Phonon Localization in Boron Nitride Nanotubes: Mixing Effect of 10 B Isotopes and Vacancies

We explored the mixing effect of 10 B isotopes and boron (B) or nitrogen (N) vacancies on the atomic vibrational properties of (10, 0) single-wall boron nitride nanotubes (BNNT). The forced oscillation technique was employed to evaluate the phonon modes for the entire range (0-100%) of 10 B isotopes and atomic vacancy densities ranging from 0 to 30%. With increasing isotope densities, we noticed a blue-shift of the Raman active A 1 phonon peak, whereas an increased density of mixed or independent B and N vacancies resulted in the emergence of a new low-frequency peak and the annihilation of the A 1 peak in the phonon density-of-states. High-energy optical phonons were localized as a result of both 10 B isotopes and the presence of mixing defects. We generated typical mode patterns for different defects to show the phonon localization processes due to the defects. We found an asymmetrical nature of the localization length with increasing 10 B isotope content, which corresponds well with the isotope inherited localization length of carbon nanotubes and mono-layer graphene. The localization length falls abruptly with the increase in concentration of both atomic vacancies (B or N) and mixing defects ( 10 B isotope and vacancies). These ndings are critical for understanding heat conduction and nanoscopic vibrational investigations like tip-enhanced Raman spectra in BNNT, which can map local phonon energies. N mixing the localization length reduces This gives a theoretical framework for isotope and vacancy-induced phonon properties, which is crucial for understanding SWBNNT thermal conductivity and Raman spectra.


Introduction
Isotopes play an essential role in manipulating the physical, chemical, and biological phenomena of a material. Physical phenomena such as nuclear ssion or fusion, liquid or solid magnetic resonance, the superconducting transition temperature of metals 1,2 are due to the isotope effect. Chemical phenomena such as chemical reactions change with isotopic contents when the substituted isotopes participate directly in the chemical bonds that are broken or formed. Biological phenomena like complex biological reactions are traced with unstable nuclei due to the isotope effect. As isotope content changes the average mass of the material, it has the most signi cant impact on the phonon (quanta of lattice vibrations) properties 3 . Especially in disordered nanotubes, the phonons show some peculiar transport characteristics like ballistic, diffusive, and localized. The change in thermal conductivity in nanotubes due to the isotope effect has also been demonstrated to have a substantial impact 4 .
The isotope effects are even more pronounced in boron nitride nanotubes (BNNT) because natural boron (B) possesses a considerable isotopic disorder. Natural BNNT consists of 80.1% 11 B and 19.9% 10 B.
Chang et al. 5 found a 50% enhancement of thermal conductivity at room temperature with 99.5% 11 B isotope in individual isolated multiwall BNNT using a microfabricated test xture with the facility of highresolution transmission electron microscopy. Ivana et al. 6 found dominant effects on the thermal conductivity reduction with isotope disorder in BNNT through ab initio calculations. They showed that diffusive scattering was the main source of the decrease in thermal conductivity, especially when the isotope concentration exceeded 10%. They concluded that localization effects could not be extracted from the thermal conductivity measurements. To demonstrate this dramatic increase of thermal conductivity in almost isotopically pure BNNT, the same group developed an independent cascade scatter model of BNNT. They modeled thermal transport in BNNT through the atomistic Green's function approach combined with rst-principles phonon calculations. They revealed that isotope-induced phonon scattering inhibited phonon transport at high frequencies. In contrast, isotope enrichment enhances thermal conductivity even in a totally diffusive environment in which scatterers act independently 7 . From these studies it is evident that the impact of isotope disruption on nanotube thermal conduction is still controversial and unclear.
On the other hand, the very rst fabrication of BNNTs was described by Chopra et al. 8 , which was based on an arc discharge method. Besides the arc discharge method 8,9 , chemical vapor deposition 10,11 and laser ablation method 12,13 are also used for synthesizing BNNT. During these synthesis processes of BNNT, there will be the introdction of ubiquitous atomic vacancies, similar to its two-dimensional counterpart 14 . Moreover, intentional atomic vacancy creation by irradiation was utilized to recognize the chirality of an independent wall in multiwall BNNT by observing the direction or the growth track of the vacancy defects 15 . Atomic vacancies alter the continuity of the regular arrangement of the hexagonal network and create dangling bonds. It has already been found that atomic vacancies may change the electrical, optical, and magnetic properties 16,17 of BNNT. Thus, atomic vacancies should have signi cant effects on the phonon properties of BNNT.
As BNNTs are structural analog of carbon nanotubes (CNT), it is regarded that they should show the same effects on the phonon properties due to alternation of the regular atomic arrangement, especially by isotopes and vacancies 18,19 . Moreover, phonon scattering by isotopes or vacancies signi cantly impact the electron-phonon interaction in BNNT 20 , leading to the possible alternation of electron transport properties. Thus, an in-depth analysis of individual isotope and vacancy, as well as combinations of these, in BNNT, is indispensable. The phonon properties of pristine BNNT were explored in literature using a valence shell model 21 , ab initio calculation 22 , the force constant model 23,24 , the continuum model 25 , Raman spectroscopy 26-29 and IR spectroscopy 30 . Even though the phonon properties in disordered BNNT have been discussed in several theoretical and experimental studies, a comprehensive investigation of this problem in natural systems with a large atomic scale is still absent. Moreover, to our knowledge, there is no work on phonon properties of BNNT with both isotopes and vacancies.
Here, a systematic detailed study of the effects of isotopes and vacancies (individual B and N atoms), as well as mixed isotopes and vacancy defects, on the phonon characteristics of BNNT is presented. The forced oscillation technique (FOT), established by Williams and Maris 31 for a disordered system with a large number of atoms, was used to determine the phonon eigenmodes of disordered BNNT. It was found that in an isolated system having a single electron and disordered potential, the eigenmodes were in exponentially localized states, whereas the eigenmodes were in spatially diffusive states with periodic potentials. This phenomenon is called Anderson localization 32 . In a disordered system, phonons should also be con ned just like electrons. In most cases, a substantial vibrational amplitude arises around defects and decays exponentially into the system's pristine area. It was perceived that the phonon modes in a disordered system shifted to the outside of the normal frequency region. 33,34 Here, we step by step inspect the properties of vibrational modes in disordered BNNT addressing the questions of how the modes shift, and the mode patterns localize.

