Thermal conductivity of amorphous and crystalline GeTe thin film at high temperature: Experimental and Theoretical study

The structural parameters are optimized via DFT calculations and the optimized lattice parameter (a = 4.23 Å) and unit cell volume (56.26 ų) of GeTe, are found to be quite consistent with the values presented in literature Campi et al. (2017); Jain et al. (2013). It is quite well established that at normal conditions GeTe crystallizes in trigonal phase (space group R3m) with 2 atoms per unit cell. This structure gives rise to a 3+3 coordination of Ge with three short stronger intrabilayer bonds and three long weaker interbilayer bonds Campi et al. (2015, 2017). The bond lengths (shorter bonds = 2.85 Å, longer bonds = 3.25 Å) are also found to be consistent with the studies done by Davide Campi et al. Campi et al. (2017).

To investigate the effect of phonons in the heat transfer processes, we study the phonon density of states and the

dispersion relation of crystalline rhombohedral (R3m) GeTe. Figure 4.(a) and (b) show the phonon density of states and the phonon dispersion relation of undoped crystalline (R3m) GeTe, calculated at ground state. The dashed line in Fig 4.(a) serves as a separator between the acoustic and optical contribution of phonons. The acoustic phonons extend from 0 to 96 cm^-1 (= 2.88 THz) in the frequency domain which is consistent with the observations of Urszula D. Wdowik et al. Wdowik et al. (2014). The frequencies of the two most prominent peaks in PDOS corresponding to acoustic ($\sim$ 80 cm^-1 = 2.40 THz) and optical phonons ($\sim$ 140 cm^-1 = 4.20 THz) respectively, are also found to be close to the values observed by Kwangsik Jeong et al. Jeong et al. (2017). Phonon dispersion relation (Fig 4.(b)) along the high symmetry direction in the Brillouin zone (BZ) also shows similar trends to that of the earlier works Wdowik et al. (2014); Bosoni et al. (2017). We observe the signature of LO-TO splitting as the discontinuities of the phonon dispersion at the $\Gamma$ point arising from the long range Coulomb interactions Wdowik et al. (2014); Bosoni et al. (2017). The approximate separator between acoustic and optical modes at 96 cm^-1 is also shown by a horizontal dashed line in the phonon dispersion relation (Fig 4.(b)). Except a small and negligible contribution of transverse optical modes at the $\Gamma$ point, all frequencies $<$ 96 cm^-1 contribute to the acoustic modes.

The calculation of the electronic thermal conductivity $\kappa_{el}$ rests generally on the use of the Wiedemann-Franz law $\kappa_{el}=L\sigma_{e}T$ where $L$ is the Lorenz number, $T$ is the temperature and $\sigma_{e}$ is the electrical conductivity. Crystalline GeTe is a p-type degenerate semiconductor Bahl and Chopra (1970); Nath and Chopra (1974) with a high hole concentration and the Fermi level lying well inside the valence band that holds the charge carriers. Therefore it behaves similar to a metal. In that case, the value for $L=\pi^{2}\left(k_{B}/e\right)^{2}/3=2.44\times 10^{-8}\,\mathrm{V^{2}.K^{-2}}$ ($k_{B}$ is the Boltzmann constant and $e$ is the electron charge) had generally been employed in most of the research works Nath and Chopra (1974). However, for highly doped materials at temperature higher than the Debye temperature $\Theta_{D}$, the Hall mobility depends on the temperature as $\mu_{H}\propto T^{-(3/2)+r}$ , where $r$ is the scattering mechanism parameter, and decreases as $\eta$ decreases with increasing temperature. Therefore, it is shown that the Lorenz number is given by Goldsmid (2010); Gelbstein et al. (2010):

where $\eta=\left(E_{F}-E_{V}\right)/k_{B}T$ is the reduced Fermi energy for $p$-type semiconductors and $r=-1/2$ when scattering by acoustic phonons is dominant Gelbstein et al. (2010). The Fermi integral is defined as: $F_{n}\left(\eta\right)=\int_{0}^{\infty}\left(x^{n}/\left(1+e^{x-\eta}\right)\right)\textit{d}x$.

