10 Condensed matter physics

10.1 Crystals & Reciprocal Space

Crystalline solids are periodic in space. Periodicity gives you enormous leverage: lattices turn sums into selection rules, Fourier transforms quantize $k$, and diffraction fingerprints structure. This section lays down the lattice–basis language, reciprocal vectors, Brillouin zones, Miller indices, and the diffraction toolkit (Bragg, Laue, Ewald, structure factors). We’ll tie together fcc/bcc duality, selection rules for common crystals, and quick ways to not get lost in indices.

10.1.1 Lattice, basis, crystal structure

A Bravais lattice is the set of points

with three linearly independent primitive vectors $\boldsymbol a_{i}$. A crystal structure = lattice plus a basis of atoms at positions $\{\boldsymbol\tau_{j}\}$ inside a primitive cell; the atomic positions are $\boldsymbol r = \boldsymbol R + \boldsymbol\tau_{j}$.

• Wigner–Seitz cell: the set of points closer to a given lattice point than any other.
• Conventional vs primitive: conventional cubic cells are nicer to look at; primitive cells are minimal-volume and great for theory.

Pro tip: never say “the diamond lattice.” Diamond has an fcc lattice with a two-point basis $\{(0,0,0),\,(\tfrac12,\tfrac12,\tfrac12)a\}$.

10.1.2 Symmetry: point groups and space groups (one-screenshot tour)

• Point group: rotations, reflections, inversion that leave at least one point fixed.
• Space group: point-group ops + translations + screw rotations and glide reflections.
• In 3D there are 14 Bravais lattices, 32 crystallographic point groups, and 230 space groups. For band theory you mostly need the lattice and the little groups of high-symmetry $k$ points.

10.1.3 Reciprocal lattice and Brillouin zone

Define the reciprocal primitive vectors $\boldsymbol b_{i}$ by $\boldsymbol a_{i}\cdot \boldsymbol b_{j}=2\pi\delta_{ij}$. In vector form

The reciprocal lattice consists of vectors $\boldsymbol G = h\,\boldsymbol b_{1} + k\,\boldsymbol b_{2} + \ell\,\boldsymbol b_{3}$ with integers $(h,k,\ell)$.

The first Brillouin zone (BZ) is the Wigner–Seitz cell of the reciprocal lattice. It is the primitive cell in $k$-space and the natural domain for Bloch wavevectors.

• Duality: fcc $\leftrightarrow$ bcc. The reciprocal of fcc is bcc and vice versa. Simple cubic is self-dual.

10.1.4 Bloch theorem in one line

In a periodic potential $V(\boldsymbol r+\boldsymbol R)=V(\boldsymbol r)$, single-particle eigenstates are Bloch waves:

Wavevectors $\boldsymbol k$ can be chosen in the first BZ. Translating $\boldsymbol k\to \boldsymbol k+\boldsymbol G$ leaves physics invariant.

10.1.5 Planes, directions, and indices (Miller crash course)

• Directions use square brackets $[u\,v\,w]$ for a lattice vector proportional to $u\,\boldsymbol a_{1}+v\,\boldsymbol a_{2}+w\,\boldsymbol a_{3}$. Families of equivalent directions: $\langle u\,v\,w\rangle$.
• Planes use parentheses $(h\,k\,\ell)$, the Miller indices, defined as reciprocals of intercepts along $\boldsymbol a_{i}$. Families: $\{h\,k\,\ell\}$.
• Spacing to plane family $(h\,k\,\ell)$ is tied to the reciprocal vector $\boldsymbol G_{hkl}$:

For cubic $a_{1}=a\,\hat x$ etc., $d_{hkl}= a/\sqrt{h^{2}+k^{2}+\ell^{2}}$.

Hexagonal often uses Miller–Bravais $(h\,k\,i\,\ell)$ with $i=-(h+k)$ to make basal-plane symmetry manifest.

10.1.6 Diffraction: Bragg, Laue, Ewald are the same vibe

Elastic scattering conserves $|\boldsymbol k|=|\boldsymbol k'|$ and transfers crystal momentum $\Delta\boldsymbol k=\boldsymbol k'-\boldsymbol k$ equal to a reciprocal vector:

Bragg condition in real space:

Laue condition in $k$-space is exactly $\boldsymbol k'-\boldsymbol k=\boldsymbol G$. Ewald sphere construction: draw a sphere of radius $|\boldsymbol k|=2\pi/\lambda$ anchored at the tail of $\boldsymbol k$; reciprocal-lattice points on the sphere give allowed reflections.

Structure factor encodes the basis:

with atomic form factors $f_{j}$. The diffracted intensity is

The lattice sum enforces $\boldsymbol G$ belongs to the reciprocal lattice; $F$ decides if it actually lights up.

