Localization of the Optical Phonon Modes in Boron Nitride Nanotubes: Mixing Effect of 10B Isotopes and Vacancies.

We explored the mixing effect of 10B isotopes and boron (B) or nitrogen (N) vacancies on the atomic vibrational properties of (10,0) single-wall boron nitride nanotubes (BNNTs). The forced oscillation technique was employed to evaluate the phonon modes for the entire range (0-100%) of 10B isotopes and atomic vacancy densities ranging from 0 to 30%. With increasing isotope densities, we noticed a blue shift of the Raman-active A1 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 A1 peak in the phonon density of states. High-energy optical phonons were localized as a result of both 10B isotopes and the presence of mixing defects. We found an asymmetrical nature of the localization length with increasing 10B isotope content, which corresponds well to the isotope-inherited localization length of carbon nanotubes and monolayer graphene. The localization length falls abruptly with the increase in concentration of both atomic vacancies (B or N) and mixing defects (10B isotope and vacancies). These findings are critical for understanding heat conduction and nanoscopic vibrational investigations such as tip-enhanced Raman spectra in BNNTs, which can map local phonon energies.

Introduction

Isotopes play an essential role in manipulating the physical, chemical, and biological phenomena of a material. Physical phenomena such as nuclear fission or fusion, liquid or solid magnetic resonance, and 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 such as 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 significant impact on the phonon (quanta of lattice vibrations) properties.[3] Especially in disordered nanotubes, the phonons show some peculiar transport characteristics such as 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 (BNNTs) because natural boron (B) possesses a considerable isotopic disorder. Natural BNNTs consist of 80.1% 11B and 19.9% 10B. Chang et al.[5] found a 50% enhancement of thermal conductivity at room temperature with 99.5% 11B isotope in individual isolated multiwalled BNNTs using a microfabricated test fixture with the facility of high-resolution transmission electron microscopy. Savić et al.[6] found dominant effects on the thermal conductivity reduction with isotope disorder in BNNTs 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 BNNTs, the same group developed an independent cascade scatter model of BNNTs. They modeled thermal transport in BNNTs through the atomistic Green function approach combined with first-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 first 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 methods[12,13] are also used for synthesizing BNNTs. During these synthesis processes of BNNTs, there will be the introduction of ubiquitous atomic vacancies, similar to their two-dimensional counterpart.[14] Moreover, intentional atomic vacancy creation by irradiation was utilized to recognize the chirality of an independent wall in multiwalled BNNTs 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 BNNTs. Thus, atomic vacancies should have significant effects on the phonon properties of BNNTs.

As BNNTs are structural analogues of carbon nanotubes (CNTs), 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 significantly impact the electron–phonon interaction in BNNTs,[20] leading to the possible alteration of electron transport properties. Thus, an in-depth analysis of individual isotopes and vacancies, as well as combinations of these, in BNNTs, is indispensable. The phonon properties of pristine BNNTs have been explored in the 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 infrared spectroscopy.[30] Even though the phonon properties in disordered BNNTs 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 BNNTs 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 BNNTs. 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 confined 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 BNNTs 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 to its length and width, respectively, were multiplied to determine the unit cell[35] of single-walled BNNTs (SWBNNTs). Figure shows the isotope and vacancy-defected ball–stick model and the Brillouin zone (BZ) of (10,0) SWBNNTs. The unit cell of a (10,0) SWBNNT consists of 20 B and 20 N atoms.

Figure 1

(a) Ball and stick model of isotope and vacancy-defected SWBNNTs. Copper-blue balls are N atoms, rosy-pink balls are 11B atoms, and light-black balls are 10B atoms. (b) BZ of the (10,0) zigzag SWBNNTs with high-symmetry points. The BZ consists of N number of straight lines through the k space of h-BN, each with a length of 2π/T, where T and N represent the BNNT unit cell length and the number of h-BN unit cells in a single BNNT unit cell, respectively. Here, T and N are 0.435 nm and 20, respectively. k and kθ are the wave vectors along the tube axis and perpendicular to it, respectively.

