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Computation of the infrared active modes in single-walled boron nitride nanotube bundles

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2012 J. Phys.: Condens. Matter 24 335304 (http://iopscience.iop.org/0953-8984/24/33/335304) View the table of contents for this issue, or go to the journal homepage for more

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IOP PUBLISHING

JOURNAL OF PHYSICS: CONDENSED MATTER

J. Phys.: Condens. Matter 24 (2012) 335304 (7pp)

doi:10.1088/0953-8984/24/33/335304

Computation of the infrared active modes in single-walled boron nitride nanotube bundles B Fakrach1 , A Rahmani1 , H Chadli1 , K Sbai1 , M Benhamou1,2 , M Bentaleb1 , J-L Bantignies3 and J-L Sauvajol3 1

Laboratoire de Physique des Mat´eriaux et Mod´elisation des Syst`emes (URAC08), Universit´e Moulay Isma¨ıl, Facult´e des Sciences, PO Box 11201, Zitoune, 50000 Mekn`es, Morocco 2 Laboratoire de Physique des Polym`eres et Ph´enom`enes Critiques, Facult´e des Sciences Ben M’sik, PO Box 7955, Morocco 3 Laboratoire Charles Coulomb (UMR CNRS 5587), Universit´e Montpellier II, 34095 Montpellier Cedex 5, France E-mail: rahmani@fs-umi.ac.ma

Received 13 April 2012, in final form 9 July 2012 Published 27 July 2012 Online at stacks.iop.org/JPhysCM/24/335304 Abstract In this work, the infrared active modes are computed for homogeneous bundles of single-walled boron nitride nanotubes (BBNNTs), using the so-called spectral moments method. The dependence of the wavenumber on these modes in terms of diameters, lengths, and numbers of tubes, is investigated. To this end, use is made of a Lennard-Jones potential for describing the van der Waals interactions between tubes in a bundle. We find that, for a finite homogeneous bundle, additional modes appear as a specific signature. Finally, these results are useful for the interpretation of the experimental infrared spectra of BBNNTs. (Some figures may appear in colour only in the online journal)

1. Introduction

chemical vapor deposition [9–12], and laser ablating [16, 17].

Recently, Guo et al [18] reported interesting results from the synthesis of BNNTs, which either appear in bundles with a width close to 50–70 nm or as a single tube with a diameter around 5–20 nm. In this context, their characterization through vibrational spectroscopies is expected to play an important role.

Raman and infrared spectroscopy are known to be the most efficient tools for investigating the vibrational properties of materials in relation to their structural and electronic properties [19, 20]. The infrared spectrum of the synthesized BNNTs is dominated by two strong characteristic peaks located around 800 and 1370 cm−1 . Lee and coworkers [13] have presented a growth approach for BNNTs. The typical BNNTs diameters and lengths range from 3 to 10 nm and greater than 10 µm, respectively [21]. The grown samples were characterized by Fourier transformed infrared (FTIR) spectroscopies [13]. The FTIR spectra are characterized by an absorption band at 1369 cm−1 assigned to the in-plane stretching modes of h-BN networks and a double peak in

Boron nitride nanotubes (BNNTs) [1] are one of the composite nanotubes and are analogous in structure to carbon nanotubes (CNTs) [2]. The structure can be obtained by rolling up a single hexagonal h-BN sheet. On the other hand, the electronic properties of BNNTs are different from those of CNTs. Unlike CNTs, BNNTs have been predicted to exhibit semiconducting properties (a uniform band gap of about 5 eV), independently of chirality and diameter [3], together with high strength and oxidation resistance [4, 5]. BNNTs are very attractive materials for application in nanoscale devices.

The stability of BNNTs was firstly studied on the basis of semi-experimental tight binding [1] and local density approximated density functional theory [3] calculations in 1994 and their synthesis was realized in 1995 with the arc-charging method using the BN electrode packed into a metal casing [6, 7]. Other preparation methods have been developed subsequently, such as arc-melting [8], hightemperature chemical reaction [9–13], ball-milling [5, 14, 15], 0953-8984/12/335304+07$33.00

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B Fakrach et al

the interval 800–815 cm−1 related to the out-of-plane radial buckling mode. These peaks are located at 810 and 1380 cm−1 (from [22]), at 809 and 1397 cm−1 (from [24]), at 785 and 1360 cm−1 (from [12]), at 813 and 1389.8 cm−1 (from [25]).

