Chirality-Pure Semiconducting CNTs

• Chirality-pure semiconducting carbon nanotubes are index-defined excitonic systems that exhibit unique optical and electronic properties.
• Advanced purification methods, including post-growth sorting and direct-growth selectivity, isolate specific (n, m) species to suppress inhomogeneous broadening.
• Chirality-specific probes like THz spectroscopy and electroabsorption enable quantitative insights into dark exciton dynamics, binding energies, and device performance.

Chirality-pure semiconducting carbon nanotubes are single-wall carbon nanotube systems in which a semiconducting species with defined chiral indices $(n,m)$ is isolated, enriched, or assembled so that optical and electronic observables are chirality-specific. In these systems, the wrapping indices fix the diameter, chiral angle, band gap, and excitonic transition energies, while chirality purity suppresses inhomogeneous broadening, isolates specific excitonic manifolds, and enables quantitative structure–property correlations in spectroscopy, transport, and device operation (Luo et al., 2014, Liu et al., 2013).

1. Geometric classification and excitonic structure

A single-wall carbon nanotube is indexed by integers $(n,m)$ that define how a graphene sheet is rolled. The standard geometric relations used across the literature are

$$d=\frac{a}{\pi}\sqrt{n^2+m^2+nm}, \qquad \theta=\tan^{-1}\!\left(\frac{\sqrt{3}\,m}{2n+m}\right),$$

with $a \approx 0.246\,\text{nm}$, and, equivalently,

$\cos\theta=\frac{2n+m}{2\sqrt{n^2+m^2+nm}}.$

Electronic type follows the usual modular rule: semiconducting tubes satisfy $(n-m)\bmod 3 \neq 0$, whereas metallic tubes satisfy $(n-m)\bmod 3 = 0$. For semiconducting tubes, one convenient family notation is $q=(n-m)\bmod 3=\pm 1$, with $q=-1$ denoted type I and $q=+1$ denoted type II (Vialla et al., 2013).

The optical structure is excitonic. In semiconducting tubes, the lowest bright transition is variously labeled $(n,m)$0 or $(n,m)$1, and the next higher bright transition $(n,m)$2 or $(n,m)$3. $(n,m)$4 dominates photoluminescence emission, whereas $(n,m)$5 is commonly used for resonant absorption and excitation mapping. In chirality-specific $(n,m)$6 tubes, THz spectroscopy further resolves the lowest dark–bright pair: the optically forbidden $(n,m)$7 dark ground state and the optically allowed $(n,m)$8 bright exciton, separated by $(n,m)$9. This dark–bright splitting is inaccessible to conventional interband probes but becomes directly observable through the internal dipole-allowed $$d=\frac{a}{\pi}\sqrt{n^2+m^2+nm}, \qquad \theta=\tan^{-1}\!\left(\frac{\sqrt{3}\,m}{2n+m}\right),$$0 transition (Luo et al., 2014).

The same indices govern higher-order family behavior and trigonal-warping corrections. In practice, chirality-pure systems are valuable precisely because they collapse the otherwise heterogeneous Kataura landscape into a single set of excitonic resonances. This makes it possible to assign $$d=\frac{a}{\pi}\sqrt{n^2+m^2+nm}, \qquad \theta=\tan^{-1}\!\left(\frac{\sqrt{3}\,m}{2n+m}\right),$$1 uniquely in devices, isolate dark-state dynamics, and compare absolute cross-sections or linewidths across species without ensemble averaging over diameter and chiral-angle distributions (Liu et al., 2013).

