SCIENCE ADVANCES | RESEARCH ARTICLE PHYSICS Copyright © 2021 The Authors, some Observation of second sound in a rapidly varying rights reserved; exclusive licensee temperature field in Ge American Association for the Advancement of Science. No claim to Albert Beardo1, Miquel López-Suárez2,3, Luis Alberto Pérez2, Lluc Sendra1, Maria Isabel Alonso2, original U.S. Government Claudio Melis3, Javier Bafaluy1, Juan Camacho1, Luciano Colombo3, Riccardo Rurali2, Works. Distributed Francesc Xavier Alvarez1, Juan Sebastián Reparaz2* under a Creative Commons Attribution Second sound is known as the thermal transport regime where heat is carried by temperature waves. Its experi- NonCommercial mental observation was previously restricted to a small number of materials, usually in rather narrow temperature License 4.0 (CC BY-NC). windows. We show that it is possible to overcome these limitations by driving the system with a rapidly varying temperature field. High-frequency second sound is demonstrated in bulk natural Ge between 7 K and room tem- perature by studying the phase lag of the thermal response under a harmonic high-frequency external thermal excitation and addressing the relaxation time and the propagation velocity of the heat waves. These results pro- vide a route to investigate the potential of wave-like heat transport in almost any material, opening opportunities to control heat through its oscillatory nature. INTRODUCTION Second sound in solids was first experimentally observed in solid The study of heat transport beyond Fourier’s regime has attracted He (6); later in NaF (7), Bi (8), and SrTiO3 (9); and most recently in renewed interest in recent years. Great efforts have been performed highly oriented pyrolytic graphite (10). Several theoretical works to unravel the physical properties of thermal waves, as well as the have also recently addressed its occurrence in low-dimensional experimental conditions that are necessary for their observation. systems (11–13). In all these experimental observations of second Applications based on such concepts have been envisioned and dis- sound, the dominance of momentum conserving phonon scattering cussed extensively already in many recent publications (1–4). The (Normal processes) with respect to resistive phonon scattering spatiotemporal propagation of the temperature field in the form of (Umklapp processes) was found to be the key mechanism leading to waves is known as “second sound,” a term that was adopted in anal- its observation. Second sound was observed almost exclusively in ogy to “first sound” (or simply “sound,” i.e., mechanical lattice the very low temperature regime (T < 5 K), with the exception of a vibrations). As pointed out in (5), first sound and second sound are recent example (10) at higher temperatures (125 K) for samples described by a similar equation where the variables have a different with low resistive phonon scattering. physical meaning, i.e., pressure and temperature, respectively. A condition for the experimental detection of second sound, The simplest differential equation that describes wave-like heat based on these experimental observations (6–10), was found to be transport from a mesoscopic perspective is the hyperbolic heat N < exp < R; i.e., the typical experimental observation times (exp) equation (HHE) according to Maxwell, Cattaneo, and Vernotte must be larger than normal phonon scattering times (N) to allow momentum redistribution but smaller than resistive phonon scat- ρ​Cp​  ​​ ( ∂t ) ​∂​​  2​T ​  + ​ ─ ∂T ​  − α​∇​​  2​T = ​ ─ ∂S​(​​r, t​)​​ 1  ​​ ​​S(r, t ) + ​τ​  ​​ ​ ─ tering times (R) to avoid decay of the phonon wave packet into the ​​​τ​  ss​​ ​ ─ ss ​​ ​​​​ (1) ∂​t​​  2​ ∂t phonon equilibrium distribution. The theoretical foundations of second sound were set in the 1960s by M. Chester, R. J. Hardy, where  is the thermal diffusivity, ss is the thermal relaxation C. P. Enz, and co-workers (5, 14, 15), who predicted the existence of time,  is the mass density, Cp is the specific heat, and S​ (r,t)​is an “drifting” second sound, i.e., a type of wave-like heat transport that external power heat source. In our context, the system is in local is triggered by the dominance of normal phonon scattering events. equilibrium, and it is well characterized by a local temperature T This type of second sound was