Characterizing an Optically Induced Sub-micrometer Gigahertz Acoustic Wave in a Silicon Thin Plate

Optically induced GHz-THz guided acoustic waves have been intensively studied because of the potential to realize noninvasive and noncontact material inspection. Although the generation of photoinduced guided acoustic waves utilizing nanostructures, such as ultrathin plates, nanowires, and materials interfaces, is being established, experimental characterization of these acoustic waves in consideration of the finite size effect has been difficult due to the lack of experimental methods with nm × ps resolution. Here we experimentally observe the sub-micrometer guided acoustic waves in a nanofabricated ultrathin silicon plate by ultrafast transmission electron microscopy with nm × ps precision. We successfully characterize the excited guided acoustic wave in frequency-wavenumber space by applying Fourier-transformation analysis on the bright-field movie. These results suggest the great potential of ultrafast transmission electron microscopy to characterize the acoustic modes realized in various nanostructures.


Section 1: Calculation of nano-plate wave dispersion
We calculated the dispersion curve and atomic displacement field in Figs. 1c, 1d, 2c, and 3c and Fig. S1 by using the partial wave technique described in the literature [2] under tractionfree conditions at the surface. We assumed the two-dimensional sample of a flat plate form with 230 nm thickness along thickness direction (Z), and infinite length along the propagation direction (X). We also assumed that the sample is homogeneous along Y direction ( ⊥ , ). We used the elastic constants of single crystalline silicon in the literature [35,36]. The whole dispersion curves of Fig. 2c and 3c are shown in Fig. S1.

Section 2: Finite-element simulation
For the finite-element method calculation, we extended the photo-induced elasto-dynamic equations in the literature [10] to a three-dimensional nanofabricated single crystalline system.
We replaced Eqs (1) and (2) is elastic constant, and ext is a photoinduced stress term. We used the elastic constant of single crystalline silicon reported in Ref. [35,36]. The expression of ext and the equations for the lattice temperature and carrier density were not changed from Ref.
[10]. We assumed that the 1.2 eV pump light homogeneously excited the thin silicon plate since the optical penetration depth (approximately 1000 nm) was considerably larger than the sample thickness. Therefore, we set the source term We used a fluence F of 1.6 mJ/cm 2 and a pump pulse duration of = 290 fs. The lateral length of the plate was set to be sufficiently large (8 m) to avoid the acoustic wave emitted from the edge. The sample thickness was set to 230 nm. Other parameters and calculation conditions used for the simulation are presented in Ref. [10].

Section 3: Out-of-plane acoustic resonance mode
We confirm the periodic modulation of the out-of-plane atomic displacement with 20 GHz frequency by the finite-element simulation. Fig. S2 sample. Finally, we mention that the third-order acoustic resonance mode at 60 GHz cannot be distinguished in the present experimental data, possibly due to low signal-to-noise ratio although it is predicted by the simulation. In this section, we describe the generation of S0 mode at 20 GHz frequency, which is not seen in the f-k distribution of nano-plate waves in Figs. 2(c) and 3(c). Figure S4(b) shows the timedependent ̅ along the whole arrow depicted in Fig. S4(a), obtained from the identical finite element calculation in the main text. By applying the Fourier-transformation to this data, we obtain the f-k distribution as shown in Fig. S4(c). Here we find that both S1 (k ≃ 2 m -1 ) and S0 (k ≃ 3 m -1 ) modes are photoexcited, although the amplitude of the S1 mode is much larger than S0. Such contribution from S0 mode is hard to distinguish in the simulated f-k image presented in the main text [ Fig. 2(c)]. This is because the simulation data used for Fig. 2  around t = 0, which shows relatively large amplitude as compared to the periodic plate waves. As discussed later, this acoustic pulse is the origin of the S1 modes extending above 20 GHz up to 40 GHz. When we apply the Fourier transformation along both spatial and temporal axes, all the dynamical contributions including acoustic pulse and periodic nano-plate waves are transformed together into the frequency-wavenumber space as described in the main text. Fig. S5(b) emphasizes the S1 mode extending above the resonance frequency up to ~60 GHz, by changing 8 the other hand, the extended continuous contribution between 20 and 60 GHz is strongly suppressed as compared to Fig. S5(b). Therefore the contribution between 20 GHz to 60GHz should be due to the pulse-like component which is generated only at the first impact of the optical-pulse irradiation.

Section 6: Thickness dependence of the photoinduced nano-plate wave
To clearly demonstrate the relation between the plate wave frequency and the plate thickness, we prepared a 450-nm thick Si single crystalline plate in the similar procedure with  Section 7: Width of the S1 mode in frequency-wavenumber space To evaluate the "width" in experimentally obtained f-k data, we cut the image in Fig. S7(b) around k ~ 2 m -1 and ~ 20 GHz as shown in Figs. S7(c,d). The gray shaded areas in Figs.
S7(c,d) demonstrate the obtained "widths". To elucidate the origin of these "widths", we also show the Fourier-transformed image [ Fig. S7(f)] of a reference data [ Fig. S7(e)] that has the same range and number-of-points with the experiment, by using a most simple function where is Heaviside step function, the sound velocity c = f/k. We used f = 20 GHz and k = 1.9 m -1 , the S1 mode at 20 GHz obtained by the dispersion curve calculation (see Sec. 1 in Supporting Information). The Heaviside step function ( − −1 ) describes the simple propagation of the wave front, which also exists in the experimental data. As shown in Figs.

S7(g,h)
, the Fourier transformation of this simple reference data already shows the similar widths along both frequency and wavenumber, reflecting the insufficient spatial and temporal ranges as The line profiles of f along Cut3 and Cut4.