Graphene
Band structure, density of states and electronic properties of graphene nanoribbons
Graphene is a single-layer arrangement of carbon atoms — essentially a one-atom-thick sheet where atoms form a honeycomb lattice. First isolated in 2004 at the University of Manchester by Andre Geim and Konstantin Novoselov — Nobel Prize in Physics 2010. Graphene is the building block of all carbon allotropes: graphite (stacked layers), carbon nanotube (rolled), buckminsterfullerene (sphere).
| a_cc | C-C bond length = 1.42 Å |
| a | Lattice constant = √3·a_cc = 2.46 Å |
| γ₀ | Hopping integral = 2.7 eV |
| v_F | Fermi velocity ≈ 10⁶ m/s (c/300) |
The graphene honeycomb lattice is not a Bravais lattice — it consists of two interpenetrating triangular sublattices (A and B). The unit cell contains 2 atoms. Lattice vectors:
High-symmetry points: Γ — Brillouin zone center, K, K' — Brillouin zone corners (Dirac points), M — edge midpoint.
Graphene electrons are primarily π-electrons (p_z orbital). Nearest-neighbor tight-binding Hamiltonian:
Dispersion relation:
+ sign — conduction band (π*), − sign — valence band (π). At K point both bands touch zero — Dirac point.
Near the K point ($\mathbf{k} = \mathbf{K} + \mathbf{q}$, $|\mathbf{q}| \ll |\mathbf{K}|$) the dispersion becomes linear:
This linear dispersion means electrons in graphene behave as massless relativistic particles — Dirac fermions. In ordinary semiconductors E ∝ k², in graphene E ∝ k. Therefore effective mass m* → 0, mobility is enormous.
Near the Dirac cone, DOS is linear in energy:
D(E=0) = 0 — DOS is zero at the Dirac point (unlike metals). Graphene is a semimetal. Van Hove singularities at E = ±γ₀ = ±2.7 eV (M point) — sharp DOS peaks where ∇_k E = 0.
A narrow strip of graphene — nanoribbon — has different electronic properties due to confinement. Edge geometry determines the electronic structure:
| Edge | Edge states | Band gap | Type |
|---|---|---|---|
| Zigzag | ✓ flat bands at E=0 | 0 eV | Metallic |
| Armchair (N=3m−1) | ✗ | 0 eV | Metallic |
| Armchair (სხვა N) | ✗ | ~0.8/W eV | Semiconducting |
| γ₀ = 2.7 eV | Nearest-neighbor hopping integral |
| v_F ≈ 10⁶ m/s | Fermi velocity = √3γ₀a/2ℏ |
| K, K' | Dirac points — valence/conduction bands touch |
| VHS | Van Hove singularity — E = ±γ₀, ±3γ₀ |
| μ > 200,000 cm²/V·s | Electron mobility (~150× higher than Si) |
| ~5000 W/m·K | Thermal conductivity (~10× higher than Cu) |
| ~97.7% | Light transmittance (single layer) |
| Property | Graphene | Si |
|---|---|---|
| Band gap | 0 eV | 1.12 eV |
| Electron mobility | ~200,000 cm²/V·s | ~1,400 cm²/V·s |
| Thermal cond. | ~5000 W/m·K | ~150 W/m·K |
| Strength | ~130 GPa | ~130 GPa |
| Transparency | 97.7% | — |
| Electronics | Transistors, sensors, flexible displays |
| Energy | Supercapacitors, LiB electrodes, solar cells |
| Biomedicine | Drug delivery, biosensors, antibacterial coating |
| Composites | Lightweight, strong materials — aerospace |
| Optics | Photodetectors, ITO replacement |
Step 1 — Band Structure
Calculate the band structure. Note: at Γ E = ±8.1 eV (= ±3γ₀), at M E = ±2.7 eV (= ±γ₀), at K E = 0 — Dirac point.
Step 2 — DOS
Switch to DOS. Note: DOS = 0 at E = 0 (semimetal). Peaks at E = ±2.7 eV — Van Hove singularities at M point.
Step 3 — Nanoribbon
Switch to nanoribbon. Compare zigzag and armchair: in zigzag you see a flat band at E=0 (edge states). Armchair N=9 opens a band gap.
Armchair N parity
Try N = 8, 9, 10, 11, 12. Note: N = 8 (= 3×3-1) — metallic, N = 9 — semiconducting, N = 11 (= 3×4-1) — metallic. N mod 3 == 2 is always metallic.
Edge states
In zigzag nanoribbon, the E=0 band spans from K to K'. These states are localized at the edges — zero in bulk. These states may carry magnetic moment (ferromagnetic stability of zigzag edge).
1. Why is graphene a semimetal rather than an ordinary semiconductor?
2. The flat band at E=0 in zigzag nanoribbon band structure:
3. Where do Van Hove singularities appear in graphene DOS?
4. Armchair nanoribbon at N = 11 (= 3×4-1):