Picosecond ultrasonic measurements of attenuation of longitudinal acoustic phonons in silicon

We report ultrafast optical measurements of the attenuation of 50 and 100 GHz longitudinal acoustic-phonon pulses in Si. Picosecond acoustic measurements were made at temperatures $50<T<300\text{K}$ on thinned ($50\text{−}\mu \text{m}$-thick) wafers. The measured phonon lifetimes at 300 K, $\approx 5–7\text{ns}$, are an order of magnitude less than expected based on three-phonon scattering rates derived from thermal conductivity data. We find instead that relaxational damping is the dominant mechanism in this frequency and temperature range. This attenuation sets an intrinsic limit on the quality factor of nanomechanical resonators that operate near room temperature.

Figure 1

$\Delta R\left(t\right)$ for a 20 nm Al-film sample. Phonon pulses are shown after one (12.3 ns) and two (24.6 ns) round trips in the Si wafer. The signals at 250 and 300 K are multiplied by a factor of two to allow visual comparison.

Figure 2

(Color online) Fourier transforms of $\Delta R$ at 50 K for two samples. The “100 GHz” plots signals from a 10 nm film sample and the “50 GHz” plots represent the signals from a 20 nm film sample. The data have been rescaled so that the magnitudes of the first round trips are the same.

Figure 3

Peak-to-peak amplitudes of the first (◼) and second (●) $\Delta R$ pulses for the 20 nm Al-film (50 GHz) sample.

Figure 4

(Color online) (a) Phonon lifetime versus frequency in Si at 300 K. The (◼) represents our measured values of the lifetime of 50 and 100 GHz longitudinal acoustic phonons. Uncertainty was estimated by an analysis of two samples with peak frequency $\approx 50\text{GHz}$. The (○), $(\times )$, (●), and (▲) represent Refs. 11, 23, 24, 25, respectively. Data from Ref. 24 is for [111] Si. The line labeled TC represents ${\tau }_{AC}={\left(B{\omega }^{2}T\right)}^{-1}$ with $B=2.4\times {10}^{-19}\text{s}{\text{K}}^{-1}$. The line labeled RD represents ${\tau }_{AC}$ based on $\alpha$ calculated using Eq. (1) with $\tau =17\text{ps}$. (b) Ideal quality factor $Q=f\tau$ for a nanomechanical resonator based on the data shown in part (a).