The infrared active modes of single BNNTs (SBNNTs) have been predicted by different theoretical approaches, namely zone-folding [26], the tight binding approach [27] and ab initio calculations [28, 29]. It must be emphasized that these approaches do not permit us to calculate the phonons in BNNT with a large diameter or finite size (length or number of tubes in a bundle) due to the significant number of atoms, which it is necessary to take into account. The spectral moment’s method [31, 33] is an alternative way to solve this latter problem. Recently, using force constant models combined with the spectral moments method (SMM), we have calculated successfully the infrared and Raman active modes of SBNNTs [20, 30]. In this work, we use the SMM to calculate the infrared spectra of bundles of single-walled BNNTs for finite and infinite sizes. These calculations were performed for a large collection of BN nanotubes bundles with various structural properties: diameter, chirality, length, and the number of (identical) tubes.

This paper is organized as follows. In section 2, we present the used model and method for the study of the system under consideration. Results and the related discussion are the aim of section 3. Finally, some concluding remarks are given in section 4.

represents the intertube spacing [4, 12, 14, 13]. The number of tubes per bundle is denoted Nt .

In the present work, we calculate the polarized infrared spectra for finite and infinite bundles. The intratube interactions are described using the same force constant as used in recent calculations of the infrared spectra of SBNNTs [20]. A Lennard-Jones potential

σ 12 σ 6 ULJ (r) = 4ε − (3) r r is used to describe the van der Waals interactions between the tubes in a bundle. In this work, the parameters for the LJ12-6 potential are εBoron = 0.004 116 eV, εNitrogen = 0.004 116 eV, σBoron = 0.341 nm, and σNitrogen = 0.337 nm [13, 22]. The boron–nitrogen and nitrogen–boron parameters were derived √ using the Lorentz–Berthelot mixing rules, εAB = εA εB and σAB = (σA + σB )/2. We found that there are no significant differences between the IR spectra of the optimized and nonoptimized BBNNT in terms of relative angles of rotation and relative translations of the individual tubes in the bundle.

The IR absorption spectrum, Iα (ω), is given by [23]: Iα (ω) =

ω X |ajα |2 (δ(ω − ωj ) − δ(ω + ωj )) nc j 2ωj

(4)

with ajα =

2. Model and method

X qk √ ej (kα) Mk k

(5)

n and c are respectively the refractive index of the material and the speed of light. qk is the effective charge tensor and Mk the mass of the kth atom. ωj and ej (kα) are the wavenumber and the (kα) component of the displacement amplitude for the kth atom (α is a Cartesian coordinate) in the jth mode. The usual method to calculate the IR spectrum consists of injecting in the previous expressions the values of ωj and ej (kα) obtained ˜ of the by direct diagonalization of the dynamical matrix D system by resolving the equation

Let us recall that the ideal nanotube structure can be obtained from a h-BN sheet by rolling it up along the straight line connecting two lattice points into a seamless cylinder in a way that the two points coincide. Following the usual terminology of [34, 35], the tube can be specified by two integers (n, m) that define the translation vector between the two points.

Alternatively, the tube can be described by its diameter, D, and the chiral angle, θ , which is the angle between the tube circumference and the nearest zigzag of B–N bonds.

According to chirality single-walled BNNTs can be classified into two types: achiral tubes, including armchair (n, n) plus zigzag (n, 0), and chiral tubes (n, m 6= n 6= 0).

The diameter D and the chiral angle θ of SBNNTs are given by ap 2 3m + nm + n2 (1) D= π with a = 0.1435 nm h√ i θ = tan−1 3m/(m + 2n) .

(2)

˜ = ωj2 |ji.

D|ji

(6)

However, when the system contains a large number of atoms, ˜ as for long tubes of finite length, the dynamical matrix D is very large and its diagonalization fails or requires long computing time. In contrast, the spectral moments method allows us to compute directly the IR spectrum of very large harmonic systems without any diagonalization of the dynamical matrix [32, 33]. Otherwise, for small samples, both approaches lead exactly to the same position and intensity for the different peaks.

Considering the symmetrical function =(ω) where a(j) is given by (equation (5)),

When SBNNTs are closely packed together, a threedimensional boron nitride bundle is formed. From the diffraction profile of the crystalline ropes of SBNNTs, it has been previously reported that these systems can be represented by two-dimensional infinite trigonal lattices of uniform cylinders. For a finite-size bundle, the nanotubes were placed one parallel to another on a finite-size trigonal array of cell parameter a0 = D + dt−t , where dt−t = 0.335 nm

=(ω) =

X |a(j)|2 j

2ωj

(δ(ω − ωj ) + δ(ω + ωj )).