2. Routes to chirality purity: direct growth, sorting, and chirality memory

Two broad strategies dominate. The first is post-growth purification, including density-gradient ultracentrifugation, gel chromatography, aqueous two-phase extraction, polymer-assisted sorting, and DNA-based sorting. The second is direct-growth selectivity, where catalyst chemistry, carbon chemical potential, and growth thermodynamics or kinetics bias the $$d=\frac{a}{\pi}\sqrt{n^2+m^2+nm}, \qquad \theta=\tan^{-1}\!\left(\frac{\sqrt{3}\,m}{2n+m}\right),$$2 distribution. The contemporary materials roadmap treats post-growth sorting as indispensable and combines it with advanced assembly, especially controlled vacuum filtration, to build macroscopic films that are simultaneously single-crystal, single-chirality, and single-wall; the same roadmap identifies gram-scale single-chirality production as a remaining bottleneck (Chang et al., 8 Apr 2026).

Direct-growth selectivity has progressed substantially, but it is not yet equivalent to true single-$$d=\frac{a}{\pi}\sqrt{n^2+m^2+nm}, \qquad \theta=\tan^{-1}\!\left(\frac{\sqrt{3}\,m}{2n+m}\right),$$3 purity. A metal-free cap-seeded vapour phase epitaxy route based on hemispherical geodesic polyarene $$d=\frac{a}{\pi}\sqrt{n^2+m^2+nm}, \qquad \theta=\tan^{-1}\!\left(\frac{\sqrt{3}\,m}{2n+m}\right),$$4 achieved $$d=\frac{a}{\pi}\sqrt{n^2+m^2+nm}, \qquad \theta=\tan^{-1}\!\left(\frac{\sqrt{3}\,m}{2n+m}\right),$$5 semiconducting purity with a $$d=\frac{a}{\pi}\sqrt{n^2+m^2+nm}, \qquad \theta=\tan^{-1}\!\left(\frac{\sqrt{3}\,m}{2n+m}\right),$$6 confidence interval $$d=\frac{a}{\pi}\sqrt{n^2+m^2+nm}, \qquad \theta=\tan^{-1}\!\left(\frac{\sqrt{3}\,m}{2n+m}\right),$$7, an average diameter of $$d=\frac{a}{\pi}\sqrt{n^2+m^2+nm}, \qquad \theta=\tan^{-1}\!\left(\frac{\sqrt{3}\,m}{2n+m}\right),$$8, and strong bias toward small-diameter semiconducting chiralities such as $$d=\frac{a}{\pi}\sqrt{n^2+m^2+nm}, \qquad \theta=\tan^{-1}\!\left(\frac{\sqrt{3}\,m}{2n+m}\right),$$9, $a \approx 0.246\,\text{nm}$0, $a \approx 0.246\,\text{nm}$1, $a \approx 0.246\,\text{nm}$2, $a \approx 0.246\,\text{nm}$3, and $a \approx 0.246\,\text{nm}$4, while armchair $a \approx 0.246\,\text{nm}$5 was not detected under $a \approx 0.246\,\text{nm}$6 excitation. Density functional theory connected this outcome to pretreatment-induced dehydrogenation, which lowers barriers for successive $a \approx 0.246\,\text{nm}$7 conversions and enables chirality drift away from the nominal $a \approx 0.246\,\text{nm}$8 cap toward nearby semiconducting species (Liu et al., 2014).

Catalyst-controlled selectivity is also described by growth models in which chirality dependence emerges only in specific gas-composition regimes. A universal kinetic model showed that low precursor supply produces minimal chirality dependence, whereas co-supply of sufficient carbon and etching agents yields a chiral-angle-dominant regime in which near-armchair chiralities grow faster; metallic tubes are more susceptible to etching, so increasing $a \approx 0.246\,\text{nm}$9 preferentially suppresses them. Complementarily, DFT on nanotube caps on Ni, Fe, and NiFe nanoparticles found that Fe-rich alloys increase electron transfer to cap edges and lower the armchair initiation barrier to $\cos\theta=\frac{2n+m}{2\sqrt{n^2+m^2+nm}}.$0 for Fe-containing catalysts, favoring enriched near-armchair semiconducting tubes such as $\cos\theta=\frac{2n+m}{2\sqrt{n^2+m^2+nm}}.$1, $\cos\theta=\frac{2n+m}{2\sqrt{n^2+m^2+nm}}.$2, and $\cos\theta=\frac{2n+m}{2\sqrt{n^2+m^2+nm}}.$3. A separate thermodynamic model showed that finite-temperature configurational entropy stabilizes chiral tubes through

