Effect of characteristic size on the collective phonon transport in crystalline GeTe

As a first step to elucidating the complex collective behavior of phonons as a function of characteristic size ($L$), it is imperative to explore the variation of $\kappa_{L}$ with $L$ and therefore to identify the effect of $L$ on the ballistic and diffusive phonon transport. Figure 1 describes this

values of $L$ as we increase the temperature. It is well known in the literature [13, 45] that ballistic conduction of phonons occurs without ph-ph scattering and displays a linear variation with $L$, whereas diffusive conduction of phonons manifests when scattered phonons carry the heat. The effect of the characteristic size on $\kappa_{L}$ can also be represented via the variation of $\kappa_{L}$ with temperatures for different $L$, as shown in Fig 2. At higher temperatures, it is well known [32] that $\kappa_{L}$ decreases with $T$, with 1/$T$ scaling due to the dominant umklapp scattering at high temperatures. As temperature is lowered, gradually $\kappa_{L}$ attains a peak following a gradual decrement at very low temperature. As we go towards higher $L$, the peaks of $\kappa_{L}$ as a function of temperature are gradually seen to be shifted towards lower temperatures (Fig. 2).

To give a more precise account of ballistic and diffusive conduction of phonons in GeTe, we further investigate the characteristic size range of ballistic and diffusive conduction as a function of temperature. The complete ballistic length regime ($L_{ball}$) is defined via the maximum value of $L$, until which $\kappa_{L}$ varies linearly with $L$. Similarly, the complete diffusive length regime ($L_{diff}$) is defined via the minimum length $L$, above which $\kappa_{L}$ reaches the thermodynamic limit and therefore reaches a plateau. In other words, $L_{diff}$ represents the longest mean free path of the heat carriers at a particular temperature [13]. Figures 3.(a) and 3.(b) represent the variations of $L_{ball}$ and $L_{diff}$ respectively, as a function of temperature. As temperature increases, we see a gradual decrement of both $L_{ball}$ and $L_{diff}$. We note here that at very high temperatures, we hardly observe any ballistic conduction of phonons and the $L_{diff}$ acquires a very low value. This is representative of the fact that at high temperatures, internal phonon-phonon scattering is so dominant that no ballistic heat conduction is seen to exist, even for very small grains of the order of 1 nm.

To delve deeper into the origin of length dependent $\kappa_{L}$ in the ballistic phonon conduction regime of GeTe, we investigate the contribution of acoustic and optical modes in the ballistic propagation of heat. Earlier, molecular dynamics simulations and experiments on suspended single-layer graphene [46, 47] suggested the ballistic propagation of long-wavelength, low-frequency acoustic phonon to be solely responsible for the length-dependent $\kappa_{L}$ in the ballistic regime. Our previous studies on GeTe [32, 28] suggested that GeTe shows a clear distinction between acoustic and optical modes in the frequency domain around 2.87 THz. The density of states goes to zero around a frequency of 2.87 THz [32], distinguishing two distinct frequency regimes: acoustic regime ($\omega$ $<$ 2.87 THz) and optical regime ($\omega$ $>$ 2.87 THz). We calculate the cumulative lattice thermal

To visualize the consequences on the mean-free path of the phonons at small $L$, we present the variation of the effective mean-free path variation with phonon frequency for different $L$ at different temperatures in Fig. 5. In the KCM nomenclature, the kinetic mean free path [$l_{k}(\omega)$] and the collective mean free path [$l_{c}(T)$] are defined as $l_{k}(\omega)$ = $v\tau_{k}$ and $l_{c}(T)$ = $\overline{v}\tau_{c}$ respectively, where $v$ is the group velocity and

