Raman-Active Modes in Finite and Infinite Double-Walled Boron

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Raman-Active Modes in Finite and Infinite Double-Walled Boron Nitride Nanotubes Brahim Fakrach, Abdelhai Rahmani, Hassane Chadli, Khalid Sbai, Patrick Hermet, and Abdelali Rahmani J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.5b02593 • Publication Date (Web): 21 May 2015 Downloaded from http://pubs.acs.org on May 28, 2015

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The Journal of Physical Chemistry

Raman-Active Modes in Finite and Infinite Double-Walled Boron Nitride Nanotubes B. Fakrach,† A.H. Rahmani,† H. Chadli,†,‡ K. Sbai,† P. Hermet,¶ and A. Rahmani∗,§ Laboratoire d’Etude des Matériaux Avancés et Applications (LEM2A), Université Moulay Isma¨ıl, Faculté des Sciences, BP 11201, Zitoune, 50000 Meknès, Morocco, Faculté Polydisciplinaire, BP 512, Boutalamine, 52000 Errachidia, Morocco, Institut Charles Gerhardt Montpellier, UMR CNRS 5253, Université de Montpellier, Place E. Bataillon, 34095 Montpellier, France, and Laboratoire d’Etude des Matériaux Avancés et Applications (LEM2A), Université Moulay Isma¨ıl, Faculté des Sciences, BP 11201, Zitoune, 50000 Meknès, Morocco, Phone:+212671934093 E-mail: [email protected]

∗

To whom correspondence should be addressed Laboratoire d’Etude des Matériaux Avancés et Applications (LEM2A), Université Moulay Isma¨ıl, Faculté des Sciences, BP 11201, Zitoune, 50000 Meknès, Morocco ‡ Faculté Polydisciplinaire, BP 512, Boutalamine, 52000 Errachidia, Morocco ¶ Institut Charles Gerhardt Montpellier, UMR CNRS 5253, Université de Montpellier, Place E. Bataillon, 34095 Montpellier, France § Laboratoire d’Etude des Matériaux Avancés et Applications (LEM2A), Université Moulay Isma¨ıl, Faculté des Sciences, BP 11201, Zitoune, 50000 Meknès, Morocco, Phone:+212671934093 †

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Abstract In this theoretical work, we study the Raman spectra of double-walled boron nitride nanotubes (DBNNTs) as a function of their diameters, chiralities and lengths. Calculations are performed using the spectral moment’s method coupled to the bond polarizability model. This original approach allows us to consider not only infinite tubes as usual in most theoretical models, but also tubes with finite lengths. We provide benchmark theoretical data to estimate the diameter and length of DBNNTs using Raman measurements.

I. Introduction Boron nitride nanotubes (BNNTs), formed by rolling a single hexagonal (h-BN) sheet, are structural analogues of carbon nanotubes (CNTs). 1 The electronic properties of BNNTs differ from those of CNTs. While the metallic or semiconducting character of CNTs is governed by their chirality and diameter, all BNNTs are semiconducting with a large band gap. 2 For the electronic structure of multi-walled BNNTs (MBNNTs), and mainly doublewalled BNNTs (DBNNTs), a hybridization between π and σ states of inner and outer tubes exists. 3,4 Thus, the fundamental energy gap of DBNNTs are smaller than in single-walled BNNTs (SBNNTs). 5 Mechanical properties of CNTs and BNNTs are also different because the B-N bond is ionic. 6 BNNTs are used in high temperature environments 7 due to their high oxidation resistance. In addition, they are also very interesting materials for application in nanoscale devic! es and are considered as alternatives to CNTs. 8 All these properties make BNNTs very attractive for innovative applications in various branches of science and technology. 9 BNNTs have been theoretically predicted in 1994 2 and synthesized into MBNNTs in 1995 10 using several methods, such as: laser ablation, 11,12 chemical vapor deposition, 13,14 ball-milling, 7 substitution reaction, 15–17 and laser heating. 18 The plasma arc method 19 showed that most of the synthesized nanotubes had a double-walled structure, and the possibility to 2 ACS Paragon Plus Environment

