Single-wall carbon nanotubes (SWCNTs) exhibit unique electronic properties that are intrinsically linked to their diameter and chirality. These one-dimensional nanostructures can be either metallic or semiconducting, depending on their geometric structure, with bandgaps that vary inversely with tube diameter. The diameter-dependent bandgap transitions in SWCNTs are a direct consequence of quantum confinement effects, where the circumferential boundary conditions imposed by the nanotube’s cylindrical geometry quantize the electronic states.

The bandgap of a semiconducting SWCNT is governed by the tight-binding model, which approximates the electronic dispersion near the Fermi level. For a given chirality vector (n,m), the bandgap Eg scales approximately as 1/d, where d is the nanotube diameter. Empirical studies have shown that Eg ≈ 0.9 eV·nm / d for diameters ranging from 0.7 nm to 2 nm. For instance, a (10,2) nanotube with a diameter of 0.94 nm exhibits a bandgap of ~0.95 eV, while a larger (20,0) nanotube with a diameter of 1.57 nm has a reduced bandgap of ~0.57 eV. This diameter dependence arises from the curvature-induced rehybridization of carbon orbitals, which modifies the π-electron network.

A critical aspect of SWCNT optics is the role of excitons—bound electron-hole pairs—which dominate the optical absorption and emission processes. Due to reduced dielectric screening in one-dimensional systems, exciton binding energies in SWCNTs are significantly larger than those in bulk semiconductors. Binding energies typically range from 0.3 eV to 0.5 eV, accounting for up to 50% of the bandgap. For example, a (6,5) SWCNT with a bandgap of 1.27 eV exhibits an exciton binding energy of ~0.4 eV, meaning the optical gap (observed in photoluminescence) is ~0.87 eV. This strong Coulomb interaction leads to stable excitons even at room temperature, influencing the performance of optoelectronic devices.

The interplay between diameter-dependent bandgaps and excitonic effects has profound implications for nanoelectronic applications. Semiconducting SWCNTs are promising candidates for field-effect transistors (FETs) due to their high carrier mobility, ballistic transport, and tunable bandgaps. A key advantage is the ability to select nanotubes with specific diameters to match the desired threshold voltage or ON-current. For instance, smaller-diameter SWCNTs (~1 nm) are suitable for high-speed logic circuits requiring larger bandgaps, while larger-diameter tubes (~1.5 nm) are better suited for low-power applications where smaller bandgaps reduce switching energy.

In photovoltaics and photodetectors, the excitonic nature of SWCNTs necessitates careful design to dissociate excitons into free carriers. Heterostructures combining SWCNTs with electron-accepting materials (e.g., C60 or polymers) have demonstrated efficient charge separation, with external quantum efficiencies exceeding 50% in some cases. The diameter-dependent optical gaps also enable wavelength-selective photodetection; arrays of SWCNTs with varying diameters can cover a broad spectral range from visible to near-infrared.

For light-emitting applications, the narrow emission linewidths (~20 meV) and high quantum yields of SWCNTs make them attractive for nanoscale lasers and single-photon sources. Electroluminescence from electrically pumped SWCNT-FETs has been achieved, with emission wavelengths tunable via diameter selection. A (7,5) tube emits at ~1050 nm, while a (8,6) tube emits at ~1130 nm, enabling applications in telecommunications and bioimaging.

Challenges remain in controlling the exact diameter and chirality distribution during synthesis, as most growth methods produce heterogeneous mixtures. Advances in selective growth or post-synthesis sorting (e.g., density gradient ultracentrifugation or DNA-based separation) have improved the availability of monodisperse SWCNTs for targeted applications. Additionally, environmental effects such as dielectric screening from substrates or adsorbates can perturb bandgaps and exciton energies, requiring encapsulation or passivation strategies for stable operation.

In summary, the diameter-dependent electronic and optical properties of SWCNTs, coupled with strong excitonic effects, underpin their utility in next-generation nanoelectronics and optoelectronics. Precise control over these parameters enables tailored device performance, from high-speed transistors to efficient photodetectors and nanoscale light sources. Continued progress in synthesis and processing will further unlock the potential of these quantum-confined systems.