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

Kanka Ghosh  Corresponding author<EMAIL_ADDRESS>University of
Bordeaux, CNRS, Arts et Metiers Institute of Technology, Bordeaux INP, INRAE,
I2M Bordeaux, F-33400 Talence, France  Andrzej Kusiak  University of Bordeaux,
CNRS, Arts et Metiers Institute of Technology, Bordeaux INP, INRAE, I2M
Bordeaux, F-33400 Talence, France  Pierre Noé  CEA-LETI, MINATEC, Grenoble,
France  Marie-Claire Cyrille  CEA-LETI, MINATEC, Grenoble, France  Jean-Luc
Battaglia  University of Bordeaux, CNRS, Arts et Metiers Institute of
Technology, Bordeaux INP, INRAE, I2M Bordeaux, F-33400 Talence, France

###### Abstract

Thermal transport properties bear a pivotal role in influencing the
performance of phase change memory (PCM) devices, in which, the PCM operation
involves fast and reversible phase change between amorphous and crystalline
phases. In this article, we present a systematic experimental and theoretical
study on the thermal conductivity of GeTe at high temperatures involving fast-
change from amorphous to crystalline phase upon heating. Modulated
photothermal radiometry (MPTR) is used to experimentally determine thermal
conductivity of GeTe at high temperatures in both amorphous and crystalline
phases. Thermal boundary resistances are accurately taken into account for
experimental consideration. To develop a concrete understanding of the
underlying physical mechanism, rigorous and in-depth theoretical exercises are
carried out. For this, first-principles density functional methods and
linearized Boltzmann transport equations (LBTE) are employed using both direct
and relaxation time based approach (RTA) and compared with that of the
phenomenological Slack model. The amorphous phase experimental data has been
described using the minimal thermal conductivity model with sufficient
precision. The theoretical estimation involving direct solution and RTA method
are found to retrieve well the trend of the experimental thermal conductivity
for crystalline GeTe at high temperatures despite being slightly overestimated
and underestimated respectively compared to the experimental data. A rough
estimate of vacancy contribution has been found to modify the direct solution
in such a way that it agrees excellently with the experiment. Umklapp
scattering has been determined as the significant phonon phonon scattering
process. Umklapp scattering parameter has been identified for GeTe for the
whole temperature range which can uniquely determine and compare umklapp
scattering processes for different materials.

##  I Introduction

Chalcogenide alloys have been evolved as excellent candidates for the purpose
of electronic nonvolatile memory storage (phase change memories-PCM)Kolobov
_et al._ (2015); Wuttig and Yamada (2007); Wang  _et al._ (2008); Lu  _et al._
(2013); Campi  _et al._ (2015, 2017); Kusiak  _et al._ (2016). This
application involves a fast and reversible alteration between amorphous and
crystalline phase on heating. PCM cells consist of a nanoscale volume of a
phase change (PC) material, normally a Tellurium (Te) based alloy, which
undergoes a reversible change between amorphous and crystalline states,
possessing a contrasting electrical resistivity and thus enabling the PCMs to
be used for binary data storage Kolobov  _et al._ (2015); Lu  _et al._ (2013).
In a PCM device, crystallization by heating the amorphous PC alloy above its
crystallization temperature with electric current pulses is called SET
operation while amorphization of the crystalline region by melting and
quenching using higher and shorter electric current pulses is called RESET
operation Kusiak  _et al._ (2016). Germanium telluride (GeTe) is one of the
promising candidates within the phase change materials due to its notably high
contrast in electrical resistance as well as a stable amorphous phase with a
higher crystallization temperature upon doping for the data retention process
Kusiak  _et al._ (2016); Wdowik  _et al._ (2014). The crystallization
temperature for GeTe is $\approx$ 180∘C (453 K) Mantovan  _et al._ (2017);
Fallica  _et al._ (2013). GeTe has also been implemented with a superlattice
configuration as GeTe-Sb2Te3 that are extensively used for their application
in optical as well as PCM storage devices Boschker  _et al._ (2018). Further,
the interface between the GeTe and Sb2Te3 in the superlattice configuration is
found to control the phase transition, accompanied by a reduced entropy loss,
which helps in making fast and efficient PCMs Simpson  _et al._ (2011). Doped
GeTe with either N or C has been studied as a way to postpone the phase change
for high temperature applications Kusiak  _et al._ (2016); Fallica  _et al._
(2013). The effect of doping is also found to reduce the thermal conductivity
($\kappa$) of GeTe significantly Fallica  _et al._ (2013). Thermal
conductivity ($\kappa$) serves as a crucial parameter for PCM operations as
heat dissipation, localization and transport can significantly affect the
SET/RESET processes and can therefore considerably influence the performance
of PCMs in terms of cyclability, switching time and data retention.

