Energy of a phonon: $E = \hbar\omega$.

Momentum of a phonon: $\vec{p} = \hbar\vec{k}$.

Density of States (DOS): $g(\omega) d\omega$ is the number of vibrational modes between $\omega$ and $\omega+d\omega$.

Specific Heat of Solids:

Dulong-Petit Law: $C_V = 3R$ at high temperatures (classical limit).
Einstein Model: Assumes all atoms oscillate at a single frequency $\omega_E$. $$ C_V = 3Nk_B \left(\frac{\hbar\omega_E}{k_BT}\right)^2 \frac{e^{\hbar\omega_E/k_BT}}{(e^{\hbar\omega_E/k_BT}-1)^2} $$ Underestimates $C_V$ at low T.
Debye Model: Treats phonons as elastic waves, valid at low T. $$ C_V = 9Nk_B \left(\frac{T}{\Theta_D}\right)^3 \int_0^{\Theta_D/T} \frac{x^4 e^x}{(e^x-1)^2} dx $$ At low T ($T \ll \Theta_D$): $C_V \propto T^3$ (Debye $T^3$ Law). $\Theta_D$: Debye temperature.

Assumes valence electrons are free to move within the crystal, forming a "Fermi gas".
Fermi Energy ($E_F$): Maximum energy of an electron at $T=0$ K. $$ E_F = \frac{\hbar^2}{2m} (3\pi^2 n)^{2/3} $$ where $n$ is electron density.
Fermi Sphere: In k-space, states up to $k_F$ are occupied. $$ k_F = (3\pi^2 n)^{1/3} $$
Density of States (DOS) for Free Electrons: $$ g(E) = \frac{V}{2\pi^2} \left(\frac{2m}{\hbar^2}\right)^{3/2} E^{1/2} $$
Specific Heat of Electrons: At low temperatures, $C_{el} \propto T$. $$ C_{el} = \frac{\pi^2}{2} n k_B \frac{T}{T_F} $$ where $T_F = E_F/k_B$ is the Fermi temperature. This is much smaller than phonon contribution at room temperature.
Electrical Conductivity: $\sigma = ne^2\tau/m$
- $n$: electron density
- $e$: electron charge
- $\tau$: relaxation time
- $m$: electron mass
Thermal Conductivity: $K = \frac{\pi^2}{3} \frac{n k_B^2 T \tau}{m}$
Wiedemann-Franz Law: $K/\sigma T = L = \frac{\pi^2}{3} (k_B/e)^2$ (Lorenz number).

Considers weak periodic potential of the lattice.
Energy Gaps: Occur at Brillouin zone boundaries (where $\vec{k} = \pm \vec{G}/2$). $$ E_{\pm} = \frac{\hbar^2 k_G^2}{2m} \pm |V_G| $$ where $k_G = |\vec{G}|/2$ and $V_G$ is the Fourier component of the potential.
Brillouin Zones: Regions in reciprocal space defined by planes that bisect reciprocal lattice vectors.
Effective Mass ($m^*$): Describes how electrons respond to external forces in a periodic potential. $$ \frac{1}{m^*} = \frac{1}{\hbar^2} \frac{d^2E}{dk^2} $$ Can be negative near band edges.
Holes: Vacancies in an otherwise filled band. Behave as positive charge carriers with positive effective mass.

Bloch's Theorem: Electron wavefunctions in a periodic potential are of the form $\psi_{\vec{k}}(\vec{r}) = e^{i\vec{k}\cdot\vec{r}} u_{\vec{k}}(\vec{r})$, where $u_{\vec{k}}(\vec{r})$ has the periodicity of the lattice.
Energy Bands: Allowed energy levels for electrons, separated by forbidden energy gaps.
Classifications of Materials:
- Conductors (Metals): Partially filled band or overlapping bands. Electrons can eas