The size and shape of graphene nanoribbons GNRs significantly affect their electronic and optical properties compared to bulk graphene. Graphene, a single layer of carbon atoms arranged in a hexagonal lattice, exhibits unique electronic properties such as high electron mobility and a zero bandgap. However, when graphene is cut into narrow strips or nanoribbons, its properties can be tuned depending on the size and edge structure of the GNRs.There are two types of edge structures in GNRs: armchair and zigzag. The electronic properties of GNRs depend on their width and edge structure, which can be classified into three categories:1. Metallic GNRs: These have zero bandgap and exhibit metallic behavior. They are typically found in armchair GNRs with a width N that is a multiple of 3, where N is the number of dimer lines across the ribbon width.2. Semiconducting GNRs: These have a finite bandgap and exhibit semiconducting behavior. They are typically found in armchair GNRs with a width N that is not a multiple of 3.3. Zigzag GNRs: These have either metallic or semiconducting behavior depending on their width and edge structure.To understand the relationship between the width of GNRs and their bandgap, we can use the tight-binding approximation model. The bandgap Eg of armchair GNRs can be calculated using the following formula:Eg = 2 * * |sin * N / 3 * a |where is the nearest-neighbor hopping energy approximately 2.7 eV , N is the number of dimer lines across the ribbon width, and a is the carbon-carbon bond length approximately 1.42 .For example, let's calculate the bandgap of an armchair GNR with a width of 5 dimer lines N = 5 :Eg = 2 * 2.7 eV * |sin * 5 / 3 * 1.42 |Eg 1.19 eVThis indicates that the armchair GNR with a width of 5 dimer lines is a semiconductor with a bandgap of approximately 1.19 eV.The optical properties of GNRs are also affected by their size and shape. The optical absorption spectrum of GNRs is determined by their electronic band structure. As the width of GNRs decreases, the bandgap increases, which leads to a blue shift in the absorption spectrum. This means that narrower GNRs absorb light at shorter wavelengths higher energies compared to wider GNRs.In summary, the size and shape of graphene nanoribbons play a crucial role in determining their electronic and optical properties. By controlling the width and edge structure of GNRs, it is possible to tune their bandgap and optical absorption properties, making them suitable for various applications in electronics and optoelectronics.