Breathing-like modes of
bundles of SWNTs and isolated MWNTs
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The valence force-field model, appended with
Lennard-Jones potential for the interlayer interactions, was applied to the
calculation of the lattice dynamics of dimers and infinite bundles of SWNTs
of armchair and zigzag type [1,2], and isolated
MWNTs with layers of armchair and zigzag type [3]. In all cases, the
non-resonant Raman intensity was derived using a bond-polarizability model. The results for the breathing-like mode (BLM)
frequencies and peak Raman intensity show that the mechanical coupling
between the layers in the bundles of SWNTs and in isolated MWNTs has major
impact on the Raman spectra of these structures with respect to those of the
isolated layers. 1) Dimers of SWNTs [1]. Fig. 1, left, shows the atomic displacements
of the two BLMs of dimers of armchair layers. The displacements for the
small-radius layers are almost unperturbed by the interlayer interactions,
and are essentially the same as for the isolated layers. However, the
interactions yield mixing of the displacements patterns of the layers, which
increases with increasing the layer’s radius. Fig. 1, right, shows the BLM frequencies for
dimers with zigzag and armchair layers in comparison with the breathing mode
(BM) frequency of the isolated layers. The interlayer interactions give rise
to splitting of the BLM frequencies up to a few tens of cm-1. The
inset shows the peak Raman intensities of the BLMs in comparison with those
of the isolated layers. It can be seen, that for some layers, the two BLMs
have comparable peak Raman intensities. The results for the RBMs derived within the
atomistic model compare well to the experimental Raman data on dimers [4],
which supports the applicability of the model with mechanical coupling to the
description of the RBMs of dimers, at least at the semi-quantitative level. 2) Infinite bundles of SWNTs [2] Fig. 2, left, shows the atomic displacements
of the BLMs of infinite bundles of armchair layers, arranged in a hexagonal
lattice. According to the atomic displacement patterns, there are (at least)
two BLMs for each bundle, BM(1) and BM(2). Fig. 2, right, shows the frequencies of the
two BLMs of infinite bundles of zigzag and armchair layers in comparison with
those of the BM of the isolated layers. The lower frequency BLM arises from
the BM of the isolated layers, modified by interlayer interactions. The
higher-frequency BLM arises from mixing of the BM with another mode of the isolated
layer with close frequency. The peak Raman intensities of both BLMs are
comparable for layer radii, larger than 1 nm. 3) Double-wall CNTs (DWNTs) [3]. Fig. 3, left, shows the atomic displacements
for the two BLMs of DWNTs. The lower-frequency BLM is an in-phase one, while
the higher-frequency BLM is an out-of-phase one. There is an obvious mixing
mainly of the higher-frequency BLM with other layer modes with close
frequencies. Fig. 3, right, shows the frequencies of the
two BLMs of DWNTs with zigzag and armchair layers in comparison with those of
the BMs of the isolated layers. The frequencies, derived within the continuum
approach (Model I) are also provided. The results for the frequencies from
the atomistic and continuum models are in agreement with each other. It
should be noted that the frequencies of the BLMs are always higher than those
of the isolated layers, in agreement with the predictions of other authors.