10.1.7 Selection rules for common structures

• Simple cubic (sc) with one atom at $(0,0,0)$:

• Body-centered cubic (bcc), basis $\{(0,0,0),\,(\tfrac12,\tfrac12,\tfrac12)a\}$:

• Face-centered cubic (fcc), basis $\{(0,0,0),\,(\tfrac12,\tfrac12,0)a,\,(\tfrac12,0,\tfrac12)a,\,(0,\tfrac12,\tfrac12)a\}$:

• NaCl (rock salt): fcc lattice with two-species basis $\{(0,0,0)\ \text{Na},\ (\tfrac12,\tfrac12,\tfrac12)a\ \text{Cl}\}$. Then

Intensity alternates with $h+k+\ell$ depending on $f$-contrast.

• Diamond: fcc lattice with basis $\{(0,0,0),\,(\tfrac14,\tfrac14,\tfrac14)a\}$. Extra phase leads to additional systematic zeros beyond fcc.

10.1.8 fcc–bcc duality and high-symmetry $k$ points

If $\{\boldsymbol a_{i}\}$ are fcc vectors, the reciprocal $\{\boldsymbol b_{i}\}$ generate a bcc lattice in $k$-space (and vice versa). Learn the standard labels:

• fcc BZ: $\Gamma$ center, $X$ at face centers, $L$ at BZ corners, $W$ mid-edges.
• bcc BZ: $\Gamma$, $H$, $N$, $P$ points.

These are where band extrema and degeneracies love to appear and where experimentalists measure ARPES and quantum oscillations.

10.1.9 Real ↔ reciprocal geometry cheats

• Volume relation:

• Plane normal: the reciprocal vector $\boldsymbol G_{hkl}$ is normal to the $(hkl)$ planes.
• Angle between planes $(h_{1}k_{1}\ell_{1})$ and $(h_{2}k_{2}\ell_{2})$:

• Spacing again: $d_{hkl}=2\pi/|\boldsymbol G_{hkl}|$.

10.1.10 Worked mini-examples

(a) Reciprocal of bcc is fcc
Start with bcc conventional cubic cell of side $a$ and primitive vectors $\boldsymbol a_{1}=\tfrac a2(\hat y+\hat z-\hat x)$ etc. Compute $\boldsymbol b_{i}$ and show the reciprocal points sit at face centers of a cubic net with spacing $2\pi/a$.

(b) $d_{hkl}$ for cubic
Using $\boldsymbol G_{hkl}=(2\pi/a)(h\,\hat x + k\,\hat y + \ell\,\hat z)$, show $d_{hkl}=a/\sqrt{h^{2}+k^{2}+\ell^{2}}$. Bragg gives $2d\sin\theta=n\lambda$; solve for the $(111)$ first-order reflection angle for given $\lambda/a$.

(c) fcc systematic absences
Evaluate $F$ for $(100)$, $(110)$, $(111)$ families and confirm only “all even” or “all odd” survive.

(d) Diamond zeros
Diamond has $F \propto \left[1 + e^{-i\pi(h+k+\ell)/2}\right]\times(\text{fcc factor})$. Show that $(222)$ is extinct but $(111)$ is allowed, consistent with X-ray tables.

(e) Ewald construction sketch
For $\lambda=1.54\ \text{\AA}$ (Cu K$\alpha$), draw a circle of radius $2\pi/\lambda$ through the origin of the reciprocal fcc lattice and mark which $\boldsymbol G$ vectors on the circle predict reflections for a given crystal orientation.

10.1.11 Common pitfalls

• Confusing lattice with structure: lattice is the periodic scaffold; the basis puts atoms on it.
• Wrong units in reciprocal space: crystallographers often use $2\pi$-free convention; here $a_{i}\cdot b_{j}=2\pi\delta_{ij}$. Pick one and stay loyal.
• Miller vs direction indices: $(hkl)$ is a plane, $[uvw]$ is a direction. For cubic, $(hkl)\perp [hkl]$; not true in general.
• Forgetting the form factor: even if $F\neq 0$, $f_{j}(\boldsymbol G)$ decays with $|\boldsymbol G|$, reshaping intensities.
• BZ confusion: the “reduced zone” scheme folds $k$ into the first BZ; if bands look doubled, you probably mixed extended vs reduced plots.

10.1.12 Minimal problem kit

• Derive the reciprocal vectors $\boldsymbol b_{i}$ from $\boldsymbol a_{i}$ and prove $V_{\mathrm{cell}}\,V_{\mathrm{cell}}^{\ast}=(2\pi)^{3}$
• Show that the Laue condition $\boldsymbol k'-\boldsymbol k=\boldsymbol G$ and Bragg’s law $2d\sin\theta=n\lambda$ are equivalent via $\boldsymbol G$ geometry
• Compute the structure factor and selection rules for NaCl and CsCl; compare which $(hkl)$ vanish in each
• For a hexagonal lattice with $a$ and $c$, write $\boldsymbol b_{i}$ and obtain $d_{hkl}$ in terms of $(h\,k\,i\,\ell)$
• Locate the high-symmetry points $\Gamma$, $X$, $L$, $W$ for fcc in Cartesian $k$-coordinates given $a$, and sketch the first BZ

10.2 Lattice Vibrations & Phonons

Atoms in a crystal aren’t statues; they wiggle. Those collective wiggles are phonons: quantized normal modes of the lattice. In this section we go from 1D chains to full 3D dynamical matrices, split acoustic vs optical branches, compute dispersions, group velocities, and densities of states, then quantize. We also hit thermal transport, anharmonicity, thermal expansion via the Grüneisen parameter, electron–phonon coupling, and quick experimental fingerprints (neutron, Raman, IR).

