Picosecond acoustic phonon pulse propagation in silicon

We report experimental and theoretical investigations of the behavior of coherent acoustic phonon pulses after propagation through millimeter-scale distances in crystalline Si. An ultrafast optical pump and probe technique is used to generate and detect the phonon pulses. The geometry of our experiment is such that we can make this measurement in either the far field or the near field of the acoustic source, allowing studies of diffraction effects in addition to dispersion and nonlinearity in the Si. We also present a rigorous derivation of the Korteweg–de Vries equation, which describes the behavior of these acoustic pulses in the nonlinear regime in one dimension. This one-dimensional (1D) model is combined with a 3D analysis of diffraction effects in anisotropic media in order to analyze far-field data.

Figure 1

Schematic diagram of pump-probe experiement. Si wafer is oriented in either [100] or [111] direction. $d_1=15\mathrm{nm}$ for both cases. $d_2=0.5\mathrm{mm}$ for [111] sample and $1.0\mathrm{mm}$ for [100] sample.

Figure 2

Ultrafast optical pump and probe setup.

Figure 3

Reflectivity change as a function of delay time for three cases: $A$, large optical pump spot, low power; $B$, small optical pump spot, low power; $C$, small optical pump spot, high power. Note that the total delay between generation of the acoustic pulse by the pump and detection of the acoustic pulse by the probe is equal to the plotted delay time plus $51.4\mathrm{ns}$.

Figure 4

Input acoustic pulse profiles for the numerical simulation. Both profiles have a characteristic width of $2.34\mathrm{ps}$. (a) Input strain profile used in the simulations discussed in the paper. (b) Input strain profile that provided improved fits to the reflectivity signals as discussed in the text. (c) Best fit to [111] Si data using input strain profile from (a). (d) Best fit to [111] Si data using input strain profile from (b).

Figure 5

Reflectivity data and simulations for [111] and [100] Si samples for the dispersion-only regime of the experiment. Total time delay has been adjusted for the [111] sample $\left(51.4\mathrm{ns}\right)$ and for the [100] sample $\left(118.6\mathrm{ns}\right)$ so that both pulses appear in the same time window. The simulations used a dispersion parameter $\gamma =1.92\times {10}^{-11}{\mathrm{cm}}^3{\mathrm{s}}^{-1}$ for [111] Si and $\gamma =1.44\times {10}^{-11}{\mathrm{cm}}^3{\mathrm{s}}^{-1}$ for [100] Si. Note that the much larger envelope of the [100] pulse is due to the longer propagation length.

Figure 6

Reflectivity data and simulations for [111] and [100] Si for the dispersion/diffraction regime of the experiment. Total time delay has been adjusted as described in Fig. 5 caption. The same dispersion parameters discussed in Sec. 4A were used in the simulation. The diffraction was determined using the third term of Eq. (9a) with ${\xi }^2=0.675$ for [111] and 1.922 for [100].

Figure 7

Normalized Fourier transform amplitude versus frequency for reflectivity in the diffracting (dashed lines) and nondiffracting (solid lines) cases. The upper two graphs are for the data and the lower two graphs are for the simulations.

Figure 8

Reflectivity and reflectivity spectra for data and simulation of off-axis wave forms for [100] Si sample. Total time delay has been adjusted as described in Fig. 5 caption. The reflectivity was measured (and simulated) for three probe beam positions: on axis (solid curves), $+4\mu \mathrm{m}$ off axis (dashed curves), and $+8\mu \mathrm{m}$ off axis (dotted curves). The lower-frequency components begin to dominate the reflectivity signal as the optical probe detection is moved farther off axis.

Figure 9

Reflectivity data and simulation for the [111] Si sample for increased pump powers. The same dispersion and diffraction parameters were used as described in Sec. 4B. $C_{NL}=3.2$ and $2\mathrm{mW}$ pumping power will give the center step-shape strain amplitude $1.57\times {10}^{-5}$.