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control their growth up to produce them in bulk amounts under certain conditions. Smith et al. 20 reported from the synthesis of BNNTs that the majority of the tubes have two (DBNNT) to five (MBNNT) walls. The synthesis of BNNTs appears in SBNNTs, DBNNTs and MBNNTs whose diameters are mostly within the 1-6 nm range. 21 The outer diameter range of the observed DBNNTs is between 2 and 4 nm. Raman spectra of MBNNTs show a strong line located in the 1356-1369 cm−1 range 23–30 that is sample-dependent and assigned to the E2g in-plane phonon mode of bulk h-BN. Raman spectra of SBNNTs are dominated by the so-called radial breathing modes (RBM) below 500 cm−1 and by the tangential modes (TM) between 1200 and 1600 cm−1 . The RBM is associated to vibration of the entire tube in the radial direction and is strongly diameter-dependent. 31–37 This phonon mode can be therefore used to determine the diameter of nanotubes. The TM is characterized by a strong Raman line and is strongly dependent of the tube chirality. 36,37 It was found that the linear relation between the RBM frequency and the inverse of diameter requires a correction in single-walled CNT bundles as well as in multiwalled CNTs 38 because the tube-tube intera! ctions cannot be neglected. Recently, Aydin 39 has calculated the nonresonant Raman and infrared spectra of zigzag-SBNNTs and zigzag-DBNNTs using density functional theory. He showed that the RBMs of the DBNNTs are upshifted with respect to that of corresponding SBNNTs. In a previous theoretical work 41 on the infrared response of DBNNTs, we derived phenomenological expressions to describe the wavenumber dependence of the radial breathinglike modes (RBLMs) with the diameter of the inner and outer tubes. Here, we follow the same methodology but in the case of the Raman spectroscopy that is more widely used than the infrared spectroscopy by experimentalists to study the vibrational properties and electronic structure of nanotubes. 22 We also provide benchmark theoretical data to estimate the length of DBNNTs. For this purpose, we study the Raman spectra of DBNNTs as a function of their diameters, chiralities and lengths. Calculations are performed using the spectral moments method coupled to the bond polarizability model. This original approach



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allows us to consider not only infinite tubes as usual in most theoretical models, but also tubes with finite lengths.

II. MODEL AND COMPUTATIONAL METHOD DBNNTs can be considered as a kind of MBNNTs for which the interlayer interaction is generally considered to be turbostratic between the inner and outer nanotubes. For the DBNNTs under consideration, the inner and outer tubes are assumed to be at a distance, d =0.34 nm ±10%, which is consistent with the experimental values 20,42 and quite close to the inter-planar distance in a bulk h-BN. The relation between the diameters of the inner (Di ) and outer (Do ) tubes is: Do = Di +2d. Van der Waals interactions, usually treated as a weak perturbation, can produce quantum phenomena in double-walled CNTs, 43 leading to a small variation in the CC-bond lengths with respect to single-walled CNTs. Because this variation is probably also small in DBNNTs compared to the calculated frequency shifts of the Raman lines, the BN-bond lengths were not relaxed in our calculations. The diameter of a given (n, m) BNNT is given by: D =

a π

q

3(n2 + m2 + mn), where a=0.1435 nm is the

nearest-neighbor B-N distance. This relation allows us to derive all the possible (k, l) outer tubes for a specific (n, m) inner tube, which leads to (n,m)@(k,l) DBNNTs. For instance, the (n,n)@(n+5,n+5) and (n,0)@(n+9,0) DBNNTs satisfy the above conditions with a separation distance close to 0.357 and 0.344 nm, respectively. For chiral DBNNTs, we consider all the combinations with a tube-tube distance located between 0.34 and 0.357 nm. The interatomic interactions of the inner and outer BNNT are described using the same force constant model we used to compute the infrared spectra of SBNNTs 40 and DBNNTs. 41 The van der Waals interactions between inner and outer tubes in DBNNTS are described by the Lennard-Jones potential, ULJ (rij ) = 4ε

