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On the question of second sound in germanium: A theoretical viewpoint

We revisit the recent work from Beardo et al. \cite{beardo2021observation} wherein the observation of second sound in germanium is claimed. We review the requirements imposed on the collision operator (or equivalently, the full scattering matrix) of the linearized phonon Boltzmann transport equation (LBTE) for the observation of driftless second sound as established by Hardy. By performing an eigendecomposition of the full scattering matrix, we show that the requirement that the smallest nonzero eigenvalue must be associated with an odd eigenvector is not satisfied. Furthermore, direct solutions to the LBTE for a frequency modulated heat source do not reveal the presence of second sound. Finally, numerical solutions to the BTE under the relaxation time approximation (RTA) in the 1D frequency-domain thermoreflectance (1D-FDTR) experimental geometry demonstrate that phase lag alone is not a suitable experimental observable for inferring second sound. We conclude by discussing the need for a second sound `smoking gun'.


I. INTRODUCTION
Second sound is a manifestation of phonon hydrodynamics. Peshkov theoretically proposed that phonon hydrodynamics can occur in crystal solids [2]. This hypothesis was subsequently confirmed by a handful of experiments reporting the observation of second sound in a few materials at low temperatures [3][4][5]. Further theoretical worked by Enz categorized the distinct speeds of second sound as 'drifting' and 'driftless' second sound [6].
Hardy subsequently showed that these two distinct manifestations of wave-like temperature responses in crystals can be obtained from different approximate solutions to the LBTE, so long as certain requirements imposed on the eigenvalues of collision operator are satisfied [7]. Hardy further noted that 'other types' of second sound may exist so long as the eigenvalues of the collision operator yield a slow decay of energy flux relative to the heating frequency.
In Ref. [1], the frequency domain thermoreflectance (FDTR) technique is used to study thermal transport in a bulk germanium crystal. In this technique, an intensity modulated optical pump beam acts as frequency modulated thermal source and a probe beam encodes changes in reflectivity (which is assumed to be directly proportional to changes in temperature) at the surface of the sample. The reported experimental observable is the phase lag between the heat source and the temperature response at the surface. Over a temperature range from 7 K to 300 K, the experimental phase lag is found deviate to from the diffusion equation at high frequencies. To interpret the data, a macroscopic hyperbolic heat equation (HHE) is invoked for which the solutions for temperature are damped waves: where τ denotes a macroscopic relaxation time, T is the temperature, α is the thermal diffusivity, ρ is density, C p is specific heat and S(r, t) is an external heat source. Reasonable fits between this equation and the experimental data is considered evidence for 'driftless' or 'other types' of second sound (due to the weak normal scattering rate relative to the Umklapp and isotopic scattering rates in germanium, drifting second sound will not appear, see Supplemental Material). However, a microscopic picture of the experimental data is still lacking. Using the BTE, we take a step in addressing this missing piece in this paper, which is organized as follows. In Sec II, an eigenvalue analysis of the collision operator is normal processes and one which decays due to U-processes. The latter part is expected to be strongly coupled to the momentum density, and this coupling then gives rise to the usual concept of drifting second sound. A formal analysis of the eigenvalues and eigenvectors of the collision operator performed by Hardy shows that mathematically it is possible that the coupling is negligible but the necessary conditions on the eigenvalue spectrum cannot be interpreted in physical terms. A definite conclusion on the feasibility of driftless second sound should, therefore, be subject to explicit results on the eigenvalues of the collision operator.' [9] In recent years, with the success of ab initio methods [10], such 'explicit results' have become obtainable.
So, in the spirit of the above statement, we investigate the eigenvalue spectrum of the collision operator of germanium to ascertain the feasibility of observing driftless second sound. The eigenvalue problem is expressed as whereΩ nn is the symmetrized collision operator, θ α n is n component of the α eigenvector with the corresponding eigenvalue 1 τα . Hardy formulated the requirements for driftless second sound in terms of the eigenvalues of the collision matrix: the smallest nonzero eigenvalue must be associated with an odd eigenvector [7] for it is the odd eigenvectors that contribute to heat flux while the even eigenvectors contribute to energy change. Following the procedure outlined in Ref. [11] using ab initio properties for germanium [12], we have obtained the eigenvalue spectrum for the collision matrix of pure germanium at T = 50 K and 300 K and find that this condition is not satisfied: the smallest odd eigenvalue is larger than the smallest even eigenvalue (see Figure 1). Similar results were previously obtained for silicon (see Figure 2c in Ref. [13]).