Following the method adopted by Gelbstein et al. Gelbstein et al. (2010), E. M. Levin and co authors Levin et al. (2013) obtained that $L$ for crystalline GeTe varies between $2.4\times 10^{-8}\,\mathrm{V^{2}.K^{-2}}$ at 320 K and $1.8\times 10^{-8}\,\mathrm{V^{2}.K^{-2}}$ at 720 K. Using this method, we calculate the variation of $L$ for GeTe as a function of temperature within the investigated range in the present study and we report the results in Table 1.

The electrical resistivity of GeTe films has been measured experimentally using the Van der Pauw technique and is found to be $\rho_{e}=1/\sigma_{e}=\left[8.5\pm 2\right]\times 10^{-6}\,\Omega.\mathrm{m}$ at the room temperature (300 K), which corresponds for the hole concentration to be 6.24 $\times$ 10¹⁹ cm^-3. The constant relaxation time approximation (CRTA) with a constant electronic relaxation time of 10^-14s is used following the work of S. K. Bahl et al. Bahl and Chopra (1970). After identifying the hole concentration, we compute the electrical conductivity ($\sigma_{e}$) by using DFT and solving the Boltzmann transport equation (BTE) for electrons, as implemented in the BoltzTaP code Madsen and Singh (2006) for the given hole concentration at different temperatures. Considering the values for $L\left(T\right)$, the calculated electronic thermal conductivity varies linearly from 0.76 W/mK at 322 K ($L=2.40\times 10^{-8}\,\mathrm{V^{2}.K^{-2}}$ at 322 K) to 0.91 W/mK at 503 K ($L=2.13\times 10^{-8}\,\mathrm{V^{2}.K^{-2}}$ at 503 K) (Table 1). These results are consistent with the experimental ones by R. Fallica et al. Fallica et al. (2013).

In Fig 2.(b), total thermal conductivity of GeTe is seen to manifest a fast change process with gradually increasing temperature, where a phase change from amorphous to crystalline phase is found to occur $\sim$ 180^∘C or 453 K. We first focus on the thermal conductivity of the amorphous phase of GeTe. Figure 5 shows lower values of $\kappa$ for amorphous phase compared to that of the crystalline phase. The values of $\kappa$ in amorphous phase are also observed to be almost constant throughout the temperature range studied in this work. Theoretically, the minimal thermal conductivity model derived by Cahill et al. Cahill et al. (1992) allows one to calculate $\kappa$ for the amorphous materials as:

where $n=\left(k_{B}\,\Theta_{D}/\hbar\right)^{-1/3}/\left(6\,\pi^{2}\,c_{s}^{3}\right)$ is the number of phonons per unit volume (which can also be calculated more rigorously from the phonon DOS apart from using the values of Table 2), $c_{s}$ is the speed of sound calculated as $c_{s}^{-3}=\left(v_{L}^{-3}+2\,v_{T}^{-3}\right)/3=1900\,\mathrm{m.s^{-1}}$ Pereira et al. (2013) and $\Theta_{i}$ is the Debye temperature per branch. When $T\gg\Theta_{D}$, this relation simplifies as

Using the required parameter values from Table 2, the minimal thermal conductivity ($\kappa_{min}$) is found to be consistent with the experimental data in amorphous phase considering the error bars involved in the experimental measurements (Fig 5). The reasonable agreement between the experimental data and $\kappa_{min}$ based on Cahill model indicates that the dominant thermal transport in the amorphous GeTe occurs in short length scales Kaviany (2014) between neighboring vibrating entities owing to the disorder present in it.

We then investigate the phonon contributions to the total thermal conductivity of crystalline GeTe. In order to evaluate the lattice thermal conductivity ($\kappa_{L}$) through the direct solution of LBTE, the method developed by L. Chaput Chaput (2013) is adopted. According to this method, lattice thermal conductivity is given as Chaput (2013)

where, $\Omega^{\sim 1}$ is the Moore-Penrose inverse of the collision matrix $\Omega$, given by Chaput (2013); Togo et al. (2015)