σ rij

12

−

σ rij

6

, where rij is the distance

between atoms i and j. The atomic parameters of this potential are: εB = 4.116 meV, εN = 6.281 meV, σB = 0.3453 nm and σN = 0.3365 nm. 45 The B-N and N-B parameters are


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derived using the Lorentz-Berthelot mixing rules: εAB =

√

εA εB and σAB = (σA + σB ) /2.

The parameters of the bond polarisability model are chosen according to Wirtz et al. 46 Raman spectra are calculated using the spectral moments method 48–50 combined with the bond polarizability model. 47 This combination is especially powerful for determining directly the Raman spectra of very large harmonic systems without any diagonalization of the dynamical matrix. The heart of the spectral moments method consists to express the response of a system into a continued fraction:

J(u) = −

1 lim Im[R(z)], π →0+

(1)

where z = u + iε, and R(z) =

b0 z − a1 −

.

b1 z−a2 −

(2)

b2 b z−a3 −... z−an n+1

The coefficients, an and bn , are given by:

an+1 =

ν¯n νn ; bn = . νn νn−1

(3)

The spectral moments, νn , and generalized moments, ν¯n , of J(u) are directly obtained from the dynamical matrix. As shown in previous works, 49,50 the calculation of Raman spectra shows that a limited number of moments (500 have been used here) are enough to get the wavenumber of active modes with a good accuracy. Within the framework of this method, the wavenumber of the Raman-active lines is directly obtained from the line position in the calculated spectrum, and we have not a direct access to the eigenvectors of modes. In all our calculations, the common axis of the inner and outer tubes is along the Z-axis, and the atoms of the DBNNT are along the X-axis of the nanotube frame. The x, y and z are the axes of the laboratory frame. Three geometrical configurations are considered: in the ZZ-configuration, both incident and scattered polarizations are along the Z-axis and, while for ZX (resp. XY) configuration, the incident and scattered pola! rizations are along the Z 5 ACS Paragon Plus Environment


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(resp. X) and X (resp. Y) axes. In our calculations, the shape of the Raman lines is assumed to be Lorentzian and their width is fixed at 1.7 cm−1 .

III. RESULTS AND DISCUSSION A. DBNNTs WITH INFINITE LENGTH In this section, calculations are performed by applying periodic conditions along the Zdirection (tube generatrice). We focus on the Raman spectra of armchair@armchair DBNNTs identified as (n,n)@(n+5,n+5). To make the comparison with the experimental results as realistic as possible, we performed an average of the Raman spectra over the DBNNT orientations with regards to the laboratory frame. The laser beam is kept along the Y axis. The calculated VV and VH-polarized spectra for the unoriented (5,5)@(10,10), (7,7)@(12,12), and (10,10)@(15,15) DBNNTs are shown in figure 1. In the VV-configuration, both the incident and scattered polarizations are along the z-axis and, for the VH-configuration, the incident and scattered polarizations are along the z and x-axes, respectively. We assigned the breathing-like modes (BLMs) of DBNNTs from the polarized Raman spectra of inner and outer SBNNTs. For armchair tubes, a single A1g and E1g line and 2E2g lines (! with a weak intensity for the lowest line) are respectively calculated in the ZZ, ZX and XY-polarizations (see the Figure 1 of Ref.