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 10B 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 SWBNNTs and the h-BN planar sheet are not the same. We have corrected the force constants of the SWBNNT by calculating the changes in the bond angles of the cylindrical tube relative to the planar h-BN sheet. The approach to achieve the curvature effect is depicted in Figure a,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′ (unfilled circle) and K′, respectively, as shown in Figure a. K and K′ rotate around the y-axis at an angle of ϕ/2, where ϕ denotes the central angle between A and B. The projection of B′ on the zx-plane and its force constant tensor are represented as B″ (unfilled circle) K″, where K″ is parallel to the x-axis. Using the new bond angles in the cylindrical SWBNNTs compared to the planar h-BN sheet, the force constants of the SWBNNTs are calculated. Table shows the modified force constants.

Figure 2

(a) View of SWBNNTs from the top. The projection of the B atom (filled circle) on the xy-plane is denoted as (unfilled circle). (b) Side view of the SWBNNT. The projection of B′ (unfilled circle) on the zx-plane is denoted as B″ (unfilled circle).

Table 1

Corrected Force Constant Parameters for (10,0) SWBNNTs in Units of 104 dyn/cm by Adding the Bending Effecta

force constant parameters	ref.[37]	corrected force constants for (10,0) SWBNNT
φr(1)	31.00	30.61557
φr(2B–B)	7.00	6.737185
φr(2N–N)	8.00	7.69964
φr(3)	1.00	0.949699
φr(4)	–1.90	–1.72893
φti(1)	18.50	18.27058
φti(2B–B)	–3.23	–3.10873
φti(2N–N)	–0.73	–0.702592
φti(3)	–3.25	–3.08652
φti(4)	1.29	1.17385
φto(1)	5.60	5.530554
φto(2B–B)	–0.70	–0.673719
φto(2N–N)	–0.55	–0.52935
φto(3)	0.65	0.6173044
φto(4)	–0.30	–0.272988

Here, r, ti, and to represent the radial, transverse in-plane, and transverse out-of-plane, respectively.

The FOT was used to estimate the eigenstate and localization phenomena of phonons for defective SWBNNTs. Usually, a large system is required to quantify the defect effects precisely. However, the dynamical matrix technique or first-principles calculations are limited by a small-scale system. Moreover, it is also challenging to extract the low-frequency modes by these techniques. The main advantage of the FOT is that if we know only the force constants and atomic mass of the system, we can determine the phonon characteristics of any disordered, complex, and large system over the whole frequency range by calculating the total energy of the system without knowing the phonon lifetime.

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 and u, respectively, are the mass and displacement of the lth atom of a lattice dynamical system consisting of N atoms, the equation of motion of the system can be described aswhere the spring constant among the lth atom and l′th atom is φ and F is the continuous periodic exterior force. The mathematical expression of F is . Here, F0 is a time-independent constant and ϕ is an arbitrary number between 0 to 2π. We can write the displacement u of the system aswhere Qλ corresponds to a normal mode and Qλ (t) and e (λ) are the magnitude and displacement values of λ, respectively. The average value of E can be obtained by averaging all possible ϕ values as

The modes whose frequencies fall within the range about ∓(2π/t) of Ω contribute to the sum in eq . t is chosen in such a way that Ωt ≫ 1, which means that the sum in eq is contributed by only a small range of frequency modes in the Ω scale. The number of such modes are greater than unity. Thus, the sum in eq can be represented as a delta (δ) function, which provides the average energy aswhere g(Ω) signifies the partial density of states (PDOS). Hence, the task is reduced to obtaining the time evolution of the equations of motion upon application of an external force. If time t is discretized with a step τ, the equations of motion can be expressed aswhere n is the number of time steps defined as t = nτ. The step size τ should be small enough to get a fine resolution. Here, τ is chosen as 0.01 × 10–13. Usually, typical atomic vibrational frequencies lie within the range of approximately 1012 to 1014 Hz. Thus, the frequency of the external applied force is chosen as 1013 Hz to extract each frequency mode of the system. The system reaches the resonant condition after a long period of external excitation. Under the resonant condition, the average total energy of the system is calculated from which the PDOS is calculated as

The eigenvector of the disordered SWBNNT is also calculated. After driving the system for a certain reasonable time period T, u(1) becomes the new displacement of an atom l. Following that time period, a new force with a new value is assigned to atom l and is written as F(1) = u(1)M. The eventual displacement after p iterations can be represented ashere, C stands for a constant. Therefore, the spatial eigenvector for the mode λ1 can be attained.