(7)

This function is identical (for ω > 0), apart from constants, to a component of the IR absorption (equation (4)), 2

J. Phys.: Condens. Matter 24 (2012) 335304

B Fakrach et al

for a given polarization. The response of the system is given by the function =(ω), which can be further written as X J(u) = |a(j)|2 δ(u − Îťj ) (8)

the IR response of nanotubes, effective charges on boron and nitrogen atoms are fixed at qB = +1 and qN = −1 for a given B–N bond. Consequently, the intensities obtained in this work cannot be compared to the experimental spectra.

However, these results can be used to analyze the dependence in intensity of a mode with diameter or chirality. Finally, the SMM predictions concerning the number and position of IR lines as a function of diameter, length, and chirality do not depend on the charge ratio (qB /qN ) and the SMM predictions can help to interpret the IR experimental data.

j

where u = ω2 and Νj = ωj2 .

J(u) can be directly carried out from the dynamical ˜ and from the effective charge tensor, without any matrix, D, diagonalization. In fact, it is easy to show that J(u) = −

1 lim Im[R(z)], π →0+

(9)

3. Results and discussion

˜ −1 |qi. − D)

where z = u + i and R(z) = hq|(zI˜ The (δn) component of the |qi vector is given by the expression qn hδn|qi = √ Mn

In this section, we report calculations for the infrared response of infinite and finite BNNTs bundles, depending upon diameter and chirality. In all of our calculations, we consider the Z nanotube axis to be along the z-axis and the X nanotube axis to be along the x-axis of the laboratory reference frame. The laser beam is kept along the y-axis.

(10)

which is supposed to be known for a given (δ) component of the IR absorption.

The spectral moments method consists of developing R(z) in a continued fraction: b0

R(z) = z − a1 −

,

b1 z−a2 −

3.1. Infinite BBNNTs Firstly, the calculations are performed on infinitely homogeneous bundles (crystal: Nt = ∞) of SBNNTs of several diameters. We recall that infinite SBNNTs are obtained by applying periodic conditions on unit cells of the studied nanotubes. The results for typical zigzag tubes (12, 0), (15, 0), (18, 0), (21, 0), and (23, 0) are reported in figure 1, where the XX (solid lines) and ZZ (dashed lines) infrared spectra are displayed in the low wavenumber 0–600 cm−1 or breathing-like mode (BLM: on the left) region, intermediate 600–1000 cm−1 region, and high wavenumber >1000 cm−1 or tangential-like mode (TLM: on the right) region. The stars (? symbol) give the position of the infrared active modes of SBNNT. In the TLM region, all spectra display one peak corresponding to A1 and E1 modes. In this region, the number of IR-active modes calculated is independent of the nanotube diameter. We can specify the mode E1 of tangential polarization XX as it moves (about 3 cm−1 ) to the tangential mode of the ZZ polarization, when the diameter increases. In the intermediate region, around 800 cm−1 spectra display a strong (a weak) peak for the E1 (A1 ) mode. We observe a downshift with increasing tube diameter, for E1 modes from 801 to 797 cm−1 and for A1 modes, from 810 to 799 cm−1 . On the low-wavenumber side of the previous modes, additional modes are predicted compared with the modes in SBNNTs (? symbol), but with weak intensity. For example, for the (18, 0) tube, two peaks around 666 and 729 cm−1 are calculated.

These additional modes upshift to a dominant mode, when the diameter increases. In addition, in the TLM and intermediate wavenumber regions, the two strong modes located around 1389–1392 (for E1 mode) and 1394 cm−1 (for A1 mode) and the modes around 800 cm−1 , are less sensitive to the diameter of tubes compared with the modes in the BLM region. In the latter, ZZ spectra exhibit two peaks denoted here as RBLM (for radial breathing-like mode) and BBLM (for bundle breathing-like mode). These modes arise from the radial breathing mode (RBM) and the doubly degenerate

(11)

b2

b z−a3 − ...3

where the coefficients an and bn are given by: an+1 = ν¯ nn /νnn ;

bn = νnn /νn−1n−1 ; b0 = 1.

(12)

The spectral generalized moments νn and ν¯ n of J(u) are Ëœ directly obtained from the dynamical matrix D. νnn = hqn |qn i;

Ëœ ni ν¯ nn = hqn |D|q

(13)

˜ − an+1 )|qn i − bn |qn−1 i; |qn+1 i = (D |q−1 i = 0; |q0 i = 1.

