# Dirac physics of graphene

> The electronic structure of graphene in six computed panels: the honeycomb lattice and its Brillouin zone, the tight-binding π bands meeting at the Dirac points, the Dirac cone, ARPES of the linear bands, the half-integer quantum Hall effect and the bilayer whose gap opens with an interlayer asymmetry.

- Page: https://www.labcanvas.com/templates/graphene-research/dirac-physics-graphene
- Open in the LabCanvas editor (LabCanvas account): https://www.labcanvas.com/editor.html?template=graphene-dirac-physics
- Updated: 2026-10-14
- Category: Graphene research
- Method: Band physics
- Starting materials: Monolayer and AB bilayer graphene
- Product: Lattice, bands, Fermi level, photoemission and quantum Hall plateaus
- Key result: Bands, cones, maps and plateaus computed by LabCanvas from the cited models; hopping, Fermi level, doping and field are settings
- Figure: 1920 × 1080 px, 45 editable layers (6 generated objects with settings)

## Steps

1. **Lattice and Brillouin zone**: A and B sublattices, the unit cell spanned by a₁ and a₂ (|a₁| = 2.46 Å), nearest neighbours 1.42 Å apart, and the reciprocal lattice with Γ, K, K′ and M (Castro Neto et al. 2009)
2. **Tight-binding bands**: Nearest-neighbour tight binding, E±(k) = ±t|1 + e^{ik·a₁} + e^{ik·a₂}| (Wallace 1947): the π and π* surfaces meet at the six Dirac points
3. **The Dirac cone**: Close to K the bands are linear; states fill up to E_F = ħv_F√(πn) for a carrier density n (Castro Neto et al. 2009)
4. **ARPES**: Angle-resolved photoemission through K shows the linear Dirac bands of a monolayer with v = 10⁶ m/s (Bostwick et al. 2007; Ohta et al. 2006)
5. **Half-integer quantum Hall**: A monolayer’s Hall conductivity σxy = ±4(N + ½)e²/h; the bilayer’s ±4N e²/h with an 8e²/h step at zero (Novoselov et al. 2005; Zhang et al. 2005; Novoselov et al. 2006)
6. **Bilayer: a tunable gap**: An AB bilayer has four bands; an asymmetry U between the layers opens a gap (Castro et al. 2010; McCann & Fal’ko 2006), tuned by a field (Zhang et al. 2009)

## Strengths

- One tight-binding model explains the cone, the density of states and the van Hove peaks
- The gate tunes the carriers through the Dirac point
- The bilayer adds a gap that a field controls

## Limitations

- Nearest-neighbour tight binding only
- Interactions and disorder are left out
- Plateau shapes are schematic

## Uses

- Teaching graphene’s band structure
- Talks on Dirac materials
- Explaining transport and photoemission data

## References

1. Wallace, “The band theory of graphite”, Phys. Rev. 1947. https://doi.org/10.1103/PhysRev.71.622 (primary)
2. Castro Neto et al., “The electronic properties of graphene”, Rev. Mod. Phys. 2009. https://doi.org/10.1103/RevModPhys.81.109 (primary)
3. Novoselov et al., “Two-dimensional gas of massless Dirac fermions in graphene”, Nature 2005. https://doi.org/10.1038/nature04233 (primary)
4. Zhang et al., “Experimental observation of the quantum Hall effect and Berry's phase in graphene”, Nature 2005. https://doi.org/10.1038/nature04235 (primary)
5. Bostwick et al., “Quasiparticle dynamics in graphene”, Nat. Phys. 2007. https://doi.org/10.1038/nphys477 (further reading)
6. Ohta et al., “Controlling the electronic structure of bilayer graphene”, Science 2006. https://doi.org/10.1126/science.1130681 (further reading)
7. Castro et al., “Electronic properties of a biased graphene bilayer”, J. Phys.: Condens. Matter 2010. https://doi.org/10.1088/0953-8984/22/17/175503 (further reading)
8. McCann & Fal'ko, “Landau-level degeneracy and quantum Hall effect in a graphite bilayer”, Phys. Rev. Lett. 2006. https://doi.org/10.1103/PhysRevLett.96.086805 (further reading)
9. Zhang et al., “Direct observation of a widely tunable bandgap in bilayer graphene”, Nature 2009. https://doi.org/10.1038/nature08105 (further reading)
10. Novoselov et al., “Unconventional quantum Hall effect and Berry's phase of 2π in bilayer graphene”, Nat. Phys. 2006. https://doi.org/10.1038/nphys245 (further reading)
11. Nair et al., “Fine structure constant defines visual transparency of graphene”, Science 2008. https://doi.org/10.1126/science.1156965 (further reading)

Credit: Computed by LabCanvas after Wallace, Phys. Rev. 1947, Castro Neto et al., Rev. Mod. Phys. 2009 (doi:10.1103/RevModPhys.81.109), Novoselov et al., Nature 2005, Zhang et al., Nature 2005, Castro et al., J. Phys. Condens. Matter 2010, and the papers in the details.

