# Structure and Raman Spectra of C60 and C70 Fullerenes Encased into Single-Walled Boron Nitride Nanotubes: A Theoretical Study

Brahim Fakrach, Fatima Fergani, M. Boutahir, Abdelhai Rahmani, Hassane Chadli, Patrick Hermet, Abdelali Rahmani

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Brahim Fakrach, Fatima Fergani, M. Boutahir, Abdelhai Rahmani, Hassane Chadli, et al., Structure and Raman Spectra of C60 and C70 Fullerenes Encased into Single-Walled Boron Nitride Nanotubes: A Theoretical Study. Crystals, 2018, 8 (3), pp.118. 10.3390/cryst8030118. hal-01722985

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Article

# Structure and Raman Spectra of C₆₀ and C₇₀ Fullerenes Encased into Single-Walled Boron Nitride Nanotubes: A Theoretical Study

Brahim Fakrach<fnref n="1" />, Fatima Fergani<fnref n="1" />, Mourad Boutahir<fnref n="1" />, Abdelhai Rahmani<fnref n="1" /> ID,
Hassane Chadli <fnref n="1" />,<fnref n="2" />, Patrick Hermet<fnref n="3" /> and Abdelali Rahmani <fnref n="1" />,*

Received: 26 January 2018; Accepted: 27 February 2018; Published: 3 March 2018

**Abstract:** We report the structures and the nonresonant Raman spectra of hybrid systems composed of carbon fullerenes (C₆₀ and C₇₀) encased within single walled boron nitride nanotube. The optimal structure of these systems are derived from total energy minimization using a convenient Lennard-Jones expression of the van der Waals intermolecular potential. The Raman spectra have been calculated as a function of nanotube diameter and fullerene concentration using the bond polarizability model combined with the spectral moment method. These results should be useful for the interpretation of the experimental Raman spectra of boron nitride nanotubes encasing C₆₀ and C₇₀ fullerenes.

**Keywords:** BN nanotube; fullerene; peapod; Raman spectroscopy; spectral moment's method; structure

**1. Introduction**

The discovery of nanomaterials such as fullerenes [1], carbon nanotubes [2] and boron nitride nanotubes [3] prevails over science and technology at nanoscale levels and becomes a hot topic for researchers and industrialists. Single walled carbon nanotubes (SWCNTs) filled with C₆₀ fullerenes are commonly called carbon peapods (C₆₀@SWCNT). They have been firstly synthesized by Smith et al. in 1998 [4] and exhibit special physical properties with different potential applications [5–11].

Similar to SWCNTs, single walled boron nitride nanotubes (SWBNNTs) can also be filled with fullerene molecules such as C₆₀ and C₇₀: they are so-called boron nitride peapods (Cₕ@SWBNNT where n = 60 or 70). The C₆₀@SWBNNT system was first theoretically proposed by Okada et al. [12] and experimentally synthesized two years later [13,14]. This system has aroused great interest in BN peapods as an ideal model to study the atoms or molecules in 1D confined conditions [13,15]. Due to its unusual electronic structure, the C₆₀@SWBNNT is an interesting candidate to study a possible superconducting state.

It was theoretically stated [15–17] that (i) C₆₀ inside SWBNNT is energetically more stable and achieved faster than C₆₀ inside SWCNT and (ii) the suction force on the C₆₀ induced by SWBNNTs is higher than that induced by the SWCNTs [12,16]. The C₆₀-C₆₀ intermolecular distance inside the SWBNNT is 0.9 nm [13,18], which is slightly smaller than the 1.0 nm intermolecular distance for

<footnote n="1">¹ Laboratoire d'Etude des Matériaux Avancés et Applications (LEM2A), Université Moulay Ismaïl, FSM-ESTK-ESTM, BP 11201, Zitoune, 50000 Meknes, Morocco; fakrach@gmail.com (B.F.); fati.fergani@gmail.com (F.F.); mourad.boutahir@gmail.com (M.B.); rahmani614@gmail.com (A.R.); h.chadli@fpe.umi.ac.ma (H.C.)</footnote>

<footnote n="2">² Equipe Physique Informatique et Modélisation des Procédés, Ecole Superieur de Technologie BP 170 ,
54000 Khénifra, Morocco</footnote>