(14)

with

Iterations enable the computation of expression (equation (8)) and then the function I(ω). Each element of the dynamical Ëœ is given by: matrix, D, 1 Dιβ (n, m) = √ φιβ (n, m) Mn Mm

(15)

with φιβ (n, m) being the force constants between n and m atoms.

The wavenumbers of the infrared active modes were directly obtained from the position of the peaks in the calculated IR spectra. In all the following calculations, the line shape of each peak is assumed to be Lorentzian and the width is fixed at 1.7 cm−1 .

The IR intensity depends on the dynamic dipolar moment variation (or effective charge ratio qB /qN ) between nitrogen and boron atoms for BNNTs systems. In the literature and to the best of our knowledge, there is no calculation of the effective charge on SBNNT. In this work, in order to enhance 3

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Figure 1. ZZ (dashed line) and XX (solid line) calculated IR spectra of (n, 0) SWBNNT crystal, for n = 12, 15, 18, 21, and 23, from bottom to top. The stars give the position of the active modes of isolated tubes.

(E symmetry) mode in SBNNTs and are full symmetric modes with breathing-like shapes. The comparison between IR spectra of SBNNTs and BBNNTs shows the appearance of a second mode (BBLM) at a wavenumber higher than that of the RBLM one. In the XX polarization, spectra dominated by one peak and additional peaks are predicted with low intensity. For all modes, we observe a downshift with increasing tube diameter, and the intensity of these modes decreases, except for the BBLM mode. At the same time, the intertube interactions largely dominate over the intratube ones and the higher wavenumber mode BBLM becomes more and more intense. The wavenumber and the vibration pattern of these modes are determined by the competitions between the intertube and intratube interactions. In this study, it is shown that the low-wavenumber modes are likely the most influenced by the range in diameter in comparison with the intermediate and high-wavenumber ones. In [28] and [29], the authors show that the effect of bundling is weak and does not exceed 10 cm−1 in most cases. These results are well reproduced in this SMM study.

In accordance with the significant dependence of the low-wavenumber modes on the tube diameter, we have reported in figure 2 the RBM for SBNNTs and RBLM, BBLM, for bundles of zigzag SBNNTs versus diameter, D. We show that the wavenumber modes obey the A/D law, with A close to the values 198.65, 205.16, and 316.26 nm cm−1 , for RBM, RBLM, and BBLM, respectively. This behavior is close to that calculated for the IR-active breathing [20] and for the Raman active breathing [30] modes, using ab initio and zone-folding methods [26, 28, 36] for SBNNTs. The obtained fitting constant values are consistent with previous RBLM calculations [36], with the value 207.6 cm−1 , for the zigzag tubes. In comparison with the carbon nanotubes, a similar

Figure 2. Diameter dependence on the wavenumber ZZ infrared active modes for isolated and bundles of SBNNTs in the low-wavenumber region.

A/D trend has been obtained [36, 37], with fitting constant values greater than those of BBNNTs. These results could be used to predict low wavenumbers for large diameter tubes.

To illustrate the symmetry dependence of the IR spectrum with chirality, the calculated ZZ (dashed lines) and XX (solid lines) IR spectra of bundles for five values of the chiral angle θ (0◦ , 10◦ , 15◦ , 20◦ , and 30◦ ), associated with (17, 0), (15, 4), (14, 5), (13, 7), and (10, 10) nanotubes, respectively, are displayed in figure 3: (1) in the low-wavenumber region (figure 3: left), weak modes depending on chirality are observed. In the XX spectrum (solid lines), two modes at 88 and 213 cm−1 are calculated for the (15, 4), (14, 5), (13, 7), and (17, 0) tube bundles, and a single mode, at 210 cm−1 , is expected for the (10, 10) armchair tube bundle. In the 4

J. Phys.: Condens. Matter 24 (2012) 335304

B Fakrach et al

Figure 3. Dependence of the BLM, intermediate, and TLM regions of BBNNT infrared spectrum, as a function of the chiral angle θ: ZZ (dashed line) and XX (solid line). The stars give the position of the modes of SBNNTs.