While considerable amount of investigations have been reported on the
electronic transport properties of GeTe Kagdada  _et al._ (2018); Levin  _et
al._ (2013), very few reports on the thermal conductivity of GeTe starting
from room temperature to high-temperature range are found to exist in the
literature. P. Nath et al. Nath and Chopra (1974) characterized the thermal
properties of thick GeTe films in both crystalline and amorphous phases. While
amorphous phase showed negligible electronic contribution in the thermal
conductivity ($\kappa$), the contribution in crystalline phases were found to
be nearly 25$\%$ of the measured value of $\kappa$ Nath and Chopra (1974) at
the room temperature. E.M. Levin et al. Levin  _et al._ (2013) observed high
thermal conductivity of GeTe at 300 K and 720 K which they attributed mostly
to free charge carriers. Non-equilibrium molecular dynamics simulation (NEMD)
studies by Davide Campi et al. Campi  _et al._ (2015) showed that a 3$\%$ of
Ge vacancies effectively reduce the bulk lattice thermal conductivity of
crystalline GeTe from 3.2 Wm-1K-1 to 1.38 Wm-1K-1 at 300 K, justifying a large
spread of the experimentally measured thermal conductivities. First principles
calculations Campi  _et al._ (2017) also revealed this large variability of
experimentally measured bulk thermal conductivity due to the presence of Ge
vacancies. Very recently, frequency domain thermoreflectance study Warzoha
_et al._ (2019) was carried out for determining thermal conductivity in GeTe
thin films as a function of film thickness for both amorphous and crystalline
phases.

Though the aforementioned studies dealt with the thermal transport of GeTe in
a broader sense, there are plenty of open questions that still remain. A
thorough and systematic understanding of the thermal conductivity of GeTe as a
function of temperatures, ranging from room temperature to a temperature that
is higher than the crystallization temperature, in the context of different
scattering processes involved, is one amongst them. This systematic
understanding involves multiple approaches: (a) An accurate experimental
estimation of the thermal conductivity of GeTe thin films by taking into
account the significant contributions of thermal boundary resistances at the
boundaries of the GeTe layer. (b) A consistent and thorough theoretical
investigation of the temperature variation of thermal conductivity starting
from first-principles density functional theory as well as solving the
linearized Boltzmann transport equations (LBTE) and (c) Using simple
phenomenological models such as Slack model, which exhibit closed-form
solutions that can easily identify the underlying physical mechanism involved
in the thermal transport. As revealed in the study by E. Bosoni et al. Bosoni
_et al._ (2017), thermal conductivity ($\kappa$) calculated from the Slack
model Morelli and Slack (2006), was indeed found in good agreement with the
experimentally measured value of $\kappa$ at room temperature.

In this article, we investigate the thermal conductivity of GeTe films from
room temperature up to 230∘C (503 K) starting from the amorphous state. The
contribution of the film thermal conductivity from the thermal resistance at
the interfaces between the GeTe film and lower and upper layers are clearly
discriminated in the whole temperature range. The modulated photothermal
radiometry (MPTR) is employed as the experimental technique for the study. The
Levenberg-Marquardt (LM) technique is used in order to identify the essential
parameters from phase measurements and a model that simulates the phase within
the experimental configuration. In a stand-alone exercise of theoretical
understanding, thermal conductivity of GeTe is calculated starting from first-
principles density functional theory (DFT) coupled with solving linearized
Boltzmann transport equations (LBTE) by both direct method and relaxation time
approach. In order to explain the measured change in $\kappa\left(T\right)$,
distinct contributions coming from various scattering mechanisms are
understood. Further, phenomenological models by Slack et al. Slack and
Galginaitis (1964) and Cahill et al. Cahill  _et al._ (1992) are also employed
to elucidate the physical mechanisms in the heat transport process. These
systematic theoretical and experimental investigations are found to provide
significant clarity and insight in understanding the variation of thermal
conductivity of GeTe for a wide range of temperature. This work is organized
as follows: Section II deals with experimental measurements using MPTR method.
Computational details involving first principles and thermal conductivity
calculations are described in Section III. Section IV presents the theoretical
results followed by summary and conclusions in Section V.

