4. Nanotubes Under Hydrostatic Pressure 65
4.4. Phonon eigenvectors of chiral tubes
4.4.1. Eigenvectors in small nanotubes
I first present the eigenvectors and frequencies of the two achiral nanotubes. The excellent agreement between my calculated and the symmetry imposed eigenvectors made me confi-dent about the calculational procedure. Figure 4.7a) and Table 4.4 show the A1g, E1g, and E2g eigenvectors and frequencies of the high-energy modes in (6,6) and b) in (10,0) nan-otubes. All high-energy modes are found to be softened compared to graphene (1660 cm−1 in my calculations) in these fairly small tubes, which is in good agreement with previous ab initio calculations and experimental findings.40, 106 The softening appears to be a little overestimated for the A1circumferential mode, which is of A1u symmetry in the (10,0) tube and at similar frequency (1550 cm−1) as the A1g mode in the (6,6) armchair tube. Note that the ordering of the frequencies is non-trivially changed in the calculated tubes as compared to simple zone folding. Both confinement and curvature effects are, however, much stronger
∗This section and parts of the following section are taken from Ref. 12. I did the calculations and the analysis presented in the reference under supervision of Pablo Ordej´on and Christian Thomsen who co authored the paper.
Table 4.4: Frequencies and angle α between the displacement and the circumferential direction for the Raman-active high-energy modes and the ra-dial breathing mode in achiral nanotubes. The largest deviation (4◦) from the symmetry imposed displacement was found for the E2g mode of the zig-zag tube.
(6,6) (10,0)
ν (cm−1) α (◦) ν (cm−1) α (◦)
RBM 285 0 289 2
A1g 1545 0 1604 88
E1g 1600 90 1606 2
E2g 1629 0 1552 86
A
1gE
1gE
2gA
1gE
1gE
2ga.)
b.)
Figure 4.7: Raman active A1g, E1g, and E2g high-energy modes a) of a (6,6) nanotubes and b) of a (10,0) nanotube.
in the calculated tubes than in real samples with a typical mean diameter≈14 ˚A. In phonon calculations for achiral tubes of different diameter Dubay and Kresse found these effects to decrease rapidly with increasing diameter.126 The calculated frequencies of the radial breathing mode in Table 4.4 are for achiral nanotube bundles. As can be seen in Fig. 4.7 the displacement patterns are in excellent agreement with the symmetry requirements. I found that the angleα between the atomic displacement and the circumference deviates by at most 5◦from the circumferentialα=0◦ or axialα =90◦direction, showing the numerical accu-racy of the calculation. I stress again that in the calculation of the force constants the mirror symmetries were not explicitly used.
In chiral nanotubes the atomic displacements are no longer, in general, along the cylindrical axes. In Fig. 4.8a.) I show an A1mode of an (8,4) tube and in b.) of a (9,3) tube. It can be seen that the displacement is along the circumference in the (8,4), but parallel to the bonds in the (9,3) nanotube. The smallest angle between the carbon-carbon bonds and the circumference in the (9,3) tube is 30◦−θ =16.1, which coincides with the displacement direction as given Figure 4.8: a) A1 high-energy eigenvector of an
(8,4) nanotube with a frequency of 1505 cm−1. The atomic displacement is parallel to the circumference, i.e.,α=3◦is close to zero. b) A1high-energy eigen-vector of a (9,3) tube (1627 cm−1). The displace-ment is parallel to the carbon-carbon bonds. The di-rection of the helix in both tubes, which is obtained from the screw axis operation, is indicated by the gray lines.
a.) (8,4)
b.) (9,3)
Table 4.5: Frequencies and displacement di-rections for the high-energy and the radial breathing mode in chiral nanotubes. If the displacement direction was found not to be constant around the tube’s circumference the mean value ofα was given.
(8,4) (9,3)
ν (cm−1) α (◦) ν (cm−1) α (◦)
RBM 275 1 292 1
A1 1670 -87 1627 16
A1 1505 3 1519 -74
E1 1610 33 1620 38
E1 1591 -59 1607 -51
E2 1626 -6 1638 7
E2 1554 85 1587 -84
in Table 4.5. In the (8,4) tube I also found vibrations along the bond direction, but they were of B1and B2symmetry and hence not optically active.
I now take a look at the degenerate modes in chiral tubes. In Fig. 4.9 I show an E1eigenvector of an (8,4) nanotube. I successively rotated the nanotube by 32◦to show the reader how the eigenvector evolves when going around the nanotube. As expected the magnitude of the atomic displacement (the length of the ticks) is modulated by a sinϕ function. Contrary to what is generally expected, however, the direction of the displacement varies as well.
