Graphene Nanoribbon Edge States

Edge Termination

Ribbon width 5 cells

Ribbon length 6 cells

Eigenstate

State index (0 = lowest energy) 0 / 0

Energy E / t

0.00

Edge weight

Localization length (participation, atoms)

0.0

How it works

A graphene nanoribbon is a strip of the honeycomb lattice cut with one of two boundary shapes: zigzag (teeth-shaped edge) or armchair (scalloped edge). Electrons hop between nearest-neighbor carbon atoms with the simplest possible tight-binding Hamiltonian:

H = -t Σ<i,j> c†ⅻ cⅻ    (sum over bonded pairs, t = hopping energy)

This simulator builds the real N×N Hamiltonian matrix for a finite ribbon (H_ij = -t if atoms i and j are bonded, 0 otherwise) and diagonalizes it numerically (cyclic Jacobi eigenvalue rotation) to get exact energies E_n and wavefunction amplitudes ψ_n(i) on every atom.

Zigzag edges support electron states that decay exponentially into the bulk and sit at (or very near) E = 0 — a signature "edge band" predicted by Fujita et al. (1996) and Nakada et al. (1996). Slide the state index near the middle of the spectrum and watch the wave pile up on the top/bottom rows.
Armchair edges generically do not host such localized zero-energy states — states near E = 0 stay spread across the whole ribbon width instead of clinging to the boundary.
Edge weight = fraction of Σ|ψ(i)|² sitting in the outer 18% of the ribbon width on each side.
Participation ratio = 1 / Σ|ψ(i)|⁴ — roughly how many atoms the state is spread across (small = tightly localized, large = delocalized like a bulk Bloch state).

Sphere height and color encode the (signed) wavefunction amplitude on each carbon site: red = positive lobe, blue = negative lobe, height ∝ |ψ(i)|.

Diagonalizing…

Positive amplitude Negative amplitude

⚠ Couldn't load the 3D engineThree.js failed to load from the CDN. Check your connection and reload.