Graphical abstract template · Graphene research · Band physics

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.

Updated 14 October 202611 cited papers45 editable layers

Dirac physics of graphene: preview of the editable figure template — Preview at reduced size. The full figure opens in the editor, where every layer can be changed.

At a glance

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 (16:9), 45 editable layers, 6 generated objects

Step by step

Lattice and Brillouin zoneA 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)
Tight-binding bandsNearest-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
The Dirac coneClose 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)
ARPESAngle-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)
Half-integer quantum HallA 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)
Bilayer: a tunable gapAn 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

Papers this template follows

Wallace, “The band theory of graphite”, Phys. Rev. 1947. doi:10.1103/PhysRev.71.622primary
Castro Neto et al., “The electronic properties of graphene”, Rev. Mod. Phys. 2009. doi:10.1103/RevModPhys.81.109primary
Novoselov et al., “Two-dimensional gas of massless Dirac fermions in graphene”, Nature 2005. doi:10.1038/nature04233primary
Zhang et al., “Experimental observation of the quantum Hall effect and Berry's phase in graphene”, Nature 2005. doi:10.1038/nature04235primary
Bostwick et al., “Quasiparticle dynamics in graphene”, Nat. Phys. 2007. doi:10.1038/nphys477further reading
Ohta et al., “Controlling the electronic structure of bilayer graphene”, Science 2006. doi:10.1126/science.1130681further reading
Castro et al., “Electronic properties of a biased graphene bilayer”, J. Phys.: Condens. Matter 2010. doi:10.1088/0953-8984/22/17/175503further reading
McCann & Fal'ko, “Landau-level degeneracy and quantum Hall effect in a graphite bilayer”, Phys. Rev. Lett. 2006. doi:10.1103/PhysRevLett.96.086805further reading
Zhang et al., “Direct observation of a widely tunable bandgap in bilayer graphene”, Nature 2009. doi:10.1038/nature08105further reading
Novoselov et al., “Unconventional quantum Hall effect and Berry's phase of 2π in bilayer graphene”, Nat. Phys. 2006. doi:10.1038/nphys245further reading
Nair et al., “Fine structure constant defines visual transparency of graphene”, Science 2008. doi:10.1126/science.1156965further reading

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.

How templates are written and checked: editorial guidelines.

Questions

What does the “Dirac physics of graphene” template show?

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.

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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).

grapheneDirac coneband structuretight bindingBrillouin zoneK pointDirac pointFermi velocitymassless Dirac fermionsARPESquantum Hall effecthalf-integerbilayer grapheneband gapcondensed matter physics