Here, $\Phi_{\lambda\lambda^{\prime}\lambda^{\prime\prime}}$ denotes the interaction strength between three phonon $\lambda$, $\lambda^{\prime}$ and $\lambda^{\prime\prime}$ scattering Togo et al. (2015). However, adopting the relaxation time approximation (RTA) in solving LBTE, lattice thermal conductivity tensor $\bm{\kappa}_{L}$ can be written in a convenient and closed form as Fugallo et al. (2013); Togo et al. (2015)

where $N$ is the number of unit cells and $V_{0}$ is the volume of unit cell. The phonon modes (q, $j$) comprising wave vector q and branch $j$ are denoted with $\lambda$. The modal heat capacity if given by

Here, $T$ denotes temperature, $\hbar$ is reduced Planck constant and $k_{B}$ is the Boltzmann constant. $\textbf{v}_{\lambda}$ and $\tau_{\lambda}$ represent phonon group velocity and phonon lifetime respectively. We consider three scattering processes, namely normal, umklapp and isotope, denoted by N, U and I respectively, in the theoretical study. For each of these processes, the phonon lifetime has been realized using Matthiessen rule as Kaviany (2014)

where $\tau_{\lambda}^{N}$, $\tau_{\lambda}^{U}$ and $\tau_{\lambda}^{I}$ are phonon lifetimes corresponding to the normal, umklapp and isotope scattering respectively.

Generally, in harmonic approximation, phonon lifetimes are infinite whereas, anharmonicity in a crystal gives rise to a phonon self energy $\Delta\omega_{\lambda}$ + $i\Gamma_{\lambda}$. The phonon lifetime has been computed from imaginary part of the phonon self energy as $\tau_{\lambda}$ = $\frac{1}{2\Gamma_{\lambda}(\omega_{\lambda})}$ fromTogo et al. (2015)

where $n_{\lambda}$ = $\frac{1}{exp(\hbar\omega_{\lambda}/k_{B}T)-1}$ is the phonon occupation number at the equilibrium. $\Delta\left(\textbf{q}+\textbf{q}^{\prime}+\textbf{q}^{\prime\prime}\right)$ = 1 if $\textbf{q}+\textbf{q}^{\prime}+\textbf{q}^{\prime\prime}=\textbf{G}$, or 0 otherwise. Here G represents reciprocal lattice vector. Integration over q-point triplets for the calculation is made separately for normal (G = 0) and umklapp processes (G $\neq$ 0). For both direct method and RTA, scattering of phonon modes by randomly distributed isotopes Togo et al. (2015) are also incorporated for comparison. The isotope scattering rate, using second-order perturbation theory, is given by Shin-ichiro Tamura ichiro Tamura (1983) as

where $g_{k}$ is mass variance parameter, defined as

$f_{i}$ is the mole fraction, $m_{ik}$ is relative atomic mass of $i$th isotope, $\overline{m}_{k}$ is the average mass = $\sum_{i}f_{i}m_{ik}$ and W is polarization vector. The database of the natural abundance data for elements Laeter et al. (2003) is used for the mass variance parameters.

For consistency check, we first simulate the lattice thermal conductivity ($\kappa_{L}$) of crystalline rhombohedral GeTe at 300 K using RTA method and compare the value of $\kappa_{L}$ with the results of the work done by D. Campi et al.Campi et al. (2017). $\kappa_{L}$, obtained from our study using RTA, is 2.29 W/mK using the PBE functional in the DFT framework, which is in good agreement with the results of D. Campi et al.Campi et al. (2017) with $\kappa_{L}$ = 2.34 W/mK. After this consistency check, we use the DFT and Boltzmann transport equation (BTE) to get the lattice thermal conductivity of GeTe at the temperature regime studied in this work (322 K $\leqslant$ $T$ $\leqslant$ 503 K).

