Defects in Crystals

9.1 Point Defects

• Vacancy: Missing atom at a lattice site.
• Interstitial: Extra atom between lattice sites.
• Substitutional: Foreign atom replacing a host atom.
• Frenkel defect: Vacancy-interstitial pair (atom moves to interstitial site).
• Schottky defect: Vacancy pair (in ionic crystals, cation and anion vacancies).

Equilibrium concentration of vacancies:

$n_v = N\,e^{-E_v/(k_B T)}$

Where $N$ is the number of lattice sites and $E_v$ is the vacancy formation energy ($\sim 1$ eV).

Derivation. Minimising the free energy $F = n_v E_v - T S_{\mathrm{config}}$ where $S_{\mathrm{config} = k_B \ln\binom{N}{n_v}}$:

$\frac{\partial F}{\partial n_v} = E_v + k_B T \ln\left(\frac{n_v}{N - n_v}\right) = 0$

For $n_v \ll N$: $n_v = N e^{-E_v/(k_B T)}$. $\blacksquare$

9.2 Dislocations

• Edge dislocation: Extra half-plane inserted into the lattice. Burgers vector $\mathbf{b}$ is perpendicular to the dislocation line.
• Screw dislocation: The lattice is sheared. $\mathbf{b}$ is parallel to the dislocation line.

Dislocations enable plastic deformation at stresses far below the theoretical shear strength. The Peach-Koehler force on a dislocation:

$\mathbf{F} = (\boldsymbol{\sigma}\cdot\mathbf{b}) \times \hat{\mathbf{t}}$

Where $\boldsymbol{\sigma}$ is the stress tensor and $\hat{\mathbf{t}}$ is the unit tangent to the Dislocation line.

9.3 Impact on Properties

Defects strongly affect electrical, mechanical, and thermal properties:

• Electrical: Donor and acceptor levels in semiconductors are substitutional defects. Vacancies act as scattering centres, reducing conductivity.
• Mechanical: Dislocations determine yield strength (Hall—Petch relation). Work hardening increases dislocation density.
• Thermal: Point defects scatter phonons, reducing thermal conductivity.