is the mean phonon velocity [31]. As the kinetic MFP

is a function of phonon frequency whereas the collective MFP is frequency independent and varies only with temperature, we present an effective MFP as $l_{eff}(\omega)$ = (1-$\Sigma$)$l_{k}(\omega)$+$\Sigma$ $l_{c}$. The separate contributions from collective and kinetic MFPs are described in Supplemental Fig. S1. Two effects can be observed from this representation. First, at low temperature, as the $L$ is increased, the optical modes at higher frequencies exhibit more scattered mean-free paths. Figure 5.(a) shows that at $T$ = 10 K, at higher frequencies in the optical modes, $L$ = 0.003 $\mu$m persists more scattered MFPs compared to the $L$ = 0.001 $\mu$m case. This feature indicates that the ballistic conduction is stronger for $L$ = 0.001 $\mu$m, where $L$ strongly controls the mean-free path than that of the $L$ = 0.003 $\mu$m case. Second, increasing temperature for fixed $L$, also leads to the gradual weakening of the ballistic conduction of phonons, as can be seen from Fig. 5. This is evident from the gradual broadening of MFPs with temperature [follow fixed color points for four different temperatures in Fig. 5.(a), (b), (c) and (d).] due to the gradually weakening control of $L$ on dictating the mean-free paths of the system.

After understanding the effect of characteristic size ($L$) on the ballistic and diffusive conduction of phonons, we turn our attention to the effect of $L$ on the collective phonon transport of crystalline GeTe. The connection between ballistic and diffusive phonon transport and the collective motion of phonons are crucial to determine the origin of the exotic hydrodynamic phonon transport in materials. In our earlier work [28], unusually, low-thermal conductivity chalcogenide GeTe emerged as a possible candidate to feature phonon hydrodynamics with the characteristic size being a dominant factor.

In this context, we start by investigating the relative strengths of the average phonon scattering rates, which is defined as

Here, $\lambda$ denotes phonon modes (q, $j$) comprising wave vector q and branch $j$. Index $i$ denotes normal, umklapp, isotope, and boundary scattering processes used, marked by N, U, I, and B respectively. $C_{\lambda}$ is the modal heat capacity, given by the following equation

where, $T$ denotes temperature, $\hbar$ is the reduced Planck’s constant, and $k_{B}$ is the Boltzmann constant. In one of the earliest works on phonon hydrodynamics, Guyer and Krumhansl [12] found that the hydrodynamic regime exists if

Further, Guyer’s condition [12] for the presence of second sound and Poiseuille’s flow reads:

In Fig 6, we explore the $L$ window that satisfies the aforementioned Guyer and Krumhansl condition of phonon hydrodynamics in crystalline GeTe. Figure 6 presents the average scattering rates due to normal (N), resistive (R) [comprised of umklapp (U) and isotope scattering (I)] and the phonon-boundary scattering as a function of temperature for GeTe. We observe a substantial width of $L$, that persists phonon hydrodynamic conditions, as the boundary scattering rates decrease gradually on increasing $L$. This is shown via the gray shaded regions in Figs. 6.(a) and 6.(b) for two representative grain sizes: $L$ = 0.08 $\mu$m and $L$ = 0.8 $\mu$m, respectively. In the scattering rate approach, we also identified the ballistic conduction region, mentioned earlier through the linear dependence of $\kappa_{L}$ with $L$, as the region where $\langle\tau_{B}^{-1}\rangle_{ave}\gg\langle\tau_{ph-ph}^{-1}\rangle_{ave}$. Similarly, the purely diffusive conduction region, mentioned earlier as the $L$-regime where $\kappa_{L}$ is independent of $L$, as the region where $\langle\tau_{B}^{-1}\rangle_{ave}\ll\langle\tau_{ph-ph}^{-1}\rangle_{ave}$. At this point, we go back to Fig. 2 to explain the cusp-like behavior of $\kappa_{L}$ as a function of temperature. This cusp-like pattern of $\kappa_{L}$ is found to present for $L$ $>$ 1 $\mu$m, as we gradually decrease the temperature. In Fig. 6.(b), this $L$ regime is identified as $L$ values for which normal scattering overpowers boundary scattering rates. At low temperatures, umklapp scattering is rare and boundary scattering acts as the dominant resistive phonon scattering. So, the effect of boundary scattering tries to reduce the $\kappa_{L}$ while the momentum conserving normal scattering tries to increase $\kappa_{L}$. Overpowering normal scattering compared to boundary scattering for $L$ $>$ 1 $\mu$m forces $\kappa_{L}$ to increase after an apparent shallow dip or a plateau and gives rise to the cusp-like pattern in Fig. 2.

