Attenuation of 7 GHz surface acoustic waves on silicon

We measured the attenuation of GHz frequency surface acoustic waves (SAWs) on the Si (001) surface using an optical pump-probe technique at temperatures between 300 and 600 K. SAWs are generated and detected by a 700 nm Al grating fabricated by nanoimprint lithography. The grating for SAW generation is separated from the grating for SAW detection by $\approx 150\mu \mathrm{m}$. The amplitude of SAWs is attenuated by coupling to bulk waves created by the Al grating, diffraction due to the finite size of the source, and the intrinsic relaxational Akhiezer damping of elastic waves in Si. Thermal phonon relaxation time and Grüneisen parameters are fitted using temperature-dependent measurement. The $fQ$ product of a hypothetical micromechanical oscillator limited by Akhiezer damping at this frequency is $\sim 3\times {10}^{13}$ Hz.

Figure 1

SEM image of the grating structure. In (a), the dark area is exposed Si. The bright area is the Al grating. Al metal bars are fabricated parallel to the edge of the gap region. Two red points illustrate schematically how the pump and probe beams are separated. In our experiments, the pump and probe beams are separated by $150\mu \mathrm{m}$. The red wavy line indicates the wave propagation from pump grating to probe grating. (b) Higher magnification image of the Al grating with a 700 nm period, $\approx 15$ nm thickness, and $\approx 50\%$ filling factor.

Figure 2

Process flow diagram for the fabrication of Al gratings using nanoimprint lithography.

Figure 3

(a) Typical in-phase signal for a close offset measurement. Pump and probe beams are offset by $10\mu \mathrm{m}$. (b) Typical out-of-phase signal for a far offset measurement. Pump and probe beams are offset by $150\mu \mathrm{m}$.

Figure 4

Slowness surface of the Rayleigh-like SAW (from the $\langle 100\rangle$ direction to $\approx {37}^{\circ }$ away, where the discontinuity is located) and the PSAW (from the discontinuity to the $\langle 110\rangle$ direction) on the Si(001) plane, and a comparison between the slowness surface close to the $\langle 110\rangle$ direction and a circle (red line).

Figure 5

Relative signal intensity when the probe beam is offset at the transverse direction at far field. Black dots are experiment results. The red line is the calculation considering phonon focusing [Eq. (12)]. The blue line is calculated diffraction for an isotropic plane [Eq. (11)]. The line labeled “No diffraction” is the calculation if no diffraction is considered.

Figure 6

(a) $\text{Im}\left(G_{33}\right)$ at the Brillouin-zone center with 0.5 grating filling factor and $\delta$ excitation. (b) Far-field signal with grating of $\approx 0.5$ filling factor. (c) $\text{Im}\left(G_{33}\right)$ at the Brillouin-zone center with 0.35 grating filling factor and $\delta$ excitation. (d) Far-field signal with grating of $\approx 0.35$ filling factor.

Figure 7

Ratio between wave amplitudes of far offset and close offset with different lengths of the grating region. The vertical axis is plotted on a log scale, and data are fitted with straight lines. Each set of data is labeled by the sample temperature. The slope can be used to calculate the attenuation coefficient of the Al grating. The inset is the measurement configuration for far offset. Red dots stand for pairs of positions of pump and probe beams. Repeating the measurement for five different gap widths while keeping the distance between the pump and probe beams constant introduces five different lengths of the grating region. Comparing them with the close offset measurement gives five data points at each temperature. The results are extrapolated to zero grating length (on the vertical axis) to get intrinsic attenuation.

Figure 8

Temperature dependence of the measured intrinsic attenuation coefficient. Red points are measurements. The black line is a fit to the data using Eq. (1) and $\tau =30$ ps and $(\langle {\gamma }^2\rangle -{\langle \gamma \rangle }^2)=0.053$.

Figure 9

Comparison between our result and prior reports and the phonon viscosity model. $L$ denotes a longitudinal bulk acoustic wave. FT and ST denote a bulk fast transverse acoustic wave and a bulk slow transverse acoustic wave, respectively. Notation like $\langle 100\rangle$ denotes the propagation direction. Data marked [a] are from Ref. [18], data marked [b] are from Ref. [15], data marked [c] are from Ref. [8], and data marked [d] are from Ref. [19]. The straight line of “SAW-Lamb viscosity” and the dashed line “SAW-Helme viscosity” correspond to the attenuation coefficient of SAW calculated using Eq. (3) and the effective viscosity of SAW from Eq. (10). They use the phonon viscosity tensor from Lamb and Richter [13] and Helme and King [16], respectively. The Akhiezer model is calculated using Eq. (1), where $\tau$ and $\langle {\gamma }^2\rangle -{\langle \gamma \rangle }^2$ are from our measurement.

Figure 10

Dispersion curve of the SAW from 0 to $\pi /\lambda$ with metal grating, drawn by plotting extreme points of $G_{33}(k_{\parallel },x_3=0,\omega )$. In both figures, the blue line is the dispersion curve of bulk fast transverse acoustic waves, and the red line is the dispersion curve of bulk longitudinal acoustic waves in the Si $\langle 110\rangle$ direction; they are not from the $G_{33}(k_{\parallel },x_3=0,\omega )$ calculation but are plotted separately. (a) The black line is $\text{Re}(G_{33}(k_{\parallel },\omega ,x_3=0))$, which shows the mode folding and band structure in the first Brillouin zone. (b) The black line is $\text{Im}(G_{33}(k_{\parallel },\omega ,x_3=0))$. The imaginary part of the Green's function appears only after the SAW couples with bulk acoustic waves. The acoustic band gap is too small to be visible in this plot.