Second Harmonic Generation of Nanoscale Phonon Wave Packets

Phonons are often regarded as delocalized quasiparticles with certain energy and momentum. The anharmonic interaction of phonons determines macroscopic properties of the solid, such as thermal expansion or thermal conductivity, and a detailed understanding becomes increasingly important for functional nanostructures. Although phonon-phonon scattering processes depicted in simple wave-vector diagrams are the basis of theories describing these macroscopic phenomena, experiments directly accessing these coupling channels are scarce. We synthesize monochromatic acoustic phonon wave packets with only a few cycles to introduce nonlinear phononics as the acoustic counterpart to nonlinear optics. Control of the wave vector, bandwidth, and consequently spatial extent of the phonon wave packets allows us to observe nonlinear phonon interaction, in particular, second harmonic generation, in real time by wave-vector-sensitive Brillouin scattering with x-rays and optical photons.

Figure 1

Preparation and observation of phonon wave packets. (a) X-rays with wave vector $k$ are scattered ($k^{\prime }$) by the reciprocal lattice vector $G$ plus a phonon with wave vector $Q$. (b) A metal film is excited by a laser-pulse train with short (blue) and long (black) pulses. (c) Short pulses excite phonon wave packets at the fundamental and higher harmonics [rectangular wave packet (blue)]. Excitation with long pulses suppresses higher harmonics and generates a sinusoidal wave (black). (d) The phonon dispersion relation connects the central frequency $\mathrm{\Omega }=2\pi /\tau$ of the excited phonon wave packet with its corresponding wave vector $Q=2\pi /\lambda$. (e) Experimental data of ultrafast x-ray diffraction from the unexcited sample shows the (002) substrate reflection (green). After excitation with short pulses (blue), peaks at $G\pm Q$ and $G\pm 2Q$ occur. These peaks originate from inelastic scattering by the induced phonon wave packet. For longer pump pulses (black), the scattering at $G\pm 2Q$ is suppressed. This evidences that a narrow band phonon wave packet without its higher harmonics was excited.

Figure 2

Transient x-ray diffraction from photoexcited phonons and their SH. (a) X-ray diffraction signals around the (002) lattice plane of the STO substrate. (b) Diffraction signals on the right shoulder of the STO substrate peak weighted by $q^2$ for better visualization. The peak around $Q=0.044{\mathrm{nm}}^{-1}$ after time zero reflects the optically excited coherent phonon wave packet with the central wave vector $Q$. After a delay of 200 ps, additional diffraction intensity around $2Q$ heralds the SH of $Q$ generated via anharmonic phonon phonon interaction. (c) The gray (black) lines quantify the transient change of the integrated peak intensities of the fundamental and its second harmonic with (and without) a correction for the x-ray absorption.

Figure 3

SHG of phonons observed by TDBS. (a) Transient relative reflectivity change at 580 nm (black line). The stepwise increase indicates the laser excitation of the metal film terminating at $t=0$. Superimposed reflectivity oscillations (green line) originate from the SH phonons. The oscillation amplitude grows with the number of phonons up-converted to their SH. (b) Measured reflectivity oscillations across the white light probe spectrum. For each wavelength, the oscillation amplitude measures the occupation of a certain phonon $q$.

Figure 4

Comparing SHG experiments to the theory. (a) The color code shows the calculated transient change of the spectral energy density ${\rho }_E$ around the SH of the excited phonons. The square of the measured relative reflectivity oscillation amplitude divided by $q^2$ in units of ${10}^{-7}{\mathrm{nm}}^{-2}$ is overlaid as contour lines. (b) Calculated ${\rho }_E$ over a broader range of wave vectors in the log scale. The photoexcited fundamental is transformed into the second and third harmonics. (c) visualizes the measured transient energy density change of the SH for different initial energy densities ${\mathcal{E}}_1$ ($\text{red}=13$, $\text{blue}=67$, and $\text{black}=235\mathrm{nJ}/{\mathrm{cm}}^3$) of the wave packet and compares this with theoretical calculations (solid lines). The wave vector for the highest energy density was $q=0.03{\mathrm{nm}}^{-1}$, somewhat larger than the $q=0.025{\mathrm{nm}}^{-1}$.