Condensed Matter Physics Expert

Crystal Structure

Bravais lattices:
  14 distinct lattice types in 3D (7 crystal systems)
  R = n₁a₁ + n₂a₂ + n₃a₃  (lattice vectors)

Common structures:
  Simple cubic (SC): 1 atom/cell
  BCC (body-centered cubic): 2 atoms/cell (Na, Fe, W)
  FCC (face-centered cubic): 4 atoms/cell (Cu, Al, Au, Ni)
  HCP (hexagonal close packed): 2 atoms/cell (Mg, Ti, Zn)
  Diamond cubic: 8 atoms/cell (Si, Ge, C)
  NaCl structure: FCC with 2-atom basis

Reciprocal lattice:
  G = m₁b₁ + m₂b₂ + m₃b₃
  bᵢ·aⱼ = 2πδᵢⱼ
  b₁ = 2π(a₂×a₃)/(a₁·a₂×a₃)

Brillouin zone:
  First BZ = Wigner-Seitz cell of reciprocal lattice
  All distinct k-vectors contained in first BZ

X-ray diffraction:
  Bragg's law: 2d·sinθ = nλ
  Structure factor: Sk = Σⱼ fⱼ exp(iG·rⱼ)
  Systematic absences → determine crystal structure

Miller indices (hkl):
  Planes with intercepts a/h, b/k, c/l
  Spacing: d = a/√(h²+k²+l²)  (cubic)

Free Electron Model

Drude model (classical):
  σ = ne²τ/m  (electrical conductivity)
  τ = mean free time between collisions
  Hall coefficient: RH = -1/ne

Sommerfeld model (quantum):
  Electrons in box: ψk = (1/√V)exp(ik·r)
  Energy: εk = ℏ²k²/2m
  Fermi energy: EF = (ℏ²/2m)(3π²n)^(2/3)
  Fermi wavevector: kF = (3π²n)^(1/3)
  Fermi temperature: TF = EF/kB

Density of states:
  g(ε) = (3n/2EF)(ε/EF)^(1/2)  (3D)
  g(EF) = 3n/2EF

Fermi-Dirac distribution:
  f(ε) = 1/[exp((ε-μ)/kBT) + 1]
  At T=0: f = 1 for ε < EF, f = 0 for ε > EF
  Chemical potential μ ≈ EF at low T

Sommerfeld expansion:
  Electronic heat capacity: Cv = (π²/3)kB²T·g(EF) = γT
  γ = π²kB²g(EF)/3  (Sommerfeld coefficient)
  Much smaller than classical Cv = 3nkB/2 ✓

Band Theory

Bloch theorem:
  ψnk(r) = unk(r)exp(ik·r)
  unk(r+R) = unk(r)  (periodic part)
  States labeled by band index n and k in BZ

Nearly free electron model:
  Weak periodic potential V(r) = ΣG VG exp(iG·r)
  Band gaps open at BZ boundaries
  Gap size ≈ 2|VG| at zone boundary k = G/2

Tight binding model:
  ψk = (1/√N) Σᵣ exp(ik·R) φ(r-R)
  εk = ε₀ - t Σ_NN exp(ik·δ)  (δ = nearest neighbor vectors)
  1D: εk = ε₀ - 2t·cos(ka)
  Bandwidth W = 4t (1D), larger in higher dimensions

Band classification:
  Metal: partially filled band OR overlapping bands
  Insulator: completely filled bands, large gap (Eg > 4eV)
  Semiconductor: completely filled bands, small gap (Eg < 4eV)
  Semimetal: tiny overlap of valence and conduction bands

Effective mass:
  1/m* = (1/ℏ²) d²ε/dk²
  Captures band curvature effect on dynamics
  Can be negative (holes at top of band)
  m* << m: light electrons (high mobility)

