 1\right)\]

where the saturation current is

\[I_0 = eA\left(\frac{D_e n_i^2}{L_e N_A} + \frac{D_h n_i^2}{L_h N_D}\right)\]

Here \(A\) is the junction area, \(D_{e,h}\) are electron and hole diffusion coefficients, and \(L_{e,h} = \sqrt{D_{e,h}\tau_{e,h}}\) are minority carrier diffusion lengths. The derivation assumes that minority carriers (electrons on the p-side, holes on the n-side) injected across the depletion region diffuse away and recombine exponentially over the diffusion length.

The exponential \(I$-$V\) characteristic is the ideal rectifier: the diode conducts readily in forward bias (where \(I \approx I_0 e^{eV/k_BT}\)) and barely in reverse bias (where \(I \approx -I_0\)). The rectification ratio at \(|V| = 0.5\) V can be \(\sim 10^8\) — an extraordinary range.

Light-Emitting Diodes and Solar Cells

In a forward-biased LED, electrons are injected into the p-side and holes into the n-side. These minority carriers recombine, and if the recombination is radiative, a photon of energy \(\approx E_g\) is emitted. Direct-gap semiconductors (GaAs, GaN, InGaAsP) have high radiative recombination efficiencies. By tuning the composition of alloys (e.g., \(\text{In}_x\text{Ga}_{1-x}\text{N}\)), the emission wavelength can be tuned across the visible spectrum — the basis of LED lighting.

A solar cell operates the p-n junction in reverse: incident photons generate electron-hole pairs throughout the device. Minority carriers that reach the depletion region are swept across by the built-in field, generating a photocurrent. The solar cell characteristic is \(I = I_\text{ph} - I_0(e^{eV/k_BT} - 1)\), where \(I_\text{ph}\) is the photocurrent. Maximum power is extracted at the maximum power point, with efficiency limited by thermodynamic and band-gap considerations. The single-junction efficiency limit (Shockley-Queisser limit) for Si is \(\approx 30\%\), achieved because photons below \(E_g\) are not absorbed and photons well above \(E_g\) lose energy to thermalization.

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Chapter 11: Magnetic Properties and Other Topics

Diamagnetism and Paramagnetism

The magnetic response of solids arises from both orbital and spin degrees of freedom of electrons. Solids with no permanent magnetic moments exhibit Langevin diamagnetism: in an applied field, Lenz’s law induces currents that oppose the field, giving a small, negative susceptibility \(\chi = M/H < 0\), independent of temperature. This is the universal magnetic response of closed-shell atoms and molecules.

Atoms or ions with unpaired spins carry a permanent magnetic moment \(\boldsymbol{\mu}\). Without an applied field, thermal fluctuations prevent alignment; the material has zero average magnetization. In a field \(\mathbf{B}\), the moments tend to align, giving a positive susceptibility — Curie paramagnetism:

\[\chi = \frac{C}{T}, \qquad C = \frac{n\mu_0\mu_B^2 g^2 J(J+1)}{3k_B}\]

This Curie law follows from treating each spin as independent (no interactions between moments) and applying the Boltzmann distribution at temperature \(T\).

Itinerant (free) electrons also show paramagnetism — Pauli paramagnetism. An applied field shifts the spin-up and spin-down Fermi levels by \(\pm\frac{1}{2}g\mu_B B\), causing a slight imbalance in spin populations. The resulting magnetization is

\[M = \mu_B^2 g(E_F) B\]

The susceptibility \(\chi_P = \mu_0\mu_B^2 g(E_F)\) is temperature-independent (unlike Curie paramagnetism) because only electrons at the Fermi level contribute. This is the same physics responsible for the suppression of the electronic heat capacity.

Ferromagnetism

When the exchange interaction between electron spins is sufficiently strong and ferromagnetic (energetically favoring spin alignment), a material can spontaneously magnetize below the Curie temperature \(T_C\). Iron, cobalt, and nickel are the elemental ferromagnets.

The Weiss mean-field theory models ferromagnetism by adding to each spin an effective “molecular field” \(\lambda M\) (where \(\lambda\) is the Weiss constant) due to its neighbors. The self-consistency equation becomes

\[M = M_s \mathcal{B}_J\!\left(\frac{g\mu_B J (B + \lambda M)}{k_BT}\right)\]

where \(\mathcal{B}_J\) is the Brillouin function. Below \(T_C = \frac{\lambda C}{1+\lambda C} \cdot \frac{C}{1} \approx \lambda C\) (for large \(\lambda\)), this equation has nonzero solutions even with \(B = 0\) — the ferromagnetic state. For \(T > T_C\), the susceptibility follows the Curie-Weiss law:

\[\chi = \frac{C}{T - T_C}\]

Summary: The Conceptual Flow

The journey through PHYS 335 has traced a coherent logical arc. We began with the geometry of crystal structure — the Bravais lattice and its description in terms of primitive vectors — because the fundamental property of a crystal is its translational periodicity. This periodicity led naturally to the reciprocal lattice, the Fourier-space counterpart of the direct lattice, and to Brillouin zones.

The reciprocal lattice immediately illuminated X-ray diffraction: the Laue condition \(\Delta\mathbf{k} = \mathbf{G}\) says that constructive interference occurs when the momentum transfer equals a reciprocal lattice vector, reducing to Bragg’s law \(2d\sin\theta = n\lambda\). X-ray diffraction is thus a direct probe of the Fourier components of the electron density.