experimentally confirmed in (6–10). (see discussion in section S8). The previous equation describes the Furthermore, the existence of a “driftless” or “high-frequency” type, propagation of a temperature wave with_ a damping term given by as well as “other types” of second sound, was also envisioned. In this ∂T/∂t and a propagation velocity υ ​​ ​  ss​​  = ​√ / ​​  ss​​ ​​. The solutions of this case, the dominance of normal scattering events is not a necessary equation lead to different heat transport regimes, depending on the condition for the existence of wave-like heat transport, but the key temporal and spatial length scales under investigation. The key to requirement is instead the slow decay of the energy flux, as predict- unlock the different regimes is the magnitude of the first term on ed by different solutions to the Boltzmann transport equation (5). the left-hand side of Eq. 1, thermal inertial term; i.e., if ss or ∂2T/∂t2 These different flavors of second sound are all contemplated in the is sufficiently large, the spatiotemporal distribution of temperature mesoscopic HHE (Eq. 1), because different microscopic mecha- field will exhibit wave-like behavior. nisms can lead to the propagation of thermal waves. We show that it is possible to observe a type of high-frequency second sound in natural bulk Ge by driving the system out of equi- 1 Departament de Física, Universitat Autònoma de Barcelona, 08193 Bellaterra, librium with a rapidly varying temperature field. Our concept is Spain. 2Institut de Ciència de Materials de Barcelona, ICMAB-CSIC, Campus UAB, based on taking advantage of the second-order time derivative in 08193 Bellaterra, Spain. 3Dipartimento di Fisica, Università di Cagliari, Cittadella Universitaria, I-09042 Monserrato (Ca), Italy. the HHE, Eq. 1, in a frequency-domain experiment. As the driving *Corresponding author. Email: jsreparaz@icmab.es frequency increases toward the hundreds of megahertz range, the Beardo et al., Sci. Adv. 2021; 7 : eabg4677 30 June 2021 1 of 6 SCIENCE ADVANCES | RESEARCH ARTICLE relative weight of this term with respect to the damping term we take advantage of the fact that for bulk Ge, ∂R/∂T ≈ 0 at T = (first-order time derivative) increases proportionally to the fre- 220 K for 532 nm of probe wavelength (see section S3), whereas quency upon a harmonic excitation, hence making the observation ∂R/∂n is expected to be almost independent of temperature between of wave-like heat propagation possible. We show that this approach 220 K and room temperature. Furthermore, ∂R/∂T exhibits a sign is robust enough to expose second sound independently, to a cer- inversion at this temperature, which is reflected in a change of the tain extent, of the phonon scattering rates of the studied material, as phase lag by an angle of . Figure 1A displays the temperature well as of temperature. Although heat transport in Ge is dominated dependence of R/R, as well as the phase lag for low (30 kHz) and by resistive phonon scattering processes, which partly originate high (100 MHz) modulation frequencies, and for a constant pump from its large isotopic diversity, we show that it is still possible to power of ≈10 mW. The reflectivity signal exhibits a minimum (19) observe second sound in the high-frequency limit. around ≈220 K, followed by a slow signal recovery at lower tem- peratures. In the lower temperature range (T < 100 K), the lattice and electronic contributions to the phase lag have opposite signs. RESULTS AND DISCUSSION Hence, the measured phase lag at high frequency in this range can- Our experiments are based on a frequency-domain optical reflec- not be explained with the electronic contribution to the optical re- tance pump-and-probe approach based on two lasers with different flectivity. On the other hand, the phase lag is shifted by  at ≈220 K wavelengths (pump = 405 nm and probe = 532 nm) focused onto the (140°→−40°). The optical reflectivity extracted from ellipsometry surface of a Ge sample to a spot size with radius Rspot ≈ 5.5 m. The experiments corresponding to T = 1 K is also shown for relative studied samples are pieces of a substrate of natural Ge. Further de- comparison. A similar behavior is observed independently of the tails are provided in the Supplementary Materials (section S1). The excitation frequency, which resembles the