Lesson 5.6 · 5. Statistical Physics

Crystals & Band Theory

Solid matter surrounds us, yet its properties are extraordinarily diverse: metals conduct electricity, diamonds do not, and silicon can be made to do either. The key to understanding this diversity lies in the quantum behavior of electrons in the periodic environment of a crystal lattice. When vast numbers of atoms arrange themselves into a regular repeating pattern, the allowed energies for electrons organize into bands separated by forbidden gaps. This single idea explains why some materials are metals, others are insulators, and others are semiconductors that power modern technology.

This lesson builds directly on quantum mechanics (lessons 6-8) and statistical physics (lessons 21-25). We will see how the wave nature of electrons, combined with the periodicity of crystals, produces the band structure that governs virtually all electronic properties of solids.

Why Band Theory Matters

A single atom of silicon has discrete energy levels. But a crystal of $10^{23}$ silicon atoms has energy levels so closely spaced that they merge into continuous bands. The structure of these bands, where they lie, how wide they are, and whether they are filled or empty, determines whether the material is a metal, an insulator, or a semiconductor. Band theory is the foundation of all solid-state electronics.

Crystal Lattices

Bravais Lattices and Unit Cells

A crystal is a solid in which atoms are arranged in a pattern that repeats periodically in three dimensions. The mathematical description of this periodicity begins with the Bravais lattice: an infinite set of points generated by three primitive translation vectors $\mathbf{a}_1$, $\mathbf{a}_2$, $\mathbf{a}_3$. Every lattice point can be written as:

\mathbf{R} = n_1 \mathbf{a}_1 + n_2 \mathbf{a}_2 + n_3 \mathbf{a}_3, \quad n_1, n_2, n_3 \in \mathbb{Z}

The unit cell is the smallest region of space that, when repeated by all lattice translations, tiles the entire crystal without gaps or overlaps. The primitive cell contains exactly one lattice point. The Wigner-Seitz cell is a particularly symmetric choice: the region of space closer to one lattice point than to any other.

In three dimensions, there are exactly 14 distinct Bravais lattices, grouped into 7 crystal systems (cubic, tetragonal, orthorhombic, hexagonal, trigonal, monoclinic, triclinic). The most important for understanding common materials are:

Simple cubic (SC): atoms only at the corners of a cube. Rare in nature (polonium is the only element).
Body-centered cubic (BCC): atoms at corners plus one at the center. Iron, chromium, tungsten.
Face-centered cubic (FCC): atoms at corners plus one at each face center. Copper, aluminum, gold.
Diamond cubic: FCC with a two-atom basis. Silicon, germanium, diamond.

The three cubic Bravais lattices: simple cubic (atoms at corners only), body-centered cubic (additional atom at center), and face-centered cubic (additional atoms at face centers)

The Reciprocal Lattice

The reciprocal lattice is a mathematical construction that is essential for understanding diffraction and electronic properties. Given primitive vectors $\mathbf{a}_1$, $\mathbf{a}_2$, $\mathbf{a}_3$ of the direct lattice, the reciprocal lattice vectors are:

\mathbf{b}_1 = 2\pi \frac{\mathbf{a}_2 \times \mathbf{a}_3}{\mathbf{a}_1 \cdot (\mathbf{a}_2 \times \mathbf{a}_3)}, \quad \mathbf{b}_2 = 2\pi \frac{\mathbf{a}_3 \times \mathbf{a}_1}{\mathbf{a}_1 \cdot (\mathbf{a}_2 \times \mathbf{a}_3)}, \quad \mathbf{b}_3 = 2\pi \frac{\mathbf{a}_1 \times \mathbf{a}_2}{\mathbf{a}_1 \cdot (\mathbf{a}_2 \times \mathbf{a}_3)}

These satisfy $\mathbf{a}_i \cdot \mathbf{b}_j = 2\pi \delta_{ij}$. The reciprocal lattice lives in momentum space (or wave-vector space). The Wigner-Seitz cell of the reciprocal lattice is called the first Brillouin zone, and it plays a central role in band theory.

Why the Reciprocal Lattice?

Any function with the periodicity of the crystal lattice can be expanded in a Fourier series whose wave vectors are reciprocal lattice vectors $\mathbf{G}$. X-ray diffraction peaks occur exactly when the scattering wave vector equals a reciprocal lattice vector (the Laue condition). The entire electronic structure of solids is naturally described in reciprocal space.