10.2.1 Harmonic lattice: setting the stage

Let $\boldsymbol R$ label Bravais sites and $\kappa$ the basis atom index inside a unit cell. The displacement of atom $(\boldsymbol R,\kappa)$ is $\boldsymbol u_{\kappa}(\boldsymbol R,t)$. In the harmonic approximation the potential energy is quadratic in displacements:

where $\Phi$ are force constants and $\alpha,\beta\in\{x,y,z\}$. Newton:

Try plane-wave solutions $u_{\kappa\alpha}(\boldsymbol R,t)=e_{\kappa\alpha}(\boldsymbol q)\,e^{i(\boldsymbol q\cdot \boldsymbol R - \omega t)}$. You get an eigenproblem

with the dynamical matrix

For $r$ atoms per cell you get $3r$ branches: 3 acoustic $\omega\to 0$ as $\boldsymbol q\to 0$, and $3r-3$ optical.

10.2.2 1D monatomic chain: the cleanest demo

Masses $m$ connected by springs $K$ with lattice spacing $a$. Let $u_{n}$ be the displacement of site $n$. Equation of motion:

Ansatz $u_{n}=u_{0}e^{i(nka-\omega t)}$ gives the dispersion

Key takeaways:

• Acoustic branch only, linear near $k=0$ with sound speed $v_{s}=a\sqrt{K/m}$
• Brillouin zone is $-\pi/a<k\le \pi/a$; $\omega(k)$ is periodic with $2\pi/a$
• Group velocity $v_{g}(k)=\partial\omega/\partial k$ controls energy flow

Density of states in 1D for a single branch:

So $g(\omega)$ diverges at BZ edges where $d\omega/dk\to 0$ (van Hove singularity).

10.2.3 1D diatomic chain: acoustic + optical

Two alternating masses $m_{1},m_{2}$, same spring $K$, unit cell length $a$ containing both atoms. With nearest-neighbor coupling, the two-branch dispersion is

• The minus sign is the acoustic branch with $\omega\to 0$ as $k\to 0$
• The plus sign is the optical branch with $\omega(k=0)=\sqrt{2K(1/m_{1}+1/m_{2})}$

At $k=0$ the acoustic mode moves the two atoms in phase; the optical mode moves them out of phase. A gap opens at the zone boundary.

10.2.4 3D crystals: polarizations and sound speeds

For $r=1$ (monatomic Bravais) you have 3 acoustic branches: one longitudinal (LA) and two transverse (TA). Near $\boldsymbol q=0$ they are sound waves with linear dispersion

In cubic crystals the speeds relate to elastic constants and mass density $\rho$ (continuum limit):

More generally, the Christoffel equation gives direction-dependent velocities.

For $r>1$ you add $3r-3$ optical branches that typically have weak $\boldsymbol q$-dispersion near the zone center. If the basis carries opposite charges, some optical modes are IR active (see §10.14).

10.2.5 Quantization: phonons as bosons

Promote each normal mode $(\boldsymbol q,s)$ to a harmonic oscillator:

with $[b_{\boldsymbol q s},b^{\dagger}_{\boldsymbol q' s'}]=\delta_{\boldsymbol q\boldsymbol q'}\delta_{ss'}$. The displacement operator is

Phonon occupation follows Bose statistics with chemical potential $\mu=0$ in equilibrium.

10.2.6 Heat capacity and Debye revisit

Total phonon energy per volume $U/V=\sum_{s}\int d\omega\,\hbar\omega\,g_{s}(\omega)\,n_{B}(\omega)$ with $n_{B}=(e^{\beta\hbar\omega}-1)^{-1}$. Low-$T$ acoustic modes $\Rightarrow$ Debye law $C_{V}\propto T^{3}$ in 3D; high $T$ $\Rightarrow$ Dulong–Petit $3Nk_{B}$ (see §9.6 for the full Debye integral). Optical modes add Schottky-like bumps when their gaps become thermally accessible.

