Modern Physics · Upper Division

Solid-State Physics

Solids are quantum many-body systems where the Pauli exclusion principle, periodic symmetry, and collective behavior produce emergent phenomena — metals, insulators, semiconductors, and superconductors — unreachable in any single-particle picture.

Prerequisites

Quantum mechanics (Ch. 20) \cdot Statistical mechanics (Ch. S) \cdot Fourier analysis (Ch. F

Learning Goals

Apply Bloch's theorem to describe electron states in a periodic potential using crystal momentum and band index.
Explain how a periodic potential opens band gaps at Brillouin zone boundaries.
Distinguish metals, semiconductors, and insulators by the position of the Fermi level relative to band gaps.
Use the Shockley equation and law of mass action to analyze p-n junction devices.
Describe the BCS theory of superconductivity, Cooper pairs, and the Meissner effect.

SS.1 Crystal Structure and Reciprocal Lattice

A crystal is a periodic arrangement of atoms. The Bravais lattice is the set of all translation vectors R = n₁a₁ + n₂a₂ + n₃a₃ (integers nᵢ, primitive vectors aᵢ). The reciprocal lattice is defined by vectors G satisfying e^(iG·R) = 1 for all lattice vectors R:

G = m_{1} b_{1} + m_{2} b_{2} + m_{3} b_{3} \qquad where \qquad b_{i} \cdot a_{j} = 2\pi \delta_{ij}

(SS.1)

X-ray diffraction from a crystal: constructive interference when the scattering vector Δk = G (a reciprocal lattice vector) — this is the von Laue condition, equivalent to Bragg's law: 2d sin θ = nλ.

SS.2 Bloch's Theorem and Band Structure

Theorem SS.1 — Bloch's Theorem

For an electron in a periodic potential V(r) = V(r+R), the eigenstates have the form

\psi_nk(r) = e^{ik\cdotr} u_{nk}(r) \qquad (Bloch wavefunction

where u_{nk} has the periodicity of the lattice: u_{nk}(r+R) = u_{nk}(r). The quantum number k (

crysta

l momentum) lives in the first Brillouin zone; n is the band index. The energy spectrum E_

s periodic in k-space with the reciprocal lattice.

The band structure E_n(k) determines all electronic properties. Near the zone boundary k = π/a, plane waves |k⟩ and |k−G⟩ become degenerate and hybridize — this opens a band gap. Materials are classified by their band structure:

Metal: Fermi level cuts through a band. Many states available for conduction at any temperature.

Insulator/Semiconductor:Fermi level in a gap. Gap > 4 eV: insulator (SiO₂: 9 eV). Gap 0.1–4 eV: semiconductor (Si: 1.1 eV, GaAs: 1.4 eV).

Semimetal: valence and conduction bands overlap slightly, but density of states at Fermi level is small (graphene, bismuth).

Example SS.1 — Nearly Free Electron Model — Band Gap

For a 1D $crystal (lattice constant a), treat the periodic potential as a small perturbation V(x) =$ $2\pix/a). Find the band gap at k = \pi/a$ .

Degenerate states:

At k = \pi/a, the states e^{i\pix/a} and e^{-i\pix/a} are degenerate (energy \hbar^{2}(\pi/a)^{2}/2m

Matrix element:

V_{12} = ⟨k|V|k-G⟩ = V_{1} (first Fourier component of potential

Diagonalize:

H = [[E_{0}, V_{1}],[V_{1}, E_{0}]]. Eigenvalues: E_{0} \pm |V_{1}|

Band gap:

E_{gap} = 2|V_{1}|. The gap equals twice the first Fourier component of the periodic potential

Larger potential \to wider gap

Bonding/antibonding:

Lower state: \psi = cos(\pix/a) (peaks at ion cores \to lower energy). Upper: sin(\pix/a) (peaks be

tween ions \to higher energy

SS.3 Semiconductors and the p-n Junction

Semiconductors can be doped to create carriers. n-type doping adds donor impurities (extra electrons); p-type adds acceptors (holes). The carrier densities satisfy the law of mass action:

np = n_{i}^{2} \qquad n_{i}^{2} = 4(k_{BT})^{3}(m_{e} m_{h})^{3/2}/(2\pi\hbar^{2})^{3} \times e^{-Eg/k_{BT}}