<footnote n="3">³ Institut Charles Gerhardt Montpellier, UMR-5253, CNRS, Université de Montpellier, ENSCM, Place E. Bataillon, 34095 Montpellier, France; patrick.hermet@unmontpellier.fr</footnote>

<footnote>* Correspondence: a.rahmani@fs-umi.ac.ma</footnote>
carbon peapods [5,19]. This difference could be due to a stronger van der Waals attraction between the SWBNNT and the $C_{60}$ or possibly to commensurability effects [13,18]. Moon et al. [20] have demonstrated from molecular dynamics simulation that the (10,10) and (17, 0) SWBNNTs are the most favorable for encapsulation of $C_{60}$ molecules. Mickelson et al. [13] have shown that the $C_{60}$'s structure inside SWBNNTs is dependent of the tube diameter, as in the case of $C_{60}$@SWCNT [21–24]. Compared to other nanostructures, nanopeapods have opened new applications such as storage materials with high capacity and stability [10,25] and considered to be potential materials for high-temperature superconductors [11]. The space inside the nanotube can be also regarded as a nanometer-sized container for chemical reactions [7].

Raman spectroscopy is the standard characterization technique for nanotubes. Raman spectrum of $C_{60}$@SWBNNT have been reported in the literature at different laser energies (488, 568 and 647 nm) [14]. All fundamental Raman lines of the encapsulated $C_{60}$ peas are observed and a structured peak at 268, 330, 355 and 466 cm$^{-1}$ is measured and assigned to the reaction products of the carbon fullerenes inside the BN nanotubes. The authors show that the radial breathing mode range reveals new spectral features which are not observed in the case of unfilled BN tubes.

The experimental Raman spectra measured on BN peapods are complex to analyze because experiments performed on macroscopic samples consist of a mixture of bundles of tubes of different diameters and chiralities. In this paper, in order to improve the comparison between the calculations and experimental data, we investigate theoretically the fullerene-fullerene and fullerene-tube interactions effect on the vibrations of $C_{60}$ and $C_{70}$ BN peapods. In contrast with $C_{60}$ molecules, the ellipsoidal shape of $C_{70}$ leads both the $C_{70}$-$C_{70}$ and $C_{70}$-SWBNNT interactions to be anisotropic. As a consequence, the electronic structure and the ground state energy of fullerene peapods depend on the configuration of the fullerene molecules inside the nanotubes. The analysis of the Raman spectra of peapods suggests that the observed shifts of the radial breathing mode (RBM) could be explained by a small increase of the tube diameter, resulting from the interaction between $\pi$-electrons of the fullerene with the interior of the nanotube [26].

In previous works [23,27,28], we studied the different possible configurations of $C_{60}$ and $C_{70}$ molecules encapsulated into SWCNTs with diameter lower than 2.28 nm. Raman spectra of $C_{60}$ carbon peapods have been calculated for $C_{60}$ linear, zigzag, double helix and layer of two molecules configurations within the bond-polarizability model. Our results were in qualitative agreement with the experimental ones. Here, we continue these studies in the case where the $C_{60}$ and $C_{70}$ molecules are now encapsulated into SWBNNTs. Our work addresses new questions as to the influence of the nanotube diameter and the $C_{60}$ and $C_{0}$ filling rate. In this context, we use a direct diagonalization of the dynamical matrix for small peapods (few hundred atoms) and the spectral moment method for larger ones. We report the structural organization of the $C_{60}$@SWBNNT and $C_{70}$@SWBNNT as a function of the diameter and the chirality of the BN nanotubes and we discuss the change in the Raman response of these hybrid systems including the influence of the fullerene concentration.

## 2. Models and Simulation Method

### 2.1. *Structure and Dynamics of C<sub>60</sub>@SWBNNT and C<sub>70</sub>@SWBNNT Peapods*

SWBNNTs are obtained by rolling a single hexagonal boron nitride sheet [29,30] and can be specified by integers $(n, m)$ which define the translation vector between two equivalent points. A BN peapod consists of fullerene molecules trapped inside a SWBNNT host. Depending on the nanotube diameter, different configurations of $C_{60}$ and $C_{70}$ can certainly exist. Here, we restrict our study by considering only a linear chain of $C_{60}$ and $C_{70}$ confined inside SWBNNT. Considering a (10,10) SWBNNT, Figure 1 shows the structures of the encapsulated $C_{60}$ and $C_{70}$. The configuration of the guest $C_{60}$ and $C_{70}$ molecules inside SWBNNTs is derived from total energy minimizations according to the procedure described in our previous works on carbon peapods [23].
<figure><img src="image_2.png" /><figcaption>Figure 1. Structure of two SWBNNT peapods: (a) C$_{60}$@(10,10) and (b) C$_{70}$@(10,10).</figcaption></figure>