Figure 5 and Table 3 present the various contributions of thermal conductivity, obtained theoretically, superimposed with the experimental data. Starting from first principles calculations, both direct solution and RTA method are used to solve the LBTE to get the lattice thermal conductivity. Since $\kappa_{L}$ is anisotropic along hexagonal c axis and a-b axes, the average lattice thermal conductivity is calculated as $\kappa_{av}$ = $\frac{2}{3}$$\kappa_{x}$ + $\frac{1}{3}$$\kappa_{z}$ Mizokami et al. (2018); Campi et al. (2017). Figure 5 and Table 3 show that the resulting theoretical $\kappa_{tot}$ for the direct solution of LBTE is slightly overestimated compared to the experimental results, specifically at the higher temperatures ($T$ $>$ 322 K). However, the direct method is found to capture the trend of $\kappa(T)$ quite well. Consistently, slightly lower values of $\kappa$ are found due to the incorporation of the effect of phonon mode scattering due to isotopes. On the other hand, the results of thermal conductivity, obtained by solving Boltzmann transport equation under the relaxation time approximation (RTA), are found to be slightly underestimated as compared to the experimental results. However, as T $>$ 372 K, RTA results seem to be in better agreement with the experimental data and the difference between experimental and RTA goes down from $\approx$ 19 $\%$ at 322 K to $\approx$ 6 $\%$ at 503 K. The trend of $\kappa(T)$ obtained through RTA, alike the direct solution, is found to be in good agreement with the experimental trend. This trend of $\kappa(T)$ implies that umklapp scattering seems to dominate the phonon-phonon scattering at higher temperatures, as predicted by experiments.

Previously, the hole concentration of GeTe is found to be 6.24 $\times$ 10¹⁹ cm^-3, indicating a significant role of vacancies for the reported overestimation of the $\kappa(T)$, obtained via direct solution of LBTE. Indeed, D. Campi et al. Campi et al. (2017) found a considerable amount of lowering of $\kappa_{L}$ of crystalline GeTe at 300 K due to the vacancy present in the sample.

The phonon scattering rate by vacancy defects are prescribed by C. A. Ratsifaritana et al. Ratsifaritana and Klemens (1987) as

where, $x$ is the density of vacancies, $G^{\prime}$ denotes the number of atoms in the crystal and $g(\omega)$ is the phonon density of states (PDOS). Using vacancies as isotope impurity, C. A. Ratsifaritana et al Ratsifaritana and Klemens (1987) denoted mass change $\Delta M$ = 3$M$, where $M$ is the mass of the removed atom. Eq. 24 states that the phonon-vacancy relaxation time is temperature independent. D. Campi et al. Campi et al. (2017) used this phonon-vacancy scattering contribution for a GeTe sample with hole concentration of 8 $\times$ 10¹⁹ cm^-3 and found an almost $\approx$ 15.6 $\%$ reduction of $\kappa_{L}$ at 300 K. As the hole concentration is almost similar to that of our work (6.24 $\times$ 10¹⁹ cm^-3) , in conjunction with the fact that $\tau_{V}^{-1}$ is temperature independent, we estimate an overall $\approx$ 15.6 $\%$ decrement of $\kappa_{L}(T)$ throughout the temperature range studied as an effect of phonon-vacancy scattering. We find that the estimated $\kappa$, incorporating the vacancy contribution, in addition to the normal, umklapp and isotope scattering, agrees excellently with the experimental data for the whole temperature range in our study. This exercise strongly depicts the significant participation of the scattering between phonons and vacancy defects at high temperatures.

It is well known that the RTA, although describes the depopulation of phonon states well, fails to rigorously account the repopulation of phonon states Fugallo et al. (2013). While at low temperatures, the applicability of RTA can be questioned due to the dominance of momentum conserving normal scattering and almost absence of umklapp scattering of phonons, in the high temperature regime of our study, RTA is found to be a good trade off between accuracy and the computational cost to describe the experimental results. This is primarily because of the higher number of scattering events at higher temperature which ensures an isothermal repopulation of the phonon modes Fugallo and Colombo (2018). However, the difference from the total solution of LBTE exists due to the over resistive nature of the scattering rates that effectively lowers the value compare to the direct solution. This feature has been discussed in literature Fugallo and Colombo (2018); Bosoni et al. (2017). The reason of this underestimation is that the RTA treats both umklapp and normal scattering processes as resistive while the momentum conserving normal scattering processes do not equally contribute to the thermal resistance as that of the umklapp scattering Bosoni et al. (2017).

