3. Raman Scattering and Tensor Invariants 45
3.3. Experiments
Before I turn to the description of the experimental setup for measuring the tensor invari-ants and the results for carbon nanotubes, I want to discuss in general the conclusions which can be drawn from experiments on unoriented samples. The only symmetry which has a non-vanishing isotropic part ¯α is the fully symmetric representation in any point or space group. Thus, phonons of A1symmetry are readily distinguished from the other species. Of particular interest is the observation of antisymmetric contributions to the Raman intensity, i.e.,γas2 6=0. If only the antisymmetric component is present the Raman peak originates from a phonon transforming as the totally antisymmetric representation, in the point group of nan-otubes this is A2(see Table 3.2). Totally antisymmetric scattering is rather rarely observed experimentally; more frequent is an antisymmetric contribution to a degenerate mode.82–84 The only possible modes in nanotubes showing mixed symmetric and antisymmetric scat-tering are E1 modes. For example, a strong incoming resonance with an optical transition which is allowed in z polarization, but forbidden in x or y polarization yields in the matrix representation of the E1Raman tensor c6=d in Table 3.2. Such antisymmetric contributions where reported by Rao et al.74, 85, 86 for multiwall nanotubes. Degenerate modes have a sym-metric anisotropyγs2 different from zero, but ¯α2=0. A large ratio ¯α2/γs2 can serve as an indicator for scattering by E1and E2symmetry modes in carbon nanotubes.
Triple-grating DILOR XY800
The setup is shown schematically in Fig. 3.2. The laser light first passes a small prism monochromator to remove the plasma lines of the laser. It is then focused onto the sample with a commercial photo objective ( f =35 mm) or a achromat-meniscus lens ( f =60 mm).
The scattered light is collected in backscattering geometry. The first two gratings of the spectrometer are used to suppress the elastically scattered light. They are arranged in sub-tractive mode – placed “back to back” – and thus act as a narrow band pass filter of variable wavelength. The light is dispersed by the last grating (1800 mm−1) and collected by a charge-coupled-device detector. In the green and blue energy range a liquid nitrogen cooled back-thinned CCD was used, in the red a Peltier cooled CCD without back thinning. The resolution achieved with a 200µm entrance slit is between 1.5 cm−1at 647 nm and 3 cm−1 at 488 nm, which is much smaller than the width of the Raman peaks observed in carbon nanotubes. The specific strengths of this spectrometer for experiments on nanotubes are its flexibility, i.e., it is not restricted to particular excitation energies, and for the observation of the radial breathing mode at ≈200 cm−1 the narrow width of the filtering by the first two gratings.
Single-grating DILOR LABRAM
The Labram is a compact micro Raman spectrometer depicted in Fig. 3.3. The plasma lines are removed by an interference filter before the notch filter reflects the laser beam into an optical microscope. An 100x objective was used to focus the light onto the sample and collect the backscattered radiation. The scattered light again reaches the notch filter, where the light
triple grating spectrometer entrance optics
CCD
Ar laser Ar /Kr
laser
+ +
prism mono
CCD (infrared) sample
Fresnel
Figure 3.2: Experimental setup for the Raman measurements with the DILOR XY800. The plasma lines of the Ar+/Kr+ or Ar laser are removed by the prism monochromator. After passing a Fresnel rhomb the light is reflected and focused onto the sample. The backscattered light is analyzed with a triple-grating spectrometer and collected by a CCD.
Ar laser HeNe laser
interf.
notch filer pin hole
spectrometer with CCD
microscope sample Fresnel
Figure 3.3: Experimental setup for Raman measurements with the DILOR LABRAM. The entrance optics of the Labram spectrometer are shown schematically. An interference filter (interf.) rejects the plasma lines of the HeNe or Ar laser. The laser is reflected by the notch filter and focused with a microscope objective onto the sample. The Raman scattered light passes the notch filter, is focused onto the entrance slit of a single-grating spectrometer. The spectrometer is equipped with a Peltier cooled CCD.
within a bandwidth of ≈3 nm of the laser line is reflected, whereas the other wavelengths pass the notch filter. To further suppress stray light and to increase the spatial resolution a pin hole with a diameter d=200µm is placed into an intermediate focus. Finally the scattered light is focused onto the entrance slits of a single grating spectrometer (1200 mm−1) and collected with a Peltier cooled CCD. By the construction of the entrance optics the Labram is restricted to excitation wavelengths for which an interference and notch filter combination is available; I used the 632 nm HeNe line and the 514 and 488 nm Ar line for excitation.
Single grating spectrometers became increasingly popular because they are easy to handle and comparatively cheap. Another advantage is their high throughput allowing the obser-vation of weak signals. On the other hand, the obserobser-vation of Raman modes close to the laser line is usually not possible. I was only able to record the radial breathing mode of single walled nanotubes when working with red excitation and using two SuperNotch filters to suppress the Raleigh scattered light. For the green and blue laser line only a single Su-perNotch and SuSu-perNotchPlus filter was available, respectively. Moreover, the difference in wavelength between the incoming and outgoing light is 1.5−2 times smaller for these exci-tation energies than for 632 nm; the light scattered by the radial breathing mode was within the band widths of the notch filter. The experimental resolution was around 3 cm−1 in the red and 5cm−1in the blue energy range.
