Figure 3 shows the estimated PDOS of 10B isotope-containing (0-100%) SWBNNT. As expected, Van Hove singularities appear in the estimated PDOS caused by the confinement effect of the reduced dimensionality of SWBNNT. The PDOS successfully generates all the distinctive peaks associated with the sp2 bonded B-N compound. The estimated PDOS is quite similar to that of the 2D *h*-BN sheet, except for tiny peaks induced by one-dimensional singularities. The high-energy in-plane tangential G-band phonon modes (longitudinal optical (iLO) and transverse optical (iTO)) are of primary attention in this study. Generally, the E2g peak in PDOS is Raman active in armchair edged BNNT, similar to 2D *h*-BN; however, the A1 peak is Raman active in zigzag edged BNNT21,22,24,27,29. We observe the A1 peak at 1376 cm−1 in our simulations, smaller than the value (E2g) obtained for the 2D *h*-BN38 due to the curvature effect18. The curvature effect of BNNT softens this Raman active mode frequency to the lower value, which is consistent with earlier results39.

A general trend of moving all vibrational modes towards the higher energy region is found when the 10B isotope level increases. The A1 peak has a considerable upward shift due to the isotope interaction. The previous studies found that adding a lighter (heavier) isotope atom generates a large upward or downward move in the high-frequency phonon mode. However, the effect was negligible in the low-frequency regime19,39,40. Isotopic substitution decreases the average atomic mass of BNNT, and even a minor amount of isotope in BNNT can drastically alter the phonon characteristics. An upward shift of the Raman active phonon modes was also detected in 10B isotope-containing BNNT in earlier stusy41. The frequency of vibrational mode \(\omega\) is related to the atomic mass \(M\) as \(\omega \propto {M}^{-1/2}\), showing that lowering the mass can raise the frequency. Using the standard harmonic oscillator theory, the linear upward shift of frequencies due to isotope defects can be described as:

$${\omega }_{{10}_{B}}\left(x\right)=\frac{{\omega }_{{11}_{B}}}{\sqrt{1-\frac{x}{11}}}$$

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Here, \({\omega }_{{11}_{B}}\) and \(x\) signify the frequency of original BNNT and the 10B isotope contents, respectively. The upward shift of the A1 peak as a function of 10B isotope concentration is depicted in Figure 4. A comparison of the values obtained from equation (4) with the results of our computation is also shown. The solid line represents the computed values of the prior relation, whereas circles reflect our computational results. We discover that our computational results and theoretical values are perfectly consistent. From 0–100% 10B isotope contents, the A1 mode shifts upward by about 42 cm−1. The system contains the same mass for both 0% and 100% 10B isotope concentrations. However, the harmonic arrangement is disrupted beyond these two extreme situations, and the mass of the nanotube should be the inverse of the square root of both 11B and 10B atoms. With the increase of 10B atoms in the system, the average atomic mass decreases, resulting in a phonon frequency shift to the high energy regime.

Next, we concentrate our investigation on the impacts of vacancies on BNNT phonon behavior. Vacancy defects have been demonstrated to induce considerable deviations in phonon and heat conduction characteristics of CNT in previous studies. The bond length and energy can be altered by vacancies, and the vibrational frequency is highly reliant on these parameters42. Because of significant changes in the phonon structure, the effect of vacancy defects on thermal conductivity is more noticeable than that of isotopes. An earlier study43 showed that the thermal conductivity of a graphene nanoribbon was reduced by 81% even at a relatively low concentration of vacancies (0.1%). Different studies on CNT44–46 manifested that 1-2% vacancy concentration resulted in about a 50% reduction in the thermal conductivity. Again, some studies on silicene nanoribbons47 and silicon nanotubes48 found around a 50% reduction in thermal conductivity with 1-2% atomic vacancy concentration. Thus, the effects of vacancies on the phonon characteristics of BNNT are exciting. The impact of various concentration of B and N vacancies on the PDOS of BNNT is presented in Figures 5**(a) and (b)**. Although the 10B isotope disorder causes an upward swing in vibrational frequencies, the PDOS peaks show a widening and downward shift as the vacancy concentration rises. For both types of vacancies, the A1 peak has totally vanished at 10% and higher densities. Using the bond-order theory, Xie et al.49 demonstrated that phonons in 2D materials are dispersed greatly by atomic vacancies. Vacancy disorder can also change the periodicity of the sp2-hybridized BNNT structure. The momentum conservation of the crystal is broken due to the disruption of the periodic order by the significant density of vacancies. Consequently, the high-energy phonon peaks no longer appear.