The C-C interaction between carbon atoms belonging to non-bonded fullerenes, and between fullerenes and the surrounding nanotube, are represented by the Lennard-Jones potential, $U_{LJ}(r) = 4\epsilon[(\frac{\sigma}{r})^{12} - (\frac{\sigma}{r})^{6}]$. For boron, nitrogen and carbon atoms, the $\epsilon_{ii}$-parameter is 0.004116, 0.006281 and 0.002635 eV, whereas the $\sigma_{ii}$-parameter is 0.3453, 0.3365 and 0.3369 nm, respectively [16,31]. The parameters $\epsilon_{ij}$ and $\sigma_{ij}$ between different atoms are calculated by the following Lorentz-Berthelot rules: $\epsilon_{ij} = \sqrt{\epsilon_i\epsilon_j}$ and $\sigma_{ij} = (\sigma_i + \sigma_j)/2$.

The intratube interactions at the surface of the SWBNNT are described by the same force constant model we recently used in the calculations of the infrared spectra of single walled boron nitride nanotubes [32]. The C-C intramolecular interactions between carbon atoms at the surface of C$_{60}$ molecules are modelized by the force constants model described by Jishi and Dresselhaus [33]. Interactions up to the fourth nearest neighbors are accounted to derive the dynamical matrix. The dynamical matrix of free C$_{70}$ molecule was calculated using the density functional theory (DFT) as implemented inside the SIESTA package [34]. The dynamical matrix (leading to the frequency of the Raman lines) is calculated by block using the previous intramolecular potentials and van der Waals potential (i.e., fullerene-SWBNNT and fullerene-fullerene interactions).

## 2.2. Methods

The time-averaged power flux of the Raman scattered light in a given direction, with a frequency between $\omega_d$ and $\omega_d + d\omega_d$ within a solid angle $d\Omega$, is related to the differential scattering cross section:

$$ \frac{d^2S}{d\Omega d\omega_d} = \frac{\omega_d^4}{16\pi^2c^4} [B(\omega) + 1] \hbar \sum_{ijkl} v_i v_k I_{ijkl}(\omega) w_j w_l \quad (1) $$
where

$$I_{ijkl}(\omega) = \sum_m a_{ij}^*(m) a_{kl}(m) \frac{1}{2\omega_m} [\delta(\omega - \omega_m) - \delta(\omega + \omega_m)] \quad (2)$$

and $\omega = \omega_0 - \omega_d$ is the Raman shift. In these equations, $(i,j,k,l)$-indices denote the Cartesian components, the star symbolizes the complex conjugation, $c$ is the speed of light in the medium, $\hbar$ is the reduced Planck constant, $\omega_0$ (resp. $\omega_d$) is the frequency of incident (resp. scattered) light, $v$ (resp. $w$) is the polarization unit vector of the incident (resp. scattered) light, $B(\omega)$ is the Bose factor, and $\omega_m$ is the frequency of the $m$th zone-center phonon mode. The Raman susceptibility tensor is defined as,

$$a_{ij}(m) = \sqrt{\Omega_0} \sum_{\kappa\gamma} \pi_{ij,\gamma}^{\kappa} u_m(\kappa\gamma) \quad (3)$$

where the sum runs over all atoms $\kappa$ and space directions $\gamma$, $\Omega_0$ is the unit cell volume, $u_m(\kappa\gamma)$ is the $(\kappa\gamma)$ component of the $m$th phonon eigendisplacement vector, and $\tilde{\pi}$ is a third-rank tensor describing the changes of the first-order optical dielectric susceptibility ($\tilde{\chi}^{\infty}$) induced by individual atomic displacements. This latter quantity is defined as,

$$\pi_{ij,\gamma}^{\kappa} = \frac{\partial \chi_{ij}^{\infty}}{\partial \tau_{\kappa\gamma}} \quad (4)$$

where $\tau_{\kappa\gamma}$ corresponds to the displacement of the $\kappa^{th}$ atom in the direction $\gamma$. As long as the phonon frequencies and eigendisplacements are known, $\tilde{\pi}$ is the central quantity to be determined to estimate Raman intensities. The first term of Equation (2), associated with the creation of a vibrational quantum, describes the Stokes scattering with a downshift in frequency ($\omega > 0$). The second term, related to an annihilation of a vibrational quantum, describes the anti-Stokes scattering with an upshift frequency ($\omega < 0$). The $m$th mode is Raman active if one component of the Raman susceptibility tensor, $a_{ij}(m)$, is nonnull.