$\cos\theta=\frac{2n+m}{2\sqrt{n^2+m^2+nm}}.$4

so that near-armchair semiconducting families $\cos\theta=\frac{2n+m}{2\sqrt{n^2+m^2+nm}}.$5 and $\cos\theta=\frac{2n+m}{2\sqrt{n^2+m^2+nm}}.$6 occupy large stability domains at moderate-to-high temperature (Otsuka et al., 2021, Dumlich et al., 2013, Magnin et al., 2018).

A distinct result is that chirality can be fixed early even when later growth conditions vary strongly. Under programmed $\cos\theta=\frac{2n+m}{2\sqrt{n^2+m^2+nm}}.$7 modulation with digital isotope labeling, $\cos\theta=\frac{2n+m}{2\sqrt{n^2+m^2+nm}}.$8 of $\cos\theta=\frac{2n+m}{2\sqrt{n^2+m^2+nm}}.$9 tubes, over a total inspected length of $(n-m)\bmod 3 \neq 0$0, preserved their $(n-m)\bmod 3 \neq 0$1 along the entire measured length, while only $(n-m)\bmod 3 \neq 0$2 chirality-change events were observed, corresponding to a length-normalized frequency of $(n-m)\bmod 3 \neq 0$3. Growth rates exhibited hysteresis because of irreversible catalyst coarsening, but chirality remained robust over lengths exceeding $(n-m)\bmod 3 \neq 0$4. This supports the view that chirality control can be established at nucleation and elongation then optimized independently (Otsuka et al., 7 Jan 2026).

3. Chirality-dependent absorption, photoluminescence, and optical metrology

A central problem in chirality-pure semiconducting nanotubes is that raw photoluminescence intensity is not a direct measure of abundance unless chirality-dependent absorption is removed. An explicit solution was developed using noncovalent tetraphenyl porphyrin functionalization, where the TPP Soret band at $(n-m)\bmod 3 \neq 0$5 provides ultra-efficient excitation-energy transfer to all semiconducting species. Time-resolved pump–probe measurements showed identical $(n-m)\bmod 3 \neq 0$6 rise and decay dynamics whether excitation occurred through nanotube $(n-m)\bmod 3 \neq 0$7 absorption or through porphyrin Soret absorption, and the transfer quantum yield was $(n-m)\bmod 3 \neq 0$8 on average with no detectable dependence on $(n-m)\bmod 3 \neq 0$9 (Vialla et al., 2013).

The key metrological quantity is the ratio

$(n-m)\bmod 3 = 0$0

with

$(n-m)\bmod 3 = 0$1

so that

$(n-m)\bmod 3 = 0$2

Because $(n-m)\bmod 3 = 0$3, $(n-m)\bmod 3 = 0$4, and $(n-m)\bmod 3 = 0$5 are taken as chirality independent, $(n-m)\bmod 3 = 0$6, and the relative variations of $(n-m)\bmod 3 = 0$7 can be extracted without knowledge of species concentration or photoluminescence quantum yield. Across $(n-m)\bmod 3 = 0$8 chiral species with diameters $(n-m)\bmod 3 = 0$9, the dominant dependence was on chiral angle and family type through $q=(n-m)\bmod 3=\pm 1$0: $q=(n-m)\bmod 3=\pm 1$1 varied by up to a factor of $q=(n-m)\bmod 3=\pm 1$2, with type I nanotubes exhibiting larger absorption than type II, especially near zigzag angles. The empirical law for the parallel-polarized cross-section was

$q=(n-m)\bmod 3=\pm 1$3

valid for $q=(n-m)\bmod 3=\pm 1$4 with approximately $q=(n-m)\bmod 3=\pm 1$5 uncertainty (Vialla et al., 2013).