Semiconductors

Intrinsic semiconductor:
  n = p = nᵢ = √(NcNv) exp(-Eg/2kBT)
  Nc = 2(2πmₑ*kBT/h²)^(3/2)  (effective DOS)
  Fermi level: μ = Eg/2 + (3/4)kBT·ln(mₕ*/mₑ*)

Doped semiconductors:
  n-type (donor atoms, e.g. P in Si): excess electrons
    n ≈ ND (donor concentration), p = nᵢ²/n
  p-type (acceptor atoms, e.g. B in Si): excess holes
    p ≈ NA, n = nᵢ²/p

Mass action law: np = nᵢ²

Carrier transport:
  Drift: J = (neμₑ + peμₕ)E  (σ = neμₑ + peμₕ)
  Diffusion: J = eDₑ∇n - eDₕ∇p
  Einstein relation: D/μ = kBT/e

p-n junction:
  Built-in potential: Vbi = (kBT/e)ln(NAND/nᵢ²)
  Depletion width: W = √(2ε₀εr·Vbi/e · (NA+ND)/(NAND))
  I-V: I = I₀[exp(eV/kBT) - 1]  (Shockley equation)

Semiconductor properties (Si at 300K):
  Eg = 1.12 eV, nᵢ = 1.5×10¹⁰ cm⁻³
  μₑ = 1400, μₕ = 450 cm²/Vs
  ε = 11.7

Lattice Dynamics & Phonons

1D monatomic chain:
  ω(k) = 2√(K/m) |sin(ka/2)|
  Acoustic branch: ω → 0 as k → 0
  vg = dω/dk = a√(K/m)cos(ka/2)

1D diatomic chain:
  Two atoms per unit cell → two branches
  Acoustic: both atoms move same direction
  Optical: atoms move in opposite directions
  Gap at zone boundary: ω = √(2K/M±m)

Phonon dispersion in 3D:
  N atoms/cell → 3N branches
  3 acoustic + 3(N-1) optical branches

Debye model:
  Linear dispersion: ωD = vsqD (Debye cutoff)
  Cv = 9NkB(T/θD)³∫₀^(θD/T) x⁴eˣ/(eˣ-1)² dx
  High T: Cv → 3NkB (Dulong-Petit)
  Low T: Cv ∝ T³ (Debye T³ law)
  θD = Debye temperature (characteristic)

Einstein model:
  All phonons same frequency ωE
  Cv = 3NkB(θE/T)² eθE/T/(eθE/T-1)²
  Works well for optical modes

Thermal conductivity:
  κ = (1/3)Cv·v·ℓ  (kinetic theory)
  ℓ = phonon mean free path
  Umklapp scattering limits κ at high T

Magnetism

Diamagnetism:
  χ < 0 (small, negative susceptibility)
  Induced moment opposes applied field
  Present in all materials (Lenz's law)
  Superconductors: perfect diamagnets χ = -1

Paramagnetism:
  χ > 0, small
  Curie law: χ = C/T  (isolated magnetic moments)
  C = nμ₀μ²/3kB  (Curie constant)
  Pauli paramagnetism (metals): χ ∝ g(EF), T-independent

Ferromagnetism:
  Spontaneous magnetization below TC (Curie temperature)
  Weiss molecular field: Bmol = λM
  Mean field theory: M = nμ·tanh(μ(B+λM)/kBT)
  TC = nμ₀μ²λ/3kB
  Above TC: Curie-Weiss: χ = C/(T-TC)

Antiferromagnetism:
  Neighboring spins antiparallel
  Neel temperature TN: transition to disorder
  χ has maximum at TN

Ferrimagnetism:
  Antiparallel but unequal moments → net magnetization
  Example: magnetite Fe₃O₄

Ising model:
  H = -J Σ_<ij> SᵢSⱼ - B Σᵢ Sᵢ
  J > 0: ferromagnetic, J < 0: antiferromagnetic
  1D: no phase transition at T > 0 (Ising 1925)
  2D: TC = 2J/kB·ln(1+√2) (Onsager 1944)
  3D: requires numerical methods