The emergence of several abrupt peaks at low-frequency with increasing vacancy concentration is another key feature of the phonon mode, as seen in Figure 5. The concentration of unsaturated bonds of B and N atoms are increased with increasing vacancy concentration, which may cause a decrease in the high-frequency phonon density. As a result, phonon modes migrate towards the low-energy region. Phonon scattering in the low-frequency region due to the defects was notable in CNT as demonstrated by Mingo et al.50. In addition to generating a substantial peak, increasing defect density also lowers the average PDOS in the low-frequency zone. Mahan et al.51 discovered the low-frequency flexural phonon modes with a quadratic dispersion in the PDOS of CNT. On the other hand, Mariani et al.52 and Ochoa et al.53 showed that the average PDOS of graphene reduces due to the stiffening of the flexural phonon modes caused by the disorder-induced strain. Jeon et al.24 found flexure phonon modes in BNNT even though the constituting atoms (B and N) are polar in nature. We believe that vacancies stiffen the flexural phonons, resulting in a drop in the average PDOS in the low-frequency region.

The effect of isotope mixing with B or N vacancies on the BNNT PDOS is depicted in Figure 6. Although the isotope disorder causes an upward change in phonon modes, the combination of the isotope and vacancies appears to have a downward shift. The vacancy disorder in BNNT, however, induces a more significant downward shift than the mixing defects. As shown in Figure 7, the softening of the A1 peak is highly influenced by defect density, with the descending shift being greater for B vacancies than for other defects. Simon et al.54 also found similar results. They created a double-walled CNT in which the inner wall was 13C enriched and the exterior wall was natural carbon (i.e., 1.1% 13C, 98.9% 12C) enriched along with inevitable vacancies. The 13C enhanced inner wall of the CNT caused a nonuniform expansion and descending shift of the tangential G mode. The inevitable point defects present in a system have a considerable impact on the phonon scattering owing to the extreme sensitivity of phonon to mass disorder. The scattering generated by point defects may cause the phonon to traverse through new states, causing the PDOS to broaden.

Furthermore, in a system having a breakdown in its lattice symmetry by defects, the atomic vibrations in that system can be identified outside of the ideal system's normal frequency range33,34. In a defect-free environment, phonons are unrestricted to move, however, in the presence of isotopes or vacancies, phonons become confined. In disordered system, the phonon mean free path changes according to the square of the localization length. As a result, the phonon energy may collect at its defective area, similar to the Anderson localization of electrons32. The static localized phonons lose their heat-carrying properties. As a result, the standard transmission of thermal energy through the sample material is hampered by these localized phonons. However, phonon localization in low dimensional materials can be beneficial in a variety of applications, including thermoelectric energy conversion and microelectronic heat management55–57. The normal linewidth of the Raman spectra is also caused by localized phonons27. To explain the experimental findings, a better understanding of isotope phonon localization and vacancy-induced BNNT is required.