Representative values make the scale explicit. For type I tubes, $q=(n-m)\bmod 3=\pm 1$6 gives $q=(n-m)\bmod 3=\pm 1$7, $q=(n-m)\bmod 3=\pm 1$8 gives $q=(n-m)\bmod 3=\pm 1$9, and $q=-1$0 gives $q=-1$1; for type II tubes, $q=-1$2 gives $q=-1$3, $q=-1$4 gives $q=-1$5, and $q=-1$6 gives $q=-1$7. By combining these absolute cross-sections with action cross-sections from the literature, the photoluminescence quantum yield was inferred to be nearly constant,

$q=-1$8

with a spread of $q=-1$9 relative. Consequently, differences in $q=+1$0-excited PL intensity are driven primarily by $q=+1$1, not by large interspecies variations in $q=+1$2. For abundance quantification, the corrective relation is

$q=+1$3

up to a proportionality constant. Without this correction, apparent enrichment in type I species can be an artifact of higher $q=+1$4 absorption rather than true compositional dominance (Vialla et al., 2013).

4. Chirality-specific excitonic probes beyond conventional PL

Chirality-pure nanotubes also make dark-state spectroscopy quantitative. In chirality-enriched $q=+1$5 SWNTs, ultrafast optical pump–THz probe measurements directly observed the internal $q=+1$6 resonance at $q=+1$7. The measured complex THz response was fit by a Lorentz-type intra-excitonic resonance plus a Drude term for coexisting $q=+1$8 plasma. The extracted oscillator strength was $q=+1$9 in the low-fluence limit; under resonant $(n,m)$00 pumping at $(n,m)$01, the majority of excitons occupied the dark ground state, with the analysis indicating $(n,m)$02 (dark) $(n,m)$03 of the total exciton density. The dark-exciton population followed bimolecular exciton–exciton annihilation kinetics with $(n,m)$04, and under off-resonant pumping the internal resonance persisted even when $(n,m)$05 reached $(n,m)$06, indicating a robust quasi-1D excitonic ground state that coexists with plasma rather than collapsing through ionization (Luo et al., 2014).

Electroabsorption provides a second chirality-resolved probe. In polymer-sorted, index-defined semiconducting SWNT films, the EA spectrum reproduced the $(n,m)$07 absorption peaks of $(n,m)$08, $(n,m)$09, $(n,m)$10, $(n,m)$11, and $(n,m)$12 one-to-one, enabling unique chirality assignment directly in biased films. In the low-field Stark regime, no measurable signal appeared at $(n,m)$13 or $(n,m)$14, the EA amplitude scaled quadratically with electric field, and no significant shift or broadening was observed from approximately $(n,m)$15 to $(n,m)$16. The empirical form reported was

$(n,m)$17

and, after normalization by absorption amplitude, the intrinsic EA response satisfied

$(n,m)$18

for $(n,m)$19. Using the Capaz-model binding energies, $(n,m)$20 decreased from approximately $(n,m)$21 for $(n,m)$22 to approximately $(n,m)$23 for $(n,m)$24. Within this diameter window, EA therefore functions as a direct in situ probe of excitonic binding in chirality-pure semiconducting tubes (Izard et al., 2015).

Taken together, THz internal spectroscopy and electroabsorption show that chirality purity is not only a spectral-cleanliness issue. It is what permits quantitative access to dark-state occupancy, oscillator strength, binding energy, phase-space filling, and interaction-driven renormalization without uncontrolled averaging over unrelated excitonic manifolds.