Results and Discussion

Figure shows the estimated PDOS of 10B isotope-containing (0–100%) SWBNNTs. As expected, Van Hove singularities appear in the estimated PDOS caused by the confinement effect of the reduced dimensionality of SWBNNTs. 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 BNNTs, similar to 2D h-BN; however, the A1 peak is Raman-active in zigzag-edged BNNTs.[21,22,24,27,29] We observe the A1 peak at 1376 cm–1 in our simulations to be smaller than the value (E2g) obtained for 2D h-BN[38] due to the curvature effect.[18] The curvature effect of BNNTs softens this Raman-active mode frequency to the lower value, which is consistent with the earlier results.[39]

Figure 3

PDOS of BNNTs with various 10B and 11B isotope concentrations.

A general trend of moving all vibrational modes toward 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.[40,41] However, the effect was negligible in the low-frequency regime.[19,39] Usually, the phonon lifetime for the low-frequency phonon modes for carbon-based and similar BN structures shows power law behavior. Moreover, in-plane and out-of-plane phonon modes have vastly different lifetime scales because of their vastly different stiffness. Consequently, isotope impurities affect the high-frequency phonon modes sensitively but not the low-frequency modes for these types of structures. Prior studies also revealed that the incorporation of heavier isotope atoms causes a significant downward shift of the high-frequency phonon mode, whereas the impact is weaker in the low-frequency regime.[42,43] Isotopic substitution decreases the average atomic mass of BNNTs, and even a minor amount of isotope in BNNTs can drastically alter the phonon characteristics. An upward shift of the Raman-active phonon modes was also detected in 10B isotope-containing BNNTs in the earlier study.[44] 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 asHere, ω11 and x signify the frequency of original BNNTs and the 10B isotope contents, respectively. The upward shift of the A1 peak as a function of 10B isotope concentration is depicted in Figure . A comparison of the values obtained from eq 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 to 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.

Figure 4

10B isotope concentration vs Raman-active A1 mode frequency. The solid line is extracted from the simple harmonic oscillator model described in eq .

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 CNTs in previous studies. The bond length and energy can be altered by vacancies, and the vibrational frequency is highly reliant on these parameters.[45] Because of significant changes in the phonon structure,[46−48] the effect of vacancy defects on thermal conductivity is more noticeable than that of isotopes. An earlier study[49] showed that the thermal conductivity of a graphene nanoribbon (GNR) was reduced by 81% even at a relatively low concentration of vacancies (0.1%). Different studies on CNTs[50−52] manifested that 1–2% vacancy concentration resulted in about a 50% reduction in the thermal conductivity.

Again, some studies on silicene nanoribbons[53] and silicon nanotubes[54] found around a 50% reduction in thermal conductivity with 1–2% atomic vacancy concentration. Thus, the effects of vacancies on the phonon characteristics of BNNTs are exciting. The impact of various concentrations of B and N vacancies on the PDOS of BNNTs is presented in Figure a,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.[55] demonstrated that phonons in 2D materials are dispersed greatly by atomic vacancies. Vacancy disorders 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.

Figure 5

PDOS values of SWBNNTs with (a) 11B vacancies and (b) N vacancies.

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 . The concentrations of unsaturated bonds of B and N atoms increase with increasing vacancy concentration, which may cause a decrease in the high-frequency phonon density. As a result, phonon modes migrate toward the low-energy region. Phonon scattering in the low-frequency region due to the defects was notable in CNTs, as demonstrated by Mingo et al.[56] In addition to generating a substantial peak, increasing defect density also lowers the average PDOS in the low-frequency zone. Mahan et al.[57] discovered the low-frequency flexural phonon modes with a quadratic dispersion in the PDOS of CNTs. On the other hand, Mariani et al.[58] and Ochoa et al.[59] 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 BNNTs 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 . 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 BNNTs, however, induces a more significant downward shift than the mixing defects. As shown in Figure , 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.[60] 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 and 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 range.[33,34] In a defect-free environment, phonons are unrestricted to move; however, in the presence of isotopes or vacancies, phonons become confined. In disordered systems, 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 beneficial in a variety of applications, including thermoelectric energy conversion and microelectronic heat management.[61−63] The normal linewidth of the Raman spectra is also caused by localized phonons.[27] To explain the experimental findings, a better understanding of isotope phonon localization and vacancy-induced BNNTs is required.