Once the Guyer and Krumhansl conditions are satisfied and a prominent $L$ window is observed to feature phonon hydrodynamics, we next investigate the spectral representation of lattice thermal conductivity ($\kappa_{L}$) in this $L$ window. In Fig. 7, using the KCM approach, we present a spectral representation of $\kappa_{L}$, distinguished by its kinetic ($\kappa_{kinetic}$) and collective contributions ($\kappa_{collective}$), as a function of phonon frequency at $T$ = 10K for four different $L$. We choose $T$ = 10 K as a representative temperature to feature collective transport of phonons. The four different $L$ values have been chosen such that it covers a wide range that traverses from ballistic transport to the hydrodynamic regime at $T$ = 10 K. As we gradually increase the $L$ [from Figs. 7(a) to 7(d)), a gradual increment of the contributions coming from the collective transport is observed (shown via the red shaded regions inside the curve). The spectral $\kappa_{L}$ goes to zero before 2.87 THz, indicating the sole contribution of acoustic phonons in thermal transport at 10 K, as was realized earlier in Fig. 4(a).

To quantify the collective motion as a function of temperature for different $L$, we investigate the variation of characteristic non-local length ($l$) in GeTe at different temperatures and grain sizes. In a complete hydrodynamic description of thermal transport, the extension of the Guyer and Krumhansl equation [11] done in the KCM framework [44], namely, the hydrodynamic KCM equation, yields

where $\tau$ is the total phonon relaxation time, Q is the heat flux, $\kappa$ is phonon thermal conductivity, and $l$ is the non-local length, that determines the non-local range in

phonon transport. If we employ the steady state, strong geometric effects, and neglect the term $2\nabla\nabla\cdot\textbf{Q}$, then the equation can be visualized as analogous to Navier-Stokes equation with $l$ resembling heat viscosity. The Knudsen number (Kn) can be obtained from $Kn=l/L$ to study the collective motion quantitatively. Lower values of Kn define the recovery of Fourier’s law whereas the hydrodynamic behavior becomes prominent when Kn gets higher values [44, 7]. Figure 8 presents the variation of Kn as a function of temperature for different $L$. As temperature is lowered, a gradual increment of Kn is observed, concomitant with the gradual prominence of non-local behavior. Kn has earlier been described [7, 9] to indicate a phonon hydrodynamic regime when 0.1 $\leq$ Kn $\leq$ 10, bearing similarities with fluid hydrodynamics. We denote this region via a shaded region in Fig. 8. In Fig. 8, we also superpose the hydrodynamic $L$-window, identified using average scattering rates following Guyer and Krumhansl conditions for three representative temperatures: $T$ = 6 K, 10 K and 20 K. We observe that both definitions match well and the hydrodynamic $L$-window obtained by scattering rate analysis falls within the Kn range for hydrodynamics.