40

), whereas a pair of lines with the A1g , E1g and 2E2g symmetry

are systematically calculated in the case of armchair@armchair DBNNTs. The VV-polarized spectra show six lines in the range of breathing-like modes (BLMs) whatever the tube chiality (figure 1). For example, in the case of the (7,7)@(12,12) DBNNT, they are centered at 72 (E1g ), 123 (E1g ), 134 (A1g ), 227 (A1g ), 276 (E2g ) and 468 cm−1 (E2g ). These lines result from the in-phase and counter-phase coupled motions of the breathing modes of the inner and outer tubes. In the VH-polarisation, only four lines are preserved as the two A1g -modes vanish. This behavior suggests that the totally symmetric character of the RBM is preserved in DBNNTs. We also observe a systematic wavenumbers upshift of 6 ACS Paragon Plus Environment

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the BLMs in DBNNTs with respect to SBNNTs. Indeed, the breathing modes in (7,7) and (12,12) SBNNTs are calculated at 207 and 121 cm−1 (A1g ), 122 and 72 cm−1 (E1g ), and 467 and 274 cm−1 (E2g ), respectively. In the range of intermediate-like modes (ILMs), there are four and two lines in the VV and VH configurations, respectively. They are associated with the in-phase and counter-phase coupled motions of the intermediate modes of the inner and outer tubes. Except the line at the lowest wavenumber, the other lines show a downshift as the tube diameter increases. Similarly to the BLMs, the wavenumber of ILMs slightly upshift from 2 to 5 cm−1 with respect to the corresponding SBNNTs. For example, the Raman lines of the (7,7)@(12,12) are located at 801 and 812 cm−1 (A1g ), and 785 and 793 cm−1 (E2g ), whereas in isolated (12,12) and (7,7) SBNNTs, they are respectively calculated at 799 and 807 cm−1 (A1g ) and 790 and 780 cm−1 (E2g ). In the range of tangential-like modes (TLMs), we observe six modes associated with the in-phase and counter-phase coupled motions of the TMs of the inner and outer tubes. As previously, we identify the TLMs from the symmetry of TMs associated with each of two SBNNTs constituting the DBNNT. For example, TLMs of (7,7)@(12,12) DBNNT are centered at 1387 and 1391 cm−1 (A1g ), 1389 and 1392 cm−1 (E1g ), and 1397 and 1399 cm−1 (E2g ), and result from the coupling of the A1g , E1g and E2g modes of the (7,7) inner tube and (12,12) outer tube. Figure 2 shows the diameter dependence of the outer tube (Do ) in armchair@armchair DBNNTs as a function of the Raman wavenumbers. The diameter dependence of the Raman lines for the related isolated SBNNTs is also displayed on this figure. We observe a systematic wavenumber upshift of the Raman lines in DBNNTs with respect to SBNNTs. For all modes, this upshift is not sensitive to the tube d! iameter except for the highest RBLM wavenumbers for which the upshift is diameter dependent. All TLMs show a wavenumber upshift as the tube diameter increases, except the E2g lines which slightly downshift. It can be emphasized that an unresolved broad TLM band is expected around 1392 cm−1 in the calculated Raman spectra of DBNNTs with diameter



