Complete Solid State Physics

Crystal Lattice

A crystal is modeled by translating a basis through a Bravais lattice. The primitive vectors $\mathbf a_1,\mathbf a_2,\mathbf a_3$ generate lattice points

$$\mathbf R=n_1\mathbf a_1+n_2\mathbf a_2+n_3\mathbf a_3,$$

with integer $n_i$. This is the starting point for using symmetry instead of tracking every atom separately.

Reciprocal Lattice

The reciprocal lattice is defined by vectors $\mathbf b_i$ satisfying

$$\mathbf a_i\cdot \mathbf b_j=2\pi\delta_{ij}.$$

Waves in a crystal are naturally organized by reciprocal vectors because crystal momentum is conserved modulo a reciprocal lattice vector.

Brillouin Zone

The first Brillouin zone is the Wigner-Seitz cell of the reciprocal lattice. It is the fundamental domain for wavevectors in a periodic solid.

Bloch Theorem

For a periodic potential $V(\mathbf r+\mathbf R)=V(\mathbf r)$, one-particle eigenstates can be written

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

where $u_{n\mathbf k}$ has the periodicity of the lattice.

Band Structure

Solving the Schrodinger equation in a periodic potential gives energy bands $E_n(\mathbf k)$. Filled and partially filled bands distinguish insulators, semiconductors, and metals.

Fermi Surface

In a metal, the Fermi surface is the surface in reciprocal space separating filled from empty electronic states at zero temperature. It controls many transport and thermodynamic properties.

Phonons

Small oscillations of a crystal lattice quantize into phonons. Acoustic phonons arise from collective translations, while optical phonons occur when atoms in the basis move relative to each other.