10.2.7 Thermal transport: $\kappa$ from kinetic theory

In a crystal, heat is mostly carried by acoustic phonons. Kinetic theory gives

where $C_{\omega s}=\hbar\omega\,\partial n_{B}/\partial T$ is the mode heat capacity, $v_{g}=\partial\omega/\partial q$ the group velocity, and $\tau$ the lifetime. Scattering channels:

• Phonon–phonon (anharmonic). Split into Normal (N) processes conserving crystal momentum and Umklapp (U) with $\boldsymbol q_{1}+\boldsymbol q_{2}=\boldsymbol q_{3}+\boldsymbol G$ that degrade heat current
• Boundary scattering (dominant in nanoscale or at very low $T$)
• Isotope/disorder scattering
• Phonon–electron and phonon–defect scattering in metals

Empirically, $\kappa(T)$ often rises as $\sim T^{3}$ at very low $T$ (ballistic/Boundary), peaks, then falls roughly as $1/T$ at high $T$ (Umklapp-limited).

10.2.8 Anharmonicity, frequency shifts, and lifetimes

Real crystals aren’t perfectly harmonic. Cubic and quartic terms cause:

• Thermal expansion (quasi-harmonic effect)
• Frequency renormalization $\omega(\boldsymbol q,T)$
• Finite lifetimes $\tau_{\boldsymbol q s}$ via three-phonon and four-phonon scattering

A handy bulk measure is the Grüneisen parameter

The volumetric thermal expansion coefficient $\alpha$ relates to $\gamma$ and the bulk modulus $B$:

(Quasi-harmonic approximation; works surprisingly well for many simple solids.)

10.2.9 Electron–phonon coupling: the vibes behind resistivity and pairing

Displacements modulate the electronic Hamiltonian $H_{e}$, producing an interaction $H_{e\text{-}ph}$. Two archetypes:

• Deformation potential (short-range): $\delta V \sim \Xi\,\nabla\cdot \boldsymbol u$ couples to LA phonons
• Fröhlich (long-range polar): in ionic crystals, LO phonons create macroscopic electric fields that couple strongly to electrons

Consequences:

• Metal resistivity: at low $T\ll \Theta_{D}$, Bloch–Grüneisen law $\rho \propto T^{5}$; at high $T$ roughly $\rho\propto T$
• Superconductivity: virtual phonon exchange provides an effective attraction that can overcome Coulomb repulsion near the Fermi surface (BCS)

10.2.10 Instabilities and soft modes: Peierls in 1D

In 1D metals the electronic susceptibility peaks at $q=2k_{F}$. Electron–phonon coupling softens the phonon at that wavevector; below a critical $T$ the $\omega(2k_{F})\to 0$ soft mode drives a lattice distortion with doubled period (dimerization), opening a gap: a Peierls transition. In higher dimensions, charge-density waves and structural transitions follow similar logic.

10.2.11 Experimental fingerprints of phonons

• Inelastic neutron scattering (INS): measures the dynamic structure factor $S(\boldsymbol q,\omega)$, mapping full $\omega(\boldsymbol q)$ and eigenvectors
• Raman spectroscopy: probes zone-center optical phonons; symmetry selection rules apply
• Infrared (IR) absorption/reflectivity: IR-active optical phonons in polar crystals split LO vs TO at $\Gamma$ (Lyddane–Sachs–Teller relation)
• Brillouin light scattering: accesses long-wavelength acoustic phonons (GHz)
• Specific heat and thermal conductivity vs $T$: bulk thermodynamic signatures of DOS and scattering

10.2.12 Worked mini-examples

(a) Group velocity in the monatomic chain
From $\omega(k)=2\sqrt{K/m}\,\left|\sin(ka/2)\right|$, compute $v_{g}(k)$ and show $v_{g}\to v_{s}=a\sqrt{K/m}$ as $k\to 0$.

(b) DOS edge singularity
Use $g(\omega)=(L/\pi)/|d\omega/dk|$ to show the 1D DOS diverges at $k=\pi/a$.

(c) Diatomic gap at $\Gamma$
Evaluate $\omega_{\text{opt}}(0)$ and $\omega_{\text{ac}}(0)$ for the diatomic chain and interpret the eigenvectors.

(d) Debye $T^{3}$
Assuming three acoustic branches with linear dispersion up to a Debye cutoff $q_{D}$, integrate to get $C_{V}\propto T^{3}$ at low $T$.

(e) Umklapp threshold
Show that an Umklapp process requires $\boldsymbol q_{1}+\boldsymbol q_{2}$ to leave the first BZ; estimate the minimum temperature where such $\boldsymbol q$ become thermally populated in a simple cubic with Debye wavevector $q_{D}$.

(f) Thermal expansion from $\gamma$
Starting from $F(T,V)=U_{\text{el}}(V)+\sum_{\boldsymbol q s}\left[\tfrac{1}{2}\hbar\omega_{\boldsymbol q s} + k_{B}T\ln(1-e^{-\beta\hbar\omega_{\boldsymbol q s}})\right]$, show that $\alpha=\gamma C_{V}/(BV)$ in the quasi-harmonic approximation.