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Do > 2.5 nm. The same behavior has been predicted by Aydin et al. using density functional theory . 39 The broad line experimentally observed 23–30 at 1365 cm−1 is therefore related to a large distribution of modes. It is assigned to the h-BN active mode E2g derived from the in-plane atomic displacement of the B and N atoms toward each other. We focus now on the diameter-dependence of the specific BLMs, and especially the inphase and counter-phase coupled motions of the RBMs of the inner and outer tubes. These modes are called LW-RBLM (in-phase motions of both tubes) and HW-RBLM (counterphase vibrations of both tubes). For instance, they are respectively centered at 157 and 304 cm−1 in (5,5)@(10,10) DBNNT. Their eigendisplacement vectors obtained from the direct diagonalization of the dynamical matrix are displayed in Figure 3 together with the RBM of the associated SBNNTs, and support their radial character. The wavenumber shift, ∆ω = ωRBLM − ωRBM , defined as the wavenumber difference between the RBLM and the RBM of the isolated inner tube is reported in Figure 4. We observe that the ∆ω shift associated to the HW-RBLM is a linear function for diameters of inner tube between 0.6 and 2.8 nm. In contrast, the shift associated to the LW-RBLM shows a more complex behavior: the wavenumber shift becomes linear for very small diameters, while for diameters of outer tube between 1.9 and 3.5 nm, it is no longer linear. This change of slope is related to an increase of the interactions between the tubes. Indeed, for small diameters, each of the two RBLMs has the RBM characteristics associated with each of the two SBNNTs constituting the DBNNT. For large diameters (above 1.9 nm), the LW- and HW-RBLMs are in-phase and counterphase collective motions of both tubes, respectively. Moreover, for these diameters, the ∆ω of the lower breathing mode tends to zero, whereas the upper one tends to 91 cm−1 which is close to the inter-layer compression mode (86 cm−1 ) of a flat double-layer of boron nitride. 44 In the case of double-walled CNTs, the dependencies of the RBLM frequencies as a function of the diameter have been reported by Popov et al. 38 The authors show that for the small ! diameters, each of the two RBLM’s has the RBM characteristics of one of the tubes. In contrast, for large diameter (above 1.7 nm), the low- and high-frequency RBLM’s are in-


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phase and counterphase collective motions of both tubes, respectively. The dependencies of the RBLM wavenumbers as a function of the diameter (Figs. 2 and 3 in Ref. 38 ) are similar to those found in our work for DBNNTs. In the 0.5-14 nm diameter range, we found that the diameter dependence of the RBLMs are well fitted by the phenomenological relations: 24 25.36 111.61 − 3 + 2 Di Di Di

(4)

188.73 150.37 160 − − 3 Do Do2 Do

(5)

ωHW −RBLM = 91.45 +

ωLW −RBLM = 0.35 +

where the wavenumbers and the diameters are given in cm−1 and nm, respectively. No physical meaning is attached to the values of the fitting parameters. It must be emphasized that these equations are directly useful to estimate the diameter of DBNNTs from the experimental RBLM wavenumbers. Similar equations have been derived for double-walled CNTs. 51 The chirality dependence of the ZZ-polarized Raman spectra is reported in Figure 5 for selected infinite armchair@armchair [(5,5)@(10,10)], zigzag@zigzag [(9,0)@(18,0)], armchair@chiral [(5,5)@(15,4)], chiral@chiral [(7,3)@(17,1)] and zigzag@chiral [(9,0)@(16,3)]. All selected DBNNTs show two lines in the range of BLMs associated with the in-phase (∼155 cm−1 ) and counter-phase (∼298 cm−1 ) coupled motions of the RBMs of the inner and outer tubes, respectively. Their wavenumbers are not dependent of the chirality of inner and outer tubes; the small wavenumber shift is associated to variations in the diameter of the selected DBNNTs. In the range of TLMs, we observe three lines for chiral@chiral, zigzag@chiral and armchair@chiral, whereas two A1g lines are evidenced for zigzag@zigzag and armchair@armchair DBNNTs. Thus, the TLMs show a chirality dependence, while the RBLMs are mainly sensitive to the tube diameter.



