Stimulated Brillouin scattering at 1 nm^-1 wavevector
by extreme ultraviolet transient gratings

Studying thermal and vibrational dynamics in nanoscale materials is critical for advancing the technological applications of faster, more efficient and more compact nanoelectronic devices, such as smartphone and computer chips, as well as for thermal barrier coatings [1], heat-assisted magnetic recording [2], nano-enhanced photovoltaics and thermoelectric energy conversion, to name a few. To achieve this, layer upon layer of very thin films are often used, with impurities added to tailor their function [3]. However, the complex structure of these materials makes it challenging to predict and characterise their thermoelastic properties.  Material properties such as elasticity, thermal conductivity and heat capacity are mostly determined by collective lattice dynamics that exhibit strong length-scale dependencies, which can drastically differ when the spatial dimensions reduce from macroscopic to microscopic scales, i.e., to sizes comparable with the characteristic length scales of nanostructures.
    Over the years, an obstacle to the full description of thermoelastic responses in the 10s of nm length-scale was given by the lack of experimental techniques capable of accessing such range [4] without the requirement of modifying or physically touching the sample. This inherently introduces limitations in the experiment design and complicates data interpretation. Collective lattice dynamics in condensed matter at wavevector $q>1$ nm^-1 can be measured by inelastic scattering of hard X-ray and thermal neutron, while Brillouin scattering and optical transient grating (TG) can be used for $q<0.1$ nm^-1. The intermediate $q=0.1-1$ nm^-1 is hardly accessible, despite efforts to expand the capabilities of Brillouin spectroscopy in the UV range [5] and for improving the performance of X-ray spectrometers [6]. In addition, these spectroscopic methods are inherently limited by the instrumental resolution when measuring narrow lines, i.e. long dynamics. This limitation does not affect time-domain techniques, such as picosecond ultrasonics and time-domain thermoreflectance. In these techniques, metal films or other nanostructures are fabricated on the sample for transducing an ultrafast optical excitation in a short wavelength thermoelastic perturbation [7, 8]. However, this intrinsically modifies the sample under investigation.
    The advent of free-electron laser (FEL) sources has recently permitted the usage of extreme ultraviolet (EUV) pulses for extending the TG approach to shorter wavelengths, i.e. in the $10-100$ nm range, enabling the excitation and probing of nanoscale thermoelasticity in a contact-less fashion [9, 10]. The EUV TG approach has been pioneered at the FERMI FEL (Trieste, Italy) with the dedicated endstation TIMER [11, 12], capable of incisively and selectively studying bulk and surface phonons [13], thermal transport kinetics [14, 9] and magnetic dynamics [15].
    In this paper, we exploit EUV TG to probe acoustic phonons in $\beta-Ga_{2}O_{3}$. In particular, by taking advantage of the phase-matching conditions imposed by the nanoscale EUV TG, we demonstrated the possibility to detect stimulated Brillouin back-scattering (SBBS) from an EUV pulse at 13.3 nm wavelength. This enabled us to probe the dynamics of phonon modes with a wavelength as short as $\approx$ 6 nm. The employed sample was an Mg-doped $\beta-Ga_{2}O_{3}$ (001)-oriented bulk crystal with monoclinic structure (space group C2/m), obtained from the Czochralski method at the Leibniz-Institut für Kristallzüchtung [16]. The excellent surface quality and well-known elastic parameters made this sample adapted for the present EUV TG experiment.

yielding a back-scattered signal, that encodes the dynamics of SBBS modes. Here, $k=2\pi n/\lambda_{\text{pr}}$ is the wavevector of the probe in the medium, where $n$ is the refractive index at $\lambda_{\text{pr}}$. Thus, the modulus of the acoustic wavevector for the SBBS signal is:

which is independent of $q_{\text{TG}}$ and is collinear with the backward diffracted signal from the TG.
    Therefore, under the current experimental conditions, the combination of the sharp penetration depth of the EUV TG pump in the material and the short wavelength EUV probe, enables the excitation and detection of phonons with $q$ as large as $q_{\text{SBBS}}\approx$ 1 nm^-1 (Fig. 1b). To further illustrate the excitation mechanism, Fig. 1c displays the EUV TG generated on the sample in the 26.6/13.3 configuration plotted against the ($x$,$z$) coordinates, taking into account the finite value of $L^{26}_{\text{abs}}$. We note that the modulation along $x$ extends in a much larger range, comparable with the width ($\text{FWHM}_{x}\approx 100$s of $\mu$m) of the excitation pulses. This is what we usually call TG. Additionally, there is a steep gradient along $z$. Such gradient launches acoustic waves in a broad range of $\Delta q$ (roughly extending up to $\sim 2\pi/L^{\text{pump}}_{\text{abs}}$). This is represented in Fig. 1e as the Fourier transform (FT) along $z$ of the EUV TG intensity profile shown in Fig. 1d. In an excitation scheme relying on a single EUV pump, there is no capability to selectively choose a specific phonon wavevector along the $z$-axis. However, in the current scenario, the phase matching condition imposed by the EUV TG (see Eq.2) selects a specific wavevector phonon with $q_{\text{SBBS}}\sim$ 1 nm^-1 from the wide range of available phonons This is illustrated by the vertical segment in Fig. 1e.
    We detected the EUV TG signal by varying the time delay ($\Delta t$) between the EUV TG excitation and the probe pulse. Measurements were conducted at both long timescales (Figs. 2a and 2e) and short timescales (Figs. 2c and 2g). As expected, at long timescales the overall signal is characterized by a slow decay, which can be attributed to the thermal relaxation of the EUV TG, modulated by phonon oscillations [9, 13]. After a few oscillations, these modulations become highly regular. For larger $\Delta t$ values, when the slow relaxation is decayed, double-frequency oscillations become visible, indicating the long-living nature of this dominant mode [13]. Conversely, the irregular shape of the initial oscillations suggests the presence of additional dynamics that damps out after some 10s of ps. The EUV TG data obtained at short timescales (Figs. 2c and 2g) were sampled with finer steps and exhibit modulations at significantly higher frequencies. These higher-frequency modulations are compatible with the previously mentioned mixing between the SBBS signal and the backward diffracted signal from the EUV TG.
    In order to quantitatively describe the waveforms at both long (blue line in Fig. 2a and 2e) and short timescale (black line in Fig. 2c and 2g) an initial fitting procedure was conducted using Eq. 3:

where the erf function accounts for a sudden rise of the signal (with $\sigma$ representing the width of the rise), followed by an exponential decay with a time constant $\tau$. Subsequently, FTs were computed on the differences between the measured traces and their respective exponential fits. The obtained results are illustrated in Figs. 2b, 2d, 2f and 2h.
    The FTs of the long timescale waveforms present a well-defined mode and its second harmonic, plus a weaker and spectrally broader feature. All frequencies in these FTs vary proportionally to $q_{\text{TG}}$, as depicted in Figs. 2b) and 2f). The presence of this broad feature confirms the existence of a damped mode, which predominantly affects the initial portion of the waveform, as already evident from the raw data.
    To comprehensively describe the signal, the complete fitting procedure incorporated these two vibrational modes, specifically a damped sinusoidal term and an undamped sinusoidal term:

The resulting best-fit results are reported as black lines in Figs. 2a and 2e. All parameters and errors mentioned further below have been obtained using Eq. 4. The values obtained from the preliminary fitting of the EUV TG signal with Eq. 3 and from the FTs were used as an initial guess for fitting the data with Eq. 4.
    The results concerning the oscillation frequencies are shown in Fig. 3a. The undamped mode is compatible with a Surface Acoustic Wave (SAW), which exhibits a linear dispersion relation as a function of $q_{\text{TG}}$. From the slope of such liner dispersion a value for the sound velocity of $c_{\text{SAW}}^{[100]}=3.15\pm 0.01$ km/s is obtained. This value is close to the estimated velocity of 3.24 km/s, as evaluated by using the transverse acoustic (TA) phonon velocity $c_{\text{TA}}^{[100]}=3.57$ km/s [18] and the Poisson’s ratio $\nu_{p}=0.2$ [19] of $\beta-Ga_{2}O_{3}$ [100], through the relation $c_{\text{SAW}}\approx c_{\text{TA}}\cdot(0.862+0.14\nu_{p})/(1+\nu_{p})$ [20]. SAW modes represent long-lived coherent surface displacements characterized by mechanical energy confined to the surface. In the employed backward diffraction geometry, these modes are expected to be the dominant contribution to the EUV TG signal, as observed experimentally.
    The damped mode also presents a liner dispersion with a velocity $c_{\text{LA}}^{[100]}=5.97\pm 0.14$, which is similar to the expected value (6.18 km/s) for longitudinal acoustic (LA) phonons [18]. Such marginal deviations between the expected and observed velocities may arise from factors such as slight misalignment of the sample relative to the [100] crystallographic direction, sample heating caused by the FEL, or the $10^{{}^{\circ}}$ tilt in the $(x,y)$-plane, necessary for collecting the backward diffracted signal [11].