10.2.13 Common pitfalls

• “Optical requires ions.” No: you only need $r>1$ atoms per cell. IR activity requires polarity; optical phonons exist regardless
• Phase vs group velocity: energy transport uses $v_{g}=\partial\omega/\partial k$, not $\omega/k$
• Forgetting selection rules: Raman sees only Raman-active symmetries; IR sees IR-active modes; neutrons couple to nuclear positions and magnetic moments
• Mixing BZ schemes: reduced vs extended zone pictures change where branches appear but not the physics
• Overusing harmonic approximation: heat transport and expansion need anharmonicity; include N vs U processes when modeling $\kappa(T)$

10.2.14 Minimal problem kit

• Derive the monatomic and diatomic 1D dispersions and sketch both within the first BZ
• Build a $6\times 6$ dynamical matrix for a 2-atom cubic basis with nearest-neighbor central forces and extract $\omega(\boldsymbol q)$ along a high-symmetry line
• Compute $g(\omega)$ for a Debye model in $d=1,2,3$ and show the corresponding low-$T$ specific heat scalings
• Use Matthiessen’s rule with boundary, isotope, and Umklapp scattering rates to fit a synthetic $\kappa(T)$ curve; identify the peak temperature scaling with sample size
• Estimate the Bloch–Grüneisen temperature and the $T^{5}$ law crossover for a metal with given $v_{s}$ and $k_{F}$

10.3 Drude–Sommerfeld Metals

Metals look messy at the atomic scale, but their electrons act surprisingly lawful. Drude gave the first kinetic picture (1900): classical electrons as a gas of charged billiard balls with occasional collisions. Sommerfeld (1928) fixed the big misses using Fermi–Dirac statistics. Together they still power a shocking amount of metal physics: DC/AC conductivity, Hall effect, magnetotransport, specific heat, Pauli paramagnetism, and the Wiedemann–Franz law.

10.3.1 Drude model: one relaxation time, huge mileage

Assume conduction electrons of (effective) mass $m^{\ast}$ and charge $-e$ experience randomizing collisions with mean time $\tau$.

Equation of motion (steady drive) for average velocity $\boldsymbol v$ under fields $(\boldsymbol E,\boldsymbol B)$:

• DC conductivity at $B=0$ with current density $\boldsymbol J = -ne\,\boldsymbol v$:

Define mobility $\mu \equiv e\tau/m^{\ast}$ so $\sigma_{0}=ne\mu$.

• AC (Drude) conductivity for field $\propto e^{-i\omega t}$:

Real part is a Lorentzian of width $1/\tau$; imaginary part switches sign at $\omega\sim 1/\tau$.

• Plasma frequency (free-electron pole in optics):

For $\omega\ll\omega_{p}$, metals reflect; for $\omega>\omega_{p}$, they transmit like a dielectric.

• Skin depth in the good-conductor limit ($\omega\tau\ll 1$):

10.3.2 Hall effect and the conductivity tensor

With $\boldsymbol B = B\hat z$, the Drude solution gives the conductivity tensor

Inverting gives the resistivity tensor. Two key outputs:

• Hall coefficient (weak field):

• Magnetoresistance in the single-band, isotropic Drude model is zero: $\rho_{xx}(B)=\rho_{0}$ despite $\sigma_{xx}$ shrinking, because the tensor inversion cancels it. Real materials show finite magnetoresistance due to multiple bands, anisotropy, or $\tau(\boldsymbol k)$ variations.

10.3.3 Drude thermal transport and Wiedemann–Franz (preview)

A simple kinetic estimate gives electronic thermal conductivity $\kappa_{e}\sim \tfrac{1}{3} C_{e} v_{F}^{2}\tau$ (once we introduce $v_{F}$). Drude naively predicts the Wiedemann–Franz ratio

But only with Sommerfeld statistics do we get the correct Lorenz number

We’ll derive this properly after quantizing the electron gas.

10.3.4 The fix: Sommerfeld free-electron gas

Electrons are fermions. Put $N$ of them in a box of volume $V$ and fill momentum states up to the Fermi sphere.

• Fermi wavevector and energy (spin degeneracy $g=2$):

• Density of states per volume for a parabolic band:

At low $T\ll T_{F}$, only a shell of width $\sim k_{B}T$ around $\epsilon_{F}$ is thermally active. This explains why most classical Drude thermodynamics failed.

10.3.5 Electronic specific heat: linear in $T$

Using Sommerfeld expansion,

Linear $T$ matches experiments and is tiny compared to $3Nk_{B}$ at room temperature because only $\sim (T/T_{F})$ of electrons participate.

10.3.6 Wiedemann–Franz, done right

Kinetic theory with Fermi statistics yields

Combine with $\sigma_{0}=ne^{2}\tau/m^{\ast}$ and $n=\int^{\epsilon_{F}} g(\epsilon)\,d\epsilon$ to obtain, for elastic scattering with energy-independent $\tau$,

Deviations of $L$ from $L_{0}$ diagnose inelastic scattering, strong energy dependence of $\tau$, or additional heat carriers (phonons, magnons).

10.3.7 Pauli paramagnetism and Landau diamagnetism (sketch)

• Pauli paramagnetism from spin polarization in a field $B$:

Small, $T$-independent to leading order.