Bloch's Theorem

The central theoretical result of band theory is Bloch's theorem. Consider an electron moving in a potential $V(\mathbf{r})$ that has the periodicity of the crystal: $V(\mathbf{r} + \mathbf{R}) = V(\mathbf{r})$ for all lattice vectors $\mathbf{R}$. The Hamiltonian commutes with all lattice translation operators, so the energy eigenstates can be chosen as simultaneous eigenstates of the Hamiltonian and all translations.

Bloch's Theorem

The eigenstates of a single electron in a periodic potential can be written as:

$$\psi_{n\mathbf{k}}(\mathbf{r}) = e^{i\mathbf{k} \cdot \mathbf{r}} \, u_{n\mathbf{k}}(\mathbf{r})$$

where $u_{n\mathbf{k}}(\mathbf{r})$ has the full periodicity of the lattice: $u_{n\mathbf{k}}(\mathbf{r} + \mathbf{R}) = u_{n\mathbf{k}}(\mathbf{r})$. The quantum number $\mathbf{k}$ is the crystal momentum (defined modulo a reciprocal lattice vector), and $n$ is the band index.

Sketch of the Proof

Translation operators: Define $\hat{T}_{\mathbf{R}} \psi(\mathbf{r}) = \psi(\mathbf{r} + \mathbf{R})$. Since $V$ is periodic, $[\hat{H}, \hat{T}_{\mathbf{R}}] = 0$ for all $\mathbf{R}$.

Simultaneous eigenstates: The translation operators also commute with each other: $\hat{T}_{\mathbf{R}} \hat{T}_{\mathbf{R}'} = \hat{T}_{\mathbf{R}+\mathbf{R}'}$. They form an abelian group, so we can find simultaneous eigenstates of $\hat{H}$ and all $\hat{T}_{\mathbf{R}}$.

Eigenvalue structure: Because $\hat{T}_{\mathbf{R}} \hat{T}_{\mathbf{R}'} = \hat{T}_{\mathbf{R}+\mathbf{R}'}$, the eigenvalues must satisfy $\lambda(\mathbf{R}) \lambda(\mathbf{R}') = \lambda(\mathbf{R} + \mathbf{R}')$. This forces $\lambda(\mathbf{R}) = e^{i\mathbf{k} \cdot \mathbf{R}}$ for some wave vector $\mathbf{k}$.

Bloch form: Writing $\psi(\mathbf{r}) = e^{i\mathbf{k}\cdot\mathbf{r}} u(\mathbf{r})$ and requiring $\hat{T}_{\mathbf{R}} \psi = e^{i\mathbf{k}\cdot\mathbf{R}} \psi$ gives $u(\mathbf{r}+\mathbf{R}) = u(\mathbf{r})$, confirming $u$ is lattice-periodic.

A crucial consequence: $\mathbf{k}$ and $\mathbf{k} + \mathbf{G}$ (where $\mathbf{G}$ is any reciprocal lattice vector) label the same physical state. Therefore, all distinct states are captured by $\mathbf{k}$ values within the first Brillouin zone.

Band Structure

From Atoms to Bands

Consider what happens as we bring $N$ atoms together to form a crystal. An isolated atom has sharp, discrete energy levels. When two atoms are brought close, each level splits into two (bonding and antibonding). With $N$ atoms, each atomic level splits into $N$ closely spaced levels that form a quasi-continuous band. Since $N \sim 10^{23}$ in a macroscopic crystal, the spacing between levels within a band is negligibly small.

The energy $E_n(\mathbf{k})$ as a function of crystal momentum $\mathbf{k}$ defines the band structure. For each band index $n$, this function is periodic in reciprocal space and is typically plotted along high-symmetry directions in the Brillouin zone.

As atoms are brought together, discrete energy levels broaden into continuous bands separated by band gaps

The Nearly Free Electron Model

The simplest approach to band structure starts with free electrons and treats the crystal potential as a weak perturbation. Free electrons have the dispersion $E = \hbar^2 k^2 / 2m$. The periodic potential opens gaps at the Brillouin zone boundaries, where the free-electron parabolas would cross.