##  II Experimental results

Amorphous GeTe films, with thicknesses of 200, 300 and 400 nm, are deposited
by magnetron sputtering in an Ar atmosphere on 200 mm silicon wafers covered
by a 500 nm thick SiO2 top layer. The thicknesses of the deposited films as
well as their homogeneity are controlled by X-Ray Reflectivity (XRR). The
detailed description of the MPTR setup had been presented elsewhere Kusiak
_et al._ (2016). The main principle consists of front face periodic heating of
the studied sample by a laser source. Since the GeTe layer is not opaque at
the laser wavelength (1064 nm) and in order to prevent oxidation or
evaporation of the GeTe at high temperature, a 100 nm thick platinum (Pt)
layer is deposited by sputtering in order to act as an optical to thermal
transducer. The periodic heat flux $\phi\left(\omega\right)$ is thus absorbed
by the Pt layer due to its high extinction coefficient at the laser
wavelength, and the optical source is then transformed into heat. On the other
hand, thanks to its high thermal conductivity and low thickness, the Pt layer
is assumed to be isothermal for the frequency range swept during the
experiment. The thermal response of the sample at the location of the heating
area by the laser is measured using an infrared detector. As the temperature
change is low at the heated area, the linearity assumption of heat transfer is
fulfilled and the emitted infrared radiation from the sample surface is
linearly proportional to the temperature at the heated area. A lock-in
amplifier is used to extract the amplitude and the phase from the signal of
the IR detector as a function of the frequency. The thermal properties are
thus obtained by fitting of the experimental phase by means of a thermal model
which allows to describe the heat transfer within the sample. According to the
film thickness and modulation frequency, the transient behaviour fulfils the
Fourier regime of heat conduction. Since the heated area is much larger (laser
spot of $\sim$ 2 mm in diameter) than the film thickness, the one-dimensional
heat transfer is considered. The explored frequency range in our experiment is
1-5 kHz. At angular frequency $\omega$, the phase is defined as
$\psi\left(\omega\right)=\arg\left[Z\left(\omega\right)\right]=\arctan\left(\mathrm{Im\left(Z\left(\omega\right)\right)}/\mathrm{Re}\left(Z\left(\omega\right)\right)\right)$,
where the transfer function $Z\left(\omega\right)$ denotes the ratio between
the periodic temperature $\theta\left(\omega\right)$ at the heated area and
$\phi\left(\omega\right)$ as:

$Z\left(\omega\right)=\frac{\theta\left(\omega\right)}{\phi\left(\omega\right)}=\frac{B}{D}$
(1)

Parameters $B$ and $D$ are calculated from the quadrupoles formalism Maillet
_et al._ (2000) as:

$\begin{bmatrix}A&B\\\ C&D\end{bmatrix}=\begin{bmatrix}A_{GeTe}&B_{GeTe}\\\
C_{GeTe}&D_{GeTe}\end{bmatrix}\begin{bmatrix}A_{SiO_{2}}&B_{SiO_{2}}\\\
C_{SiO_{2}}&D_{SiO_{2}}\end{bmatrix}\begin{bmatrix}A_{Si}&B_{Si}\\\
C_{Si}&D_{Si}\end{bmatrix}$ (2)

Where

$\displaystyle A_{j}$ $\displaystyle=$ $\displaystyle
1+\exp\left(-2\,\gamma_{i}\,e_{i}\right);\,B_{j}=\frac{\left(1+\exp\left(-2\,\gamma_{i}\,e_{i}\right)\right)}{\gamma_{i}\kappa_{i}^{*}}$
(3) $\displaystyle C_{j}$ $\displaystyle=$
$\displaystyle\left(1+\exp\left(-2\,\gamma_{i}\,e_{i}\right)\right)\,\gamma_{i}\,\kappa_{i}^{*};\,D_{j}=A_{j}$
(4)

With $\gamma_{i}=\sqrt{j\,\omega/a_{i}^{*}}$, where
$a_{i}^{*}$($=\kappa_{i}^{*}/C_{p_{i}}$), $\kappa_{i}^{*}$, $C_{p_{i}}$ and
$e_{i}$ are the effective thermal diffusivity, effective thermal conductivity,
specific heat per unit volume and thickness of layer $i$ respectively. For the
SiO2 and Si layers, the effective thermal conductivity is equal to the real
thermal conductivity, i.e., $\kappa_{SiO_{2}}^{*}=\kappa_{SiO_{2}}$ and
$\kappa_{Si}^{*}=\kappa_{Si}$. On the other hand, the effective thermal
resistance for GeTe ($R_{GeTe}^{*}$) accounts with the intrinsic thermal
conductivity $\kappa_{GeTe}$ of the GeTe layer and the thermal resistance at
the two interfaces with Pt and SiO2 as:

$R_{GeTe}^{*}=\frac{e_{GeTe}}{\kappa_{GeTe}^{*}}=\frac{e_{GeTe}}{\kappa_{GeTe}}+\underbrace{R_{Pt-
GeTe}+R_{GeTe-SiO_{2}}}_{R_{i}}$ (5)

where $R_{i}$ is the total interfacial thermal resistance. It must be
precisely mentioned here that the GeTe layer cannot be considered as thermally
resistive for the highest frequency values and especially for the high
thickness of the layer. This is the reason why we consider heat diffusion
within the quadrupole model. This can be easily demonstrated by calculating
the Fourier related quantity $\sqrt{2\,a_{GeTe}/\omega}$, using the known
value of $a_{GeTe}$ at room temperature and comparing it to the thickness
$e_{GeTe}$. Considering the measured value $Y_{\phi}\left(\omega_{i}\right)$
of the phase at different frequencies $\omega_{i}\,\left(i=1..N\right)$, the
value of $\kappa_{GeTe}^{*}$ is estimated by minimizing the objective function
$J=\left\|\mathbf{Y_{\phi}}-\mathbf{\mathbf{\Psi}}\right\|_{2}$, where
$\mathbf{Y_{\phi}=\mathit{Y_{\phi}\left(\omega_{i}\right)_{i=1..N}}}$ and
$\mathbf{\Psi}=\mathbf{\mathbf{\psi}}\left(\omega_{i}\right)_{i=1..N}$ are
vectors with length $N$, related respectively to the measured and simulated
phase at all the investigated frequencies. This minimization is achieved by
implementing the Lavenberg-Marquardt (LM) algorithm Aster  _et al._ (2019).
Then, the effective thermal resistance $R_{GeTe}^{*}$ of GeTe thin film is
plotted as a function of film thicknesses ($e_{GeTe}$) for different
temperatures as shown in Fig 1.

A linear regression
$R_{GeTe}^{*}=\alpha\,e_{GeTe}+\beta=e_{GeTe}/\kappa_{GeTe}+R_{i}$ is found
that allows to extract

Figure 1:  Variation of thermal resistance ($R^{*}_{GeTe}$) of GeTe thin film
as a function of film thickness ($e_{GeTe}$) for different temperatures
ranging from amorphous to crystalline phase change.

$\kappa_{GeTe}=1/\alpha$ from the slope (shown in Fig 2.(b)) and the sum of
the two interfacial resistances $R_{i}=\beta$ by extrapolation to $e_{GeTe}=0$
(shown in Fig 2.(a)). The standard deviation of $\kappa_{GeTe}^{*}$ is
calculated from the covariance matrix at the end of the iterative minimization
process as:
$\sigma\left(\kappa_{GeTe}^{*}\left|Y\right.\right)^{2}\sim\mathbf{cov}\left(\mathbf{\Theta}\right)\mathbf{E}/\sqrt{N}$
where the covariance matrix is:
$\mathbf{cov}\left(\mathbf{\Theta}\right)=\left(\mathbf{S}^{T}\mathbf{S}\right)^{-1}$
with vector $\mathbf{S}=\left[S_{Q}\left(\alpha_{i}\right)\right]_{N}$ with
$S_{Q}\left(\alpha_{i}\right)=\left[\partial\mathbf{\mathbf{\psi}}\left(\omega_{i}\right)_{i=1,N}/\partial\kappa_{GeTe}^{*}\right]_{\kappa_{GeTe}^{*}=\hat{\kappa}_{GeTe}^{*}}$
denoting the sensitivity function of the phase according to
$\kappa_{GeTe}^{*}$ calculated for $\kappa_{GeTe}^{*}=\hat{\kappa}_{GeTe}^{*}$
where $\hat{\kappa}_{GeTe}^{*}$ is the optimal value for $\kappa_{GeTe}^{*}$.
Finally the residual vector is
$E=\mathbf{Y_{\phi}}-\mathbf{\Psi}\left(\kappa_{GeTe}^{*}=\hat{\kappa}_{GeTe}^{*}\right)$.
The standard deviation on $R_{GeTe}^{*}$ is obtained starting from
$\sigma(\kappa_{GeTe}^{*})$ by application of the law of propagation of
uncertainties. Finally, the standard deviations on $\kappa_{GeTe}$ and $R_{i}$
are expressed according to residual variance of linear fitting on
$R_{GeTe}^{*}$ points. We mention here that the grain size of GeTe at the time
of the crystallization is found to be 40 nm as reported in our earlier work
Kusiak  _et al._ (2016) and it increases while increasing the annealed
temperature (Fig 4 in Kusiak  _et al._ (2016)). Since the MPTR investigates a
very large area, it is thus expected to measure the average thermal
conductivity that is given by $\kappa_{L}^{av}$ \+ $\kappa_{el}$, where
$\kappa_{L}^{av}$ = $\frac{2}{3}$$\kappa_{x}$ \+ $\frac{1}{3}$$\kappa_{z}$,
where, $\kappa_{x}$ and $\kappa_{z}$ stand for the lattice thermal
conductivities along hexagonal a and c axes respectively.

