Polarization dependence of coherent phonon generation and detection in highly-aligned single-walled carbon nanotubes

We have investigated the polarization dependence of the generation and detection of radial breathing mode (RBM) coherent phonons (CPs) in highly-aligned single-walled carbon nanotubes. Using polarization-dependent pump-probe differential-transmission spectroscopy, we measured RBM CPs as a function of angle for two different geometries. In Type I geometry, the pump and probe polarizations were fixed, and the sample orientation was rotated whereas, in Type II geometry, the probe polarization and sample orientation were fixed, and the pump polarization was rotated. In both geometries, we observed an almost complete quenching of the RBM CPs when the pump polarization was perpendicular to the nanotubes. For both Type I and II geometries, we have developed a microscopic theoretical model to simulate CP generation and detection as a function of polarization angle and found that the CP signal decreases as the angle goes from 0${}^{ˆ}$ (parallel to the tube) to 90${}^{ˆ}$ (perpendicular to the tube). We compare theory with experiment in detail for RBM CPs created by pumping at the $E_{44}$ optical transition in an ensemble of single-walled carbon nanotubes with a diameter distribution centered around 3 nm, taking into account realistic band structure and imperfect nanotube alignment in the sample. We theoretically determined a $\cos {}^8(\theta )$ dependence for Type I CP spectroscopy experiments and a $\cos {}^4(\varphi )$ polarization dependence for Type II CP spectroscopy experiments and, after including misalignment effects to our fitting, we determined the nematic order parameter of our sample to be 0.81.

Figure 1

Absorption spectra of aligned single-walled carbon nanotubes in the (a) near-infrared to visible and (b) far-infrared (or terahertz) ranges for parallel ($A_{\parallel }$) and perpendicular ($A_{\perp }$) polarization.

Figure 2

Typical differential transmission of the pump-probe signal for highly-aligned single-walled carbon nanotubes. The large electronic component is fit from $t=$ 0.5 ps to $t=$ 2 ps with a number of exponentials, which was then subtracted from the trace. The inset displays the extracted coherent phonon oscillations.

Figure 3

Scanning electron microscopy image of aligned single-walled carbon nanotubes, and the two experimental configurations employed in the current pump-probe spectroscopy work. (a) Type I: pump and probe polarizations are fixed and sample orientation is rotated. (b) Type II: probe and sample orientations are fixed and pump polarization is rotated.

Figure 4

Experimental differential transmission data in the time domain showing coherent phonon oscillations of the radial breathing mode for different polarization angles in (a) Type I and (b) Type II configurations (see Fig. 3). The traces are vertically offset for clarity.

Figure 5

Coherent phonon spectra for different polarization angles in Type II configuration obtained through a Fourier transform of the time-domain data in Fig. 4b. The traces are vertically offset for clarity.

Figure 6

Spectrally integrated coherent phonon intensity as a function of angle, measured in (a) Type I and (b) Type II configurations.

Figure 7

(a) Theoretical band structure for (38,0) nanotubes showing the strong $E_{44}$ and $E_{77}$ transitions for a polarization vector parallel to the tube. The wave vector $k$ is expressed in units of $\pi /T$ where $T=4.31\hspace{0.5em}$Å is the length of the translational unit cell. (b) Computed absorption spectra for polarization angles varying from 0${}^{ˆ}$ (parallel to tube) to 90${}^{ˆ}$ (perpendicular to tube). (c) Coherent phonon spectra as a function of pump energy and pump polarization angle $\varphi$ for RBM coherent phonons ($\omega =10.91\hspace{0.5em}\text{meV}$). Coherent phonon spectra are Fourier transforms of computed time dependent differential transmission for Type II pump-probe experiments. (d) Integrated power (black dots) obtained by taking the area under the computed $E_{44}$ peaks in lower left panel. We fit (red curve) the theoretical calculations to $A\cos {}^p(\varphi )$ where $A$ and $p=4.017$ are fitting parameters.

Figure 8

In (b) we plot the amplitude of RBM coherent phonon oscillations as a function of the pump polarization angle ($\theta$ or $\varphi$) for a (38,0) nanotube photoexcited by a $50\hspace{0.5em}\text{fs}$ pump at the theoretical $E_{44}$ transition. In (a) we plot the amplitude of the resulting differential transmission signal for Type I and II experiments as a function of angles $\theta$ and $\varphi$ respectively. Our results are fit to $A\cos {}^p(\theta )$ $[A\cos {}^p(\varphi )]$ where numerical values of the best fit $p$ are indicated in the figure.

Figure 9

Integrated RBM coherent phonon power as a function of (a) $\theta$ for Type I and (b) $\varphi$ for Type II experiments pumping close to the $E_{44}$ transition. Experimental data points are red downward-pointing triangles fit by a solid red line. Theoretical data points for a (38,0) nanotube are black upward-pointing triangles and are fit by a black dashed line. The fitting functions are of the form $A\cos {}^p(\theta )$ and $A\cos {}^p(\varphi )$ where the numerical values of the best fit $p$ are indicated in the figure. The experimental curves have undergone background subtraction to obtain complete quenching at an angle of ${90}^{°}$ and then have been rescaled so that the integrated power at an angle of $0^{°}$ is equal to unity.

Figure 10

Effects of tube misalignment on integrated coherent phonon power for an ensemble of nanotubes with tube axis orientation angles following a Gaussian distribution. (a) Type I fitting function $A\cos {}^p(\theta )$ with $p=8$ and (b) Type II fitting function $A\cos {}^p(\varphi )$ with $p=4$.

Figure 11

Experimental integrated RBM coherent phonon power for the $E_{44}$ transition as a function of $\theta$ for Type I (black upward-pointing triangles) and $\varphi$ for Type II (red downward-pointing triangles) experiments. Type I experiments are fit to $A\cos {}^p(\theta +\Delta \theta )+B$ where $p=8$ (black dashed line) and Type II experiments are fit to $A\cos {}^p(\varphi +\Delta \varphi )+B$ for $p=4$ (solid red line). $A$ and $B$ are background subtraction and rescaling parameters and the standard deviations for the random tube axis misalignment $\Delta \theta$ and $\Delta \varphi$ are restricted to be the same for both Type I and II.