Simple phenomenological models can also serve a fast and efficient way to decipher the underlying physical mechanism. The Slack model Morelli and Slack (2006) expresses the lattice thermal conductivity, when $T>\Theta_{D}$ and the heat conduction happens mostly by acoustic phonons, starting from the analytical expression of the relaxation time related to umklapp processes as:

with:

where $n_{c}$ is the number of atoms per unit cell, $\delta^{3}$ is volume per atom ($\delta$ is in angstrom in the relation), $\overline{M}$ is average atomic mass of the alloy and $G$ is the Gruneisen parameter (see Table 2). This relation between lattice thermal conductivity and Gruneisen parameter in a solid is valid within a temperature range where only interactions among the phonons, particularly, anharmonic umklapp processes are dominant Morelli and Slack (2006). E. Bosoni et al. Bosoni et al. (2017) found a good agreement between the lattice thermal conductivity of crystalline GeTe coming from the full solution of BTE and that of the Slack model at room temperature. Lattice thermal conductivity due to Slack model ($\kappa_{L}$(Slack)) is presented in Fig 5. Total thermal conductivity is then realized by adding $\kappa_{L}$(Slack) with $\kappa_{el}$. We retrieve an almost identical trend of both lattice ($\kappa_{L}$) and total thermal conductivity ($\kappa_{tot}$) in Slack model as that of the RTA based solutions. This almost identical values of $\kappa_{L}$(Slack) and $\kappa_{L}$(RTA) depicts that the optical phonons contribute very little to $\kappa_{L}$(RTA) as $\kappa_{L}$(Slack) takes into account only acoustic phonon contributions. The values obtained for $\kappa_{tot}$(Slack) are found to be lower than the experimental data for $T$ $<$ 412 K and gradually seem to agree well for $T$ $\geqslant$ 412 K (Fig 5 and Table 3).

Though RTA based solutions underestimate the experimental data, the trend of $\kappa(T)$, which is almost identical to phenomenological Slack model, is what needs attention to elucidate the underlying heat transport mechanism at high temperatures. Further, RTA based solutions are straightforward and can reveal the distinct role of each contributing parameters appearing in Eq.18. Consequently, in the following sections, we systematically study the thermal transport properties of GeTe using both frequency and temperature variations using the results obtained from RTA solutions.

To investigate lattice thermal conductivity ($\kappa_{L}$) in a more comprehensive manner, we calculate the cumulative lattice thermal conductivity as a function of phonon frequency for the high temperature regime defined as Togo et al. (2015); Mizokami et al. (2018)

where $\kappa_{L}$ ($\omega^{\prime}$) is defined as Togo et al. (2015); Mizokami et al. (2018)

with $\frac{1}{N}$ $\sum_{\lambda}\delta(\omega^{\prime}-\omega_{\lambda})$ is weighted density of states (DOS).

Figure 6 shows the cumulative $\kappa_{L}$ of crystalline rhombohedral GeTe, along hexagonal c axis ($\kappa_{z}$), along its perpendicular direction ($\kappa_{x}$) and their average $\kappa_{av}$ = (2$\kappa_{x}$ + $\kappa_{z}$)/3, as a function of phonon frequencies for four different temperatures using RTA framework. The derivatives of the cumulative values of $\kappa_{z}$ and $\kappa_{x}$ with respect to frequencies are also shown for each temperature. It is found that the lattice thermal conductivity ($\kappa_{L}$) of GeTe is anisotropic with the value along $z$, parallel to the c axis in the hexagonal notation, is found to be smaller with respect to that of the $xy$ plane for the whole range of temperature studied. This picture is consistent with the recent theoretical findings Campi et al. (2017). More details to the anisotropic aspect of $\kappa_{L}$ will be discussed in the next subsection.

Figure 6 shows some distinct features in the cumulative lattice thermal conductivity ($\kappa^{c}_{L}$) of GeTe as a function of both phonon frequency and temperature. As the temperature is increased, $\kappa^{c}_{L}$ is found to saturate at gradually

lower values, indicating gradual decrement of the lattice thermal conductivity with temperature. Further, the derivatives of $\kappa^{c}_{L}$ with respect to phonon frequencies indicate the density of heat carrying phonons with respect to the phonon frequencies Linnera and Karttunen (2017) and their contribution to the $\kappa^{c}_{L}$. We note that this density of modes go to zero at a frequency where $\kappa^{c}_{L}$ reaches a plateau. This frequency ($\sim$ 2.87 THz) has been found to be identical to the frequency where phonon DOS marks the separation between acoustic and optical modes ($\sim$ 96 cm^-1 = 2.88 THz in Fig 4). This correspondence imply that the phonon density of states play a crucial role as a deciding factor to the $\kappa_{L}$. While in Fig 6, a significantly higher contribution of these modes correspond to $\kappa_{x}$ is observed compared to that of the $\kappa_{z}$ in the acoustic frequency regime (frequency $<$ 2.87 THz), an almost equal contribution of d$\kappa_{x}$/d$\omega$ and d$\kappa_{z}$/d$\omega$ are found in the optical frequency regime (frequency $>$ 2.87 THz).