The inset shows the peak Raman intensity of the BLMs of the two layers of a
DWNTs in comparison with that of the isolated layers. The atomistic model of the lattice dynamics
of the DWNTs has been applied to the study of the BLMs of DWNTs with a narrow
inner layer and various outer layers [5,6,7]. The
comparison of the calculated BLM frequencies with the extensive experimental
data shows that the atomistic model with mechanical coupling can describe the
interlayer interactions at the semi-quantitative level [8]. 4) Triple-wall CNTs (TWNTs) [3]. Fig. 4, left, shows the atomic displacements
for the three BLMs of TWNTs. The lowest-frequency BLM is an in-phase one,
while the higher-frequency BLMs are out-of-phase ones. There is a small
mixing of the BLMs with other layer modes with close frequencies. Fig. 4, right, shows the frequencies of the
three BLMs of TWNTs with zigzag and armchair layers in comparison with those
of the BMs of the isolated layers. The frequencies, derived within the
continuum approach (Model I) are also provided. The results for the
frequencies from the atomistic and continuum models are in agreement with
each other. It should be noted that the frequencies of the BLMs are always
higher than those of the isolated layers, in agreement with the predictions
of other authors. The inset shows the peak Raman intensity of the BLMs of the
three layers of a TWNTs. 5) MWNTs [3]. Fig. 5 shows the calculated frequencies of
the BLMs of MWNTs for a given inner layer radius Rin. It is clearly
seen that: -the frequency of the lowest-frequency BLM
(totally in-phase BLM) decreases with the increase of the number of layers N and tends to zero in the limit N → ∞. In this
limit, the atoms of the layers are displaced in-phase perpendicularly to the
layer and the mode tends to the transverse acoustical phonon of multilayer
graphene with zero frequency. -the frequency of the highest-frequency BLM
(totally out-of-phase BLM) tends to a finite value in the limit N → ∞. In this
limit, the atoms of adjacent layers are displaced out-of-phase and the mode
tends to the transverse optical phonon of multilayer graphene with calculated
frequency of 127 cm-1, which corresponds well to the experimental
value. References: 1. L. Henrard, V. N. Popov, and A. Rubio, Phys. Rev. B 64
(2001) 205403. 2. V. N.
Popov and L. Henrard, Phys. Rev. B 63 (2001)
233407. 3. V. N.
Popov and L. Henrard, Phys. Rev. B 65 (2002)
235415. 4. A. Débarre, M. Kobylko, A. M. Bonnot, A. Richard, V. N. Popov, L. Henrard,
and M. Kociak, Phys. Rev. Lett. 101 (2008) 197403. 5. R.
Pfeiffer, Ch. Kramberger, F. Simon, H. Kuzmany, V. N. Popov, and H. Kataura,
Eur, Phys. J B 42 (2004) 345. 6. R.
Pfeiffer, F. Simon, H. Kuzmany, and V. N. Popov,
Phys. Rev. B 72 (2005) 161404(R). 7. H.
Rauf, T. Pichler, R. Pfeiffer, F. Simon, H. Kuzmany, and V. N. Popov, Phys. Rev. B 74 (2006) 235419. 8. D. Levshov, T. Than, R. Arenal, V. N. Popov, R. Parret, M. Paillet, V. Jourdain, A. A. Zahab, T. Michel, Yu. I. Yuzyuk, and J.-L. Sauvajol, NanoLett. 11 (2011) 4800-4804. |
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Fig. 1. Left: The atomic displacement patterns of the BLMs of dimers of
armchair layers. Right: Frequencies of the two BLMs of dimers of zigzag and
armchair layers in comparison with those of the breathing mode of the
isolated layers (BM). Inset: the corresponding peak Raman intensities [1]. |
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Fig. 2. Left: The atomic displacement patterns of the BLMs of infinite bundles of armchair layers. Right: Frequencies of the two BLMs of infinite bundles of zigzag and armchair layers in comparison with those of the breathing mode of the isolated layers (BM). Inset: the corresponding peak Raman intensities [2]. |
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Fig. 3. Left: The atomic displacement patterns of the two BLMs of DWNTs of armchair layers. Right: Frequencies of the two BLMs of DWNTs with zigzag and armchair layers in comparison with those of the BMs of the isolated layers. Inset: the corresponding peak Raman intensities [3]. |
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Fig. 4. Left: The atomic displacement patterns of the three BLMs of TWNTs of armchair layers. Right: Frequencies of the three BLMs of TWNTs with zigzag and armchair layers in comparison with those of the BMs of the isolated layers. Inset: the corresponding peak Raman intensities [3]. |
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Fig. 5. Frequencies of the BLMs of MWNTs for a given inner layer radius and
different number of layers [3]. |
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Valentin Popov
May 14, 2006