% miko:type: complete
% miko:title: Complete Solid State Physics
% miko:summary: A structured path from crystal symmetry to bands, Fermi surfaces, and phonons.
% miko:tags: solid
% miko:difficulty: undergrad
% miko:related: complete-quantum-mechanics, complete-statistical-mechanics
% miko:keywords: crystal, lattice, band structure, phonon

\documentclass{article}
\usepackage{amsmath}
\title{Complete Solid State Physics}
\begin{document}
\maketitle

\section{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.

\section{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.

\section{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.

\section{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.

\section{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.

\section{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.

\section{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.

\end{document}
