Dynamical diffraction formalism for imaging time-dependent diffuse scattering from coherent phonons with Dark-Field X-ray Microscopy

Abstract

Coherent acoustic phonons, whose damping sets the upper bound of quality factors in acoustic resonators, play a critical role in advanced telecommunication and quantum information technologies. Yet, probing their decay in the GHz regime remains challenging using conventional surface-based techniques. Dark-field X-ray microscopy (DFXM) offers a solution by enabling through-depth, non-destructive and full-field imaging of strain fields and dislocations inside bulk materials with high spatial and angular resolution. We previously used kinematic diffraction theory to describe DFXM signals based on how the Bragg peak shifts due to the strain wave, allowing us to reconstruct the frequency spectrum of coherent phonons as a function of depth through the sample. The approach of tracking the Bragg peak shifts to study phonon dynamics, however, places an upper-bound to the highest phonon frequency that can be studied, determined by the spatial resolution of the measurement. In this work, we discuss how coherent phonon dynamics can be studied with DFXM from time-dependent intensity oscillation sidebands. This approach simultaneously allows studying coherent phonon dynamics in real and reciprocal space, overcoming frequency resolution limits imposed by the real-space resolution of Bragg-peak tracking. Using Takagi-Taupin dynamical diffraction formalism, we establish the spatial and reciprocal space resolution achievable for studying the coherent phonon dynamics and evaluate conditions for observing long-lived intensity oscillations. We close by proposing experimental strategies to optimize excitation bandwidths and reciprocal-space selectivity. The formalism in the paper enables the design of DFXM experiments for quantitative, frequency-resolved measurements of acoustic phonon decay and phonon-defect interactions in bulk crystalline materials.

Figures (6)

• Figure 1: (a) The schematic of the experimental geometry used in this paper. The sample is a (100) silicon deposited with a gold transducer with a thickness much smaller than the characteristic electron- phonon coupling length scale in gold ($\lambda_{e-p}$). A 800 nm femtosceond laser excitation launches a strain wave along the plane normal [100]. A sheet x-ray beam is incident at an angle $\theta$ near the Bragg angle $\theta_B$ of the [400] diffraction plane. The scattered light at the scattering angle $2\theta$ is collected using an objective lens with the real space image obtained in a detector placed in the imaging condition. (b) The frequency corresponding to the maximum weight of the frequency spectrum of x-ray reflectivity recorded at different positions ($\Delta\theta$) away from the Bragg peak. The strain used to simulate the time-dependent reflectivity is shown in Fig 2c.
• Figure 2: The comparison of the frequency spectra of the applied strain to the frequency spectra of the reflectivity. (a) shows the time-domain behavior of a strain wave whose frequency spectra is plotted in (b). (c) shows the time-domain behavior of a strain wave whose frequency spectra is plotted in (d)
• Figure 3: Magnitude of intensity oscillations as a function of changing sample thickness. The Takagi-Taupin simulations resulting in the intensity oscillations are performed for strain waves with a primary phonon wavelength of 50 nm in Si. (a) and (c) shows cases where the wave undergoes a 1/e decrease in amplitude over one full cycle while (b) and (d) show cases where the wave undergoes a 1/e decrease in amplitude over five full cycles. (a) and (b) show the cases where we begin sampling the oscillations when the rightmost front of the strain waves is 1125 nm away from the surface of the sample. (c) and (d) show the cases where we begin sampling the oscillations when the rightmost front of the strain waves is 200 nm away from the surface of the sample. The red, blue and green shades in the plots show Region I, II and III discussed in Section 3 respectively.
• Figure 4: Comparison of the frequency spectrum of the populated phonons (solid red) in the material to the DFXM measurable frequency spectrum determined by the reciprocal space resolutions described in Section 4. R1 (solid blue) shows the frequency spectrum of the scattering side bands when using a energy bandwidth of $10^{-4}$ and an incoming beam divergence of $10^{-4}$ rad. R2 (solid green) shows the frequency spectrum of the reflectivity when using a energy bandwidth of $6 \times10^{-6}$ and an incoming beam divergence of $6 \times10^{-6}$ rad.
• Figure 5: Time and frequency behavior of the strain wave in a Si sample generated by ultrafast excitation of a gold transducer of different film thicknesses.The strain wave is simulated using udkm1dsim toolbox. (a) and (b) show the respective time and frequency behavior for a 5 nm gold transducer, (c) and (d) show the behavior for a 10 nm transducer, (e) and (f) for a 20 nm transducer and (g) and (h) for a 40 nm transducer. The red lines in the frequency-domain plots represent the expected frequency based on the theory described in Section 5 and the sky-blue transparent region represents the FWHM for each spectrum.