Intrinsic phonon properties of double-walled carbon nanotubes^*

The RBLM frequencies depend on the diameter of the constituent layers and on the mechanical coupling between the inner and outer layers [9]. It was shown experimentally that an up-shift of both in-phase and out-of-phase RBLM frequencies in comparison with the RBM frequencies of related individual SWNTs is observed [10, 11]. It was also experimentally established that the amplitude of the upshift depends on the inter-layer distance between concentric layers [11, 12].

In order to understand these results, different models of van-der-Walls coupled harmonic oscillators were developed in order to predict the dependence of the RBLM frequencies on the diameters of the inner and outer layers, associated to a given wall-to-wall distance [11, 13, 14]. In the simplest approach (see Liu et al [11]), the frequencies of the in-phase (ω_L) and out-of-phase (ω_H) RBLMs are given by

$$\begin{eqnarray}&&{{\omega}_{\text{L}}}=\frac{1}{2\pi c}{{\omega}_{\text{c}}}~\sqrt{\frac{1}{2}\left[\left({{z}_{1}}+{{z}_{2}}\right)-\sqrt{{{\left({{z}_{1}}-{{z}_{2}}\right)}^{2}}+4}\right]},\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&{{\omega}_{\text{H}}}=\frac{1}{2\pi c}{{\omega}_{\text{c}}}~\sqrt{\frac{1}{2}\left[\left({{z}_{1}}+{{z}_{2}}\right)+\sqrt{{{\left({{z}_{1}}-{{z}_{2}}\right)}^{2}}+4}\right]~},\end{eqnarray} \tag{ 2 }$$

where $\omega _{\text{i}}^{2}=\frac{{{k}_{\text{i}}}}{{{m}_{\text{i}}}}$, $\omega _{\text{o}}^{2}=\frac{{{k}_{\text{o}}}}{{{m}_{\text{o}}}}$, $\omega _{\text{c}}^{2}=\frac{{{k}_{\text{c}}}}{\sqrt{{{m}_{\text{i}}}{{m}_{\text{o}}}}}$, ${{z}_{1}}=\frac{\omega _{\text{i}}^{2}}{\omega _{\text{c}}^{2}}+\sqrt{\frac{{{m}_{\text{o}}}}{{{m}_{\text{i}}}}}$ and ${{z}_{2}}=$ $\frac{\omega _{\text{o}}^{2}}{\omega _{\text{c}}^{2}}+\sqrt{\frac{{{m}_{\text{i}}}}{{{m}_{\text{o}}}}}$, m_i (m_o) are inner-wall (outer-wall) unit-length mass of the constituent SWNT and they are determined from the knowledge of the diameters of inner and outer SWNTs which are directly related to their chiral indices (n_i, m_i) and (n_o, m_o), respectively [11].

The intrinsic force constant k_i (k_o) for inner (outer) constituent SWNT can be accurately determined from the individual radial breathing mode (RBM) frequency-diameter relationship [15]. It should be emphasized that we have to use the RBM frequency-diameter relations adapted to the experimental environment of each layer. In our experimental conditions (DWNT in air and room temperature), the frequencies of the RBM of the inner tube is given by ω_i  =  228/d_i (nm) and that of the outer tube by ω_o  =  228 (cm^–1 nm) /d_o (1  +  C_e$d_{\text{o}}^{2}$)^1/2, with C_e  =  0.065 nm^–2 arising from the interaction of the outer-wall with the ambient atmosphere. The unit-length coupling force constant, k_c, characterizes the van-der-Waals interaction between the two concentric carbon nanotube layers. It can be approximated using the average unit-area inter-layer van-der-Waals potential U_vdw as

$$\begin{eqnarray}&&{{k}_{\text{c}}}=\frac{{{\partial}^{2}}{{U}_{\text{vdw}}}}{\partial {{r}^{2}}}\pi \frac{\left({{d}_{\text{o}}}+{{d}_{\text{i}}}\right)}{2},\end{eqnarray} \tag{ 3 }$$

where (∂²U_vdw/∂r²) is the unit-area force constant between the layers. The dependence of the unit-area force constant (∂²U_vdw/∂r²) on the inter-walls distance, δr, was recently established from measurements on index-identified DWNTs (see figure 3(d) of reference [11]). Therefore given wall- to-wall distance, δr, accurately derived from the assignment of the DWNT, one can use the figure 3(d) of reference [11] to estimate the value of (∂²U_vdw/∂r²), and then k_c, for each index-identified DWNT.