As mentioned above, for finite-size bundles, the nanotubes were placed parallel to each other on a finite-size trigonal array to form concentric shells. In figure 4, calculations of the average (non-oriented) IR spectrum have been performed on bundles of (9, 9) SBNNTs. The number of tubes was Nt = 1, 7, 19, and 61, in comparison with the crystal system (Nt = ∞). Bundles formed with nanotubes of infinite (figure 4: left) and finite L = 12.3 nm lengths (figure 4: right) were considered. In the following, we only discuss the results obtained in the intermediate and TLM ranges of the IR spectrum. In the intermediate range of IR spectrum of an isolated tube, a dominant mode located at 798 cm−1 is predicted, independently of the length of the nanotubes. This mode upshifts to 800 cm−1 , for an infinite bundle. On the low-wavenumber side of the previous mode, additional lines are observed in a bundle of finite size. In a real sample, the number of tubes in a bundle varies from a few tubes to hundreds of tubes. As a consequence, a broad band located in the intermediate range is expected in the experimental infrared spectrum of a SBNNTs macroscopic sample. In the TLM range, lines located at 1392 and 1393 cm−1 dominate the spectrum, independently of the number of tubes in the bundle nanotube length. The most important difference between infinite and finite tube lengths is the appearance of additional modes, located at 1389 and 1394 cm−1 in bundles formed by tubes of finite length. As a consequence, a non-symmetric profile extending to the low-wavenumber side of the maximum band located around 1393 cm−1 is expected to be experimentally observed. More generally, since the sample is a set of bundles of different sizes, formed of tubes of finite length having different diameters and chiralities, a broadening of the TLM band is expected in the IR spectrum measured on SBNNTs macroscopic samples.

ZZ spectrum (dashed lines), no mode appears for armchair bundle tubes, and a single peak centered at 158 cm−1 is found for zigzag and chiral tubes. Other modes are predicted with weak intensity. For each symmetry, let us call ωz and ωa the wave numbers of zigzag and armchair modes, respectively. We observe for both polarized infrared spectra an oscillatory wavenumber behavior that is related to the variations in diameter of the selected tubes: 1.34 nm (17, 0), 1.37 nm (15, 4), 1.35 nm (14, 5), 1.39 nm (13, 7), and 1.37 nm (10, 10). On the ZZ spectrum, the increase of θ from 0◦ (zigzag) to 30◦ (armchair) leads to the progressive vanishing of the mode located at a wavenumber close to ωz , and for θ = 30◦ , no mode was observed. The XX spectrum is dominated by a single mode observed for zigzag and armchair tubes. The increase of θ from 0◦ to 30◦ leads to the progressive appearance of two peaks at wavenumbers close to ωz and ωa , respectively. When θ increases, the intensity of the mode located around ωz , is transferred to that of the mode located around ωa . (2) In the intermediate wavenumber range (figure 3: middle), a strong (weak) mode located around 800 cm−1 is observed on the XX (ZZ) spectra as predicted. For all chiralities additional modes are predicted with weak intensities. (3) In the TLM region (figure 3: right), the XX and ZZ spectra are dominated by a single strong peak located around 1392 and 1393 cm−1 respectively for all chiralities. These results show that, in contrast with the radial region behavior, the IR spectrum of BBNNTs is weakly dependent on the chirality, in the intermediate and TLM regions.

3.2. Finite BBNNTs To investigate the effects of finite size on IR-active modes of homogeneous BBNNTs, we developed calculations of IR spectra of bundles of different numbers of infinite nanotubes.

5

J. Phys.: Condens. Matter 24 (2012) 335304

B Fakrach et al

Figure 4. Dependence of the average infrared spectrum (non-oriented sample) on the number of tubes in a bundle. Bundles are formed by (9, 9) SBNNTs of infinite length (left) and finite length L = 12.3 nm (right). All spectra are displayed with the same intensity scale.

4. Concluding remarks

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We recall that, in this work, we have investigated infrared spectra of homogeneous bundles of SBNNTs. Thanks to SMM, we have investigated bundles consisting of more than 100 SBNNTs of several sizes. Unlike in the low-wavenumber region, the spectrum of BBNNTs is weakly dependent on chirality and diameter, in the intermediate and TLM regions.

Finite size (length and number of tubes) effects are observed, essentially in the intermediate region with additional modes, which can be considered for the analysis of experimental data. The number of experimental data on the infrared active modes of boron nitride nanotubes is limited [12, 13, 22, 25].

In agreement with our results, IR-active modes are clearly identified in the intermediate range 650–850 cm−1 and in the TLM one 1350–1450 cm−1 . Finally, we think that the calculated IR spectra reported in the present work can be useful to understand future IR data.

Acknowledgment The work was supported by a CNRS-France/CNRSTMorocco agreement.

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