Methods
The translation vector (T) and the chiral vector (C) of the nanotube related with its length and width, respectively, were multiplied to determine the unit cell 35 of single-wall boron nitride nanotube (SWBNNT). Figure 1 shows the isotope and vacancy defected ball-stick model and the Brillouin zone (BZ) of (10, 0) SWBNNT. The unit cell of a (10, 0) SWBNNT consists of 20 B and 20 N atoms. In this analysis, we used 125-unit cells with a total of 5000 atoms and a length of 54.37 nm. Ballistic transport of charge carriers will be possible with such a length since it is smaller than 100 nm 36 . Site percolation theory was employed to introduce 10 B isotope and atomic vacancies into the structure. The maximum bond probability of ~70% can be achieved for the honeycomb site percolation structure. Thus, we investigate a wide range of vacancy densities up to 30% and isotope levels up to 100%. The fourth nearest neighbor atomic interactions are considered using the force constants derived by Xiao et al. for the planar h-BN sheet 37 . Because of the curvature effect, the force constants of SWBNNT and h-BN planar sheet are not be the same. We have corrected the force constants of the SWBNNT, calculating the changes of the bond angles of the cylindrical tube relative to the planar h-BN sheet. The approach to achieve the curvature effect is depicted in Figures 2(a) and 2(b). Considering a force constant K, between two atoms A (B atom) and B (N atom), the projection of B on the xy-plane and its force constant tensor are denoted as B′ (un lled circle) and K′, respectively, as shown in Figure 2(a). K and K′ rotate around the y-axis at an angle of ϕ/2, where ϕ denotes the center angle between A and B. The projection of B′ on the zx-plane and its force constant tensor are represented as B′′ (un lled circle) K′′, where K′′ is parallel to the x-axis. Using the new bond angles in the cylindrical SWBNNT compared to the planar h-BN sheet, the force constants of the SWBNNT are calculated. Table I shows the modi ed force constants.
The FOT was used to estimate the eigenstate and localization phenomena of phonons for defective SWBNNT. This method has the advantage of being able to determine the phonon characteristics of any disordered system over the whole frequency range using only the values of atomic mass and force constants. The principal idea of the FOT comes from the resonance of a mechanical system upon application of excitation. A resonant state with frequency Ω can be acquired in a lattice dynamical system if an uninterrupted excitation having an exterior continuous periodic force of frequency Ω is imparted to the system for a long time. If M l and u l , respectively, are the mass and displacement of the l th atom of a lattice dynamical system consisting of N atoms, the equation of motion of the system can be described as where the spring constant among the l th atom and l 'th atom is ϕ ll ' and F l is the continuous periodic exterior force. The mathematical expression of F l is: F l = F 0 M l cos φ l . Here F 0 is a timeindependent constant and φ l is an arbitrary number between 0 to 2π. We can write the displacement u l of the system as where Q λ corresponds to a normal mode, Q λ (t) and e l (λ) are the magnitude and displacement values of here C stands for a constant. Therefore, the spatial eigenvector for the mode λ 1 can be attained. Figure 3 shows the estimated PDOS of 10 B isotope-containing (0-100%) SWBNNT. As expected, Van Hove singularities appear in the estimated PDOS caused by the con nement effect of the reduced dimensionality of SWBNNT. The PDOS successfully generates all the distinctive peaks associated with the sp 2 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 E 2g peak in PDOS is Raman active in armchair edged BNNT, similar to 2D h-BN; however, the A 1 peak is Raman active in zigzag edged BNNT 21,22,24,27,29 . We observe the A 1 peak at 1376 cm −1 in our simulations, smaller than the value (E 2g ) obtained for the 2D h-BN 38 due to the curvature effect 18 . The curvature effect of BNNT softens this Raman active mode frequency to the lower value, which is consistent with earlier results 39 .