ellipsometry data for which pump laser (thermal excitation) is modulated between 30 kHz and n = 0. From the reflectivity signal ratio between the minimum 200 MHz with a sinusoidal power output waveform, leading to a observed in Fig. 1A and the value of the reflectivity at room tem- dynamic modulation of the optical reflectivity of the surface of the perature, we can estimate an upper boundary for (1/R)[∂R/∂n] ≈ −2 × sample, which is also well described by a harmonic waveform. A 10−29 m3, assuming that the residual signal at ≈220 K has a pure- frequency-dependent phase lag gradually develops, defined as a ly electronic origin. The inset of Fig. 1A displays calculations of the phase difference between the harmonic thermal excitation, S​ (r,t)​, electronic (solving the electron recombination diffusion equation) and the response of the sample, T ​ (r,t)​(sections S2 and S7), which and lattice (solving HHE) contributions to the optical reflectivity as can be modeled using Eq. 1 in the present experimental conditions. a function of frequency (section S3). Whereas the lattice contribu- The choice of Ge as a candidate for the observation of second sound tion to the optical reflectivity is at least 20-fold larger than the elec- is not arbitrary, and it is mostly based on the large optical absorp- tronic component at the higher frequencies, for lower frequencies, tion coefficient of this material for the wavelengths used in this ex- this ratio is substantially increased. periment. The optical penetration depth of the pump and probe Figure 1B displays the experimental phase lag as a function of lasers is 405nm = 15 nm and 532nm = 17 nm, respectively (section frequency between 30 kHz and 200 MHz at room temperature. The S3). These particular conditions make Ge an ideal material for this complex thermal response of the specimen was, at first, computed study, because the small penetration depth of both lasers ensures numerically within Fourier’s model, solving the parabolic approxi- that the measured phase lag is local and, thus, accurately describes mation to the three-dimensional (3D) HHE (diffusive case), which the oscillations of the thermal waves. The penetration depth of the is obtained when the first term of Eq. 1 can be neglected. In the pump and probe lasers should be compared with the thermal wave- range between 30 kHz and 1 MHz, the agreement between Fourier’s length υss/f. At room temperature and the highest excitation fre- solution and the experimental data is excellent, although deviations quency considered, υss/f = 840 nm, thus considerably larger than the are already observed around 1 MHz. Above 30 MHz, the difference optical penetration depth. We note that wave-like effects are not ob- between the experimental phase lag and Fourier’s predictions is served in the presence of a metallic transducer (section S9), because evident. For the higher-frequency range, the experimental data show the thermal interface between the transducer and the Ge substrate, that the phase lag (absolute value) decreases with increasing fre- as well as the transducer itself, dominates the system response in the quency. This trend cannot even be qualitatively reproduced by frequency range where wave-like effects are expected. Fourier’s model, which predicts that as frequency increases, the In the present experiments, the changes R of the optical reflec- phase lag approaches −/4 and even lower values (see section S7 tivity upon the pump laser excitation are driven by the lattice tempera- and fig. S7-3). The full 3D solution of the HHE based on the finite-­ ture variation, hereafter referred to as T, and by the modulation element method (FEM) was used to fit the experimental data n of the carrier concentration in the conduction band. The result- through the entire frequency range, and it is shown in Fig. 1B. A ing R at the energy corresponding to the probe wavelength is pro- detailed description of the fitting procedure is presented in section vided by the first-order expansion R = (∂R/∂T)T + (∂R/∂n)n S7. We obtained ​ ​​ exp ss​  ​​= 500 ps and  exp = 3 × 10−5 m2/s, thus leading exp (16). Interestingly enough, the contribution provided by the varia- to a propagation velocity ​​υ​ss​  ​  = 250​m/s, all at room temperature. tion of the carrier concentration is expected to dominate the way These experimental observations