To get a clear quantitative picture, we evaluate the separate contributions of acoustic and optical modes to the $\kappa_{L}$. The values of $\kappa^{c}_{L}$ for phonon frequencies $<$ 2.87 THz have unambiguously been considered to be the contribution from acoustic modes while the same for frequencies $>$ 2.87 THz have been taken as optical modes contribution. Table 4 shows the percentage contributions of both acoustic and optical modes to the lattice thermal conductivity ($\kappa_{L}$) as a function of temperature. We observe that a dominant 77 $\%$ of the contribution comes from acoustic modes compared to only around 23 $\%$ from the optical modes for the whole temperature range studied.

As mentioned in the previous subsection, the lattice thermal conductivity ($\kappa_{L}$) of crystalline GeTe shows anisotropic behavior. The resulting $\kappa_{L}$ along $z$ direction ($\kappa_{L}(z)$), parallel to the c axis in the hexagonal notation (along axis c in Fig 3) is found to be smaller than that of the $xy$ plane (a-b plane in Fig 3),($\kappa_{L}(x)$), as shown in Fig 7.

To investigate the anisotropy, we study the ratio $\kappa^{c}_{L(x)}(\omega)$/$\kappa^{c}_{L(z)}(\omega)$, where $\kappa^{c}_{L(x)}(\omega)$ and $\kappa^{c}_{L(z)}(\omega)$ represent the cumulative lattice thermal conductivities along

$xy$ plane (a-b plane in Fig 3) and $z$ direction (along c axis in Fig 3) respectively. Figure 8.(a) shows this ratio as a function of phonon frequencies for the four different temperatures. We observe that despite the wide temperature range studied, the shape of the curves remain independent of the temperature. The ratio $\kappa^{c}_{L(x)}(\omega)$/$\kappa^{c}_{L(z)}(\omega)$, initially starts with a low value, becomes maximum to 1.5 at around 1.4 THz. With further increase of frequencies, the ratio decreases and settles at a value of 1.37.

The temperature independence of the anisotropy ratio $\kappa^{c}_{L(x)}(\omega)$/$\kappa^{c}_{L(z)}(\omega)$ can be further understood by studying the cumulative outer product of the phonon group velocities $\textbf{v}_{\lambda}\otimes\textbf{v}_{\lambda}$, defined as

and $W^{c}_{x}$($\omega$)/$W^{c}_{z}$($\omega$), which is the ratio between the cumulative outer product of the phonon group velocities along $x$ and $z$ direction.

Since phonon group velocities are almost temperature independent, following that feature, we find that the ratio of $W^{c}_{x}$($\omega$)/$W^{c}_{z}$($\omega$) is strongly correlated with $\kappa^{c}_{L(x)}(\omega)$/$\kappa^{c}_{L(z)}(\omega)$. From zero frequency, $W^{c}_{x}$($\omega$)/$W^{c}_{z}$($\omega$) starts increasing and reaches the ratio 1 around 0.7 THz, close to that of the $\kappa^{c}_{L(x)}(\omega)$/$\kappa^{c}_{L(z)}(\omega)$ (0.4 THz). Further increasing frequency increases the anisotropy between the in-plane ($xy$) and out-of the plane ($z$) components of the cumulative outer product of phonon group velocities and finally saturates to a value of 1.23 at higher frequencies which is also comparable to the saturation value of $\kappa^{c}_{L(x)}(\omega)$/$\kappa^{c}_{L(z)}(\omega)$, that is 1.37. Thus, the anisotropy associated with the phonon group velocities, or more specifically, the cumulative outer product of phonon group velocities determines the anisotropy in the cumulative lattice thermal conductivity.