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B. DBNNTs WITH FINITE LENGTH BNNT samples have experimentally a finite length above 10 µm. 20,28 In this context, the spectral moments method is especially suitable in contrast to most theoretical methods which are restricted to study DBNNTs with an infinite length because of periodic boundary conditions. Thus, we considered (n,n)@(n+5,n+5) DBNNTs with open ends without any termination. Their length are ∼250 nm, without applying periodic conditions on unit cells. In the following, we note Li and Lo the relative lengths of the inner and outer tubes, respectively. The dependence of the ZZ-polarized Raman spectrum of (5,5)@(10,10) DBNNT as a function of the Li /Lo ratio is displayed in Figure 6 up to Li /Lo = 2. In the BLM range and for Li = Lo , the spectrum is dominated by two lines located at 157 and 304 cm−1 respectively assigned to the in-phase and out-of-phase BLMs. For Li < Lo , the line at 304 cm−1 remains unaltered whereas the line at 157 cm−1 becomes a doublet where its lowwavenumber component centered at 145 cm−1 is assigned to the radial mode of the part of the outer tube which does not interact with the inner tube. For Li > Lo , we have the opposite trend: the line at 157 cm−1 remains unaltered whereas the line at 304 cm−1 becomes a doublet where its low-wavenumber component centered at 290 cm−1 is assigned to the radial mode of the part of the inner tube which does not interact with the outer tube. The additional lines at 145 and 290 cm−1 are ! the same as those of the radial mode in (10,10) and (5,5) SBNNTs, respectively. The ILM range shows the same dependence of the Li /Lo ratio than the BLM range and the above discussion is also valid. The intensity of the lines at 145 and 801 cm−1 (resp. 290 and 820 cm−1 ) decreases when the Li /Lo ratio increases (resp. decreases) and vanish for Li = Lo . As expected, all RBLMs and ILMs are upshifted with respect to the RBM of the corresponding SBNNTs: for Li < Lo , the LW-RBLM (resp. ILM) shifts by ∆ω=12 cm−1 (resp. ∆ω=2 cm−1 ), and for Li > Lo , the HW-RBLM (resp. ILM) shifts by ∆ω=14 cm−1 (resp. ∆ω=5 cm−1 ). Thus, the measurement of the RBLMs and ILMs modes can be used to estimate the length of the inner and outer SBNNT using 10 ACS Paragon Plus Environment

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Raman spectroscopy. The TLMs slightly depend on the Li /Lo ratio. They are calculated at 1381 and 1390 cm−1 in the (5,5)@(10,10) DBNNT for Li = Lo (Figure 6, right). For Li > Lo , the intensity of mode at 1381 cm−1 increases and is assigned to the TM of the part of the inner tube which does not interact with the outer tube (the TM of the corresponding (5,5) SBNNT is calculated at 1381 cm−1 ). For Li < Lo , the intensity of mode at 1390 cm−1 increases and is assigned to the TM of the part of the outer tube which does not interact with the inner tube (the TM of the corresponding (10,10) SBNNT is calculated at 1390 cm−1 ). The TLMs have a slight wavenumber dependence with respect to the relative lengths of the inner and outer tubes. Consequently, they are not useful to estimate the length of tubes, and in the following we only focus on the range of the BLMs and ILMs. Now, we consider the case of (5,5)@(10,10), (7,7)@(12,12) and (10,10)@(15,15) DBNNTs for Li =0.5Lo (Figure 7, left) and Li =2Lo (Figure 7, right). Regardless of the length and the diameter of the tube, the Raman spectra show three lines in the range of BLMs: two that results from the in-phase and out-of-phase coupled motions of radial modes of the inner and outer tubes, while the third originates from the radial mode of the individual biggest nanotube. For the smallest diameters [(5,5)@(10,10)], each of the two RBLMs at 157 and 304 cm−1 has similar feature to the RBMs of the related isolated (5,5) and (10,10) SBNNT located at 145 and 290 cm−1 respectively. In fact, the wavenumbers of the RBMs are significantly different, and the mixing of these modes is therefore weak. In contrast, for the largest diameters [(10,10)@(15,15)], due the closeness of the wavenumbers of the RBMs (97 and 145 cm−1 ), the mixing of the LW- and HW-RBL! Ms located at 109 and 174 cm−1 , which are corresponding to the in-phase and out-of-phase collective motions, is more important. In comparison with the Raman spectrum of the related SBNNTs, it can be pointed out that the RBLMs of DBNNTs are upshifted. For Li =0.5Lo , the upshift of the LW- and HW-RBLMs is slightly dependent on tube diameter (∆ω '12 cm−1 ). For Li = 2Lo : ∆ω=14 cm−1 for (5,5)@(10,10), ∆ω=20 cm−1 for (7,7)@(12,12) and ∆ω=29 cm−1 for (10,10)@(15,15).