The phase change occurs well within the range of the expected temperature
($\sim$ 180∘C = 453 K). We note that $R_{i}$ for amorphous state is difficult
to estimate because of the very low values of $\kappa_{GeTe}$. For crystalline
phase, we retrieve a consistent behavior with the value of $R_{i}$ being
increased as the temperature is lowered (Fig 2.(a)). In the high temperature
regime, diffuse mismatch model (DMM) has been found to describe $R_{i}$ quite
satisfactorily for crystalline solids Reifenberg  _et al._ (2010). According
to DMM model,

Figure 2:  (a) Variation of total interfacial thermal resistance $R_{i}=R_{Pt-
GeTe}+R_{GeTe-SiO_{2}}$ with temperature. Phase change occurs around 180 ∘C =
453 K, which is shown via dotted line. (b) Experimentally measured thermal
conductivity $\kappa_{GeTe}\left(T\right)$ of the GeTe thin film as a function
of temperature. The rightward arrows denote the forward cycle exhibiting phase
change from amorphous to crystalline phase while the leftward arrow defines
the backward cycle where GeTe exists in crystalline phase.

asymptotic behavior of $R_{i}$ at high temperatures depends inversely on the
heat capacity as Reifenberg  _et al._ (2010)

$R_{i}=\left(\frac{\sum_{j}c_{2,j}^{-2}}{12\left(\sum_{j}c_{1,j}^{-2}+\sum_{j}c_{2,j}^{-2}\right)}\sum_{j}c_{1,j}\right)^{-1}\frac{1}{C_{1}\left(T\right)}$
(6)

where, $C_{1}(T)$ is the heat capacity of material 1 at $T$ and $c_{l,j}$ is
the velocity of phonon mode $j$ in material $l$. Here all the parameters
except $C_{1}(T)$ are temperature independent. Since $C_{1}(T)$ increases with
temperature, the above relation shows that $R_{i}$ decreases with increasing
temperature. In Fig 2.(b), we observe a monotonically decreasing trend of
$\kappa_{GeTe}$ as a function of T. Indeed, at high temperatures (T $\gg$
$\Theta_{D}$), where $\Theta_{D}$ is Debye temperature, umklapp scattering is
the dominating phonon-phonon scattering process associated with high momentum
change in phonon-phonon collisions Kittel (1986). Glen A. Slack et al.Slack
and Galginaitis (1964) approximated umklapp relaxation time as $\tau_{U}^{-1}$
= $AT\omega^{2}exp\left(-\Theta_{D}/3T\right)$ which becomes $\tau_{U}^{-1}$ =
$AT\omega^{2}$ when T $\gg$ $\Theta_{D}$. This relaxation time estimation
leads to $\kappa$ $\propto$ 1/$\tau_{U}^{-1}$ $\propto$ 1/$T$.

##  III Computational details

Phonon density of states (PDOS) of crystalline GeTe (space group R3m) has been
calculated employing the density functional perturbation theory (DFPT) Baroni
_et al._ (2001) using the QUANTUM-ESPRESSO Giannozzi  _et al._ (2009) suite of
programs. As the first step, self-consistent calculations, within the
framework of density functional theory (DFT), are carried out to compute the
total ground state energy of the crystalline R3m-GeTe. For this purpose,
Perdew-Burke-Ernzerhof (PBE) Perdew  _et al._ (1996) generalized gradient
approximation (GGA) is used as the exchange-correlation functional. The spin-
orbit interaction has been ignored due to its negligible effects on the
vibrational features of GeTe as mentioned in literature Shaltaf  _et al._
(2009); Campi  _et al._ (2017). Electron-ion interactions are represented by
pseudopotentials using the framework of projector-augmented-wave (PAW) method
Blochl (1994). The Kohn-Sham (KS) orbitals are expanded in a plane-wave (PW)
basis with a kinetic cutoff of 60 Ry and a charge density cutoff of 240 Ry as
prescribed by the pseudopotentials of Ge and Te. The Brillouin zone
integration for self consistent electron density calculations are performed
using a 12$\times$12$\times$12 Monkhorst-Pack (MP) Monkhorst and Pack (1976)
k-point grid.