The Raman intensities can be calculated within the framework of the nonresonant bond polarizability model [35]. In this model, we consider that the optical dielectric susceptibility ($\tilde{\chi}^{\infty}$) of the crystal can be decomposed into individual contributions, arising only from the polarizability ($\tilde{\alpha}^b$) of bonds $b$ between nearest-neighbor atoms,

$$\chi_{ij}^{\infty} = \frac{1}{\Omega_0} \sum_b a_{ij}^b \quad (5)$$

and the polarizability of a particular bond $b$ is assumed to be given by the empirical equation:

$$a_{ij}^b = \frac{1}{3}(\alpha_l + 2\alpha_p)\delta_{ij} + (\alpha_l - \alpha_p)(\hat{r}_i\hat{r}_j - \frac{1}{3}\delta_{ij})\hat{r}_\gamma \quad (6)$$

where $\hat{r}$ is the unit vector along the bond $b$. The parameters $\alpha_l$ and $\alpha_p$ correspond to the longitudinal and perpendicular bond polarizability, respectively. A further assumption is that the parameters of this model are functions of the bond lengths $r$ only, so that the derivative of the polarizability tensor with respect to the displacement of atom $\kappa$ in direction $\gamma$ is given by:

$$\frac{\partial \alpha_{ij}^b}{\partial \tau_{\kappa\gamma}} = \frac{1}{3}(2\alpha'_p + \alpha'_l)\delta_{ij}\hat{r}_\gamma + (\alpha'_l - \alpha'_p)(\hat{r}_i\hat{r}_j - \frac{1}{3}\delta_{ij})\hat{r}_\gamma + \frac{\alpha_l - \alpha_p}{r}(\delta_{i\gamma}\hat{r}_j + \delta_{j\gamma}\hat{r}_i - 2\hat{r}_i\hat{r}_j\hat{r}_\gamma) \quad (7)$$

where $\alpha'_l = (\frac{\partial \alpha_l}{\partial r})_{r=r_0}$, $\alpha'_p = (\frac{\partial \alpha_p}{\partial r})_{r=r_0}$ and $r_0$ is the equilibrium bond distance. The values of these $\alpha'$-parameters are usually fitted with respect to the experiments [35].

Raman spectra are calculated using a direct diagonalization of the dynamical matrix for small samples (few hundreds atoms). When the system contains a large number of atoms, the dynamical matrix is very large and its diagonalization fails or requires long computing time. In contrast,
the spectral moments method allows to compute directly the Raman spectrum of very large harmonic systems without any diagonalization of the dynamical matrix [36–39].

In all our calculations, the nanotube axis is along the Z-axis and a carbon atom is along the X-axis of the nanotube reference frame. The laser beam is kept along the Y-axis of the reference frame. We consider that both incident and scattered polarizations are along the Z-axis to calculate the ZZ-polarized spectra.

### 3. Results and Discussion

#### 3.1. Optimized Structure of C₆₀@SWBNNT and C₇₀@SWBNNT

The optimal configuration of the inserted fullerene molecules inside the tube is calculated by minimizing the energy of the C₆₀–SWBNNT and C₆₀–C₆₀ interactions described by the Lennard-Jones potential. Results are listed in Table 1 for tubes whose diameter is between 1.30 and 1.45 nm. The linear chain is the optimal configuration for the tubes having a diameter $D_{tube} = 2(R + d)$, where R is the C₆₀ fullerene radius and d is the interlayer C₆₀-SWBNNT distance. The optimal C₆₀–C₆₀ gap inside SWBNNTs is calculated around 1.0 ± 0.05 nm for all optimized peapods. This value is larger than the one observed by Zettl et al. [18] (≈0.89 nm). This difference could be due to a stronger van der Waals attraction between the nanotube and the C₆₀.