The atomic scale localization behavior of phonon modes in SWBNNT has also been investigated in the presence of isotope and vacancy defects. In a perfect system, the wave of a phonon is disturbed. However, phonons can go through ballistic, diffusive, or confined regions in a disordered structure. Defects cause the disruption of the momentum conservation of the system. As a result of the interruption of momentum conservation of the regular atomic arrangement, the wave vectors of phonons cannot occupy regular quantum numbers. The phonons disperse into various phonon states and become spatially confined. There have been numerous attempts to estimate and display the phonon localization events in isotope defected nanostructures. Savic et al.6 performed a first-principles analysis to investigate phonon transmission in 14C isotope enriched CNT and 10B isotope disordered BNNT, claiming that the thermal conductivity decrease in isotope disordered CNT and BNNT is owing to diffusive scattering, whereas localization of phonons were not considered. Because heat transport happens with ballistic and diffusive phonons, they reasoned that phonon localization could not be seen. They projected that localization effects would arise in optical modes with short wavelengths. As a result, to see any localization effects, separate experimental techniques competent for exploring these high-energy modes are necessary. Using molecular dynamics modeling, Li et al.58 examined the heat transport of isotope disordered graphene nanoribbon and CNT. They discovered that the loss in thermal conductivity in CNT is more significant than in GNR owing to strong optical phonon localization. They calculated the phonon dispersion relation in an attempt to depict the localized phonon modes but were unable to obtain the actual image of the localization effect. The mixing effect of 10B isotope and vacancies as well as their independent effect on the phonon localization in unfolded SWBNNT has been explored here. We are particularly interested in high-frequency tangential phonon modes.

Figure 8 depicts the atomic vibrational patterns for the A1 phonon mode at 1376 cm−1. The vibrational patterns displayed here are based on 460 sites only. A sufficiently localized mode can be retained in this site area, which reduces the need for a significant number of atoms. The circle in Figure 8 specifies an atom; larger circles denote 11B atoms, smaller circles indicate 10B atoms, and different hues represent the strength of the displacement. The mode pattern for pure BNNT is shown in Figure 8**(a)**. The modes present an unequal distribution across the entire sample area. The mode patterns are determined by imposing a random force proportionate to the displacement of atoms at each time step iteratively. As a result, the eigenmode positions are expected to change over time. The randomness of the eigenmodes could be due to other factors. Usually, optical phonons experience more frequent scattering due to their high density compared to the acoustic phonon in the PDOS. The range of possible scattering outcomes is increased since scattering activities are generally inelastic59. Consequently, the coherent backscattering strength decreases owing to the reduced possibility of a specific mode reversing its scattering order. However, to achieve a high coherent backscattering effect, scattering events should be entirely elastic60. As a result, eigenmodes in the sample may appear at random.

The presence of the 10B isotope or vacancies in the SWBNNT structure induces a spatially localized eigenmode in the displacement pattern. Figure 8 **(b-d)** shows the mode patterns with a 20% randomly oriented 10B isotope and B or N vacancy defected sample. Few modes are observed to be confined and scattered irregularly across the sample area. The displacement pattern for isotope (10%) mixing with vacancies (10%) is shown in Figure 8 **(e-f)**. The vibrational modes are not adequately diffused in this example, and some vibrational modes are severely confined. Only a few atoms vibrating with the highest amplitude are discovered concentrated near the vacancy defects, indicating a significant localization. The location of localized atoms remains unevenly distributed near the vacancies, and it evolves through time. With increasing defect concentrations, the localization impact becomes more substantial. Prior investigation revealed that the thermal conductivity of nanotube structures reduces with cumulative defect concentrations. The localization of phonons due to the defects provides important evidence of earlier findings because phonons are the primary heat carrier in semiconducting nanostructures.

To further illustrate the degree of localization, the inverse participation ratio *IPR* of the system is derived. It is possible to express the *IPR* in the following way:

$$IPR=\frac{\sum _{l=1}^{N}{\left|{u}_{l,\lambda }\right|}^{4}}{{\left({\sum }_{l=1}^{N}{\left|{u}_{l,\lambda }\right|}^{2}\right)}^{2}}$$