At a zone boundary $k = \pm \pi/a$ (in 1D), the electron wave is Bragg-reflected: it forms standing waves rather than traveling waves. The two standing waves $\sin(\pi x/a)$ and $\cos(\pi x/a)$ have different spatial distributions relative to the ions, and therefore different potential energies. This energy difference is the band gap.

$$E_g = 2|V_G|$$

where $V_G$ is the Fourier component of the crystal potential at reciprocal lattice vector $G$.

The Tight-Binding Model

The complementary approach starts from isolated atomic orbitals and adds the hopping of electrons between neighboring sites. In the simplest 1D case with one orbital per site and nearest-neighbor hopping amplitude $t$:

E(k) = \epsilon_0 - 2t \cos(ka)

This produces a single band of width $4t$ centered at $\epsilon_0$. The bandwidth is proportional to the overlap between neighboring orbitals, which means that tightly bound (core) electrons form narrow bands, while weakly bound (valence) electrons form wide bands.

Metals, Insulators, and Semiconductors

What Determines Conductivity?

The distinction between a metal, an insulator, and a semiconductor is determined entirely by the band structure and the filling of electrons:

Metal: the highest occupied band is partially filled. Electrons near the Fermi energy can easily move to nearby empty states when an electric field is applied.
Insulator: the highest occupied band (valence band) is completely filled, and there is a large energy gap ($E_g > 3$ eV) to the next empty band (conduction band). At room temperature, virtually no electrons can be thermally excited across the gap.
Semiconductor: same structure as an insulator, but with a smaller gap ($E_g \sim 0.1$ to $3$ eV). At room temperature, a small but significant number of electrons are thermally excited into the conduction band.

Band filling determines material properties: metals have partially filled bands, insulators have a large gap, and semiconductors have a small gap allowing thermal excitation

The Fermi energy $E_F$ is the energy of the highest occupied state at absolute zero. In a metal, $E_F$ lies within a partially filled band. In an insulator or semiconductor, $E_F$ lies within the band gap (specifically, for an intrinsic semiconductor, near the middle of the gap at $T = 0$).

A Deep Question

Diamond and silicon have exactly the same crystal structure (diamond cubic) and the same number of valence electrons per atom (four). Yet diamond is a transparent insulator and silicon is a semiconductor. What accounts for this difference? The answer lies entirely in the band gap: 5.5 eV for diamond versus 1.1 eV for silicon, a consequence of the different atomic sizes and orbital overlaps.

Effective Mass and Holes

Effective Mass

An electron in a crystal responds to external forces differently from a free electron because the periodic potential modifies its inertia. Near a band extremum, the energy can be expanded as:

E(\mathbf{k}) \approx E_0 + \frac{\hbar^2}{2m^*}(\mathbf{k} - \mathbf{k}_0)^2

where the effective mass $m^*$ is defined by:

\frac{1}{m^*} = \frac{1}{\hbar^2} \frac{d^2 E}{dk^2}

At the bottom of the conduction band, $d^2E/dk^2 > 0$, so $m^* > 0$. At the top of the valence band, $d^2E/dk^2 < 0$, so $m^* < 0$. The effective mass can be much smaller or larger than the free electron mass. In GaAs, $m^* \approx 0.067 m_e$, which is why electrons move so fast in that material.

Holes

A nearly full valence band with one missing electron behaves exactly like a nearly empty band with one positive charge carrier. This fictitious particle is called a hole. Holes have:

Positive charge $+e$
Positive effective mass (the negative of the electron's effective mass at the top of the valence band)
Momentum opposite to the missing electron's momentum

The hole picture is not just a convenience. It is the natural description for transport in the valence band, and it leads to the concept of electron-hole pairs that are central to semiconductor physics.

Connection: Dirac Sea and Antiparticles

The concept of holes in solid-state physics is directly analogous to Dirac's prediction of the positron. Dirac proposed that the vacuum is a "sea" of filled negative-energy electron states; a missing electron in this sea appears as a particle with positive charge and positive energy. In both cases, the absence of a particle in a filled band behaves as a particle with opposite quantum numbers.