In contrast with C₆₀ molecules, the ellipsoidal shape of C₇₀ leads both the C₇₀–C₇₀ and C₇₀-SWBNNT interactions to be anisotropic. In our calculations, we considered a number of C₇₀ molecules initially placed such that their long axis was parallel to the nanotubes axis. The energy minimization of the different C₇₀@SWBNNTs is performed for any molecular orientation characterized by three Euler angles (following the terminology used by Bradley et al. [40]). Only the rotation angle $0 ≤ θ ≤ 90°$ over the tube-axis can vary during our structural relaxation procedures. Thus, the optimum C₇₀ packing can be characterized by the degree of inclination $θ$: $θ = 0°$ for lying orientation, $θ = 90°$ for standing orientation and $0 < θ < 90°$ for tilted orientation. Our calculations show that this angle increases with the diameter of nanotubes (see Table 2) and no correlation is clearly observed between this angle and their chirality.

Both for the C₆₀@SWBNNTs and C₇₀@SWBNNTs, the interlayer C₆₀-SWBNNT and C₇₀-SWBNNT distance is between 0.301 and 0.351 nm (see Tables 1 and 2). These values are consistent with the experimental ones [41–46] and quite close to the interplanar distance in a bulk hexagonal-BN.

<table><caption>**Table 1.** Optimized structural parameters of the C₆₀ molecules inside SWBNNT in diameter range where the C₆₀ molecules adopt a linear chain inside nanotubes.</caption><thead><tr><th>SWBNNT<br>Index<br>(n,m)</th><th>SWBNNT<br>Diameter<br>(nm)</th><th>SWBNNT<br>Chirality<br>(deg)</th><th>C<sub>60</sub>-SWBNNT<br>Distance<br>(nm)</th><th>C<sub>60</sub>-C<sub>60</sub><br>Distance<br>(nm)</th></tr></thead><tbody><tr><td>(11,8)</td><td>1.307</td><td>24.80</td><td>0.303</td><td>1.010</td></tr><tr><td>(12,7)</td><td>1.316</td><td>21.36</td><td>0.307</td><td>1.007</td></tr><tr><td>(13,6)</td><td>1.330</td><td>18</td><td>0.314</td><td>1.007</td></tr><tr><td>(17,0)</td><td>1.344</td><td>0</td><td>0.321</td><td>1.010</td></tr><tr><td>(14,5)</td><td>1.351</td><td>14.70</td><td>0.324</td><td>1.000</td></tr><tr><td>(10,10)</td><td>1.370</td><td>30</td><td>0.334</td><td>1.008</td></tr><tr><td>(15,4)</td><td>1.372</td><td>11.52</td><td>0.335</td><td>1.010</td></tr><tr><td>(12,8)</td><td>1.379</td><td>23.41</td><td>0.339</td><td>1.009</td></tr><tr><td>(14,6)</td><td>1.406</td><td>17</td><td>0.351</td><td>1.009</td></tr><tr><td>(18,0)</td><td>1.424</td><td>0</td><td>0.350</td><td>1.003</td></tr><tr><td>(11,10)</td><td>1.439</td><td>28.42</td><td>0.349</td><td>1.005</td></tr><tr><td>(13,8)</td><td>1.452</td><td>22.17</td><td>0.334</td><td>1.012</td></tr></tbody></table>
<table><caption>**Table 2.** Optimized structural parameters of the C<sub>70</sub> molecules inside BNNT in diameter range where the C<sub>70</sub> molecules adopt a linear chain inside nanotubes.</caption><thead><tr><th>SWBNNT<br>Index<br>(n,m)</th><th>SWBNNT<br>Diameter<br>(nm)</th><th>SWBNNT<br>Chirality<br>(deg)</th><th>C<sub>70</sub>-SWBNNT<br>Distance<br>(nm)</th><th>Angle<br>θ<br>(deg)</th><th>C<sub>70</sub>-C<sub>70</sub><br>Distance<br>(nm)</th></tr></thead><tbody><tr><td>(13,6)</td><td>1.330</td><td>18</td><td>0.307</td><td>0</td><td>1.1273</td></tr><tr><td>(17,0)</td><td>1.344</td><td>0</td><td>0.314</td><td>0</td><td>1.1276</td></tr><tr><td>(14,5)</td><td>1.351</td><td>14.70</td><td>0.317</td><td>0</td><td>1.1280</td></tr><tr><td>(10,10)</td><td>1.370</td><td>30</td><td>0.327</td><td>0</td><td>1.1281</td></tr><tr><td>(15,4)</td><td>1.372</td><td>11.52</td><td>0.3028</td><td>0</td><td>1.1280</td></tr><tr><td>(14,6)</td><td>1.406</td><td>17</td><td>0.305</td><td>2</td><td>1.1260</td></tr><tr><td>(18,0)</td><td>1.424</td><td>0</td><td>0.309</td><td>40</td><td>1.1079</td></tr><tr><td>(11,10)</td><td>1.439</td><td>28.42</td><td>0.301</td><td>47</td><td>1.1060</td></tr><tr><td>(18,1)</td><td>1.465</td><td>2.68</td><td>0.304</td><td>65</td><td>1.0821</td></tr><tr><td>(17,3)</td><td>1.478</td><td>8</td><td>0.325</td><td>90</td><td>1.0039</td></tr></tbody></table>