• Landau diamagnetism from orbital motion in quantizing fields gives

for a free-electron gas. In real metals the net $\chi$ also includes core diamagnetism and possible Van Vleck terms.

10.3.8 Thermopower and the Mott formula (one-liner)

The Seebeck coefficient $S$ (voltage from a temperature gradient) vanishes in Drude’s classical picture but, with Fermi statistics,

This Mott formula makes $S$ a sensitive probe of particle–hole asymmetry and scattering near $\epsilon_{F}$.

10.3.9 What Drude gets wrong (and why Sommerfeld wins)

• Heat capacity: classical Drude predicts $C_{e}\sim Nk_{B}$; data show $C_{e}\propto T\ll Nk_{B}$. Sommerfeld fixes it.
• Hall sign: single-band Drude gives $R_{H}=-1/ne$ (electrons). Many materials have positive $R_{H}$ due to hole-like Fermi surfaces or multiband compensation.
• Magnetoresistance: zero in single-band Drude, but finite in reality from anisotropy, multiple pockets, or $\tau(\boldsymbol k)$.
• High-frequency optics: Drude alone misses interband absorption and bound-charge contributions; add Lorentz oscillators for realistic $\varepsilon(\omega)$.

10.3.10 Connecting to band structure

Replace $m^{\ast}$ by the band effective mass from the dispersion $\epsilon(\boldsymbol k)$ near the Fermi surface:

Anisotropic $m^{\ast}$ makes $\sigma$ a tensor even at $B=0$. Real transport uses Boltzmann theory linearized around the Fermi surface with $\tau(\boldsymbol k)$ (see §10.7).

10.3.11 Optical sum rule and spectral weight

The f-sum rule ties the integrated Drude weight to carrier density:

In correlated systems, Drude weight can be suppressed and shifted to mid-infrared interband-like features—often read as a fingerprint of reduced quasiparticle weight.

10.3.12 Worked mini-examples

(a) Numbers you can trust
Take $n=8.5\times 10^{28}\ \mathrm{m^{-3}}$, $\tau=3\times 10^{-14}\ \mathrm{s}$, $m^{\ast}=m_{e}$. Compute $\sigma_{0}$, $\rho_{0}$, $\omega_{p}$, and $\delta$ at $\omega/2\pi=10\ \mathrm{GHz}$.

(b) Hall bar
For a sample with thickness $t$, width $w$, current $I$ along $x$, and field $B\hat z$, show the Hall voltage $V_{H} = I\,B/(ne\,t)$ in the Drude single-band limit. Estimate $V_{H}$ for copper with the numbers from (a).

(c) Sommerfeld $C_{e}$
Derive $C_{e}=\gamma T$ using the Sommerfeld expansion and express $\gamma$ in terms of $n$ and $m^{\ast}$.

(d) Wiedemann–Franz from Boltzmann
Linearize the Boltzmann equation with elastic impurity scattering and show $L=L_{0}$ when $\tau(\epsilon)$ is constant near $\epsilon_{F}$.

(e) Mott thermopower sign
Assuming $\sigma(\epsilon)\propto \epsilon^{s}$ near $\epsilon_{F}$, show $S = -(\pi^{2}/3)(k_{B}^{2}T/e)\,s/\epsilon_{F}$ and interpret $S>0$ vs $S<0$.

10.3.13 Common pitfalls

• Forgetting effective mass: always use $m^{\ast}$ from the band structure, not bare $m_{e}$, in transport formulae.
• Reading magnetoresistance from $\sigma_{xx}$: the measured $\rho_{xx}$ can be $B$-independent in single-band Drude even though $\sigma_{xx}$ shrinks; invert the tensor.
• Assuming $R_{H}=-1/ne$ universally: multiband compensation and warped Fermi surfaces can flip the sign or reduce magnitude.
• Overusing constant $\tau$: energy- or angle-dependent scattering changes $L$, $S$, and MR; be cautious outside low-$T$ impurity-limited regimes.
• Mixing optical and transport masses: cyclotron, thermodynamic, and band masses can differ in anisotropic or interacting systems.

10.4 Bloch Theorem & Band Structure

Periodic potentials take messy atomic chaos and compress it into clean momentum-space rules. The headline: Bloch’s theorem says electrons in a crystal move as plane waves modulated by a lattice-periodic envelope. From that, gaps pop open at Brillouin-zone boundaries, dispersions $E_{n}(\boldsymbol k)$ define velocities and effective masses, and both “nearly free” and “tight-binding” lenses become the same story, zoomed in from opposite sides.

10.4.1 Bloch states and the central equation

For a single electron in a periodic potential $V(\boldsymbol r+\boldsymbol R)=V(\boldsymbol r)$,

Expand the lattice-periodic part in reciprocal vectors $\boldsymbol G$:

and the potential as $V(\boldsymbol r)=\sum_{\boldsymbol G} U_{\boldsymbol G}\,e^{i\boldsymbol G\cdot \boldsymbol r}$. Insert into Schrödinger’s equation to get the central equation

It’s an infinite linear system in the plane-wave coefficients $c$. Truncate sensibly and you get bands.