5. Single-tube and device-level characterization

High-throughput identification of chirality-pure semiconducting CNTs in realistic device stacks became practical with cross-polarized reflection microscopy. In this approach, the substrate-reflected field $(n,m)$25 is suppressed while the CNT-scattered field $(n,m)$26 is preserved, so the contrast enhancement factor is approximately $(n,m)$27, routinely exceeding $(n,m)$28 for small $(n,m)$29. The method provides video-rate imaging at $(n,m)$30 frames per second with approximately $(n,m)$31 exposure, a field of view of approximately $(n,m)$32 diameter, and greater than $(n,m)$33 intensity contrast for single SWNTs on fused silica, quartz, $(n,m)$34, $(n,m)$35, and partially under top dielectrics such as $(n,m)$36. Broadband spectroscopy over $(n,m)$37 is acquired in approximately $(n,m)$38 per tube, enabling $(n,m)$39 assignment for approximately $(n,m)$40 of tubes with diameters $(n,m)$41; in one growth condition, $(n,m)$42 individual SWNTs were profiled, including $(n,m)$43 semiconducting and $(n,m)$44 metallic tubes. In active FETs, the same platform showed that high-order resonances broaden dramatically with electrostatic hole doping up to approximately $(n,m)$45, consistent with strong interband electron–electron scattering and linewidth evolution $(n,m)$46; for the $(n,m)$47 $(n,m)$48 transition, the undoped exciton–phonon dephasing time was $(n,m)$49, with an added exciton–electron decay contribution $(n,m)$50 at high doping (Liu et al., 2013).

Chirality-sorted light emission was demonstrated directly in monochiral $(n,m)$51 devices assembled by dielectrophoresis on $(n,m)$52-thick $(n,m)$53 membranes. For $(n,m)$54, the geometric parameters are $(n,m)$55 and $(n,m)$56. Photoluminescence excitation spectroscopy identified $(n,m)$57 emission at $(n,m)$58 ($(n,m)$59) and $(n,m)$60 excitation at $(n,m)$61 ($(n,m)$62) on $(n,m)$63. Electrically driven devices emitted a single Lorentzian line at $(n,m)$64 ($(n,m)$65), assigned to the excitonic $(n,m)$66 transition by direct comparison with the PL excitation map. EL intensity increased with current beyond a threshold of approximately $(n,m)$67, the EL peak width increased linearly with current, and extrapolation to zero current yielded a width comparable to the PL $(n,m)$68 linewidth, consistent with current-induced lifetime broadening attributed to exciton–exciton annihilation. The estimated $(n,m)$69 photon yield was approximately $(n,m)$70 photons per injected charge at $(n,m)$71, and, using an integrated $(n,m)$72 intensity ratio of approximately $(n,m)$73, the total photon yield was approximately $(n,m)$74 per charge. The work also showed that $(n,m)$75 surfaces are comparatively inert with respect to nanotube optical properties, in contrast to $(n,m)$76, which induced a phonon-mediated dark exciton feature near $(n,m)$77 (Pfeiffer et al., 2011).

These device-level studies establish that chirality purity is operationally meaningful. It determines not only which resonances are present, but also how linewidths evolve under gating, how emission color is fixed by $(n,m)$78, and which substrate or contact perturbations can be separated from intrinsic excitonic behavior.

6. Macroscopic single-chirality films, aligned photonics, and emerging heterostructures

The most direct macroscopic consequence of chirality purity in transport was shown in single-chirality SWCNT films comprising large-gap semiconducting $(n,m)$79 and $(n,m)$80, narrow-gap semiconducting $(n,m)$81 and $(n,m)$82, and armchair metallic $(n,m)$83. All films followed the quasi-1D Mott variable-range-hopping form

$(n,m)$84

with experimentally fitted $(n,m)$85 for $(n,m)$86, $(n,m)$87 for $(n,m)$88, $(n,m)$89 for $(n,m)$90, $(n,m)$91 for $(n,m)$92, and $(n,m)$93 for $(n,m)$94. Theory gave localization lengths of $(n,m)$95, $(n,m)$96, $(n,m)$97, $(n,m)$98, and $(n,m)$99, respectively, establishing three categories: large-gap semiconducting, narrow-gap semiconducting, and armchair metallic. Thus, even in random junction networks, chirality purity preserves a strong band-structure dependence of localization and low-temperature conductivity (Gao et al., 2021).