## 3.2. Raman Spectra of Completely Filled Peapods

We study the Raman active modes in C<sub>60</sub>@SWBNNTs and C<sub>70</sub>@SWBNNTs. For this purpose, a peapod filled to saturation has been considered. The calculated ZZ-polarized Raman spectra of C<sub>60</sub>@(10,10) and C<sub>70</sub>@(10,10), with their corresponding unfilled (10,10) armchair SWBNNT (tube diameter close of 1.37 nm), and the unoriented C<sub>60</sub> and C<sub>70</sub> molecules are displayed in Figure 2. Raman lines can be divided into three frequency ranges: (i) below 200 cm<sup>-1</sup> where the breathing-like modes (BLM) dominate, (ii) an intermediate range between 200 and 600 cm<sup>-1</sup>, and (iii) above 1350 cm<sup>-1</sup> where the tangential-like modes (TLM) are located. Infinite peapods have been obtained by applying periodic conditions along the tube axis.

<figure><img src="image_3.png" /><figcaption>**Figure 2.** Calculated ZZ-polarized Raman spectra of infinite C<sub>60</sub>@(10,10) and C<sub>70</sub>@(10,10) peapods. Spectra of infinite empty (10,10) SWBNNT, C<sub>60</sub> and C<sub>70</sub> molecules are also reported for references.</figcaption></figure>
The main modes of the (10,10) SWBNNT, and the C₆₀ and C₇₀ molecules, are almost not affected by the encapsulation in the TLM region. Indeed, in this region, the position and the relative intensity of the Raman lines observed in peapods is a superposition of the Raman lines of each of the sub-systems. For the BLM region, an upshift of the RBM of the tube is observed when the C₆₀ and C₇₀ molecules are encapsulated into the tube. For instance, the RBM of the unfilled (10,10) SWBNNT shifts from 145 cm⁻¹ in empty (10,10) to 148 and 149 cm⁻¹ in completely filled C₆₀@(10,10) and C₇₀@(10,10), respectively. In the two kinds of peapods, these modes are assigned to the radial breathing like mode (RBLM) of the tube where all atoms in the tube and in the surface of fullerene adjacent to the tube move in phase along the radial direction (see Figure 3a,b). In contrast, the atoms in adjacent surfaces of fullerenes move in counter-phase along the axial direction. In the intermediate range between 200 and 600 cm⁻¹, only phonon modes of the C₆₀ and C₇₀ molecules can be observed. The line centered at 492 cm⁻¹ and assigned as the symmetric A<sub>g</sub> mode of C₆₀ shifts to 496 cm⁻¹ in C₆₀@(10,10) peapod. Similarly, the line centered at 453 cm⁻¹ in C₇₀ and assigned as the symmetric A'<sub>1</sub> mode shifts to 457 cm⁻¹ with the encapsulation. These calculated upshifts can be explained by the van der Waal intermolecular interactions acting between the fullerenes and the SWBNNTs. Concerning the atomic motions modified by the encapsulation, we displayed in Figure 3, the eigendisplacement vectors of these two Raman modes obtained in peapods (Figure 3f,g) from the direct diagonalization of the dynamical matrix, together with the RBM of the unfilled (10,10) SWBNNTs (Figure 3d) and the pure C₆₀ (Figure 3c) and C₇₀ (Figure 3e) molecules. We observe in peapods, a counter-phase coupled motion of the breathing modes of the encapsulated molecule and the tube, and all atoms of the C₆₀ and C₇₀ molecules vibrate in-phase in the radial direction (see Figure 3f,g).

