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(A) TTG signal at 85 K for a range of grating periods. In the inset, circles represent the measured second sound frequency as a function of the wavevector and the solid line is a linear fit corresponding to a phase velocity of 3200 m/s. (B) Absolute value of complex frequency-domain Green's functions vs frequency at 80 K for a range of TTG periods. (C) Simulated thermal grating amplitude vs time at 80 K.
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Wavelike thermal transport in solids, referred to as second sound, has until now been an exotic phenomenon limited to a handful of materials at low temperatures. This has restricted interest in its occurrence and in its potential applications. Through time-resolved optical measurements of thermal transport on 5-20 {\mu}m length scales in graphite,...
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... the temperature is lowered to 85 K, strikingly different behavior is observed, as shown in Fig. 2A. Unlike the signals at 300 K which decay monotonically, signal waveforms at 85 K yield damped oscillations, with the signal falling below zero (26). In the heterodyne detection scheme, this sign flip of the TTG signal means that the spatial phase of the grating has shifted by (21), i.e. the local maxima of the surface displacements ...
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... and minima cannot switch places because heat can only move from hotter to colder regions. With increasing TG period, the negative dip in the response becomes shallower and eventually disappears. The position of the dip shifts to longer times as the period increases, indicating that the frequency of the wave-like dynamics decreases. The inset in Fig. 2A shows this frequency, determined from the position of the first minimum of the response (corresponding to ½ of the oscillation period) as a function of the wavevector q. The nearly linear dependence indicates a velocity of about 3200 m/s (determined from the slope of the linear fit multiplied by 2). It should be noted that TTG signals ...
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... regime (9,19,20). However, previous studies dealt with the stationary BTE which is unable to capture transient phenomena such as second sound. Recently, Chiloyan et al. (32) described a technique for calculating frequency- domain Green's functions for the non-stationary and spatially non-uniform BTE. Such Green's functions, illustrated in Fig. 2B describe a response of the phonon population to a heat source having the form of a harmonic plane wave, exp(it-qr). In our case, the heat source is sinusoidal in space and can be modeled as impulsive in time since the laser pulse duration is short compared to the observed dynamics. Consequently, temporal dynamics of the TTG can be ...
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... periods yielded exponential TTG decays in agreement with the experiment. The decay times of the simulated waveforms yielded the dependence of the apparent diffusivity on the TTG period reproducing the trend seen in the experiment, as shown in Fig. 1C. In contrast, at 85 K the frequency-domain Green's functions yield a resonant peak as seen in Fig. 2B. This peak is a hallmark of second sound and gives rise to damped oscillations in the simulated time-domain waveforms shown in Fig. 2C. The simulated waveforms agree qualitatively with the experimental data (see Supplementary material for a detailed comparison of measured and simulated waveforms) and yield trends with respect to the ...
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... of the apparent diffusivity on the TTG period reproducing the trend seen in the experiment, as shown in Fig. 1C. In contrast, at 85 K the frequency-domain Green's functions yield a resonant peak as seen in Fig. 2B. This peak is a hallmark of second sound and gives rise to damped oscillations in the simulated time-domain waveforms shown in Fig. 2C. The simulated waveforms agree qualitatively with the experimental data (see Supplementary material for a detailed comparison of measured and simulated waveforms) and yield trends with respect to the TTG period that are consistent with the observations, including the disappearance of the second sound signature at large TTG periods. The ...
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... material for a detailed comparison of measured and simulated waveforms) and yield trends with respect to the TTG period that are consistent with the observations, including the disappearance of the second sound signature at large TTG periods. The calculated second sound velocity determined from the peak position of the frequency resonance in Fig. 2C is 3650 m/s, which is somewhat higher than the measured velocity. The theory correctly predicts that the second sound velocity falls in between the slow and fast transverse acoustic velocities. In contrast to graphite, in other materials second sound was found to be slower than the slowest phonon velocity. The peculiarity of graphite ...
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... TTG decay slower: normal phonon-phonon scattering processes facilitate heat transport by redistributing energy to modes with higher group velocity. It is instructive to determine the second sound domain in the temperature-TTG period parameter space. We use the ratio of the maximum at the peak of the magnitude of frequency- domain Green's function (Fig. 2B) and the minimum between the peak and zero frequency as a metric for the strength of the second sound effect. As one can see in Fig. 4, second sound is predicted to occur between 50 and 250 K, with higher temperatures corresponding to shorter thermal transport length scales: while for L=10 m the temperature window closes at ~150 K in ...
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... low-temperature measurements shown in Figs. 2A and 3 were performed with a pump spot size of 168 µm in diameter and a pump pulse duration of 15 ps. The probe and reference beams both had spot sizes of 135 µm in diameter, and unlike the room-temperature measurements the probe was not electro-optically modulated. The repetition rate of the pump and the mean acquisition rate of the ...
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... verified that changing either the excitation energy or probe power by a factor of two has no effect on the shape of the measured signal waveforms at 85 K. We had to increase the excitation energy by as much as a factor of 10 to see a noticeable change in the normalized signal, see Fig. S2. Fig. S2. Measured TTG signals at 85 K and a period L=10 µm for pump pulse energies 10 and 100 nJ normalized to unity at the maximum. We had to increase the pump energy by a factor of 10 to see an appreciable change in the normalized ...