Whereas the tick on the highlighted atom is perpendicular to one of the carbon-carbon bonds in the first picture, i.e., α ≈40◦, they are almost parallel to the bonds withα ≈10◦ in last two pictures; Table 4.5 lists the mean valueα =33◦. For the E2eigenvector in the (8,4) tube no such variation is seen. The same sequence as in Fig. 4.9 is presented in Fig. 4.10 for the E2mode with a frequency of 1626 cm−1. The atomic displacement is roughly parallel to the circumference in all four pictures.
The angular dependence of the direction of the eigenvectors becomes more obvious when plotting the atomic displacement along the z axis versus the circumferential displacement (Fig. 4.11). The first two diagrams at the top in Fig. 4.11 corresponds to the A1high-energy
(8,4) E
10° 32° 64° 96°
Figure 4.9: Doubly-degenerate E1eigenvector of an (8,4) tube with a frequency of 1610 cm−1. The sequence shows the change in displacement when going around the tube in steps of 32◦. The atoms which are highlighted by the small circles are connected by the screw symmetry of the tube.
modes in the (8,4) and (9,3) tube. For simplicity I considered only the displacement of the atoms belonging to the same graphene sublattice. Totally symmetric modes then show up as a single point in such a plot. The dashed lines point along the carbon-carbon bonds; the co-incidence between the displacement and the bond direction in the (9,3) nanotube is obvious.
A B symmetry mode (not shown) would be seen as two points in the z−xy displacement diagram, because the character of the screw axis generator is−1.45 Open ellipses describe E symmetry eigenvectors with a varying or “wobbling” displacement direction. The principal axis of the ellipse gives the average angle of the eigenmode, i.e., α. The open ellipse with α ≈33◦ of the (8,4) E1 symmetry mode corresponds to the eigenvectors in Fig. 4.9. The degenerate eigenmode has the same ellipse. In general, the displacement of a degenerate eigenmode is obtained from a given eigenvector by a 90◦rotation of the coordinate system around the z axis. The relationship between the axial and the circumferential displacement is thus the same for degenerate phonons. Two ellipses perpendicular two each other repre-sent degenerate modes of the same symmetry but different frequencies. Correspondingly, the middle diagram of the (8,4) tube in Fig. 4.11 shows both the 1610 (α =33◦) and the 1591 cm−1(−59◦) E1eigenvector.
In the middle panel to the right I depict the displacement for the E1 modes in the (9,3) nanotube. The xy and the z displacment are again of similar magnitude, i.e., α ≈40◦ and 50◦, but they are now in phase and yield a constant direction of the atomic displacement around the tube. Two other examples of almost closed ellipses are the E2 eigenmodes in both tubes (lowest panels in Fig. 4.11). Although the eigenvectors are in fact wobbling, the magnitude of the, e.g., the z displacement for the 1626 cm−1 E2 (8,4) mode is very small.
Therefore, the variation is not observable in the full eigenmode plot of Fig. 4.10.
(8,4) E
20° 32° 64° 96°
Figure 4.10: E2high-energy mode of an (8,4) nanotube with a calculated frequency of 1626 cm◦. The atomic displacement is along the circumference and, in contrast to Fig. 4.9, no wobbling is evident.
The modulation of the displacement magnitude by a sin 2ϕ around the circumference is nicely seen when following the highlighted atom.
Figure 4.11: z component of the displace-ment versus the xy circumferential compo-nent. Top: A1 high-energy eigenvectors of an (8,4) tube (left) and a (9,3) tube (right).
The dashed lines point into the direction of the carbon-carbon bonds, the full lines are parallel to aaa111 and aaa222. Middle: E1 eigenvec-tors of the two chiral nanotubes. The atomic displacement direction is strongly varying in the (8,4) nanotube resulting in an open dis-placement ellipses. In the (9,3) nanotube the z and circumferential component are similar in magnitude but with an almost vanishing phase difference. Bottom: E2 eigenvectors of the chiral tubes. The small z or xy component re-sult in a closed ellipse almost parallel to the circumference or the nanotube axis.
-8 -4 0 4 8 12
-12 -8 -4 0 4 8 12
-10 -5 0 5 10 -12
-8 -4 0 4 8
-10 -5 0 5 10
(8,4)
A
1(9,3)
E
1z Displacement
E
2xy Displacement
The symmetry of the phonon together with its displacement ellipse fully determine the phonon eigenvectors. The symmetry yields the sin mϕ function of the axial and circumfer-ential components, whereas the principle axes of the ellipses specify the relative magnitude and the phase shift between the two components.
In Section 2.4. I discussed that symmetry arguments are not able to predict the phonon eigen-vectors of chiral nanotubes. The absence of mirror planes and the low-symmetry confinement wave vectors in chiral tubes affect their vibrational properties in a fundamental manner. In particular, a classification of the high-energy modes into LO and TO vibrations is not appli-cable to chiral nanotubes.