III. LBTE SOLUTIONS
In previous studies, several different ansatzs for the phonon distribution functions have been used to anticipate these 'other types' of second sound. Ceppellotti et al. assume plane wave forms for the non-equilibrium distribution [14]. Sendra et al. assume a form where the heat flux and its first derivatives in time and space are considered independent variables [15].
Recently, Shang et al. used a first order correction to the drifted Bose-Einstein distribution under the Callaway approximation to the collision operator to obtain a dispersion relation for second sound [16]. Here, we do not impose any ansatz on the phonon distribution function and in doing so retain the ability to capture Hardy's three types of second sound without assuming a priori their occurrence. Assuming the temperature rise due to the heating is small compared to the background temperature such that a local temperature exists, we directly solve the linearized BTE as previously presented in Ref. [17].
where g n is the deviational phonon energy density for mode n, v n is the phonon group velocity, Ω n,n is the unsymmetrized collision operator, ω n is the phonon frequency, g 0 n is the deviational equilibrium phonon energy density and Q n is the spectral heat source. It can be argued that at very high heating frequencies (ωτ >> 1), regardless of the specifics of the collision operator, that wave-like transport can exist. This is the temporal equivalent to the ballistic limit. Wave-like ballistic transport has been observed in the original second sound heat pulse experiments in NaF [4]. Indeed, we do observe resonances in the ballistic limit of graphite ( Figure III at f = 25 GHz as well as the Supplemental  Material and Ref. [19]). These ballistic temperature waves can be distinguished from second sound oscillations through inspection by making a suitable correspondence to the first sound speeds of the material.

IV. DIRECT SIMULATIONS OF THE FDTR GEOMETRY
The solutions of the full scattering matrix LBTE are presently restricted to unbound geometries or specularly reflecting boundaries [20]. To simulate a geometry representative of the FDTR experiment with a diffuse boundary, we turn to the Discrete Unified Gas Kinetic Scheme (DUGKS) that is capable of solving the BTE under the RTA and Callaway approximations [21] with ab initio inputs (see Supplemental Material).
A sizable discrepancy between the phase lag of temperature obtained from solutions to RTA-BTE and HHE is found in germanium at T = 50 K and 300 K ( Figure 4). This discrepancy, in part, is due to the failure of the HHE at high heating frequencies. As shown in Figure 2, the BTE solutions begin to deviate from the diffusion and HHE solutions at First, it should be emphasized that temperature waves have not been directly observed, but rather a proxy metric (i.e., phase lag) is interpreted within a model that assumes a priori wave-like behavior that is limited to ωτ < 0.1. Past FDTR and time domain thermoreflectance (TDTR) experiments have been modeled within the BTE [22][23][24][25] and the solutions to the BTE obtained here contain the flexibility to predict (or fit) the phase lag.
Second, the 'spectrality' of the heat source is not captured in the current HHE. Nonthermal sources have been shown to impact the temperature response [26]. In previous analyses of FDTR experiments, it was shown that the inclusion of a two temperature model for the metal transducer removed much of the non-diffusive behaviour of the substrate [27,28].
Notably, by modeling of the transducer, the frequency domain representation of TDTR measurements on silicon were captured by a diffusive model up to a modulation frequency of 1 GHz [29]. While there is no metal transducer in Ref. [1], the thermalization of hot electrons requires careful treatment [30]. The computational cost of numerically simulating the 3D experimental configuration remains impractical, nonetheless, it has not been ruled out that the inclusion of such non-thermal sources without invoking second sound or phonon hydro-dynamics in the DUGKS-based solutions to the RTA-BTE would recover the experimental data.
Finally, and most crucially, our BTE solutions demonstrate the absence of a microscopic mechanism for the occurrence of measurable second sound in germanium: the lack of strong normal scattering eliminates the occurrence of drifting second sound, the eigenvalues of the collision matrix eliminate the possibility of driftless second sound and the lack of resonance in temperature response in the mid to high heating frequencies eliminates the 'other type' of second sound.  [32]. The germanium properties have been experimentally verified in Ref. [33]. For graphite, the details of the ab initio calculations can be found in Ref. [34].