Knudsen number calculation also reveals the Ziman hydrodynamic regime for GeTe. Looking at the vertical dashed lines corresponding to $T$ = 6 K and $T$ = 10 K in Fig. 8, a small $L$-region is observed which does not fall into the rectangles, defined to indicate a hydrodynamic regime using scattering rate hierarchy. However, they fall inside the regime of 0.1 $<$ Kn $<$ 10, especially in the regime where Kn is close to 0.1. This corresponds to the Ziman hydrodynamic regime which corresponds to a regime where N scattering dominates but dissipates mostly by R scattering contrary to the Poiseuille hydrodynamic regime where N scattering dissipates mostly by the boundary scattering of the phonons. On the other hand, looking at $L$ values that lie inside 0.1 $<$ Kn $<$ 10 but with values close to 10, also sometimes do not lie inside the rectangular region (see the case of $L$ = 0.04 and 0.1 $\mu$m at $T$ = 10 K in Fig 8). Recalling Fig 6.(a), we observe that $L$ = 0.04 $\mu$m at $T$ = 10 K designates a scattering rate hierarchy, where $\langle\tau_{B}^{-1}\rangle_{ave}>\langle\tau_{N}^{-1}\rangle_{ave}>\langle\tau_{R}^{-1}\rangle_{ave}$, but $\langle\tau_{B}^{-1}\rangle_{ave}$ is not $\gg$ $\langle\tau_{ph-ph}^{-1}\rangle_{ave}$. Therefore, though it follows the prescribed hierarchy for hydrodynamics, the $L$ values do not enable a complete ballistic propagation and a competition between ballistic and diffusive phonons operates. This competition makes it difficult to distinguish sharp boundaries between different regimes. We will discuss more about this competition later. To characterize the repopulation of phonons in a different way, following Markov et al. [9], we extract a length scale related to the propagation of heat wave before being dissipated, called the heat wave propagation length ($L_{hwpl}$), defined as a length at which the completely diffusive thermal conductivity decays 1/e times:

where $L_{diff}$ is the minimum length $L$, above which $\kappa_{L}$ reaches the thermodynamic limit, as mentioned earlier in Fig. 3. $L_{hwpl}$ is connected to second sound, a typical characteristic for hydrodynamic heat transport phenomenon, which demonstrates the heat propagation as damped waves in a crystal [12, 1, 48] as a result of coherent collective motion of phonons due to the domination of N scattering. In this context, drift velocity of phonons ($\overline{v}$) and phonon propagation length ($\lambda_{ph}$) are defined as

and

where, $C_{\alpha}$ is heat capacity of mode $\alpha$, $\mathbf{v}_{\alpha j}^{g}$ is phonon group velocity of mode $\alpha$ and $j$ can be either the component along the $a$ axis ($x$) or the hexagonal $c$ axis ($z$). Heat transfer of GeTe is anisotropic, as can be recalled from our earlier studies [32, 28], featuring different group velocities along the hexagonal $c$ axis and its perpendicular ($a$ axis) direction and therefore yields different drift velocities and different phonon propagation lengths along $x$ and $z$. Figure 9 presents the variation of heat wave propagation length ($L_{hwpl}$) with temperature along with the variation of phonon propagation lengths along $x$ and $z$. Phonon propagation lengths are distinguished [9] as $\lambda_{hydro}$ and $\lambda_{gas}$ via

Figure 9 shows the variation of heat wave propagation length ($L_{hwpl}$), superimposed with phonon propagation lengths with temperature along both $a$ and $c$ axis directions of GeTe. We observe that $L_{hwpl}$ follows well the trend of $\lambda_{hydro}$ as a function of temperature in the whole temperature range studied. $\lambda_{gas}$, the phonon propagation length corresponds to the uncorrelated phonon gas where both N and R scattering processes contribute to the damping of heat wave, on the other hand, seems to diverge from $L_{hwpl}$ as the temperature is lowered. This feature is an indication of gradual prominence of hydrodynamic behavior of phonons as the temperature is lowered. Similarly, the reasonable match between $L_{hwpl}$ and $\lambda_{hydro}$ predicts that heat wave propagation length is well captured by phonon flow with resistive damping caused by umklapp and isotope scattering. At very low temperature ($T$ = 4 K), a slight deviation is observed between $L_{hwpl}$ and $\lambda_{hydro}$ which can be attributed to the importance of boundary scattering as a significant damping process at very low temperature.

Therefore, $L_{hwpl}$ can lead to the identification of the length scale at different temperatures at which phonon hydrodynamics can exist and therefore Poiseuille’s flow and second sound phenomena can be observed. Interestingly, comparing $L_{hwpl}$ and characteristic size ($L$) of the sample, we can define Knudsen number in another approach as [9] Kn = $L_{hwpl}$/$L$. The variation of Kn obtained using $L_{hwpl}$, is presented as a function of temperature in the Supplemental Material (Fig. S2). The variation of Kn with $T$ is found similar to our earlier evaluation of Kn using nonlocal length (Fig. 8).