For phonon calculations, a hexagonal 2$\times$2$\times$1 supercell, consisting
of 24 atoms, is used. A representative

Figure 3: 2$\times$2$\times$1 supercell of GeTe crystal (R3m) and its trigonal
primitive cell. The structure of GeTe can be seen as a stacking of bilayers
along c axis of hexagonal unit cell as mentioned in Campi  _et al._ (2015,
2017). Here red spheres denote Ge atoms and blue spheres denote Te atoms. This
representation is realized through VESTA Momma and Izumi (2011). Here, Ge
atoms at the boundary take into account the additional Te atoms outside the
cell.

structure of crystalline GeTe as stacking bilayers is shown in Fig 3. To study
the phonon density of states, linear response theory is applied via DFPT, to
the Kohn-Sham equations to solve the electronic charge density ($\rho_{n}$)
under small perturbations. As the force constants are connected to the
derivatives of $\rho_{n}$ with respect to atomic displacements, harmonic force
constants are calculated by diagonalizing the dynamical matrix in reciprocal
space. Phonon density of states (PDOS) are then evaluated by the inverse
Fourier transform of the interatomic force constants (IFC) to real space from
that of the dynamical matrices, using a uniform 5$\times$5$\times$5 grid of
q-vectors.

The thermal conductivity of GeTe can be separated into two distinct
contributions, one from electronic transport and the other from the phonon
transport or the lattice contribution, such that $\kappa$ = $\kappa_{el}$ \+
$\kappa_{L}$, where $\kappa_{L}$ is the lattice thermal conductivity and
$\kappa_{el}$ is the electronic thermal conductivity. $\kappa_{el}$ has been
obtained from first-principles calculations by solving semi-classical
Boltzmann transport equation (BTE) for electrons. Constant relaxation time
approximation (CRTA) and rigid band approximation (RBA) are employed as
implemented in BoltzTraP code Madsen and Singh (2006). The energy projected
conductivity tensor is calculated using:

$\sigma_{\alpha\beta}(\epsilon)=\frac{1}{N}\sum_{i,\textbf{k}}\sigma_{\alpha\beta}(i,\textbf{k})\frac{\delta(\epsilon-\epsilon_{i,\textbf{k}})}{d\epsilon}$
(7)

Therefore, the transport tensors, or more specifically the electrical
conductivity tensor in this study, can be obtained from

$\sigma_{\alpha\beta}(T;\mu)=\frac{1}{\Omega}\int\sigma_{\alpha\beta}(\epsilon)\left[-\frac{\partial
f_{\mu}(T;\epsilon)}{\partial\epsilon}\right]d\epsilon$ (8)

where, N is the number of k-points sampled, $i$ is the band index,
$\epsilon_{i,\textbf{k}}$ are band energies, $\Omega$ is the volume of unit
cell, $f_{\mu}$ is the Fermi distribution function and $\mu$ is the chemical
potential. The code computes the Fermi integrals and returns the transport
coefficients for different temperature and Fermi levels.

For getting lattice thermal conductivity $\kappa_{L}$, linearized phonon
Boltzmann transport equation (LBTE) is solved using both direct method
introduced by L. Chaput et al. Chaput (2013) as well as the single mode
relaxation time approximation or relaxation time approximation (RTA) employing
PHONO3PY Togo  _et al._ (2015) software package. Initially, the supercell
approach with finite displacement of 0.03 Å is applied to calculate the
harmonic (second order) and the anharmonic (third order) force constants,
given by

$\Phi_{\alpha\beta}(l\kappa,l^{\prime}\kappa^{\prime})=\frac{\partial^{2}\Phi}{\partial
u_{\alpha}(l\kappa)\partial u_{\beta}(l^{\prime}\kappa^{\prime})}$ (9)

and

$\Phi_{\alpha\beta\gamma}(l\kappa,l^{\prime}\kappa^{\prime},l^{\prime\prime}\kappa^{\prime\prime})=\frac{\partial^{3}\Phi}{\partial
u_{\alpha}(l\kappa)\partial u_{\beta}(l^{\prime}\kappa^{\prime})\partial
u_{\gamma}(l^{\prime\prime}\kappa^{\prime\prime})}$ (10)

respectively. First principles calculations using QUANTUM-ESPRESSO Giannozzi
_et al._ (2009) are implemented to calculate the forces acting on atoms in
supercells. Using finite difference method, harmonic force constants are
approximated as Togo  _et al._ (2015)

$\Phi_{\alpha\beta}(l\kappa,l^{\prime}\kappa^{\prime})\simeq-\frac{F_{\beta}[l^{\prime}\kappa^{\prime};\textbf{u}(l\kappa)]}{u_{\alpha}(l\kappa)}$
(11)

where F[$l^{\prime}$$\kappa^{\prime}$; u($l$$\kappa$)] is atomic force
computed at r($l^{\prime}$ $\kappa^{\prime}$) with an atomic displacement
u($l\kappa$) in a supercell. Similarly, anharmonic force constants are
obtained usingTogo  _et al._ (2015)