The macroscopic relaxation time, τ , was calculated using the obtained ab initio inputs following the procedure described in Ref. [1]. Relevant data and source code can be found at https://github.com/schuberm/.  boundary scattering rates respectively and denotes thermal averages, to estimate the temperature and length window for second sound in germanium. For pure germanium, a window (shaded pink region) emerges at T < 15 K for lengths greater than 1 mm. However, once isotope scattering is included, the hydrodynamic window disappears.
Assuming the temperature rise due to the heating is small compared to the background temperature T ref such that a local temperature T exists, and the phonon Boltzmann transport equation (BTE) under the single-mode relaxation time approximation (RTA) could be written as [36] ∂g n ∂t where g n is the deviational phonon energy density for mode n, v n is the phonon group velocity, τ n = τ n (T ref ) is the effective relaxation time, g 0 n = C n (T − T ref ) is the deviational equilibrium phonon energy density and C n is the mode-dependent specific heat. Q n is the mode-dependent heat source, which depends on the spatial position x and time t. The local energy density U , temperature T and heat flux q could be obtained by taking the moment of phonon distribution functions, where dK is an integral over the whole wave vector space. A pseudo-temeprature T p is introduced to ensure the energy conservation of the phonon scattering term, so that The discrete unified gas kinetic scheme (DUGKS) [37, 38] is introduced to solve Eq. (4) with ab initio input. The phonon dispersion and scattering in the whole first Brillouin zone are used, and no isotropic wave vector space is assumed, which is different from previous studies [38 -40]. The finite volume method is used to discretize the spatial space, the trapezoidal quadrature is used for the time integration of the phonon scattering term and heat source term, while the mid-point rule is used for the flux term. Then Eq. (4) in integral form over a control volume can be written as follows, where V i is the volume of the cell i, N (i) denotes the sets of neighbor cells of cell i, ij denotes the interface between cell i and cell j, S ij is the area of the interface ij, n ij is the normal unit vector of the interface ij directing from cell i to cell j; ∆t is the time step from time t m to t m+1 = t m + ∆t, where m is an index of the time steps. In order to remove the implicitness, two new distribution functions are introduced and defined as Then Eq. (10) can be expressed as where x ij = x ij − v n ∆t/2. Equation (14) can be reformulated as follows, g +,m n (x ij ) in Eq. (15) is reconstructed by numerical interpolation, where σ c is the spatial gradient of the distribution functionḡ +,m n (x c ) in the cell c. If v n ·n ij > 0, c = i; else c = j. The van Leer limiter is adopted to determine the gradient to ensure the numerical stability and accuracy.
According to the energy conservation principle of the phonon scattering term, g n + ∆t 2 Q n − g 0 n 2τ n + ∆t dK = g 0 n − g n 2τ n dK = 0.
Then the pseudo-temperature at the cell center T m+1 i,p can be updated, Once T m+1 When the mode-dependent BTE with ab initio calculations input is simulated, the heat source is mode-dependent whereṠ is the heat source at the macroscopic level. When the linear phonon dispersion and gray model is used, the specific heat, group velocity, heat source and relaxation time are all frequency-independent. For three-dimensional materials, we have v = | v|s, where s = (cos θ, sin θ cos ϕ, sin θ sin ϕ) is the unit directional vector, θ ∈ [0, π], ϕ ∈ [0, 2π] is the azimuthal angle. The cos θ ∈ [−1, 1] is discretized with the N θ -point Gauss-Legendre quadrature, while the azimuthal angular space ϕ ∈ [0, π] (due to symmetry) is discretized with the Nϕ 2 -point Gauss-Legendre quadrature. For all quasi-1D periodic steady problems in this paper, the simulated system size is large enough to ensure the semi-infinite boundary. Non-uniform cells are used to discrete the spatial domain, and the cell size increases form the left to the right with a fixed ratio r size . When < 10 −6 , the thermal system reaches periodic steady state, where where N cells is the total discretized spatial cells, t p is the heating period, and x i is the spatial position of the cell i.
For quasi-1D frequency domain thermoreflectance (FDTR) geometry [41], we havė where A is the amplitude of the heat source. The left boundary is the diffusely reflecting adiabatic boundary [38] and the right is assumed semi-infinite with temperature T ref .
The linear phonon dispersion and gray model is used, and the parameters of Germanium at 300 K in Ref. [41] are adopted, where C = 1.60 ×  The temperature at the left boundary is fixed and changed with time [42], i.e., T (x = 0, t) = T ref + ∆T cos(2πt/t p ),