The blurry regions of transitions between ballistic, hydrodynamic, and diffusive transport are intriguing to understand the competition between different phonons with a wide range of mean free paths. Ideally, phonons with a wide spectrum of mean-free paths can be distinguished as either ballistic (MFP $>$ $L$) or diffusive (MFP $<$ $L$) phonons. However, the relative strength between ballistic and diffusive phonons are crucial to realize the competition between these two kind of phonons which eventually plays a decisive role in dictating the visible hydrodynamic phenomena. The phonon Knudsen minimum is such an indicator for the transition between ballistic and hydrodynamic phonon propagation regimes and had been used for several materials including graphene [16], graphite [20], SrTiO₃ [23], black phosphorus [18] to detect phonon hydrodynamics. Figures 10.(a) and (b) present the the variation of normalized thermal conductivity ($\kappa_{L}^{*}$ = $\kappa_{L}/L$), a quantity that is similar to dimensionless $\kappa_{L}$, as a function of inverse Knudsen number, calculated using nonlocal length and heat wave propagation lengths respectively. Figure 10.(b) shows a wider range of 1/Kn as the Kn obtained using heat wave propagation length reaches higher values at low temperatures compared to that of the non-local length calculation from hydrodynamic KCM method. However, we observe almost similar trends of $\kappa_{L}^{*}$ with the variation of 1/Kn coming out of the two different approaches in obtaining the Knudsen number. At $T$ = 300 K, a steep linear decreasing trend is observed which is associated with the diffusive phonon scattering events as phonons behave as uncorrelated gas particles and resistive scattering is prominent and dominating at this temperature.

Starting from $T$ = 20 K, a gradual onset of a horizontal regime is visible before the linearly decreasing trend of $\kappa_{L}^{*}$ as the temperature is lowered. At $T$ = 4 K, surprisingly, a cusp-like trend, resembling that of a shallow minimum followed by a prominent maximum is observed before a linearly decreasing $\kappa_{L}^{*}$ at higher 1/Kn. The cusp-like shallow minimum at $T$ = 4 K indicates the phonon Knudsen minimum and predicts the presence of prominent transition from ballistic to hydrodynamic regime. Further, a prominent maximum in $\kappa_{L}^{*}$ has only been observed at $T$ = 4 K, which designates the strong presence of hydrodynamic phonon transport in GeTe. Similar observation can be found by Li et al. [16] for suspended graphene, where the increasing trend of $\kappa_{L}$, normalized by sample width, was attributed to the strong presence of hydrodynamic phonon transport.

The behavior of phonon Knudsen minimum of GeTe convinces us to understand the competition between ballistic and diffusive phonons in the quasi-ballistic regimes of phonon transport. We specifically turn our attention toward the reason behind the strong presence of hydrodynamics at $T$ = 4 K visible through Knudsen minimum in Fig 10. We recall that even $T$ = 6 K, $T$ = 8 K persist in phonon hydrodynamics, realized via the average scattering rate analysis and Knudsen number variation with temperature. To perceive the reason behind the difference between strong and weak phonon hydrodynamics, we investigate the scaling relation between $\kappa_{L}$ and $L$ in the intermediate regime of transport, where the transport is neither fully ballistic nor fully diffusive.

Figure 11 describes the variation of $\kappa_{L}$ with $L$ at $T$ = 4 K, 6 K, and 10 K. Three phonon propagation regimes have been identified. At lower values of $L$, ballistic phonons dominate the transport and therefore a linear dependency of $\kappa_{L}$ on $L$ is observed. At very high $L$, the phonon transport is completely diffusive and a plateau-like regime is observed, denoting an independence of $\kappa_{L}$ over $L$. The intermediate regime where the phonon propagation shifts from complete ballistic to complete diffusive, plays a crucial role in determining the strong or weak presence of hydrodynamic propagation of phonons. Figure 11.(c) indicates a sublinear variation in the intermediate regime at $T$ = 10 K. At $T$ = 6 K [Fig. 11(b)], a minute superlinear behavior is observed while at $T$ = 4 K [Fig. 11(a)], an enhanced superlinear behavior is perceived in the intermediate regime.