10.4.2 Zone folding and reduced vs extended zone

Free electrons have parabolas $E=\hbar^{2}|\boldsymbol k|^{2}/2m$ that repeat every reciprocal vector: $\boldsymbol k$ and $\boldsymbol k+\boldsymbol G$ are equivalent. You can:

• Extended-zone: keep $E(\boldsymbol k)$ as endlessly repeated parabolas in $k$-space
• Reduced-zone: fold all momenta back into the first Brillouin zone; copies become separate bands indexed by $n$

When $V\neq 0$, states that were degenerate under folding mix and gaps open at the crossings.

10.4.3 Gaps from Bragg reflection: a 2×2 demo

At a boundary where $\boldsymbol k$ and $\boldsymbol k'=\boldsymbol k-\boldsymbol G$ are degenerate for the free electron, keep only those two plane waves in the central equation. The eigenproblem is

with $\epsilon_{\boldsymbol q}=\hbar^{2}|\boldsymbol q|^{2}/2m$. At the exact zone boundary where $\epsilon_{\boldsymbol k}=\epsilon_{\boldsymbol k'}$, the two energies split as

so the band gap is

Moral: stronger Fourier components of the lattice potential at reciprocal vectors $\boldsymbol G$ carve bigger gaps at the corresponding Brillouin-zone planes.

10.4.4 Kronig–Penney in one line (1D intuition)

For a 1D periodic array of barriers (lattice constant $a$), the dispersion satisfies a transcendental relation

where $\alpha=\sqrt{2mE}/\hbar$ in the wells and $P$ encodes barrier strength. Regions where the right-hand side has magnitude $\le 1$ are allowed bands; where it exceeds 1 are forbidden gaps. Tuning $P$ moves you smoothly from nearly free ($P\to 0$) to tight-binding ($P\gg 1$).

10.4.5 Group velocity, effective mass, and DOS

Band dispersion is not just eye candy; its derivatives are physics:

• Group velocity

• Effective mass tensor near an extremum

Curvature $\uparrow$ ⇒ light mass; curvature $\downarrow$ ⇒ heavy mass. Negative curvature near a valence-band maximum means holes with positive charge and positive mass run the transport.

• Density of states for a single band in 3D

The DOS diverges where the gradient vanishes along the constant-energy surface (van Hove singularities).

10.4.6 Nearly free vs tight-binding: two zoom levels, same film

• Nearly free electron (NFE): start from parabolas and open gaps perturbatively where they cross. Good when $|U_{\boldsymbol G}| \ll \epsilon_{F}$.
• Tight-binding (TB): start from localized orbitals with hopping $t$ between sites; bands emerge from bonding/antibonding combinations. Best when orbitals are compact and $t$ is small compared to on-site energies.

Both are limits of the same Bloch machinery. NFE explains gap openings and light masses in simple metals; TB nails band shapes in covalent/ionic solids and will be our §10.5 focus (Wannier functions live here).

10.4.7 $k\cdot p$ method: local expansion near a band edge

Near a high-symmetry $\boldsymbol k_{0}$, expand the Hamiltonian as $H(\boldsymbol k)\approx H(\boldsymbol k_{0})+\frac{\hbar}{m}\,\boldsymbol k\cdot \boldsymbol p + \cdots$. Second-order perturbation yields

where $\{|m0\rangle\}$ are Bloch states at $\boldsymbol k_{0}$. This connects effective mass to interband matrix elements and symmetry selection rules (e.g., heavy vs light holes in semiconductors).

10.4.8 Symmetry, degeneracies, and spin–orbit spice

• Crystal symmetry protects degeneracies at high-symmetry $k$ points and along lines. Breaking symmetry (strain, fields) splits them.
• Inversion and time reversal together enforce Kramers doublets with spin.
• Spin–orbit coupling (SOC) mixes spin and orbital character; with broken inversion you can get Rashba-like splittings. We stay light here and return to topology in §10.11.

10.4.9 Fermi surfaces and transport pointers

The Fermi surface is the constant-energy surface $E_{n}(\boldsymbol k)=\epsilon_{F}$. Its curvature and connectivity control:

• Cyclotron masses and quantum oscillations (de Haas–van Alphen, Shubnikov–de Haas)
• Anisotropic conductivity and Hall effect (via velocities and lifetimes on the FS)
• Nesting vectors that can amplify density-wave instabilities

Bloch dispersion + scattering time is the whole transport game in §10.7.

10.4.10 Wannier functions: real-space avatars of bands

A unitary transform of Bloch states over the BZ gives Wannier functions $|n,\boldsymbol R\rangle$ localized near cell $\boldsymbol R$. Smooth gauge choices yield maximally localized Wanniers that act like tight-binding orbitals and are perfect for model building, interpolation, and computing real-space responses. We build on this in §10.5.