From the crystal structure we inferred the bonding — ionic, van der Waals, covalent, or metallic — which determines the interatomic force constants. These forces control the phonon dispersion, derived from the equations of motion of the crystal lattice. The quantum mechanics of harmonic oscillators then gives phonons as the quantized normal modes, described by the Bose-Einstein distribution. The Debye model of phonons explains the universal \(T^3\) heat capacity of insulators at low temperature.

The electrons in a crystal are governed by the Pauli principle and the periodic potential of the lattice. The Sommerfeld free electron model — electrons as a Fermi gas — correctly predicts the linear electronic heat capacity, the Wiedemann-Franz law, and many transport properties of simple metals. The key refinement is Bloch’s theorem: the eigenstates in a periodic potential are not plane waves but Bloch waves \(\psi_{n\mathbf{k}} = e^{i\mathbf{k}\cdot\mathbf{r}}u_{n\mathbf{k}}(\mathbf{r})\), leading to the band structure picture. Band gaps open at Brillouin zone boundaries where standing waves form. The question of how the band structure fills up with electrons determines whether a material is a metal, insulator, or semiconductor.

Finally, semiconductor physics applies band theory to technologically vital materials. Doping shifts the Fermi level to create n-type or p-type semiconductors; p-n junctions are the basis of diodes, transistors, LEDs, and solar cells. The physics of these devices, from the built-in potential to the exponential \(I$-$V\) curve, all follow from the basic statistical mechanics of electrons and holes in bands separated by a gap.

This progression — from geometry to forces to vibrations to electrons to bands to devices — is one of the most beautiful narratives in physics, and it stands as a monument to the power of combining symmetry, quantum mechanics, and statistical mechanics to understand the material world.

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Appendix A: Mathematical Tools

Fourier Series in Crystals

For any function \(f(\mathbf{r})\) with the periodicity of a Bravais lattice:

\[f(\mathbf{r}) = \sum_{\mathbf{G}} f_{\mathbf{G}} e^{i\mathbf{G}\cdot\mathbf{r}}, \qquad f_{\mathbf{G}} = \frac{1}{V_c}\int_{\text{cell}} f(\mathbf{r}) e^{-i\mathbf{G}\cdot\mathbf{r}} d^3r\]

The Parseval relation: \(\frac{1}{V}\int |f(\mathbf{r})|^2 d^3r = \sum_{\mathbf{G}} |f_{\mathbf{G}}|^2\).

Density of States

In \(d\) dimensions, the free-electron density of states per unit volume per unit energy is

\[g(\epsilon) = \frac{1}{(2\pi)^d} \cdot 2 \cdot \frac{d\Omega_d(k)}{d\epsilon} = \begin{cases} \frac{1}{\pi}\sqrt{\frac{2m}{\hbar^2\epsilon}} & d=1 \\ \frac{m}{\pi\hbar^2} & d=2 \\ \frac{1}{2\pi^2}\left(\frac{2m}{\hbar^2}\right)^{3/2}\epsilon^{1/2} & d=3 \end{cases}\]

Sommerfeld Expansion

For any smooth function \(H(\epsilon)\):

\[\int_{-\infty}^{\infty} H(\epsilon) f(\epsilon) \, d\epsilon = \int_{-\infty}^{\mu} H(\epsilon) \, d\epsilon + \frac{\pi^2}{6}(k_BT)^2 H'(\mu) + \frac{7\pi^4}{360}(k_BT)^4 H'''(\mu) + \cdots\]

This asymptotic expansion (valid for \(T \ll T_F\)) is the workhorse of metallic electron calculations.

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Appendix B: Key Formulas

Crystal structure:

• Lattice: \(\mathbf{R} = n_1\mathbf{a}_1 + n_2\mathbf{a}_2 + n_3\mathbf{a}_3\)
• Reciprocal primitive vectors: \(\mathbf{b}_i \cdot \mathbf{a}_j = 2\pi\delta_{ij}\)
• Interplanar spacing: \(d_{hkl} = a/\sqrt{h^2+k^2+l^2}\) (cubic)

Diffraction:

• Bragg’s law: \(2d\sin\theta = n\lambda\)
• Laue condition: \(\Delta\mathbf{k} = \mathbf{G}\)
• Structure factor: \(S_{\mathbf{G}} = \sum_j f_j e^{i\mathbf{G}\cdot\mathbf{d}_j}\)

Phonons:

• Monatomic chain: \(\omega = 2\sqrt{C/M}|\sin(ka/2)|\)
• Debye frequency: \(\omega_D = v_s(6\pi^2 n)^{1/3}\)
• Low-\(T\) heat capacity: \(C_V = \frac{12\pi^4}{5}Nk_B(T/\Theta_D)^3\)

Free electrons:

• Fermi wavevector: \(k_F = (3\pi^2 n)^{1/3}\)
• Fermi energy: \(E_F = \hbar^2k_F^2/2m\)
• Electronic heat capacity: \(C_{\text{el}} = \frac{\pi^2}{2}Nk_B(T/T_F)\)
• Drude conductivity: \(\sigma = ne^2\tau/m\)
• Wiedemann-Franz: \(\kappa/(\sigma T) = \pi^2k_B^2/(3e^2)\)

Semiconductors:

• Mass action law: \(np = n_i^2\)
• Intrinsic carriers: \(n_i = \sqrt{N_cN_v}\exp(-E_g/2k_BT)\)
• Diode equation: \(I = I_0(e^{eV/k_BT}-1)\)