Armchair Graphene Nanoribbon Band Gap Width: The 3p, 3p+1, 3p+2 Rule Explained
The band gap of an armchair graphene nanoribbon (AGNR) depends in part on its width. In the nearest-neighbor tight-binding model, the number of carbon dimer lines across the ribbon, (N), divides ribbons into three families:
- (N = 3p): semiconducting, generally with the larger gap of the two gapped families.
- (N = 3p+1): semiconducting, generally with the smaller gap.
- (N = 3p+2): gapless in this model.
Here, (p) is an integer. This classification uses (N) as the number of dimer lines across the ribbon. It describes the idealized model, not a fixed gap value for real ribbons.
Why the width family matters
Electrons in a narrow ribbon are confined across its width, which restricts the allowed electronic states. In the nearest-neighbor tight-binding model, a ribbon in the (3p+2) family has an allowed state at graphene’s Dirac point, where the conduction and valence bands meet. The (3p) and (3p+1) families have gaps in this model, with the (3p) family generally having the larger gap.
Calculations have examined this family-dependent pattern for ideal armchair ribbons Physical Review B. Scanning tunneling spectroscopy has measured electronic properties of atomically precise ribbons, including band gaps in semiconducting ribbons Nature.
The family label does not specify an exact gap. Ribbons in the same gapped family can have different gap sizes because their widths differ. Wider semiconducting ribbons generally have smaller gaps, though the size also depends on the ribbon’s structure and physical conditions.
How to classify a ribbon
Use the number of dimer lines across the width, written as (N). Divide (N) by three and use the remainder:
- Remainder 0 means (3p).
- Remainder 1 means (3p+1).
- Remainder 2 means (3p+2).
For example, (N=10) has a remainder of 1 and belongs to the (3p+1) family. It is semiconducting in the idealized model. A ribbon with (N=9) belongs to the (3p) family and is also semiconducting; one with (N=11) belongs to the (3p+2) family and is gapless in the nearest-neighbor model.
| Width index (N) | Remainder when divided by 3 | Family | Character in the nearest-neighbor model |
|---|---|---|---|
| 6, 9, 12 | 0 | (3p) | Semiconducting, generally larger gap |
| 7, 10, 13 | 1 | (3p+1) | Semiconducting, generally smaller gap |
| 8, 11, 14 | 2 | (3p+2) | Gapless |
In common AGNR naming, the number in a label such as 7-AGNR refers to the width index (N), the dimer-line count. It does not refer to the ribbon’s length.
What the rule does and does not predict
The modulo-three classification identifies the idealized family. It does not give a numerical band gap. Estimating a particular ribbon’s gap requires considering its width and structure. In general, semiconducting ribbons have smaller gaps as their width increases.
The gapless label for the (3p+2) family applies to the nearest-neighbor tight-binding model. It does not guarantee that a measured ribbon has a perfectly gapless spectrum or conducts with zero resistance. Finite length, edge structure, substrate effects, and other departures from the ideal geometry can change measured properties. Calculations have also examined how spin arrangements at ribbon edges may affect electronic structure under some conditions IOPscience.
Use the family rule as an initial classification. A real sample may differ from the ideal model.
Key Takeaways
- Classify an AGNR using (N), the number of dimer lines across its width.
- In the nearest-neighbor tight-binding model, (3p+2) ribbons are gapless; (3p) and (3p+1) ribbons are semiconducting.
- The (3p) family generally has a larger gap than the (3p+1) family.
- Semiconducting gaps generally decrease as ribbon width increases.
- Real ribbon properties can be affected by edge structure, finite length, and substrate effects.