are in qualitative agreement with the reflectivity is affected in experiments, like the present ones, in- the computational experiments (see schematic illustration in Fig. 1C) volving pulsed laser sources, i.e., for high electronic excitation den- by nonequilibrium molecular dynamics (NEMD) and are shown in sities (17, 18). However, in our excitation conditions, the electronic the inset of Fig. 1B (see also section S5 and fig. S5-1). Although a contribution to the optical reflectivity can be neglected; hence, the quantitative agreement cannot be expected because of differences optical reflectivity is dominated by the temperature of the lattice between the experimental and the computational setups (reduced (T) for all excitation frequencies. We demonstrate this by studying size of the sample and purely 1D heat flux in NEMD), the NEMD the relative magnitude of (∂R/∂T)T and (∂R/∂n)n. In particular, results show a notable similarity with the experimental results, with Beardo et al., Sci. Adv. 2021; 7 : eabg4677 30 June 2021 2 of 6 SCIENCE ADVANCES | RESEARCH ARTICLE Fig. 1. Optical reflectivity, experimental and simulated phase lag versus frequency, and thermal penetration depth for a Ge sample. (A) The top panel displays the optical reflectivity change as a function of temperature for 30 kHz (red) and 100 MHz (blue). The black dots were obtained from ellipsometry measurements to a temperature rise of 1 K (full symbols→dR/dT < 0, open symbols→dR/dT > 0). The bottom panel displays the phase lag of the signal with respect to the pump excitation. The phase lag is directly obtained from the measurements and must be corrected by ±180° to normalize it to the [0;±45°] as in (B). The inset displays calculations accounting for the thermal and electronic contributions to the reflectivity at 300 K. (B) The experimental phase lag as a function of the pump excitation frequency is shown in black open symbols. The inset displays numerical experiments using NEMD. In dashed red line, we display the prediction based on Fourier’s law. The solutions based on the 3D HHE are shown in a black line with a resulting fitted ss = 500 ps. (C) Schematic illustration of the geometry used for the NEMD numerical experiments. (D) Frequency-­ dependent thermal penetration depth calculated using the solution of the HHE (ʌHHE), the diffusive case (ʌdiff), and the penetration depth (ʌss) obtained in the high-­ frequency limit. the phase lag initially decreasing, hitting a minimum, and then re- expected and indeed experimentally observed for lower tempera- covering. These results are of particular interest because, within the tures. Wave-like effects are already present below fc, as can be ob- NEMD approach, no assumption is made regarding the heat trans- served comparing the experimental data with the corresponding fits port regime. Thus, the numerical experiments are an independent using the HHE to the Fourier predictions, as shown in Fig. 2A. The confirmation of the appearance of second sound in the high-frequency onset frequency can be estimated as f > 0.1fc, corresponding to a limit. The observed large reduction of the phase lag with respect to phase lag difference between the experimental data and the Fourier the Fourier prediction (see Fig. 1B) indicates the emergence of a solution >5°. The onset of wave-like effects is also evidenced by the more efficient heat transport mechanism as compared to diffusion deviations between diff and HHE as shown in Fig. 1D. The mini- at high frequencies. Ballistic behavior of phonons (20–22) cannot mum observed on the phase lag curves and the position of the crit- be invoked to rationalize the experimentally observed dependence ical frequency fc relative to the frequency of the minimum originate of the phase lag with frequency. As discussed previously in (21), a from the relation between a, ss, and Rspot (see discussion in section reduction of the thermal conductivity at high frequency would lead S7). The frequency-dependent phase lag in Fig. 2A was fitted at to an increase in the phase lag (see fig. S7-4), rather than a decrease each temperature using the HHE, as described for the room tem- (absolute value), as we observe. Conversely, the observed frequency-­ perature case, and the results for ​​​exp exp ss​  ​​ and ​​υ​ss​  ​​are shown in Fig. 2B dependent phase lag is well