Superconductivity

Discovery: Onnes 1911 (mercury, 4.2 K)
Meissner effect: perfect diamagnetism (B = 0 inside)
Critical temperature TC, critical field HC(T)

London equations:
  ∂J/∂t = (nse²/m)E
  ∇×J = -(nse²/m)B
  London penetration depth: λL = √(m/μ₀nse²)
  Magnetic field decays inside: B(x) = B₀exp(-x/λL)

BCS Theory (Bardeen, Cooper, Schrieffer 1957):
  Cooper pairs: two electrons bound via phonon exchange
  Binding energy gap: Δ = 2ℏωD exp(-1/N(0)V)
  TC = 1.13 ℏωD/kB exp(-1/N(0)V)
  Energy gap: 2Δ(0) = 3.52 kBTC  (BCS universal ratio)

Coherence length: ξ = ℏvF/πΔ
Type I vs Type II:
  κ = λL/ξ < 1/√2: Type I (complete Meissner, single HC)
  κ > 1/√2: Type II (vortex phase, HC1 < H < HC2)

Josephson effect:
  Current through insulating barrier: I = IC sin(φ)
  DC Josephson: supercurrent with no voltage
  AC Josephson: V = ℏ/2e · dφ/dt = hf/2e
  SQUID: superconducting quantum interference device

High-temperature superconductors:
  Cuprates (YBCO): TC ~ 90-130 K
  Iron-based: TC ~ 55 K
  MgB₂: TC = 39 K
  Mechanism not fully understood (not BCS)
  Record: LaH₁₀ at high pressure, TC ~ 250 K

Phase Transitions & Critical Phenomena

Order parameter η:
  η = 0 in disordered phase, η ≠ 0 in ordered phase
  Magnetization (magnetic), density difference (liquid-gas)

Landau theory:
  F = a₀ + a₂(T-TC)η² + a₄η⁴ + ...
  a₄ > 0: second order transition
  a₄ < 0: first order transition

Critical exponents:
  M ∝ |T-TC|^β         β ≈ 0.326 (3D Ising)
  χ ∝ |T-TC|^(-γ)      γ ≈ 1.237
  Cv ∝ |T-TC|^(-α)     α ≈ 0.110
  ξ ∝ |T-TC|^(-ν)      ν ≈ 0.630
  Mean field: β=1/2, γ=1, α=0, ν=1/2

Scaling and universality:
  Critical exponents depend only on:
  - Dimensionality d
  - Symmetry of order parameter
  NOT on microscopic details!

Renormalization group (Wilson, Nobel 1982):
  Systematic method to calculate critical exponents
  Key idea: integrate out short-wavelength fluctuations
  Fixed points → universality classes

Topological Materials

Integer Quantum Hall Effect (IQHE):
  2D electron gas in magnetic field
  Hall conductance: σxy = ne²/h  (n = integer)
  Chern number: topological invariant
  Robust against disorder!

Topological insulators:
  Bulk insulating gap, but metallic surface states
  Protected by time-reversal symmetry
  Surface states: Dirac cone, spin-momentum locking
  Examples: Bi₂Se₃, Bi₂Te₃, HgTe quantum wells

Topological invariants:
  Z₂ invariant (time-reversal invariant systems)
  Chern number (breaks time-reversal)
  Calculated from Bloch wavefunctions in BZ

Weyl semimetals:
  Linear crossing of two bands in 3D (Weyl points)
  Topological charge (chirality) ±1
  Fermi arc surface states connecting Weyl points
  Examples: TaAs, WTe₂

Majorana fermions:
  Particles that are their own antiparticles
  Predicted in topological superconductors
  Non-Abelian anyons — topological quantum computing

Common Pitfalls

Related Skills

• quantum-mechanics-expert: Foundation of band theory
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• statistical-mechanics: Phase transitions and thermodynamics
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• materials-science-expert: Crystal structure and properties