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Finally, we calculate the Raman response of DBNNTs as a function of their lengths, L. Here,The tubes have open ends without any terminations and their lengths, L = Li = Lo , are up to 100 nm. These lengths are close to the experimental observations. 20,28 Because the Raman spectra of DBNNTs show the same behavior as a function of the nanotube length whatever their chiralities, only the case of (9,6)@(15,10) reported in Figure 8 will be discussed. As in the case of SBNNTs 40 and bundle of SBNNTs, 52 a large number of additional lines are observed, especially in the BLM range. The intensity of these lines rapidly decreases as the length of the tube increases. For a length of ∼30 nm, the position and intensity of the lines are very close to those calculated for DBNNTs with infinite length. In consequence, the effect of the tube length on the Raman spectrum could be only experimentally observed on very small tubes. Because i! n most experiments DBNNTs have more than 30 nm length, our calculations for the infinite tubes can be used to interpret the Raman experiments.

IV. CONCLUSIONS In conclusion, we have calculated the Raman spectrum of isolated DBNNT using the spectral moment’s method coupled to the bond polarisability model. We found that the position of the RBMs and ILMs in DBNNTs shift with respect to their corresponding ones in isolated SBNNTs. The dependence of the Raman spectra as a function of the diameter and the chirality of the inner and outer tubes has been analyzed. Mathematical expressions have been derived to describe the dependence with the diameter of the RBLMs in isolated infinite DBNNTs. These expressions can be used to experimentally derive the inner and outer tube diameters from the measurement of the RBLM wavenumber. Important finite-size effects (length of nanotubes) are observed, mainly in the BLM and ILM region, with additional Raman lines predicted. These lines can be used to estimate the length of DBNNTs using Raman spectroscopy.


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

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A

E

1 g

1 2 3 A

E 1 g

2 g

E 1 g

E

2 2 7

2 g

A

7 8 5 A

E

1 g

7 2

0

E

2 0 0

2 g

1 3 9 1 E

1 g

4 6 8

1 3 4 A

1 g

Page 20 of 33

E

2 7 6

4 0 0

6 0 0

2 g

1 g

1 g

1 g

8 1 2

1 3 9 2

1 3 8 9

8 0 1 A

7 9 3

7 8 0

8 1 0

W a v e n u m b e r (c m

1 g

1 3 8 0 -1

E

1 3 8 7 E

2 g

2 g

1 3 9 9

1 3 9 7

1 3 9 2

1 4 0 4

)

Figure 1: VV (solid line) and VH (dashed line) polarized Raman spectra of unoriented samples of infinite (5,5)@(10,10) (bottom), (7,7)@(12,12) (middle), and (10,10)@(15,15) (top) DBNNTs. Wavenumber are divided into three ranges: BLMs (left), ILMs (middle) and TLMs (right).

1 4 0 0

1 3 9 5

a v e n u m b e r (c m -1 )

4 0 0

1 3 9 0 3 0 0 1 3 8 5

2 0 0

8 2 0

W

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

R a m a n In te n s ity (a .u .)


8 0 0 1 0 0

7 8 0 0 1 .0

1 .5

2 .0

2 .5

D O

3 .0

3 .5

1 .0

1 .5

2 .0

2 .5

D

(n m )

O

3 .0

3 .5

(n m )

Figure 2: Diameter dependence of the Raman wavenumbers for (n,n)@(n+5,n+5) DBNNTs (solid symbols), (n,n) and (n+5,n+5) SBNNTs (open symbols)


Page 21 of 33

LW-RBLM (157 cm-1)

LW- RBM (145 cm-1)

HW-RBLM (304 cm-1)

HW- RBM (289 cm-1)

Figure 3: Calculated atomic motions of selected phonon modes in (5,5)@(10,10) DBNNTs and (5,5) and (10,10) SBNNTs. Arrows are proportional to the amplitude of the atomic motion.