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where \({u}_{l,\lambda }\) represents the eigenmode's displacement of the \(l\)th atom. In the localized state, a limited number of atoms oscillate with the largest displacement. Considering the normalization of the eigenvector among the *n* number of atoms, the amplitude the localized atom is \(u=1/\sqrt{n}\). \(IPR=1/n\) can now be used to represent the inverse participation ratio. It is worth noting that the strong localization can be achieved when *n* =1 (i.e., *IPR* =1) and only one atom oscillates in that mode. With *n = N* and \(IPR=1/N\), a dispersed oscillation pattern with all atoms vibrating at the same amplitude of \(u=1/\sqrt{N}\) should be noticed. We calculated the localization \({L}_{\lambda }\) length from the value of *IPR*, which is correlated to the *IPR* as \({L}_{\lambda }\propto {IPR}^{-\frac{1}{2}}\),61 and the \({L}_{\lambda }\)of the state, \(\lambda\) can be stated as

$$\frac{{L}_{\lambda }}{{L}_{0}}={\left(\frac{{IPR}_{0}}{{IPR}_{\lambda }}\right)}^{\frac{1}{2}}$$

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where \({L}_{0}\) is the unfolded SWBNNT size, i.e., the 2D *h*-BN sheet and \({IPR}_{0}\) is the average value of \({IPR}_{\lambda }\).

Figure 9 shows the fluctuation of the \({L}_{\lambda }\) with the variation of isotope concentration. The simulation was run with 5000 atoms at = 1376 cm−1 for the A1 mode phonon. The filled circles in Fig. 9 denote the average of ten eigenmodes. The \({L}_{\lambda }\) is associated to the \({IPR}_{\lambda }\) as \({L}_{\lambda }\infty {IPR}_{\lambda }^{-1/2}\). It is projected that strong localization will be obtained with the shortest localization length and vice versa. According to the calculations, the localization length decreases dramatically when isotope contents elevate from 0–60%, then rise again from 60–100%. At 60% isotope content, the minimum localization length was discovered. Despite the asymmetrical behavior, this finding is consistent with the earlier work on single-layer graphene by Rodriguez-Nieva et al.62. A variety of reasons could cause this erratic character. The influence of decreasing the mass of a system (by adding 10B isotopes to a 11B lattice) differs from that of increasing the system's mass (by adding 11B isotopes to a 10B lattice). The \({L}_{\lambda }\) is calculated from the *IPR*, which is derived from the eigenstate amplitudes of atoms. The total mass of the system decreases as 10B atoms are added to the 11B lattice. This eigenstate amplitude of the atoms with reduced mass will be larger than the pristine structures' eigenstate amplitude. The same thing happens when 11B impurities are introduced into a 10B lattice. In both circumstances, the eigenstates of the pristine structure are smaller than the *IPR*, causing a lower \({L}_{\lambda }\). In the reduced mass case, the effect of isotope addition is stimulatingly abrupt. As a result, the localization length may have an asymmetrical characteristic. There could be additional explanations for the asymmetry. The localization of phonons in the flat zones of dispersion relation can readily happen as described by Savic et al.6. Besides, they6 also revealed that optical phonons in the high-frequency region are predominantly localized6. The defect-related backscattering is considerable in the optical phonons in the high-frequency region, as observed in CNT63. An island with different frequency is developed for the localized eigenstate compared to the rest of the atoms, which may be the physical origin of localization.

The \({L}_{\lambda }\) for different combinations of defects has also been extracted, as shown Figure 10. The values of \({L}_{\lambda }\) decays noticeably with the rise of mixing defects or separate B or N point vacancies, as shown in Figure 10. For the same defect density, the combined defects demonstrate a sudden decrease in behavior in comparison to isotope disorder. There is because the bond length and energy changes when vacancies form in the lattice structure, which modifies the force constant parameters. The force constants for the 10B isotope-containing sample, on the other hand, stay unchanged since the bonding chemistry is unaffected by the additional neutron in each nucleus. However, because of the mass change of the ion, the isotope defect affects the dynamics. Because the frequency *f* is related with the system mass *m* as the inverse of the square root (i.e., *f*\(\propto {m}^{-\frac{1}{2}}\)), hence, changes in 10B isotope mass can significantly modify the PDOS. The isotope mixing with vacancies, however, changes the PDOS more suddenly. Thus, the comparison of the \({L}_{\lambda }\)for different impurity concentrations and different types of defects is not straightforward. As a result, there may be a sharp drop in the \({L}_{\lambda }\).