Doping and p-n Junctions

Doping

The conductivity of a semiconductor can be controlled by introducing impurity atoms. This process, called doping, is the foundation of all semiconductor technology.

n-type doping: replacing a silicon atom (4 valence electrons) with a phosphorus atom (5 valence electrons) adds an extra electron. Phosphorus acts as a donor: it introduces a filled energy level just below the conduction band. At room temperature, this electron is easily promoted to the conduction band. The majority carriers are electrons (negative, hence "n-type").

p-type doping: replacing silicon with boron (3 valence electrons) creates a missing electron (a hole). Boron acts as an acceptor: it introduces an empty level just above the valence band. An electron from the valence band fills this level, leaving a hole that acts as a positive charge carrier. The majority carriers are holes (positive, hence "p-type").

n_e \approx N_D \exp\left(-\frac{E_D}{2k_B T}\right) \quad \text{(n-type, low T)}

where $N_D$ is the donor concentration and $E_D$ is the donor ionization energy (typically 10-50 meV, much smaller than the band gap).

The p-n Junction

When p-type and n-type semiconductors are brought into contact, the resulting p-n junction is the most important device in electronics. At the junction:

Electrons from the n-side diffuse into the p-side; holes from the p-side diffuse into the n-side.
This diffusion leaves behind fixed ionized donors (positive) on the n-side and ionized acceptors (negative) on the p-side.
The resulting electric field creates a depletion region free of mobile carriers.
In equilibrium, the diffusion current exactly balances the drift current from the electric field.

A forward voltage bias reduces the barrier and allows current to flow exponentially; a reverse bias increases the barrier and blocks current. This asymmetry gives the p-n junction its rectifying (diode) behavior:

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

This simple equation describes the current-voltage characteristic of an ideal diode. Solar cells, LEDs, and transistors are all built from p-n junctions.

Connection: Quantum Statistics in Action

The entire theory of semiconductor devices relies on the Fermi-Dirac distribution (lesson 23). The occupation probability $f(E) = 1/(e^{(E-\mu)/k_BT}+1)$ determines how many electrons populate the conduction band and how many holes exist in the valence band. Without quantum statistics, we could not explain why silicon works as a semiconductor.

Exercises

Reciprocal lattice: Show that the reciprocal lattice of an FCC lattice with cube side $a$ is a BCC lattice with cube side $4\pi/a$. What is the volume of the first Brillouin zone?
Tight-binding band: For the 1D tight-binding model $E(k) = \epsilon_0 - 2t\cos(ka)$, compute the effective mass at the band bottom ($k=0$) and band top ($k = \pi/a$). Express your answers in terms of $t$ and $a$.
Semiconductor statistics: For an intrinsic semiconductor with band gap $E_g = 1.1$ eV at $T = 300$ K, estimate the fraction of valence electrons excited to the conduction band. Use the approximation $n \sim \exp(-E_g/2k_BT)$, with $k_BT \approx 0.026$ eV at room temperature.
Bloch's theorem: A Bloch state has the form $\psi_k(x) = e^{ikx} u_k(x)$ with $u_k(x+a) = u_k(x)$. Show that $\psi_k(x+a) = e^{ika}\psi_k(x)$. What does this imply about the probability density $|\psi_k|^2$?
Doping calculation: Silicon has about $5 \times 10^{22}$ atoms/cm$^3$. If we dope it with phosphorus at a concentration of 1 part per million, what is the donor concentration $N_D$? Compare this to the intrinsic carrier concentration $n_i \approx 1.5 \times 10^{10}$ cm$^{-3}$ at room temperature.

Key Takeaways

Crystals are periodic arrangements of atoms described by Bravais lattices; the reciprocal lattice and Brillouin zone are essential for understanding their electronic properties.
Bloch's theorem states that electrons in a periodic potential have wave functions of the form $\psi = e^{i\mathbf{k}\cdot\mathbf{r}} u(\mathbf{r})$, where $u$ is lattice-periodic, leading to a band structure $E_n(\mathbf{k})$.
Bands arise from the splitting of atomic levels; the distinction between metals, insulators, and semiconductors depends entirely on band filling and gap size.
The effective mass captures how the crystal potential modifies an electron's response to external forces; holes are the natural description of missing electrons in a nearly full band.
Doping introduces controlled impurities that create n-type (electron) or p-type (hole) conductivity; the p-n junction, the most important semiconductor device, results from joining these two types.

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