Results And Discussion
A general trend of moving all vibrational modes towards the higher energy region is found when the 10 B isotope level increases. The A 1 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 lowfrequency regime 19,39,40 . 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 10 B isotope-containing BNNT in earlier stusy 41 . The frequency of vibrational mode ω is related to the atomic mass M as ω ∝ 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: Here, ω 11 B and x signify the frequency of original BNNT and the 10 B isotope contents, respectively. The upward shift of the A 1 peak as a function of 10 B 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 re ect our computational results. We discover that our computational results and theoretical values are perfectly consistent. From 0-100% 10 B isotope contents, the A 1 mode shifts upward by about 42 cm −1 . The system contains the same mass for both 0% and 100% 10 B 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 11 B and 10 B atoms. With the increase of 10 B 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 parameters 42 . Because of signi cant changes in the phonon structure, the effect of vacancy defects on thermal conductivity is more noticeable than that of isotopes. An earlier study 43 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 CNT [44][45][46] manifested that 1-2% vacancy concentration resulted in about a 50% reduction in the thermal conductivity. Again, some studies on silicene nanoribbons 47  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  al. 53 showed that the average PDOS of graphene reduces due to the stiffening of the exural phonon modes caused by the disorder-induced strain. Jeon et al. 24 found exure phonon modes in BNNT even though the constituting atoms (B and N) are polar in nature. We believe that vacancies stiffen the exural 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 signi cant downward shift than the mixing defects. As shown in Figure 7, the softening of the A 1 peak is highly in uenced 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 13 C enriched and the exterior wall was natural carbon (i.e., 1.1% 13 C, 98.9% 12 C) enriched along with inevitable vacancies. The 13 C 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 identi ed outside of the ideal system's normal frequency range 33,34 . In a defect-free environment, phonons are unrestricted to move, however, in the presence of isotopes or vacancies, phonons become con ned. 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 electrons 32 . 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 bene cial in a variety of applications, including thermoelectric energy conversion and microelectronic heat management [55][56][57] . The normal linewidth of the Raman spectra is also caused by localized phonons 27 . To explain the experimental ndings, 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 con ned 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 con ned. There have been numerous attempts to estimate and display the phonon localization events in isotope defected nanostructures. Savic et al. 6 performed a rst-principles analysis to investigate phonon transmission in 14 C isotope enriched CNT and 10 B 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 signi cant 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 10 B 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. However, to achieve a high coherent backscattering effect, scattering events should be entirely elastic 60 . As a result, eigenmodes in the sample may appear at random.
The presence of the 10 B 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 10 B isotope and B or N vacancy defected sample. Few modes are observed to be con ned 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 con ned. Only a few atoms vibrating with the highest amplitude are discovered concentrated near the vacancy defects, indicating a signi cant 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 ndings 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: where u l , λ 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/ √ 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/ √ N should be noticed. We calculated the localization L λ length from the value of IPR, which is correlated to the IPR as L λ ∝ IPR − 1 2 , 61 and the L λ of the state, λ can be stated as where L 0 is the unfolded SWBNNT size, i.e., the 2D h-BN sheet and IPR 0 is the average value of IPR λ . Figure 9 shows the uctuation of the L λ with the variation of isotope concentration. The simulation was run with 5000 atoms at = 1376 cm −1 for the A 1 mode phonon. The lled circles in Fig. 9 denote the average of ten eigenmodes. The L λ is associated to the IPR λ as L λ ∞IPR − 1 / 2 λ . 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 nding 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 in uence of decreasing the mass of a system (by adding 10 B isotopes to a 11 B lattice) differs from that of increasing the system's mass (by adding 11 B isotopes to a 10 B lattice). The L λ is calculated from the IPR, which is derived from the eigenstate amplitudes of atoms. The total mass of the system decreases as 10 B atoms are added to the 11 B lattice. This eigenstate amplitude of the atoms with reduced mass will be larger than the pristine structures' eigenstate amplitude. The same thing happens when 11 B impurities are introduced into a 10 B lattice. In both circumstances, the eigenstates of the pristine structure are smaller than the IPR, causing a lower L λ . 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 at zones of dispersion relation can readily happen as described by Savic et al. 6 . Besides, they 6 also revealed that optical phonons in the high-frequency region are predominantly localized 6 . The defect-related backscattering is considerable in the optical phonons in the high-frequency region, as observed in CNT 63 . 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 λ for different combinations of defects has also been extracted, as shown Figure 10. The values of L λ 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 modi es the force constant parameters. The force constants for the 10 B 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 ∝ m − 1 2 ), hence, changes in 10 B isotope mass can signi cantly modify the PDOS. The isotope mixing with vacancies, however, changes the PDOS more suddenly. Thus, the comparison of the L λ for different impurity concentrations and different types of defects is not straightforward. As a result, there may be a sharp drop in the L λ .