Using the previous expressions, we plot in figure 3 the calculated RBLM frequencies as a function of the diameter of inner (outer) nanotubes for wall-to-wall distance, δr  =  ▵D/2, ranging from 0.32 to 0.38 nm. As expected, (i) RBLM are upshifted with respect to the RBM of the constituent SWNT, (ii) the RBLM frequencies decrease with increasing diameter of inner (outer) tubes, (iii) for the same inner (or outer) diameter, the RBLMs strongly depend on the wall-to-wall distance.

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Figure 4 summarizes the experimental dependence of the RBLMs recently obtained from Raman experiments performed on several index-identified DWNTs.

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It is confirmed that RBLM are upshifted with respect to the RBM of the constituent SWNT (the values of the shifts are given in parentheses in the figure 4). On the other hand, we clearly observe that: (i) for close inter-layer distance, both RBLM downshift when the tube diameters increase (figure 4, vertical arrow); (ii) for close average tube diameters and an interlayer distance δr larger than 0.335 nm (close to the equilibrium distance in graphite: 0.34 nm), the out-of-phase RBLM downshifts when the interlayer distance increases (figure 4, horizontal arrow). Both behaviors are in good agreement with the prediction of our simple model (see figure 3). As shown recently [11, 16], a given value of δr can be associated to an effective pressure between the layers: an interlayer distance larger (smaller) than 0.34 nm leads to an effective negative (positive) pressure between the layers. Negative (positive) pressure means that the inner-outer interaction causes a slight contraction (expansion) of the outer tube concomitant with the expansion (contraction) of the inner tube [11, 16]. In other words, the dependence of the shift of the RBLM on δr can be understood as a dependence on the effective pressure between the layers. It should be emphasized that the effective pressure (positive or negative) between the tubes reaches gigapascals owing to variations in the wall-to-wall distance [11]. Note that a negative pressure in the gigapascal range is difficult to achieve using conventional approaches.

On the basis of the results established from Raman experiments performed on individual index-identified SWNTs [7], one expects to observe, in the range of the G modes, 4 (chiral@chiral), 3 (chiral@achiral or achiral@chiral) or 2 (achiral@achiral) components in the (// //) polarized Raman spectrum measured on an individual DWNT. Each component is assigned to the G⁺ or G⁻ component of the coupled inner and outer constituent SWNTs. We remind that the G⁺ component is assigned to the LO (TO) G mode and the G⁻ component to the TO (LO) G mode in SC (M) SWNT [7, 17]. However, in DWNTs, some components can appear at close frequencies and thus cannot be experimentally resolved.

Figure 5 compares the profile of G modes measured on index-identified SC@SC, M@SC, and SC@M DWNTs. We clearly establish that, as in SWNTs, the profile of the G⁻ component of each layer is broad (narrow) when the layer is metallic (SC) (figure 5, vertical direction). As a consequence, in terms of characterization, the observation of a broad component unambiguously identifies the presence of a metallic nanotube within the DWNT.

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Recently the dependence of the G-modes frequencies on the diameter and inter-layer distance in SC@SC DWNT was reported [18]. It was found that the TO (G^– component) and LO (G⁺ component) frequencies of the inner and outer tubes shift significantly in SC DWNTs with respect to the frequencies of the same modes in each constituent SWNT. Such dependence is illustrated on figure 5 (horizontal direction). The direction and values of the shift depend on the inter-layer distance [18]. Especially, as shown in figure 5 (inset), the shift of the G modes of the inner layer with respect to the G modes of the related SWNT increases with the inter-layer distance. We unambiguously establish that the larger the interlayer distance is, the larger is the shift of the LO and TO modes. As for RBLM, this shift is due to the negative effective pressure experienced by the inner and outer nanotube which occurs when the inter-layer distance is larger than the equilibrium distance evaluated at 0.335 nm in DWNT (close to the inter-layer distance in graphite, 0.34 nm).

In figure 6 we plot together the frequencies of the TO G-mode of the inner tube measured on index-identified DWNT (blue symbols) and, on the other hand, the ones obtained on (6,5)@SC DWNTs by Villalpando-Paez et al (red symbols) [19]. It must be emphasized that the index-assignment of the outer tubes of these later DWNTs, and then their δr values, was reexamined by comparing the experimental RBLMs of reference [19] with the results of the model presented in the previous subsection of this paper. A shift of the TO mode is observed in a large interlayer distance range. An upshift is found for δr smaller than 0.335 nm (positive pressure effect) and a downshift for δr larger than 0.335 nm (negative pressure effect). Theoretical calculations are in progress to relate these shifts to the amplitude of the effective pressure due to the structural relaxation (contraction or expansion) of the layers and including the interaction of the relaxed layers [20].

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