Quantum Materials
Crystal and reciprocal lattices, Bloch theorem, band structure, tight binding, quasiparticles, correlated materials, topological insulators, superconductivity, 2D systems, defects, and materials for qubits. Expanded with original in-page technical chapters, mathematical proofs, independently checked calculation baselines and directly embedded credited original instructor videos where relevant.
Published syllabus
Course Units
Crystal lattices and symmetry
Crystal symmetry acts through translations and point-group operations on atomic arrangements. A primitive cell generates the periodic lattice; a basis specifies additional atoms inside it. A material's space group constrains band degeneracies, selection rules and phonon modes. Defects and surfaces break some bulk symmetries locally.
Reciprocal lattice and Brillouin zones
Reciprocal lattice vectors satisfy a_i·b_j=2πδij, and the Brillouin zone is a primitive cell in reciprocal space. Diffraction selects momentum transfers equal to reciprocal vectors, linking abstract lattice geometry to experimental scattering peaks.
Bloch theorem
Bloch's theorem states that eigenfunctions of a single-particle Hamiltonian with periodic potential can be written e^{ik·r}u_nk(r), with u periodic under lattice translations. This separates crystal momentum from cell-periodic orbital structure and enables band calculations.
Band structure
Electronic bands arise from eigenvalues of a periodic single-particle or quasiparticle Hamiltonian across crystal momentum. Whether a material conducts depends on chemical potential, temperature, scattering and available states, not merely a colorful band plot. Effective masses depend on band curvature near an extremum.
Tight-binding models
Tight-binding models project electronic states onto localized orbitals and encode hopping, onsite energies and symmetries in a finite matrix H(k). They provide transparent limiting cases and control over lattice geometry. Long-range hopping and spin–orbit coupling can change gaps and topological invariants.
Phonons and quasiparticles
Phonons are quantized normal modes of lattice vibration; quasiparticles more broadly describe collective excitations with effective dispersion and lifetime. The dynamical matrix determines harmonic phonon frequencies. Anharmonicity causes scattering and finite lifetime, while electron-phonon coupling can influence resistivity and superconductivity.
Correlated materials and Mott physics
A Mott insulator is insulating because strong interactions can localize charge even when noninteracting band counting suggests a metal. Competition between hopping bandwidth and onsite repulsion controls many correlated lattice models. Magnetic order and orbital physics complicate the phase diagram.
Topological insulators
Topological insulators have insulating bulk states with boundary modes protected under stated symmetries and gap conditions. Berry phases, Chern invariants and Z2 indices classify relevant band structures. Disorder or interactions may require many-body descriptions beyond an elementary noninteracting Bloch calculation.
Superconductivity
Superconductivity combines zero-resistance current response with phase coherence and magnetic effects. In BCS-type descriptions, paired fermions open an excitation gap and form a coherent condensate. Strongly correlated or unconventional superconductors may require pairing symmetries and mechanisms beyond phonon-mediated weak coupling.
Two-dimensional and moire systems
Two-dimensional materials and moiré superlattices create narrow bands and tunable interactions through stacking angle, lattice relaxation and electrostatic control. The large moiré length scale can amplify correlation effects, but quantitative phase diagrams require device-specific disorder, strain, screening and twist-angle calibration.
Defects and quantum emitters
Point defects, color centers and localized excitons can serve as optical quantum emitters with spin or orbital degrees of freedom. Optical brightness, spectral diffusion, photon indistinguishability and spin coherence depend strongly on the local crystalline and electromagnetic environment.
Materials for qubits and quantum sensors
Materials for quantum devices are selected against thermal, microwave, dielectric, magnetic and fabrication constraints. Superconducting circuits need low-loss surfaces and stable Josephson junction interfaces, while spin qubits emphasize isotope disorder and electrostatic control. Cryogenic packaging and reproducible process characterization can dominate performance.