3.3.1. Polarized measurements
The tensor invariants of the Raman scattered light are obtained from a linear combination of the intensities in linear and circular polarization as shown in Section 3.2.. To determine them experimentally I need to record the Raman intensities in parallel, perpendicular, corotating, and contrarotating polarization without changing the illumination level or removing any po-larizing elements in the light path between the measurements.76, 77, 82 The basic setup I used – a combination of two linear polarizing elements (a Fresnel rhomb and a polarization filter) and a λ/4 wave plate – is shown in Fig. 3.4. With the position of the polarizing elements as indicated in the figure a backscattering Raman spectrum under corotating polarization is recorded: The light coming from the laser is vertically polarized. After passing the Fres-nel rhomb (90◦ position) the light is horizontally linear polarized. The angle between the principle axis of the λ/4 wave plate and the horizontal plane is 45◦; the light is circularly polarized after passing the wave plate. It is then focused onto the sample and the scattered beam consists of a left and right hand circularly polarized part. Going back through the λ/4 wave plate the circular polarization is converted into linear polarization. The corotating part is polarized vertical and the contraroting part horizontal (the circular polarizations are specified in the lab frame). Only vertically polarized light can pass the analyzer and thus be recorded by the spectrometer and the CCD.
In the experiments the polarization direction of the analyzer was chosen according to the larger sensitivity of the spectrometer, which is horizontal in the blue and vertical in the red energy range. The intensities in the four different polarizations are then obtained by rotating the Fresnel rhomb and theλ/4 wave plate. Let us assume that the analyzer is vertical, the
Ar/Kr Laser Fresnel rhomb
(90°)
l/4 wave plate (45°)
Spectrometer with CCD
Analyser (vertical) Nanotubes
Figure 3.4: Setup for the measurements of the tensor invariants. The polarization direction of the incoming light is chosen by the Fresnel rhomb. The laser then passes aλ/4 zero-order wave plate and is focused onto the sample. The scattered light comes back through theλ/4 plate, is analyzed with a polarization filter, and focused onto the entrance slits of the spectrometer.
Figure 3.5: Raman spectra of CCl4 a) under linear and b) circular polar-ization. Next to the fully symmetric mode at 460 cm−1 and the mode at 314 cm−1the measured depolariza-tion ratioρ and reversal coefficient P are given.
200 400 600 800 200 400 600
P = 5.0 P = 0.04
b)
Raman Shift (cm
-1)
ρ = 0.75
ρ = 0.01
CCl
4a)
Raman intensities are then given by
parallel Ik Fresnel rhomb 0◦ λ/4 wave plate 0◦ analyzer vertical
perpendicular I⊥ 90◦ 0◦ vertical
corotating I 0◦ 45◦ vertical
contrarotating I 90◦ 45◦ vertical
I used high-precision zero-order wave plates for 488 and 647 nm excitation wavelength; only a multi-order wave plate was available for 514 nm. Theλ/4 wave plate was placed right be-fore the focusing lenses or the microscope objective and aligned perpendicular to the laser beam. The angular positions of all polarizing elements I verified by looking at the minima and maxima in the intensity of the fully polarized laser light reflected at a metall surface and by measuring the known tensor invariants of CCl4. Figure 3.5 shows the Raman spectra of CCl4in a) the two linear and b) the circular polarizations. In this molecule the fully symmet-ric mode is characterized by only the isotropic invariant ¯α2 being different from zero. For the A1g peak at 460 cm−1 a depolarization ratioρ =0 and a reversal coefficient P=0 are expected, which is in excellent agreement with the measured values given in Fig. 3.5. All other modes should showρ =0.75 and P=6 ( ¯α2=0,γas2 =0). Deviations from the theo-retical value are usually found for the reversal coefficient, because the circular polarizations are much stronger affected by the non-ideal backscattering.70, 82 Whereas the theoretical de-polarization ratio is the same regardless of the scattering geometry, the reversal coefficient, e.g., in forward scattering is the inverse of the backscattering value. In CCl4 the measured P for the non-fully symmetric modes varies between 3.6 at 780 cm−1 and 5.3 at 220 cm−1. A better (more robust) indicator for the symmetry of a Raman mode than the raw values ofρ and P are the ratios between the tensor invariants. In particular, the ratio between the symmetric anisotropy and the isotropic invariant γs2/α¯2=0.4 for the 460 cm−1 mode, but γs2/α¯2is well above 100 for all other modes.
Figure 3.6: Raman tensor in-variants of the high-energy modes in single walled nanotubes excited with 488 nm. a) Parallel and per-pendicular linear polarization yield-ing ρ=0.35 for all Raman peaks.
b) Corotating and contrarotating po-larization with P=1.01.