1 0 0

H W -R B L M L W -R B L M

8 0

-1

)

6 0

D w (c m

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


4 0

2 0

0 3

6

9

1 2

1 5

D ia m e te r (n m )

Figure 4: Wavenumber shift, ∆ω = ωRBLM − ωRBM , between the RBLM and the RBM of the isolated inner tube as a function of the tube diameter for (n,n)@(n+5,n+5) DBNNTs.



3 0 4 1 3 9 0

1 5 7 1 3 8 0 (5 ,5 )@ (1 0 ,1 0 ) 3 0 3

1 3 9 3

1 5 6


1 3 8 0 1 3 9 0

(5 ,5 )@ (1 5 ,4 ) 2 9 8

1 5 6

1 3 9 4 1 3 8 0 1 3 9 0 (7 ,3 )@ (1 7 ,1 )

2 9 3

1 3 9 3

1 5 3

1 3 9 5 (9 ,0 )@ (1 6 ,3 )

1 3 9 0

2 8 7

1 3 9 4

1 4 6

1 3 9 5 (9 ,0 )@ (1 8 ,0 )

0

1 0 0

2 0 0

3 0 0

4 0 0

1 3 7 6


-1

1 3 8 4

1 3 9 2

1 4 0 0

)

Figure 5: Calculated ZZ-polarized Raman spectra of selected infinite DBNNTs in the range of BLMs (left) and TLMs (right).

2 9 0 1 5 7

8 2 5

8 2 0

3 0 4 8 0 3

L i= 2 L o

L i= 1 . 5 L


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 22 of 33

8 2 5 1 3 8 1

8 0 3

L i= L o

L i= 0 . 5 L 1 4 5

8 2 5 8 0 3

1 2 0

o

8 0 1 3 0 4

1 5 7

0

o

1 3 9 0

3 0 4

1 5 7

2 4 0

3 6 0

7 8 0

7 9 5

L i= 0 . 2 5 L 8 1 0


8 2 5 -1

8 4 0

o

1 3 6 5

1 3 8 0

1 3 9 5

1 4 1 0

)

Figure 6: Calculated ZZ-polarized Raman spectra of (5,5)@(10,10) DBNNTs as a function of the Li /Lo ratio. Li and L0 are the length of the inner and outer tube, respectively.


Page 23 of 33

L i= 0 . 5 L

L i= 2 L

(1 0 ,1 0 )@ (1 5 ,1 5 ) o

o

1 0 9 1 1 0

9 7

1 4 5

R a m a n In te n s ity ( a .u .)

1 7 4

1 7 4

1 3 4

1 3 4

2 9 0 (5 ,5 )@ (1 0 ,1 0 )

1 0 0

2 2 7

3 0 4

1 5 7 1 4 5

0

2 0 7

(7 ,7 )@ (1 2 ,1 2 )

2 2 7

1 2 1

2 0 0

3 0 0

4 0 0 0

3 0 4

1 5 7

1 0 0 -1


2 0 0

3 0 0

4 0 0

)

Figure 7: Wavenumber dependence of (5,5)@(10,10), (7,7)@(12,12), and (10,10)@(15,15) DBNNTs for Li = 0.5Lo (left) and Li = 2Lo (right) in the range of BLMs.

1 2 7

1 3 9 1

2 1 1 1 3 8 7

1 3 9 3

L = ∝


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


L = 3 0 n m

L = 1 5 n m

L = 7 .5 n m 6 0

1 2 0

1 8 0

2 4 0

1 3 6 0


1 3 8 0 -1

1 4 0 0

)

Figure 8: Dependence of the ZZ-polarized Raman spectra of (9,6)@(15,10) DBNNT as a function of its length L in the range of BLMs (left) and TLMs (right).

Li>>Lo

LiLo

Li

Raman-Active Modes in Finite and Infinite Double-Walled Boron

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