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... verified that changing either the excitation energy or probe power by a factor of two has no effect on the shape of the measured signal waveforms at 85 K. We had to increase the excitation energy by as much as a factor of 10 to see a noticeable change in the normalized signal, see Fig. S2. Fig. S2. Measured TTG signals at 85 K and a period L=10 µm for pump pulse energies 10 and 100 nJ normalized to unity at the maximum. We had to increase the pump energy by a factor of 10 to see an appreciable change in the normalized ...
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... S5 is a reproduction of the data plotted in Figs. 2A,C of the main text, but now each measured waveform is plotted separately alongside the corresponding simulated waveform. ...
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... the temperature is lowered to 85 K, strikingly different behavior is observed, as shown in Fig. 2A. Unlike the signals at 300 K which decay monotonically, signal waveforms at 85 K yield damped oscillations, with the signal falling below zero (26). In the heterodyne detection scheme, this sign flip of the TTG signal means that the spatial phase of the grating has shifted by (21), i.e. the local maxima of the surface displacements ...
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... and minima cannot switch places because heat can only move from hotter to colder regions. With increasing TG period, the negative dip in the response becomes shallower and eventually disappears. The position of the dip shifts to longer times as the period increases, indicating that the frequency of the wave-like dynamics decreases. The inset in Fig. 2A shows this frequency, determined from the position of the first minimum of the response (corresponding to ½ of the oscillation period) as a function of the wavevector q. The nearly linear dependence indicates a velocity of about 3200 m/s (determined from the slope of the linear fit multiplied by 2). It should be noted that TTG signals ...
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... regime (9,19,20). However, previous studies dealt with the stationary BTE which is unable to capture transient phenomena such as second sound. Recently, Chiloyan et al. (32) described a technique for calculating frequencydomain Green's functions for the non-stationary and spatially non-uniform BTE. Such Green's functions, illustrated in Fig. 2B describe a response of the phonon population to a heat source having the form of a harmonic plane wave, exp(it-qr). In our case, the heat source is sinusoidal in space and can be modeled as impulsive in time since the laser pulse duration is short compared to the observed dynamics. Consequently, temporal dynamics of the TTG can be ...
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... periods yielded exponential TTG decays in agreement with the experiment. The decay times of the simulated waveforms yielded the dependence of the apparent diffusivity on the TTG period reproducing the trend seen in the experiment, as shown in Fig. 1C. In contrast, at 85 K the frequency-domain Green's functions yield a resonant peak as seen in Fig. 2B. This peak is a hallmark of second sound and gives rise to damped oscillations in the simulated time-domain waveforms shown in Fig. 2C. The simulated waveforms agree qualitatively with the experimental data (see Supplementary material for a detailed comparison of measured and simulated waveforms) and yield trends with respect to the ...
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... of the apparent diffusivity on the TTG period reproducing the trend seen in the experiment, as shown in Fig. 1C. In contrast, at 85 K the frequency-domain Green's functions yield a resonant peak as seen in Fig. 2B. This peak is a hallmark of second sound and gives rise to damped oscillations in the simulated time-domain waveforms shown in Fig. 2C. The simulated waveforms agree qualitatively with the experimental data (see Supplementary material for a detailed comparison of measured and simulated waveforms) and yield trends with respect to the TTG period that are consistent with the observations, including the disappearance of the second sound signature at large TTG periods. The ...
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... material for a detailed comparison of measured and simulated waveforms) and yield trends with respect to the TTG period that are consistent with the observations, including the disappearance of the second sound signature at large TTG periods. The calculated second sound velocity determined from the peak position of the frequency resonance in Fig. 2C is 3650 m/s, which is somewhat higher than the measured velocity. The theory correctly predicts that the second sound velocity falls in between the slow and fast transverse acoustic velocities. In contrast to graphite, in other materials second sound was found to be slower than the slowest phonon velocity. The peculiarity of graphite ...
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... TTG decay slower: normal phonon-phonon scattering processes facilitate heat transport by redistributing energy to modes with higher group velocity. It is instructive to determine the second sound domain in the temperature-TTG period parameter space. We use the ratio of the maximum at the peak of the magnitude of frequencydomain Green's function (Fig. 2B) and the minimum between the peak and zero frequency as a metric for the strength of the second sound effect. As one can see in Fig. 4, second sound is predicted to occur between 50 and 250 K, with higher temperatures corresponding to shorter thermal transport length scales: while for L=10 m the temperature window closes at ~150 K in ...
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... low-temperature measurements shown in Figs. 2A and 3 were performed with a pump spot size of 168 µm in diameter and a pump pulse duration of 15 ps. The probe and reference beams both had spot sizes of 135 µm in diameter, and unlike the room-temperature measurements the probe was not electro-optically modulated. The repetition rate of the pump and the mean acquisition rate of the ...
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... verified that changing either the excitation energy or probe power by a factor of two has no effect on the shape of the measured signal waveforms at 85 K. We had to increase the excitation energy by as much as a factor of 10 to see a noticeable change in the normalized signal, see Fig. S2. Fig. S2. Measured TTG signals at 85 K and a period L=10 µm for pump pulse energies 10 and 100 nJ normalized to unity at the maximum. We had to increase the pump energy by a factor of 10 to see an appreciable change in the normalized ...
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... verified that changing either the excitation energy or probe power by a factor of two has no effect on the shape of the measured signal waveforms at 85 K. We had to increase the excitation energy by as much as a factor of 10 to see a noticeable change in the normalized signal, see Fig. S2. Fig. S2. Measured TTG signals at 85 K and a period L=10 µm for pump pulse energies 10 and 100 nJ normalized to unity at the maximum. We had to increase the pump energy by a factor of 10 to see an appreciable change in the normalized ...
