Observation of second sound in a rapidly varying temperature field in Ge

Abstract

Second sound is known as the thermal transport regime where heat is carried by temperature waves. Its experimental observation was previously restricted to a small number of materials, usually in rather narrow temperature 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 temperature 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 provide a route to investigate the potential of wave-like heat transport in almost any material, opening opportunities to control heat through its oscillatory nature.

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 τ_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 (ʌ_HHE), the diffusive case (ʌ_diff), and the penetration depth (ʌ_ss) obtained in the high-frequency limit.

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 experimental relaxation times (τ_ss), as well as the propagation velocity (v_ss), 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, $f=f_{\mathrm{\lambda }}^{\text{eq}}+{\mathbf{\beta }}_{\mathrm{\lambda }}\bullet \mathbf{q}+{\mathbf{\gamma }}_{\mathrm{\lambda }}\bullet (\mathrm{\partial }\mathbf{q}/\mathrm{\partial }t),$ combined with DFT simulations as described in sections 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.