Aligned chirality-pure films also produce optical states unavailable in mixed networks. In near-monolayer aligned $$d=\frac{a}{\pi}\sqrt{n^2+m^2+nm}, \qquad \theta=\tan^{-1}\!\left(\frac{\sqrt{3}\,m}{2n+m}\right),$$00 films of thickness approximately $$d=\frac{a}{\pi}\sqrt{n^2+m^2+nm}, \qquad \theta=\tan^{-1}\!\left(\frac{\sqrt{3}\,m}{2n+m}\right),$$01, Mueller-matrix micro-ellipsometry resolved an in-plane Type-I excitonic hyperbolic regime just above $$d=\frac{a}{\pi}\sqrt{n^2+m^2+nm}, \qquad \theta=\tan^{-1}\!\left(\frac{\sqrt{3}\,m}{2n+m}\right),$$02, where $$d=\frac{a}{\pi}\sqrt{n^2+m^2+nm}, \qquad \theta=\tan^{-1}\!\left(\frac{\sqrt{3}\,m}{2n+m}\right),$$03 and $$d=\frac{a}{\pi}\sqrt{n^2+m^2+nm}, \qquad \theta=\tan^{-1}\!\left(\frac{\sqrt{3}\,m}{2n+m}\right),$$04. Electrostatic gating shifted the hyperbolic window by approximately $$d=\frac{a}{\pi}\sqrt{n^2+m^2+nm}, \qquad \theta=\tan^{-1}\!\left(\frac{\sqrt{3}\,m}{2n+m}\right),$$05 in total, with a $$d=\frac{a}{\pi}\sqrt{n^2+m^2+nm}, \qquad \theta=\tan^{-1}\!\left(\frac{\sqrt{3}\,m}{2n+m}\right),$$06 shift of the high-energy transition and a $$d=\frac{a}{\pi}\sqrt{n^2+m^2+nm}, \qquad \theta=\tan^{-1}\!\left(\frac{\sqrt{3}\,m}{2n+m}\right),$$07 shift of the low-energy transition. The maximum momentum enhancement increased from $$d=\frac{a}{\pi}\sqrt{n^2+m^2+nm}, \qquad \theta=\tan^{-1}\!\left(\frac{\sqrt{3}\,m}{2n+m}\right),$$08 to $$d=\frac{a}{\pi}\sqrt{n^2+m^2+nm}, \qquad \theta=\tan^{-1}\!\left(\frac{\sqrt{3}\,m}{2n+m}\right),$$09, the propagation angle was modulated by approximately $$d=\frac{a}{\pi}\sqrt{n^2+m^2+nm}, \qquad \theta=\tan^{-1}\!\left(\frac{\sqrt{3}\,m}{2n+m}\right),$$10, and finite-difference simulations yielded a Purcell factor $$d=\frac{a}{\pi}\sqrt{n^2+m^2+nm}, \qquad \theta=\tan^{-1}\!\left(\frac{\sqrt{3}\,m}{2n+m}\right),$$11 for a dipole $$d=\frac{a}{\pi}\sqrt{n^2+m^2+nm}, \qquad \theta=\tan^{-1}\!\left(\frac{\sqrt{3}\,m}{2n+m}\right),$$12 above the film, with electrical modulation of approximately $$d=\frac{a}{\pi}\sqrt{n^2+m^2+nm}, \qquad \theta=\tan^{-1}\!\left(\frac{\sqrt{3}\,m}{2n+m}\right),$$13 and no optical cavity (Lynch et al., 29 Sep 2025).