$\Phi_{\alpha\beta\gamma}(l\kappa,l^{\prime}\kappa^{\prime},l^{\prime\prime}\kappa^{\prime\prime})\simeq-\frac{F_{\gamma}[l^{\prime\prime}\kappa^{\prime\prime};\textbf{u}(l\kappa),\textbf{u}(l^{\prime}\kappa^{\prime})]}{u_{\alpha}(l\kappa)u_{\beta}(l^{\prime}\kappa^{\prime})}$
(12)

where F[$l^{\prime\prime}$$\kappa^{\prime\prime}$; u($l$$\kappa$),
u($l^{\prime}$ $\kappa^{\prime}$)] is atomic force computed at
r($l^{\prime\prime}$ $\kappa^{\prime\prime}$) with a pair of atomic
displacements u($l\kappa$) and u($l^{\prime}\kappa^{\prime}$) in a supercell.
These two sets of linear equations are solved using Moore-Penrose
pseudoinverse as is implemented in PHONO3PY Togo  _et al._ (2015).

We use a 2$\times$2$\times$2 supercell of GeTe for our first-principles
calculations of anharmonic force constants. Using the supercell and finite
displacement approach, 228 supercells are obtained, having different pairs of
displaced atoms, for the calculations for the anharmonic force constants. A
larger 3$\times$3$\times$3 supercell is employed for calculating the harmonic
force constants. For all the supercell force calculations, the reciprocal
space is sampled using a 3$\times$3$\times$3 k-sampling MP mesh shifted by a
half grid distances along all three directions from $\Gamma$\- point. The
total energy convergence threshold has been kept at 10-10 a.u. for supercell
calculations. For lattice thermal conductivity calculations employing both the
direct solution of LBTE and that of the RTA, q-mesh of 24$\times$24$\times$24
are used. The imaginary part of self-energy have been calculated using
tetrahedron method from which phonon lifetimes are obtained .

##  IV Theoretical results and Discussions

###  IV.1 Phonon density of states

The structural parameters are optimized via DFT calculations and the optimized
lattice parameter (a = 4.23 Å) and unit cell volume (56.26 Å3) 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

Figure 4:  (a) Calculated phonon density of states (PDOS) for GeTe crystal
(R3m). The dotted line denotes the separation frequency between acoustic and
optical modes ($\sim$ 96 cm-1 = 2.88 THz) which is consistent with previous
studies (see text). (b) Phonon dispersion relation of the rhombohedral (R3m)
GeTe. Transverse and longitudinal phonon modes are denoted via solid and
dashed lines, respectively. The approximate separation frequency between
acoustic and optical modes ($\sim$ 96 cm-1) are shown via green dashed line.

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.

###  IV.2 Electronic thermal conductivity

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):

$L=\left(\frac{k_{B}}{e}\right)^{2}\left[\frac{\left(r+\frac{7}{2}\right)\left(r+\frac{3}{2}\right)\,F_{r+5/2}\left(\eta\right)\,F_{r+1/2}\left(\eta\right)-\left(r+\frac{5}{2}\right)^{2}\,F_{r+3/2}^{2}\left(\eta\right)}{\left(r+\frac{3}{2}\right)^{2}\,F_{r+1/2}^{2}\left(\eta\right)}\right]$
(13)

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:
$F_{n}\left(\eta\right)=\int_{0}^{\infty}\left(x^{n}/\left(1+e^{x-\eta}\right)\right)\textit{d}x$.

Table 1: Electronic thermal conductivity ($\kappa_{el}$) of the crystalline R3m-GeTe is presented (column 4) as a function of temperature (T) using Wiedemann-Franz law. The variation of electrical conductivity ($\sigma_{e}$) and the Lorenz number (L) with temperature are also shown in column 2 and column 3 respectively. T (K) |  $\sigma_{e}$ ($\Omega^{-1}m^{-1}$) | L (W$\Omega$ $K^{-2}$) |  $\kappa_{el}$ = $L\sigma_{e}T$ (W/mK)
---|---|---|---
322 | 9.78$\times 10^{4}$ | 2.40$\times 10^{-8}$ | 0.76
372 | 9.35$\times 10^{4}$ | 2.33$\times 10^{-8}$ | 0.81
412 | 9.14$\times 10^{4}$ | 2.27$\times 10^{-8}$ | 0.85
452 | 8.84$\times 10^{4}$ | 2.20$\times 10^{-8}$ | 0.88
503 | 8.45$\times 10^{4}$ | 2.13$\times 10^{-8}$ | 0.91

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).