10.4.11 ARPES and band mapping (how we see $E(\boldsymbol k)$)

Angle-resolved photoemission (ARPES) measures the occupied part of $E(\boldsymbol k)$ directly. In 2D materials it’s almost literal; in 3D there’s a $k_{z}$ wrinkle. Combine with quantum oscillations and optical conductivity to reconstruct the low-energy band structure that transport actually feels.

10.4.12 Worked mini-examples

(a) Gap size from a single Fourier component
Take a 1D cosine potential $V(x)=2U_{G}\cos(Gx)$ with $G=2\pi/a$. Do the 2×2 central-equation truncation at $k=G/2$ and show $E_{g}=2|U_{G}|$.

(b) Effective mass near a band minimum
For $E(k)\approx E_{0}+\alpha k^{2}+\beta k^{4}$ in 1D, extract $m^{\ast}=\hbar^{2}/(2\alpha)$ and discuss how the $k^{4}$ term skews DOS but not $m^{\ast}$ at the minimum.

(c) DOS singularity in 2D
On a square lattice TB band $E(\boldsymbol k)=-2t(\cos k_{x}a+\cos k_{y}a)$, locate the saddle points and show the logarithmic van Hove singularity in $g(E)$.

(d) Zone-folding check
Start from a free-electron parabola and fold by $\pm G$ into the first BZ. Draw where degeneracies occur and mark which get lifted when $U_{G}\neq 0$.

(e) $k\cdot p$ for a direct-gap semiconductor
At $\Gamma$, use symmetry to argue that conduction and valence states have opposite parity so the linear-in-$k$ coupling vanishes; show why the conduction-band mass is then dominated by second-order coupling to remote bands.

10.4.13 Common pitfalls

• Mixing reduced and extended zones: keep track of whether a “crossing” is real or just a folded copy.
• Equating curvature with speed: velocity is first derivative, not curvature; flat band $\Rightarrow$ heavy mass and slow carriers.
• Forgetting the basis: selection rules and gap sizes depend on the Fourier components $U_{\boldsymbol G}$, which encode the atomic basis and bonding.
• Assuming parabolic everywhere: effective-mass fits work only near extrema; far away, nonparabolicity and anisotropy matter.
• Ignoring SOC where it’s large: in heavy elements or broken-inversion crystals, SOC reshapes band order and splitting—critical for topology.

10.4.14 Minimal problem kit

• Derive the central equation from the Bloch expansion and write it as a matrix eigenproblem for a finite set of $\boldsymbol G$
• Perform the 2×2 gap-opening calculation at a generic zone plane with normal $\boldsymbol G$; obtain $E_{g}=2|U_{\boldsymbol G}|$
• Compute $m^{\ast}$ tensor for an ellipsoidal constant-energy surface and relate it to cyclotron mass in a field along an arbitrary axis
• For a 1D Kronig–Penney with delta barriers, plot allowed bands for several $P$ and identify the NFE and TB limits
• Using a simple TB model on honeycomb, show how Dirac cones arise and how a staggered sublattice potential opens a mass gap

10.5 Tight-Binding & Wannier: From Local Orbitals to Bands

When electrons are more “homebody” than “frequent flyer,” tight-binding (TB) is the lens. Start with localized atomiclike orbitals, let them hop between sites, and you get bands with shapes and symmetries carved by real-space chemistry. Wannier functions are the clean, orthonormal real-space avatars of Bloch bands, perfect for building models, computing responses, and adding fields or disorder. This section builds TB from scratch, adds multi-orbital and Slater–Koster rules, handles magnetic fields via Peierls phases, and shows how Wanniers tie to polarization and edge physics (SSH preview).

10.5.1 From orbitals to Bloch sums

Pick orbitals $\{\phi_{\alpha}(\boldsymbol r-\boldsymbol R-\boldsymbol\tau_{\alpha})\}$ centered on basis sites $\boldsymbol\tau_{\alpha}$ in cell $\boldsymbol R$, with orbital label $\alpha$ (e.g., $s,p_{x},p_{y},p_{z}$).

Form Bloch sums

Expand eigenstates as $\psi_{n\boldsymbol k}=\sum_{\alpha} c_{n\alpha}(\boldsymbol k)\,\varphi_{\alpha\boldsymbol k}$. Matrix elements

with hopping $t_{\alpha\beta}(\boldsymbol R)\equiv\langle \phi_{\alpha 0}|H|\phi_{\beta\boldsymbol R}\rangle$, overlap $s_{\alpha\beta}(\boldsymbol R)\equiv\langle \phi_{\alpha 0}|\phi_{\beta\boldsymbol R}\rangle$. Solve the generalized eigenproblem

In orthogonal TB we approximate $S=\mathbb I$.

10.5.2 1D monatomic chain (orthogonal TB)

One orbital per site, lattice constant $a$, on-site energy $\varepsilon_{0}$, nearest-neighbor hopping $-t$.

Bandwidth $W=4t$. Effective mass near an extremum at $k=0$ is