Recalling Equation 18, in an alternate way to understand the frequency dependence of the different parameters that contribute to the lattice thermal conductivity ($\kappa_{L}$), we study the variation of modal heat capacity ($C_{\lambda}$), phonon group velocity ($\textbf{v}_{\lambda}$) and the phonon lifetime ($\tau$) as a function of phonon frequency for the high temperature regime studied in this work. The dominant contribution of phonon group velocities ($\textbf{v}_{\lambda}$) are found to arise from the acoustic phonons and the optical modes are found to exhibit substantially lower group velocities than the former. Furthermore, as expected, increasing temperatures does not change the dependence of group velocities on phonon frequencies. Similar behavior is expected from the modal heat capacity (C_λ) as a function of phonon frequencies for different temperatures. Indeed, C_λ stays nearly constant with a small $\sim$ 2-5$\%$ deviation (2$\%$ for 503 K and 5$\%$ for 322 K) from $k_{B}$ which is consistent with the classical limit of C_λ at high temperatures.

This almost non varying patterns of phonon group velocities and mode heat capacity with temperature prompt us to look closely on the frequency variation of phonon lifetimes ($\tau_{\lambda}$) at these temperatures. We note here that $\tau_{\lambda}$ defines modal phonon relaxation time or phonon lifetime with $\lambda$ denotes each mode. Figure 9 depicts the phonon lifetimes, coming from TA, LA and optical modes as a function of phonon frequency. We observe that increasing phonon frequency from 0 THz gives an initial rise and then a quick decay of lifetimes within $\sim$ 3 THz and with relatively smaller values afterwards. This clearly indicates that the dominant contribution of phonon lifetime and in turn, the lattice thermal conductivity is coming from the acoustic modes, which operate in the frequencies $<$ 2.87 THz.

Considering the temperature variation, we find that acoustic modes induce smaller values of phonon lifetimes with increasing temperature (Fig 9). Two prominent observations evolve through this: (a) the trends in phonon lifetimes ($\tau_{\lambda}$) as a function of frequency reassure the fact that acoustic phonons are the dominant carriers of heat which contributes to $\kappa_{L}$, (b) as the phonon lifetime is directly proportional to $\kappa_{L}$, the reduced contribution of phonon lifetime with increasing temperature directs towards a gradual decrement of $\kappa_{L}$ with temperature. The gradually reducing values of phonon lifetime with increasing temperature can be understood more prominently considering the mean free path picture. The modal mean free path of phonons ($\Lambda_{\lambda}$) can be written as

The transport of heat through phonons in the diffusive regime ($\Lambda$ $\ll$ $L$, $L$ = linear dimension of the medium of travelling phonons) undergoes several scattering processes namely scattering by electrons, other phonons, impurities or grain boundaries Kaviany (2014). At high temperatures, phonon-phonon scattering dominates along with impurity scattering to some extent. It is well understood that anharmonic coupling on thermal resistivity leads to $\Lambda$ $\propto$ $1/T$ at high temperatures Kittel (1986). To elaborate it further, we study the mean free paths of phonons ($\Lambda_{\lambda}$) as a function of phonon frequencies for different temperatures. As can be seen from Fig 10, the mean free paths decay quickly within $\sim$ 3 THz with gradually increasing phonon frequency starting from 0 THz and saturate at lower values afterwards. This is precisely due to the dominance of the acoustic modes in thermal transport of GeTe. With increasing temperature, the mean free paths are observed to exhibit lower values validating $\Lambda$ $\propto$ $1/T$. The anisotropic nature of $\kappa_{L}$ has also been understood by means of the mean free paths along a and c axes of R3m-GeTe. Separate contributions of the mean free paths along a-axis and c-axis are shown in Fig 10. Throughout the temperature range studied, phonon mean free paths correspond to the a-axis show higher values compared to that of the c-axis, giving rise to an enhanced heat transfer along a-axis with higher values of $\kappa_{L}(x)$ compared to the c-axis with lower values of $\kappa_{L}(z)$. Thus, the anisotropic mean free path distribution of phonons is found to be the key reason for exhibiting an anisotropic heat transfer in crystalline R3m-GeTe.