In the remainder of this paper, we focus on the low frequency range of the ZZ-polarized Raman spectrum of BN peapods. We investigated the dependence of some specific Raman active modes of infinite peapods as a function of the diameter of the nanotube. We already reported that the RBM is the most influenced mode by the nanotube filling. This mode is also the most appropriate for extracting structural and dynamical information of SWBNNT. A few years ago, we found [32,47], using the bond polarizability model combined with the spectral moments method, a relation for isolated tubes between their diameter *D* and their RBM frequency: $\omega_{RBM} = a/D$, where *a* = 198 nm.cm⁻¹. This relation was in agreement with the different approaches reported in the literature [48–50]. Figure 4 shows the RBM frequency in SWBNNTs and the RBLM frequency in C₆₀@SWBNNTs and C₇₀@SWBNNTs as a function of the inverse tube diameter. For the two kinds of peapods, the evolution of the RBLMs is qualitatively the same as those of the RBM: their frequencies decrease when the tube diameters increase. However, the RBLMs show deviation from the scaling law stated for the RBM frequency and this deviation is even more important when the diameter of the tubes is small (below 1.35 nm). Both for the C₆₀ peapods and C₇₀ peapods, we found that the frequency of the RBLM follows a law in *a/D* + *b* with *a* = 363.6 nm.cm⁻¹ and *b* = −116.7 cm⁻¹ for C₆₀ peapods and *a* = 320.9 nm.cm⁻¹ and *b* = −85 cm⁻¹ for C₇₀ peapods.

### 3.3. Raman Spectra of Incompletely Filled Peapods

In real peapod samples, it is reasonable to consider that all the nanotubes are not completely filled with C₆₀ or C₇₀ and the highest filling rates range from 70 to 90% [51,52]. The exact degrees of filling is still under discussion and ranges from a certain percent to almost 100% filling. In this section, we make the hypothesis of partial filling of the tubes with a quasi infinite long chain. This hypothesis is supported by observations reported in Ref. [53]. Very long tubes are considered (more than 100 cells) and a defined number of molecules C₆₀ and C₇₀ are inserted. To avoid finite size effects, we applied periodic conditions along the tube axis.
<figure><img src="image_4.png" /><figcaption>**Figure 3.** Calculated atomic motions (eigendisplacement vectors) in C<sub>60</sub>, C<sub>70</sub>, (10,10) SWBNNT, C<sub>60</sub>@(10,10) and C<sub>70</sub>@(10,10). Arrows are proportional to the amplitude of the atomic motions.</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>**Figure 4.** Calculated RBLM frequencies (squares) in isolated infinite nanopeapods and RBM (circles) frequencies in infinite isolated SWBNNTs. Only tubes with a diameter below 1.42 and 1.45 nm are considered for C₆₀@SWBNNT and C₇₀@SWBNNT, respectively.</figcaption></figure>

We found that the number of molecules inside the SWBNNT has no significant effect on the Raman spectrum except on the RBLM mode which is the most sensitive to the degree of filling of the SWBNNTs. Figure 5 displays the ZZ-polarized Raman spectra below 600 cm⁻¹ of C₆₀@(10,10) and C₇₀@(10,10) linear peapods as a function of their filling rate. Five values of the filling rate have been considered: F = 20, 40, 60, 80 and 100% corresponding to 2, 4, 8, 12 and 16 encapsulated molecules (C₆₀ or C₇₀). We observe that all Raman lines undergo a frequency upshift more or less significant as the filling level increases. Similarly, the intensity of the C₆₀ and C₇₀ characteristic lines increases. A single RBLM peak characterizes all spectra for F = 20 and 100% of the filling factor. For F = 100% (tubes completely filled) the RBLM peak is located at 148 and 149 for C₆₀@(10,10) and C₇₀@(10,10) respectively and at 145 for F = 20% (low filling level). Let us call ωₕ and ωᵢ the frequency of the heavy and light filling factor, the increase of F from 20 to 100% leads to the appearance of two peaks at frequency close to ωₕ and ωᵢ. The intensity of these peaks shifts from the one located around ωᵢ to that located around ωₕ when increasing F. This last behaviour is observed for carbon peapods [54].