(SS.2)

A p-n junction forms when p and n regions contact. Electrons diffuse from n to p, holes from p to n, creating a depletion region with a built-in electric field that opposes further diffusion. At equilibrium, the Fermi level is flat across the junction. Under forward bias: reduces the barrier → exponential current increase (Shockley diode equation):

I = I_{0} (e^{eV/k_{BT}} - 1) \qquad (Shockley equation)

(SS.3)

This exponential behavior is the basis of diodes, transistors (BJT, MOSFET), solar cells, and LEDs. A solar cell is a reverse-biased p-n junction illuminated with photons — the photocurrent drives the junction into forward bias.

SS.4 Superconductivity

Below a critical temperature T_c, some metals lose all electrical resistance (superconductivity, discovered by Kamerlingh Onnes, 1911) and expel magnetic fields (Meissner effect). The BCS theory (Bardeen, Cooper, Schrieffer, 1957) explains this via Cooper pairs: electrons near the Fermi surface pair through phonon-mediated attraction despite Coulomb repulsion, forming a macroscopic quantum state described by a single wavefunction:

|BCS⟩ = \prod_k (u_{k} + v_{k} c^\dagger_{k↑} c^\dagger_{-k↓}) |0⟩ \qquad (BCS ground state)

(SS.4)

Key signatures: energy gap Δ(T) at the Fermi surface (2Δ ≈ 3.52 k_BT_c), quantized magnetic flux Φ = n × Φ₀ (Φ₀ = h/2e = 2.07×10⁻¹⁵ Wb), and the Josephson effect (supercurrent across a tunneling barrier). High-T_c superconductors (cuprates, T_c up to 135 K; hydrogen sulfide under pressure: 203 K) are not fully explained by BCS.

Definition SS.1 — Common Traps

Band gaps are collective crystal effects: they are not atomic energy levels copied unchanged.
Holes behave as quasiparticles: they represent missing electrons in nearly full bands.
Effective mass can differ from electron mass: band curvature controls carrier dynamics.
Conductivity depends on carriers and scattering: high carrier density alone is not enough.

Exercises — SS.1–SS.4 Solid-State Physics

State the band gap of silicon at room temperature and classify it as a metal, semiconductor, or insulator.

Straightforward

In t

he nearly free electron model, the first Fourier component of the periodic potential is V_{1}

eV. Find the band gap.

Straightforward

3.For an FCC crystal (lattice constant a), find the reciprocal lattice vectors and identify the first Brillouin zone shape. Which high-symmetry points are conventionally labeled?

Straightforward

Apply the Drude model to copper: find \tau and the mean free path. Why does this mean free pa

th require a quantum mechanical explanation?

Intermediate

5.Derive the phonon dispersion relation for a monatomic linear chain. What happens at the zone boundary? How does a diatomic chain differ?

Intermediate

Explain the integer quantum Hall effect. Why is \sigma_xy = \nue^{2}/h quantized exactly? How does t

he fractional QHE differ?

Challenging

Key Takeaways

$Bloch theorem: \psi_nk = e^{ik\cdotr} u_{nk}. Crystal momentum k in Brillouin zone; bands E_{n}(k$ .
$Band gap at zone boundary: 2|V_{G}| (Fourier component of potential). Metal vs. insulator vs$ . semiconductor.
$Law of mass action: np = n_{i}^{2}. Shockley diode: I = I_{0}(e^{eV/kT} - 1$ .
$BCS: Cooper pairs (phonon-mediated). Energy gap 2\Delta \approx 3.52 k_BT_c. Meissner effect; flux qu$ antization.
$Phonon dispersion: \omega = 2\sqrt(K/m)|sin(ka/2)|. Sound at small k; flat at zone boundary$ .
$QHE: \sigma_xy = \nue^{2}/h exactly (Chern number). Resistance standard. Fractional QHE: composite f$ ermions.