In the intermediate quasi-ballistic regime of phonon propagation, where both ballistic and diffusive phonons

operate and compete with each other, seems to be a marker to designate sample sizes ($L$) with strong hydrodynamic phonon transport characteristics. To further quantify the intermediate nonliearity (both sub and superlinearity), we evaluate and present the scaling exponent[20] $\alpha$ = $\partial log(\kappa_{L})$/$\partial log(L)$ as a function of $L$ for different temperatures in Fig. 12.

$\alpha$ = 0 indicates the size-independent behavior of $\kappa_{L}$ and therefore describes the completely diffusive phonon propagation regime. On the other hand, $\alpha$ = 1 reveals the linear size dependency and henceforth the complete ballistic phonon conduction regime. The superlinear dependence of $\kappa_{L}$ on $L$ in the intermediate regime can be captured by the the condition $\alpha$ $>$ 1. From Fig. 12 we observe that at low $L$, for low temperatures, $\alpha$ goes to 1. for higher temperatures, as expected almost no ballistic regime is observed with $\alpha$ $<$ 1. As we increase $L$, in the intermediate regime, a gradual departure from $\alpha$ = 1 is observed. For $T$ = 4 K and $T$ = 6 K, this departure leads to a regime with $\alpha$ $>$ 1, while for $T$ = 8K and 10 K this deviation leads to sublinear or $\alpha$ $<$ 1 scaling. At high $L$ values gradually all phonons become diffusive and $\alpha$ goes to zero.

There are several striking features to point out from Fig. 12. First, prominent contribution of drifting phonons at 4 K leads to an enhanced superlinear scaling with $\alpha$ $>$ 1, representing the signature of phonon Poiseuille flow [20] and therefore prominent phonon hydrodynamics which assists in featuring the Knudsen minimum seen in Fig. 10. Here we mention that even for $T$ = 4 K, the exponent $\alpha$ gradually starts from 1, reaches a maximum value around $L$ = 0.8 $\mu$m, and goes sublinear with $\alpha$ $<$ 1 thereafter before going to zero at very high $L$ values. Therefore, sublinear scaling is universal in the intermediate regime. For $T$ = 4 K, however, the sublinear scaling precedes a superlinear behavior displaying strong hydrodynamic feature. Second, a minute superlinear scaling, observed in Fig. 11(b) inset for $T$ = 6 K, can be realized in a better way by observing the small $L$-window for which $\alpha$ $>$ 1 for $T$ = 6 K. At $T$ = 8K and 10 K, though sublinear scaling is observed in the intermediate regime, it decays to zero in different rates. After $L$ = 10 $\mu$m, the decay rate seems faster than that of the cases below $L$ = 10 $\mu$m.

We understand that although average scattering rate and Knudsen number variation with temperature indicates phonon hydrodynamics to present in GeTe for several temperature and characteristic size window, low-$\kappa_{L}$ material GeTe needs several factors to manifest a strong hydrodynamic response by phonons. In this context, superlinear scaling of $\kappa_{L}$ on $L$ plays a crucial role in the transition from complete ballistic to complete diffusive propagation regimes. To understand and investigate the reason behind superlinear and sublinear scaling at the intermediate quasi-ballistic regime of phonon transport, we calculate the ratio $\gamma$ as a function of $L$ for three temperatures: $T$ = 4 K, 6 K and 10 K. We define $\gamma$ as

where $\left\langle\tau_{N}^{-1}\right\rangle$, $\left\langle\tau_{R}^{-1}\right\rangle$, and $\left\langle\tau_{B}^{-1}\right\rangle$ are average scattering