The lines at 291, 496 and 510 cm⁻¹ respectively upshift to 296, 497 and 512 cm⁻¹ and increases in intensity from 20 to 100% for C₆₀@(10,10). The same the lines centered at 266, 493 and 457 cm⁻¹ increases in intensity from 20 to 100% for C₇₀@(10,10). The observed frequency shift can be associated with C₆₀-SWBNNT, C₇₀-SBNN, C₆₀-C₆₀ and C₇₀-C₇₀ van der Waals interaction effects.

Using the spectral moment's method, we investigate the evolution of the average nonresonant intensity ratios between Raman mode of C₆₀ molecules and BN nanotube as a function of the concentration of C₆₀s inside the tubes. In the case of C₆₀ carbon peapods, we have found that the dependence of the Raman spectrum with the filling factor, the relative intensity ratio between a Raman mode of C₆₀ and a mode of SWBNNT, normalized by the same ratio calculated for a reference filling factor, gives a useful method to derive from Raman experiments the relative concentration of C₆₀ in peapods samples prepared with the same batch as that of the reference filling factor sample.

First, we calculate the Raman spectrum at different filling factors and for each filling factor. Then, for each filling factor, the integrated intensity ratios between Raman mode of C₆₀ molecules and the RBLM (for C₆₀ phonon modes: Hg(2), Ag(1), Hg(3) and Hg(4)) or G-mode (for C₆₀ phonon modes: Hg(7) and Ag(2)) are calculated. These calculated intensity ratios are normalized with respect to the same intensity ratios calculated for the 60% filling factor sample. The obtained relative
concentrations derived by this way in the case C₆₀@(10,10) are shown in Figure 6. As expected, the relative concentrations calculated for each C₆₀ mode are close especially for low frequency modes expect the Hg(7) and Ag(2) high frequency modes that could showing a overestimated concentration.

<figure><img src="image_6.png" /><figcaption>**Figure 5.** ZZ-polarized Raman spectra in RBLM regions for infinite nanopeapods as a function of their filling rate: C₆₀@(10,10) (left) and C₇₀@(10,10) (right).</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>**Figure 6.** Calculated average intensity ratios, normalized on the 60% filling factor intensities, between modes of the C₆₀ in the C₆₀@(10,10) peapod crystal and the RBLM (Hg(2), Ag(1), Hg(3), Hg(4)) or G-mode (Ag(2)) of the (10,10) BN nanotube.</figcaption></figure>

**4. Conclusions**

We have calculated the nonresonant Raman spectrum of isolated boron nitride peapods. The C₆₀ and C₇₀ molecules adopt a linear arrangement for SWBNNTs diameter lower than 1.45 and 1.42 nm, respectively. Both for the obtained C₆₀ and C₇₀ peapods, the dependence of the Raman spectrum as a
function of the tube diameter and the filling rate have been analyzed. We showed that the configuration of the C₆₀ and C₇₀ molecules significantly impact the Raman line frequency as well as their intensities in the BN peapod structure. In particular, the line position is modulated by both fullerene-fullerene and fullerene-SWBNNT interactions, whereas the angle dependence of C₇₀'s affects their intensities in polarized spectra. We found that the behaviour of the RBLM mode with the pod diameter was clearly modified in peapod. This involves that the linear relation linking the RBLM frequency and the inverse of the tube diameter found in SWBNNT has to be modified in the case of peapod sample. We discussed the Raman signatures of the C₆₀ and C₇₀ peapods according to their filling rate and for the RBLM in particular. The relative intensity ratio between a Raman mode of C₆₀ and C₇₀ and a mode of SWBNNT, normalized by the same ratio calculated for a reference filling factor, could be a useful method to derive from Raman experiments the relative concentration of fullernes in peapods samples prepared with the same batch as the reference filling factor sample. Finally, we think that the calculated Raman spectra reported in the present work can be useful to understand future Raman data.

**Acknowledgments:** The work was supported by the CNRS-France/CNRST-Morocco agreement.

**Author Contributions:** Brahim Fakrach, Fatima Fergani and Mourad Boutahir conceived and carried out the calculations. Abdelhai Rahmani and Hassane Chadli supervised the simulations and contributed to the text. Patrick Hermet and Abdelali Rahmani shared in the writing of the paper.

**Conflicts of Interest:** The authors declare no conflicts of interest.

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