Фізика топологічних матеріалів

🌀 Топологічні ізолятори

2D Топологічні ізолятори

HgTe quantum wells: topological phase transition. Edge states: spin-momentum locked. σ_edge = e²/h.

3D топологічні ізолятори

Bi₂Se₃, Bi₂Te₃: surface Dirac cone. Spin texture. ARPES measurements. Protected surfaces.

Z₂ invariant

Topological invariant: ν ∈ {0,1}. Time-reversal symmetry. Strong vs weak. Edge robustness.

Applications

Quantum computing, spintronics, sensors. Topological protection. Low dissipation.

⚡ Quantum Hall effect

Integer QHE

σ_xy = νe²/h, ν = 1,2,3... Landau levels E_n = (n+½)ℏω_c. Chiral edge states.

Fractional QHE

ν = 1/3, 2/3, 5/3... Laughlin wavefunctions. Anyons, fractional charges. Electron correlation.

Edge states

Chiral, unidirectional. Backscattering forbidden. Robust to disorder. Topological protection.

Chern number

C = (1/2π) ∫ d²k·Ω(k): topological invariant. Integer. Band structure.

🔬 Topological semimetals

Dirac semimetals

Na₃Bi, Cd₃As₂: Dirac points. Linear dispersion E ~ k. Four-fold degeneracy.

Weyl semimetals

TaAs: Weyl points. Chiral charges ±1. Fermi arcs: surface states. Anomalous Hall.

Nodal line

Band crossing along lines. Higher-dimensional degeneracies. Drumhead surface states.

Topology

Berry phase, monopole charges. Robustness. Quantum phase transitions.

🌊 Berry phase

Geometric phase

γ_n = ∮ ⟨u_n(k)|i∇_k u_n(k)⟩·dk. Adiabatic evolution. Gauge invariant.

Berry curvature

Ω_n = ∇_k × A_n. Anomalous velocity, Hall effect. Chern number.

Chern number

C = (1/2π) ∫ d²k·Ω(k): integer topological invariant. Robust to perturbations.

Applications

Topological classification, quantum Hall, Chern insulators. Band theory.

🔬 Edge states

Chiral edge states

Unidirectional propagation. Backscattering prohibited. Quantized conductance.

Helical edge states

Spin-momentum locking. Topological insulator edges. Time-reversal protection.

Robustness

Protected topology: disorder, defects. Local perturbations не destroy.

Scattering

Protected от backscattering. Edge mode quantization. Transport signatures.

🌊 Topological transitions

Gap closing

Bulk bandgap → 0: transition. Topological → trivial. Dirac point emergence.

Phase diagrams

Normal → topological. Parameter space mapping. Critical points.

Symmetry breaking

Time-reversal, inversion. Topological protection removal. Phase transition.

Probes

ARPES, transport, optics. Edge state detection. Chern number measurements.

📊 Графіки та діаграми

Chern number

Topological invariant:

C quantized integer. QHE: σ_xy = Ce²/h. Robust index.

Dirac cone

Edge states:

Topological insulator surface. Spin-momentum locking. Fermi velocity.

C = (1/2π) ∫ d²k·Ω(k) Де Ω = Berry curvature C ∈ Z: integer Gap: C ≠ 0

🧪 Практичні приклади

Приклад 1: HgTe QW

2D topological insulator: quantum well. Edge states σ_edge = e²/h. Spin-momentum locking.

Приклад 2: Bi₂Se₃

3D TI: surface Dirac cone. ARPES measurements. Surface states protected.

Приклад 3: Quantum Hall

Integer ν: σ_xy = νe²/h. Chiral edge states. Robust to disorder. Precision: 10⁻⁹.

Приклад 4: TaAs Weyl

Weyl semimetal: chiral charges ±1. Fermi arcs surface. Anomalous Hall conductivity.

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