rates for normal, resistive, and boundary scattering respectively. Figure 13.(a) shows that $\gamma$ increases gradually and reaches a plateau as we increase $L$. In the ballistic phonon conduction regime, we observe $\gamma(T=4K)$ $<$ $\gamma(T=6K)$ $<$ $\gamma(T=10K)$. This is due to the effect of strong boundary scattering at low temperature and low $L$. However, in the regime of complete diffusive propagation of phonons a reverse trend is observed: $\gamma(T=4K)$ $>$ $\gamma(T=6K)$ $>$ $\gamma(T=10K)$ as in this regime, $\gamma$ is independent of size. We define these saturation values as $\gamma_{diff}$. Again, the crucial crossover is observed in the intermediate $L$-regime. We also mark the difference between $\gamma$ = 1 and $\gamma_{diff}$ in Fig. 13(a) via double headed arrows. This difference characterizes the relative strength of normal scattering compared to the dissipative resistive scattering and therefore indicates the strength for persisting coherent phonon flow.

However, we tend to understand the reason behind the nonlinear behavior of $\kappa_{L}$ at the intermediate $L$-regime. For this purpose, we present the variation of the exponent of $\gamma$ by calculating $\partial log(\gamma)/\partial log(L)$ as a function of $L$ in Fig. 13(b). We observe that the exponent for $T$ = 4 K is higher and for both $T$= 4 K and 6 K, it stays around 1 ($\gamma$ being linearly increasing with $L$) in the intermediate regime. However, for $T$ = 10 K, $\partial log(\gamma)/\partial log(L)$ drops up to several orders (at $L$ = 10 $\mu$m, it drops around 10 times) compared to the $T$ = 4 K and $T$ = 6 K cases. Therefore, the higher the exponent $\partial log(\gamma)/\partial log(L)$ in the intermediate regime and closer to 1, the higher the chances of the collective phonon flow due to strong normal scattering. This eventually can lead to the strong appearance of phonon hydrodynamics with superlinear $L$ dependence of $\kappa_{L}$ and prominent Knudsen minimum apart from other signatures born out of the assessment of Knudsen number and average scattering rate as a function of temperature.

Figure S1 presents the variation of effective mean free paths of GeTe as a function of phonon frequency for two different $L$ values at $T$ = 10 K. In KCM approach, mean free paths can be distinguished as kinetic and collective mean free paths. As described in the main text, the kinetic mean free paths ($l_{k}(\omega)$) are different for different modes but the collective mean free paths ($l_{c}(T)$) are same for all modes and it is only a function of temperature. The effective mean free path has been realized using $l_{eff}(\omega)$ = (1-$\Sigma$)$l_{k}(\omega)$+$\Sigma$ $l_{c}$. In Fig S1.(a), we observe that at $T$ = 10 K, the dominating contribution comes from the kinetic mean free path for $L$ = 0.003 $\mu$m. We recall from the main text that this $L$ corresponds to ballistic phonon conduction regime for GeTe at 10 K. However, $L$ = 0.8 $\mu$m satisfies the Guyer and Krumhansl condition for phonon hydrodynamics and consistently the collective mean free path is seen to dominate over the kinetic mean free path (Fig S1.(b)).

Figure S2 presents the variation of Knudsen number (Kn) as a function of temperature for different $L$ values. Kn has been realized via the heat wave propagation length ($L_{hwpl}$) as $L_{hwpl}$/$L$, where $L_{hwpl}$ is obtained as a characteristic length at which the lattice thermal conductivity in the completely diffusive limit correspond to bulk sample reduces to 1/e times. The hydrodynamic regime follows 0.1 $<$ Kn $<$ 10 and therefore has been marked with the shaded region. The hydrodynamic $L$-regimes obtained using average scattering rates are also shown via rectangular boxes for $T$ = 6 K, 10 K and 20 K. Except the ballistic hydrodynamic boundary regimes for $T$ = 6 K, the regimes evaluated by Kn and average scattering rates are found to be consistent. The transition between ballistic and hydrodynamic regimes are often found to be blurry and without sharp demarcation in low-thermal conductivity materials. This has been discussed in the main text.