## Questions

**What does the “Dirac physics of graphene” template show?** The electronic structure of graphene in six computed panels: the honeycomb lattice and its Brillouin zone, the tight-binding π bands meeting at the Dirac points, the Dirac cone, ARPES of the linear bands, the half-integer quantum Hall effect and the bilayer whose gap opens with an interlayer asymmetry.

**What are the steps shown in “Dirac physics of graphene”?** 1. Lattice and Brillouin zone: A and B sublattices, the unit cell spanned by a₁ and a₂ (|a₁| = 2.46 Å), nearest neighbours 1.42 Å apart, and the reciprocal lattice with Γ, K, K′ and M (Castro Neto et al. 2009). 2. Tight-binding bands: Nearest-neighbour tight binding, E±(k) = ±t|1 + e^{ik·a₁} + e^{ik·a₂}| (Wallace 1947): the π and π* surfaces meet at the six Dirac points. 3. The Dirac cone: Close to K the bands are linear; states fill up to E_F = ħv_F√(πn) for a carrier density n (Castro Neto et al. 2009). 4. ARPES: Angle-resolved photoemission through K shows the linear Dirac bands of a monolayer with v = 10⁶ m/s (Bostwick et al. 2007; Ohta et al. 2006). 5. Half-integer quantum Hall: A monolayer’s Hall conductivity σxy = ±4(N + ½)e²/h; the bilayer’s ±4N e²/h with an 8e²/h step at zero (Novoselov et al. 2005; Zhang et al. 2005; Novoselov et al. 2006). 6. Bilayer: a tunable gap: An AB bilayer has four bands; an asymmetry U between the layers opens a gap (Castro et al. 2010; McCann & Fal’ko 2006), tuned by a field (Zhang et al. 2009).

**What are the starting materials and the product?** Starting materials: Monolayer and AB bilayer graphene. Product: Lattice, bands, Fermi level, photoemission and quantum Hall plateaus.

**Which papers is “Dirac physics of graphene” based on?** Wallace, “The band theory of graphite”, Phys. Rev. 1947 (doi:10.1103/PhysRev.71.622); Castro Neto et al., “The electronic properties of graphene”, Rev. Mod. Phys. 2009 (doi:10.1103/RevModPhys.81.109); Novoselov et al., “Two-dimensional gas of massless Dirac fermions in graphene”, Nature 2005 (doi:10.1038/nature04233); Zhang et al., “Experimental observation of the quantum Hall effect and Berry's phase in graphene”, Nature 2005 (doi:10.1038/nature04235); Bostwick et al., “Quasiparticle dynamics in graphene”, Nat. Phys. 2007 (doi:10.1038/nphys477); Ohta et al., “Controlling the electronic structure of bilayer graphene”, Science 2006 (doi:10.1126/science.1130681); Castro et al., “Electronic properties of a biased graphene bilayer”, J. Phys.: Condens. Matter 2010 (doi:10.1088/0953-8984/22/17/175503); McCann & Fal'ko, “Landau-level degeneracy and quantum Hall effect in a graphite bilayer”, Phys. Rev. Lett. 2006 (doi:10.1103/PhysRevLett.96.086805); Zhang et al., “Direct observation of a widely tunable bandgap in bilayer graphene”, Nature 2009 (doi:10.1038/nature08105); Novoselov et al., “Unconventional quantum Hall effect and Berry's phase of 2π in bilayer graphene”, Nat. Phys. 2006 (doi:10.1038/nphys245); Nair et al., “Fine structure constant defines visual transparency of graphene”, Science 2008 (doi:10.1126/science.1156965).

**Can I edit the “Dirac physics of graphene” template?** Yes. Open it in the LabCanvas editor: all 45 layers are editable, including 6 generated objects (structures, plots, equipment) with their own settings. Change the text, colours and layout, then export PNG, TIFF, SVG or PDF.

**Do I need an account to use this template?** Not to view it here. To open and edit it, sign in to LabCanvas or create a free account (the free plan keeps up to 3 designs). Downloading the figure is part of LabCanvas Pro (US$19 a month or US$199 a year).