reproduced by the HHE, i.e., consider- and table S2. We note that the accuracy of the fits reduces for lower ing the contribution of wave-like heat transport. temperatures due to the frequency-dependent temperature rise in- The frequency window where second sound is expected can be duced by the pump laser. estimated comparing the thermal penetration depth of the diffusive To understand the origin of these observations, we have devel- and wave-like regimes. Figure 1D displays the penetration depth, oped a rather simple model (see derivation in section S6) based on HHE, calculated using the exact solution of the HHE (see section _ the expansion of the perturbed phonon distribution function (23), S7), as well as_ the diffusive and wave-like limits, ​  ​​ diff​​  = ​√  / (f ) ​​ i.e., the intermediate state assumed by the nonequilibrium distribu- and ​​​  ss​​  = 2 ​√ ​​  ss​​ ​​, respectively. A critical frequency fc is obtained tion before it decays to the equilibrium one by means of dissipative when diff = ss, thus providing an estimation of the frequency for resistive processes, as: ​​f​ λ​​  = ​f eq ​eq​  ​​ is λ​ ​  ​ + ​𝛃​  λ​​  ∙ q + ​𝛄​  λ​​  ∙ (∂ q / ∂ t)​, where ​​f which the diffusive and wave-like contributions to heat transport the equilibrium phonon distribution function, ​q​is the heat flux, 𝛃​  ​​ λ​​​ are similar. The temperature dependence of ss, υss, and fc was stud- and ​​𝛄​  λ​​​are mode-dependent functions to be determined, t is the ied between room temperature and 7 K, and it is shown in Fig. 2A temporal coordinate, and  denotes each phonon mode. We note (full set in section S2). As temperature decreases, the ratio between that the case with ​​𝛄​  λ​​  = 0​leads to the same propagation velocity for the penetration _ depth of the wave-like and diffusive contributions is second sound as in (5) (section S6). In our case, however, the pres- ​​​  ss​​  / ​​  diff​​  = ​√4f ​​  ss​​(T) ​​, which implies that lower temperatures fa- ence of rapidly varying temperatures leads to rapidly varying ​q​, sug- vor the spatial propagation of the thermal waves because larger ss is gesting that the expansion of the perturbed phonon distribution Beardo et al., Sci. Adv. 2021; 7 : eabg4677 30 June 2021 3 of 6 SCIENCE ADVANCES | RESEARCH ARTICLE Fig. 2. Phase lag curves versus temperature, relaxation time, and velocity of second sound (theory and experiment), and simulated spatial propagation profiles for the diffusive and wave-like cases. (A) Phase lag versus frequency for the higher-frequency range as a function of temperature. We plot three points at 300, 100, and 15 K with the corresponding fits to the data point using the 3D HHE. In dashed lines, we display the prediction based on Fourier’s law at each temperature. (B) The exper- imental relaxation times (ss), as well as the propagation velocity (vss), are shown as a function of temperature. The dashed lines are guides to the eye. The full lines are the predictions based on the expansion of the perturbed phonon distribution function, ​f = ​ f eq λ​ ​ ​ + ​𝛃​ λ​​  ∙ q + ​𝛄​ λ​​  ∙ (∂ q / ∂ t ),​combined with DFT simulations as described in sec- tions S6 and S4, respectively. (C) Finite-element simulations of the spatial distribution of the temperature field at a function of temperature in the direction perpendicular to the surface of the sample at the highest excitation frequency of ≈300 MHz for an arbitrary time. The parabolic and hyperbolic solutions are shown in dashed and full lines, respectively. function in terms of ∂​ q/∂t​is a reasonable assumption. The previous and the measured values at very low temperatures are expected (see ansatz for f was then introduced into the linearized Boltzmann section S8 for details). We note that similar experiments (25, 26) transport equation (BTE) to find the solution for ​​𝛃​  λ​​​ and ​​𝛄​  λ​​​. It can were explained in terms of nonlocality as described by the Guyer-­ be shown that within this framework, it is possible to derive the Krumhansl equation, which reduces to Eq. 1 in the absence of non- HHE, with a corresponding explicit expression for ss, in terms of local effects. The present experiments, however, are beyond the individual phonon relaxation times () as shown in section S6 applicability of this equation because the heating region is much smaller than the nonlocal length (27). Future work should address ∂ ​f​eq​  ​ ​ ​ ​​  ℏ ​​  ​​ ​2​ ​ ​​  ​2 ​ ​​  _ ∑ ∂T  ​ the attenuation of nonlocal effects under the present conditions and ​​​  ss​​ = ​─    ∂ ​ f eq ​​ (2) its influence in the propagation of thermal waves.  ​ ​ ​  ​∑ ​ ​​  ℏ ​​  ​​ ​2​  ​ ​​  ​  ​​ ​ _ ∂T ​ The spatial dependence of the temperature field in the parabolic (diffusive) and the hyperbolic (wave-like) cases was simulated using where ℏ is the Planck constant,  is the phonon energy, and  FEMs in the direction perpendicular to the surface of the sample at is the phonon group velocity. We computed , , and  from the an arbitrary time. Figure 2C displays the normalized temperature solution of the BTE based on density functional theory (DFT) profiles for 15, 100, and 300 K at the highest experimental excitation interatomic force constants (IFCs) (section S4 and table S4). We frequency of ≈300 MHz (see fig. S7-5 for simulations of the tem- restricted ourselves to the relaxation time approximation (RTA) perature field at fc). As expected, the wave-like behavior of the tem- after verifying that the full iterative BTE picture does not alter the perature field exhibits a strong temperature dependence. The observed prediction of the theory. Using the RTA has the additional benefit propagation depth, particularly at lower temperatures, is especially of providing a comparison on equal footing with previous theoreti- interesting if considering the possibility of high-frequency modu- cal descriptions (5) and allowing unambiguous definition of the re- lated thermal interference. We think that the present approach laxation times (24). We observe that the corrections to the RTA could open interesting possibilities for the experimental observa- provided by a full iterative solution of the BTE are very small (with- tion of wave-like heat transport in other materials and lead to the in 4% in the thermal conductivity) for Ge at temperatures as low as development of strategies to control heat transport. 50 K (fig. S4-1). The resulting _ values were inserted into Eq. 2, which yielded ​​​theo theo ss​  ​​ and ​​υss ​ ​  ​ = ​√​​​  theo​  / ​theo ​ ​  ​ ​​as a function of temperature, ss as shown in Fig. 2B. The agreement of the predicted values with MATERIALS AND METHODS those obtained from the experiments is remarkable for T > 100 K, Samples considering that the values of , , and  are evaluated within a All samples were pieces cleaved from a 2-inch-diameter nominally fully ab initio scheme. We note that the model leading to Eq. 2 is undoped Ge wafer with (100) crystallographic orientation, high re- expected to be valid for ∣T∣<< T, where ∣T∣ is the amplitude of sistivity (>40 ohm·cm), and etch pit density <3000/cm2. The wafer the laser-induced thermal oscillations and T is the absolute tem- was purchased from International Wafer Service Inc. (USA). We perature as set by the cryostat. In our experiments, ∣T∣max < 20 K; have studied three different types of samples: (i) clean Ge (no native thus, the observed deviations between the theoretical predictions oxide), (ii) Ge + native oxide + 60 nm of Au, and (iii) a 60-nm-thick Beardo et al., Sci. Adv. 2021; 7 : eabg4677 30 June 2021 4 of 6 SCIENCE ADVANCES | RESEARCH ARTICLE Au transducer was evaporated onto the surface of a Ge piece similar Nosé-Hoover thermostatting, while equations of motions have to (i). Additional details on the etching and evaporation processes been integrated by the velocity-Verlet algorithm (more details can of the samples are given in section S1. be found in section S5). Amplified frequency-domain thermoreflectance SUPPLEMENTARY MATERIALS We developed a low-noise custom-built frequency-domain thermo­ Supplementary material for this article is available at http://advances.sciencemag.org/cgi/ reflectance setup to measure the thermal response of the samples. content/full/7/27/eabg4677/DC1 The thermal excitation was provided by a pump laser diode with a wavelength of 405 nm. The probe laser used was a continuous-wave REFERENCES AND NOTES (CW) laser of 532-nm wavelength and with a maximum output 1. N. Li, J. Ren, L. Wang, G. Zhang, P. Hänggi, B. Li, Colloquium: Phononics: Manipulating power of 100 mW. The output power was controlled with neutral heat flow with electronic analogs and beyond. Rev. Mod. 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