Conclusions
In summary, the FOT was used to investigate the effects of 10 B isotope mixing with B or N point vacancies on SWBNNT phonon modes. The PDOS analysis revealed that the Raman active A 1 phonon peak is highly prone to both 10 B isotope and vacancy disorder. With increasing isotope levels, the A 1 mode exhibits a fairly linear downshift to the low-frequency region. A stronger downshift in the A 1 mode was observed for vacancy defects than for the isotope, and it vanishes when defect concentrations reach a certain level. In the low-energy regime, the mixing defects generate a new sharp peak. We created typical mode patterns for different types of disordered-structure to demonstrate the phonon localization processes caused by the defects. Both 10 B isotope and mixing defects cause high-energy optical phonons to be localized. With increasing 10 B isotope levels, we discovered an asymmetrical character of the localization length, which corresponds well to the isotope inherited localization length of carbon nanotubes and mono-layer graphene. With the inclusion of separate B or N point vacancies as well as mixing defects, the localization length reduces abruptly. This paper gives a theoretical framework for isotope and vacancy-induced phonon properties, which is crucial for understanding SWBNNT thermal conductivity and Raman spectra.           Localization length as a function of 10 B isotope concentration.

Figure 10
Impurity concentration versus localization length of BNNT with a mixture of (a) 10 B isotope and B vacancies and (b) 10 B isotope and N vacancies.

Supplementary Files
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