A related but symmetry-distinct manifestation is second-order nonlinearity in aligned single-enantiomer films. Centimeter-scale, densely packed, aligned $$d=\frac{a}{\pi}\sqrt{n^2+m^2+nm}, \qquad \theta=\tan^{-1}\!\left(\frac{\sqrt{3}\,m}{2n+m}\right),$$14 films with enantiomeric purity $$d=\frac{a}{\pi}\sqrt{n^2+m^2+nm}, \qquad \theta=\tan^{-1}\!\left(\frac{\sqrt{3}\,m}{2n+m}\right),$$15, global linear dichroism $$d=\frac{a}{\pi}\sqrt{n^2+m^2+nm}, \qquad \theta=\tan^{-1}\!\left(\frac{\sqrt{3}\,m}{2n+m}\right),$$16, and thickness approximately $$d=\frac{a}{\pi}\sqrt{n^2+m^2+nm}, \qquad \theta=\tan^{-1}\!\left(\frac{\sqrt{3}\,m}{2n+m}\right),$$17 exhibited giant SHG. The dominant tensor element reached

$$d=\frac{a}{\pi}\sqrt{n^2+m^2+nm}, \qquad \theta=\tan^{-1}\!\left(\frac{\sqrt{3}\,m}{2n+m}\right),$$18

at a pump wavelength of $$d=\frac{a}{\pi}\sqrt{n^2+m^2+nm}, \qquad \theta=\tan^{-1}\!\left(\frac{\sqrt{3}\,m}{2n+m}\right),$$19, resonant with $$d=\frac{a}{\pi}\sqrt{n^2+m^2+nm}, \qquad \theta=\tan^{-1}\!\left(\frac{\sqrt{3}\,m}{2n+m}\right),$$20. The SHG vanished in aligned racemic controls and was below $$d=\frac{a}{\pi}\sqrt{n^2+m^2+nm}, \qquad \theta=\tan^{-1}\!\left(\frac{\sqrt{3}\,m}{2n+m}\right),$$21 of the aligned single-enantiomer signal in random single-enantiomer films, demonstrating that macroscopic inversion-symmetry breaking requires both enantiomer selection and alignment. Many-body Bethe–Salpeter calculations reproduced both the magnitude and the spectral shape, including the dominant $$d=\frac{a}{\pi}\sqrt{n^2+m^2+nm}, \qquad \theta=\tan^{-1}\!\left(\frac{\sqrt{3}\,m}{2n+m}\right),$$22-enhanced peak and a second feature near $$d=\frac{a}{\pi}\sqrt{n^2+m^2+nm}, \qquad \theta=\tan^{-1}\!\left(\frac{\sqrt{3}\,m}{2n+m}\right),$$23 attributed to $$d=\frac{a}{\pi}\sqrt{n^2+m^2+nm}, \qquad \theta=\tan^{-1}\!\left(\frac{\sqrt{3}\,m}{2n+m}\right),$$24 (Xu et al., 2024).

These aligned-film results motivate a broader heterostructure framework in which post-growth chirality purification and controlled vacuum filtration are combined to create “Single$$d=\frac{a}{\pi}\sqrt{n^2+m^2+nm}, \qquad \theta=\tan^{-1}\!\left(\frac{\sqrt{3}\,m}{2n+m}\right),$$25” structures: single-crystal, single-chirality, single-wall films with $$d=\frac{a}{\pi}\sqrt{n^2+m^2+nm}, \qquad \theta=\tan^{-1}\!\left(\frac{\sqrt{3}\,m}{2n+m}\right),$$26, deterministic thickness from monolayer to many tens of nanometers, and nanometer-scale stacking control. In that framework, chirality-pure semiconducting CNTs become building blocks for artificial bilayers, quantum wells, and superlattices with tunable band alignment, exciton funneling, and cavity-enhanced emission, while residual surfactants, contact resistance, and sorting throughput remain the dominant materials bottlenecks (Chang et al., 8 Apr 2026).

Chirality-pure semiconducting carbon nanotubes therefore occupy a distinctive position in nanoscience: they are simultaneously an index-defined excitonic system, a metrological standard for chirality-dependent absorption and emission, a platform for dark-state and Stark spectroscopy, and a scalable thin-film semiconductor whose macroscopic transport, nonlinear optics, and anisotropic electrodynamics remain traceable to a single one-dimensional band structure.