# Grafeno bicamada torcido em ângulo mágico

Twistronics interativa — torça duas camadas de grafeno e observe a superrede moiré se formar, as bandas de Dirac se achatarem no ângulo mágico de 1,05°, e a densidade de estados explodir numa fase correlacionada.

> Página canônica: https://elysiatools.com/pt/visualizations/magic-angle-graphene

- **Categoria:** Physics

## Visão geral

Interactive twistronics — distinct from the existing crystal-structures (3D Bravais lattices), bloch-sphere-gate-visualizer (qubit gates), and moire-patterns (classical optical interference); this is the only case treating the electronic band-structure engineering of twisted van der Waals bilayers and the magic-angle flat-band physics that drives correlated phases. The model implements the Bistritzer–MacDonald (BM) continuum framework: the moiré superlattice period L = a/(2·sin(θ/2)) (a = 0.246 nm graphene lattice constant), the moiré reciprocal momentum k_θ = 2·k_D·sin(θ/2), and the dimensionless coupling α = w/(ħ·v_F·k_θ). The renormalized Fermi velocity follows the BM second-order perturbation result v*_F/v_F = (1−3α²)/(1+6α²), calibrated so the first magic-angle zero (α = 1/√3) lands at θ ≈ 1.05°. The flat-band bandwidth W ≈ ħ·v*_F·k_θ collapses from ~hundreds of meV at 5° to ~few meV at the magic angle, and the van Hove singularity (VHS) energy at the moiré M-points tracks W. The band structure E(k) along the moiré BZ path Γ–K–M is modeled with a Dirac crossing at K (slope = ħ·v*_F) and a saddle (VHS) at M, and the 2D density of states is computed by histogramming the band energy on a k-grid plus analytic log-divergent VHS peaks that sharpen and collapse toward E = 0 as θ → magic angle. Four visualization panels: (1) Real-space moiré lattice rendering two rotated graphene honeycomb layers (blue layer 1, gold layer 2) as dot patterns with the magenta moiré unit-cell hexagon of side L overlaid — the period balloons from sub-nm at 30° to ~13 nm at the magic angle, with the atom count per cell (~11 000 at 1.05°) shown. (2) Band-structure plot E(k) along Γ–K–M showing the conduction (cyan) and valence (gold) mini-bands meeting at the Dirac point K, steep at large θ and nearly horizontal (flat) at the magic angle, with live bandwidth annotation. (3) Fermi-velocity-vs-θ curve v*_F/v_F showing the dramatic BM plunge to ~0 at the magic angle, with the magic-angle reference line and a pulsing current-point marker. (4) Density-of-states plot with Fermi-level and VHS reference lines, the DOS peak exploding at E = 0 as θ → magic angle. Adjustable parameter: twist angle θ (0.3°–30°). Five angle presets: Magic 1.05° (flat band), 2.0° (above), 0.5° (sub-magic), 5.0° (weak coupling), 30° (decoupled/commensurate). A 'Scan to magic' button animates θ toward the magic angle. Real-time diagnostics: regime (magic/sub-magic/near/weak/decoupled), moiré period L, coupling α, velocity ratio v*_F/v_F, bandwidth W, VHS energy, atoms per moiré cell, and the magic-angle constant. Educational content covers the moiré superlattice and giant unit cell, the BM continuum model and the α coupling, flat bands and the magic angle (v*_F → 0, Coulomb interaction wins), van Hove singularities and DOS divergence, correlated phases (Mott insulation at half-filling, superconductivity on doping, Cao et al. 2018), and applications to twistronics, quantum computation, tunable sensors, and materials-by-design from the van der Waals library. Multi-language support (zh, en, es, fr, de, ru, pt).

## Conteúdo relacionado

- [Reação de Ordem Zero - Visualização Interativa](https://elysiatools.com/pt/visualizations/zero-order-reaction): Visualização interativa da cinética de reação de ordem zero e mudanças de concentração ao longo do tempo
- [Reação de Primeira Ordem - Visualização Interativa](https://elysiatools.com/pt/visualizations/first-order-reaction): Visualização interativa da cinética de reação de primeira ordem e do decaimento exponencial de concentração
- [Reação de Segunda Ordem - Visualização Interativa](https://elysiatools.com/pt/visualizations/second-order-reaction): Visualização interativa da cinética de reação de segunda ordem e da dinâmica de colisão bimolecular
- [Equação de Arrhenius](https://elysiatools.com/pt/visualizations/arrhenius-equation): Visualização interativa do efeito da temperatura na velocidade da reação - Explore energia de ativação, fator pré-exponencial e constante de velocidade
- [Reação Reversível](https://elysiatools.com/pt/visualizations/reversible-reaction): Visualização interativa da cinética de reação reversível A ⇌ B - Explore taxas de reação, constante de equilíbrio e mudanças de concentração
- [Reação Consecutiva](https://elysiatools.com/pt/visualizations/consecutive-reaction): Visualização interativa da cinética de reação consecutiva A → B → C - Explore picos de concentração intermediária
- [Reação em Cadeia](https://elysiatools.com/pt/visualizations/chain-reaction): Visualização interativa da polimerização radical em cadeia
- [Princípio de Le Chatelier](https://elysiatools.com/pt/visualizations/le-chateliers-principle): Visualização interativa do princípio de Le Chatelier - Explore como mudanças de concentração, pressão e temperatura afetam o equilíbrio químico