To investigate the contribution of mean free paths of different lengthscales to $\kappa_{L}$, cumulative lattice thermal conductivity (considering the $\kappa_{ave}$ = (2$\kappa_{x}$ + $\kappa_{z}$)/3) is studied as a function of phonon mean free paths for the four different temperatures (Fig 11). We observe from Fig 11 that the maximum values of the phonon mean free paths ($\Lambda_{max}$) gradually decrease with temperature (T = 322 K, $\Lambda_{max}$ $\sim$ 152 Å; T = 372 K, $\Lambda_{max}$ $\sim$ 131 Å; T = 452 K, $\Lambda_{max}$ $\sim$ 108 Å; T = 503 K, $\Lambda_{max}$ $\sim$ 97 Å), consistently with $\Lambda$ $\propto$ $1/T$. Moreover, a dominant contribution ($\approx$ 67 $\%$) to the lattice thermal conductivity is found to be coming from the phonon mean free paths $\leqslant$ 60 Å. As temperature increases, the contributions from the phonon mean free paths $\leqslant$ 60 Å are increased to almost 93 $\%$ (Fig 11).

Increasing temperature, thus, manifests in a way of increasing phonon-phonon scattering processes which acts

on the lowering of mean free path of phonons. With almost similar $\textbf{v}_{\lambda}$, the lowering of mean free path of phonons can be associated with the decrements of $\tau_{\lambda}$.

Up till now we understood the dominant contributions of the acoustic modes to the thermal transport mechanisms of crystalline rhombohedral GeTe. In this section, we further systematically discriminate the relative contributions of transverse and longitudinal acoustic phonons to the lattice thermal conductivity of crystalline GeTe for the temperature range investigated in this study. It is observed that the transverse modes (TA) contribute to almost 75 $\%$ while the longitudinal counterpart (LA) adds up the rest of 25 $\%$ to the lattice thermal conductivity for the whole temperature range.

Figure 12 presents the lattice thermal conductivities, obtained via Ab-initio DFT calculations coupled with RTA, due to both transverse (TA) and longitudinal (LA) acoustic phonons as a function of temperature. For consistency, we also plot the $\kappa_{L}$ values for acoustic modes from the experimentally measured values of $\kappa$. This has been evaluated by subtracting the electronic thermal conductivity ($\kappa_{el}$), measured from the simulation, as well as the $\kappa_{L}$ due to optical phonons, computed from Ab-initio DFT and RTA, from the experimentally measured values of $\kappa$. Indicating a consistent picture from both experiment and theoretical calculations, as discussed earlier, an identical trend is found (Fig 12) between the data calculated from experiment and the simulated values of $\kappa_{L}$ due to acoustic phonons (TA+LA) despite the differences

between the values, mostly for T $<$ 412 K.

Throughout this work, the trend of simulated $\kappa(T)$ using RTA has been found to be a straightforward and convenient framework to describe the temperature dependence of the experimental results of $\kappa$ for crystalline GeTe. Further, the exact similarity between RTA and the phenomenological Slack model, indicates that umklapp scattering plays an important and significant phonon-phonon scattering mechanism in the high temperature range ($T$ $\gg$ $\Theta_{D}$), $\Theta_{D}$ being the Debye temperature. To identify the umklapp scattering parameter involved in the process, we fit the phonon relaxation time correspond to the umklapp scattering, obtained from RTA, with the analytical expression given by Glen A. Slack et al. Slack and Galginaitis (1964)

with $A$ $\propto$ $\frac{\hbar G^{2}}{\overline{M}c^{2}\Theta_{D}}$ and $B$ $\sim$ 1/3, where $\overline{M}$ is average atomic mass of the alloy and $G$ is the Gruneisen parameter. Being an empirical equation, it is necessary to find the parameter $A$ from the fitting procedure. Figure 13 shows the phonon relaxation times as a function of phonon frequency along with the fitted curve for all the four temperatures. We observe that at higher frequencies the trend of phonon relaxation time indeed shows $\omega^{-2}$ dependence for all temperatures. The values of $A$, thus retrieved from the fitting, are found to be independent of temperature with an average value of 1.1$\times$ 10^-4 ps K^-1.

This quantification, via the parameter $A$, serves as an important generic identification as this parameter can uniquely distinguish and compare the umklapp scattering processes for different crystalline materials ranging from $\Theta_{D}$ to an arbitrary high temperatures, within the operational regime of umklapp scattering process.