Figure 6

PDOS values of BNNTs with a (a) mixture of 10B isotope and B vacancies and a (b) mixture of 10B isotope and N vacancies.

Figure 7

Impurity concentration vs A1 mode frequency.

The atomic-scale localization behavior of phonon modes in SWBNNTs 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. Savić et al.[6] performed a first-principles analysis to investigate phonon transmission in 14C isotope-enriched CNTs and 10B isotope-disordered BNNTs, claiming that the thermal conductivity decrease in isotope-disordered CNTs and BNNTs is due to diffusive scattering, whereas localization of phonons was 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.[64] examined the heat transport of isotope-disordered GNRs and CNTs. They discovered that the loss in thermal conductivity in CNTs 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 isotopes and vacancies, as well as their independent effect on the phonon localization in unfolded SWBNNTs, has been explored here. We are particularly interested in high-frequency tangential phonon modes.

Figure 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 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 BNNTs is shown in Figure a.

Figure 8

Typical mode patterns at ω = 1376 cm–1 for unfolded SWBNNTs with (a) pristine, (b) 20% B vacancy, (c) 20% N vacancy, (d) 20% 10B isotope, 20% combined (e) 10% 10B isotope and 10% B vacancies, and (f) 10% 10B isotope and 10% N vacancy-defected structures. Each circle denotes an atom (smaller circles are 10B atoms and larger circles are 11B atoms), and the colors denote the displacement, which is linearly normalized by the maximum value. The intensity of the displacement of each atom is presented by the color bar.

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 inelastic.[65] 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 elastic.[66] 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 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 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 for 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 waywhere u represents the eigenmode’s displacement of the lth 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 length Lλ from the value of IPR, which is correlated with the IPR as Lλ ∝ IPR–1/2,[67] and the Lλ of the state λ can be described aswhere L0 is the unfolded SWBNNT size, that is, the 2D h-BN sheet, and IPR0 is the average value of IPRλ.

Figure shows the fluctuation of Lλ 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 Figure denote the average of 10 eigenmodes. Lλ is associated with IPRλ as Lλ ∞ IPRλ–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 to 60% and then rise again from 60 to 100%. At a 60% isotope content, the minimum localization length was discovered.

Figure 9

Localization length as a function of 10B isotope concentration.

Despite the asymmetrical behavior, this finding is consistent with the earlier work on single-layer graphene by Rodriguez-Nieva et al.[43] 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). Lλ 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 Lλ. 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 Savić 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 CNTs.[68] An island with a different frequency is developed for the localized eigenstate compared to the rest of the atoms, which may be the physical origin of localization.

The Lλ value for different combinations of defects has also been extracted, as shown in Figure . The values of Lλ decay noticeably with the rise of mixing defects or separate B or N point vacancies, as shown in Figure . For the same defect density, the combined defects demonstrate a sudden decrease in behavior in comparison to isotope disorder. This is because the bond length and energy change 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 to the system mass m as the inverse of the square root (i.e., f ∝ m–1/2), 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 Lλ for different impurity concentrations and different types of defects is not straightforward. As a result, there may be a sharp drop in Lλ.

Figure 10

Impurity concentration vs localization length of BNNTs with a mixture of 10B isotope and (a) B and (b) N vacancies.

Conclusions

In summary, the FOT was used to investigate the effects of 10B isotope mixing with B or N point vacancies on SWBNNT phonon modes. The PDOS analysis revealed that the Raman- active A1 phonon peak is highly prone to both 10B isotope and vacancy disorders. With increasing isotope levels, the A1 mode exhibited a fairly linear downshift to the low-frequency region. A stronger downshift in the A1 mode was observed for vacancy defects than for the isotope, and it vanished when defect concentrations reached a certain level. In the low-energy regime, the mixing defects generated a new sharp peak. We created typical mode patterns for different types of disordered structures to demonstrate the phonon localization processes caused by the defects. Both 10B isotope and mixing defects caused high-energy optical phonons to be localized. With increasing 10B isotope levels, we discovered an asymmetrical character of the localization length, which corresponds well to the isotope-inherited localization length of CNTs and monolayer graphene. With the inclusion of separate B or N point vacancies as well as mixing defects, the localization length reduced 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.