###  IV.3 Lattice thermal conductivity

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:

$\textstyle\kappa_{min}\left(T\right)=\left(\frac{\pi}{6}\right)^{1/3}\,k_{B}\,n^{2/3}\,\sum_{i=1}^{3}v_{i}\,\left(\frac{T}{\Theta_{i}}\right)^{2}\int_{0}^{\Theta_{i}/T}\frac{x^{3}\,e^{x}}{\left(e^{x}-1\right)^{2}}\,\mathrm{d}x$
(14)

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

$\kappa_{min}=\frac{1}{2}\left(\pi\,n^{2}/6\right)^{1/3}\,k_{B}\,\left(v_{L}+2\,v_{T}\right).$
(15)

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)

$\kappa_{\alpha\beta}=\frac{\hbar^{2}}{4k_{B}T^{2}NV_{0}}\sum_{\lambda\lambda^{\prime}}\frac{\omega_{\lambda}\upsilon_{\alpha}(\lambda)}{sinh(\frac{\hbar\omega_{\lambda}}{2k_{B}T})}\frac{\omega_{\lambda^{\prime}}\upsilon_{\beta}(\lambda^{\prime})}{sinh(\frac{\hbar\omega_{\lambda^{\prime}}}{2k_{B}T})}(\Omega^{\sim
1})_{\lambda\lambda^{\prime}}$ (16)

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

$\Omega_{\lambda\lambda^{\prime}}=\delta_{\lambda\lambda^{\prime}}/\tau_{\lambda}+\pi/\hbar^{2}\sum_{\lambda^{\prime\prime}}\mid\Phi_{\lambda\lambda^{\prime}\lambda^{\prime\prime}}\mid^{2}\frac{[\delta(\omega_{\lambda}-\omega_{\lambda^{\prime}}-\omega_{\lambda^{\prime\prime}})+\delta(\omega_{\lambda}+\omega_{\lambda^{\prime}}-\omega_{\lambda^{\prime\prime}})+\delta(\omega_{\lambda}-\omega_{\lambda^{\prime}}+\omega_{\lambda^{\prime\prime}})]}{sinh(\frac{\hbar\omega_{\lambda^{\prime\prime}}}{2k_{B}T})}$
(17)

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)

Table 2:  Theoretical and experimental values of different parameters used for thermal conductivity calculation using Slack Morelli and Slack (2006) and Cahill Cahill  _et al._ (1992) model. GeTe (R3m) | Parameter description(s) | Value(s)
---|---|---
$V_{0}$ [Å3] | Volume of the elementary cell | 56.26 Jain  _et al._ (2013),[calculated from DFT]
$\rho$ [kg m-3] | Density | 5910 Jain  _et al._ (2013),[calculated from DFT]
$M_{Ge}$ [g mol-1] | Molar mass of Ge | 72.63
$M_{Te}$ [g mol-1] | Molar mass of Te | 127.6
$\Theta_{D}$ [K] | Debye temperature | 180 Campi  _et al._ (2017)
$\upsilon_{L}$ [m s-1] | Phonon group velocity (longitudinal) | 2500 Pereira  _et al._ (2013)
$\upsilon_{T}$ [m s-1] | Phonon group velocity (Transverse) | 1750 Pereira  _et al._ (2013)
$G$ | Gruneisen parameter | 1.7 Bosoni  _et al._ (2017)
$E_{F}$ [eV] | Fermi energy | 7.2552 [calculated from DFT]
$N$ [kg-1] | number of phonon per unit mass |  $5.6723\times 10^{24}$ [calculated from PDOS]

${}\bm{\kappa}_{L}=\frac{1}{NV_{0}}\sum_{\lambda}C_{\lambda}\textbf{v}_{\lambda}\otimes\textbf{v}_{\lambda}\tau_{\lambda}$
(18)

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

$C_{\lambda}=k_{B}\left(\frac{\hbar\omega_{\lambda}}{k_{B}T}\right)^{2}\frac{exp(\hbar\omega_{\lambda}/k_{B}T)}{[exp(\hbar\omega_{\lambda}/k_{B}T)-1]^{2}}$
(19)

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)

$\frac{1}{\tau_{\lambda}}=\frac{1}{\tau_{\lambda}^{N}}+\frac{1}{\tau_{\lambda}^{U}}+\frac{1}{\tau_{\lambda}^{I}}$
(20)

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)

$\Gamma_{\lambda}(\omega_{\lambda})=\frac{18\pi}{\hbar^{2}}\sum_{\lambda^{\prime}\lambda^{\prime\prime}}\Delta\left(\textbf{q}+\textbf{q}^{\prime}+\textbf{q}^{\prime\prime}\right)\mid\Phi_{-\lambda\lambda^{\prime}\lambda^{\prime\prime}}\mid^{2}\\{(n_{\lambda^{\prime}}+n_{\lambda^{\prime\prime}}+1)\delta(\omega-\omega_{\lambda^{\prime}}-\omega_{\lambda^{\prime\prime}})+(n_{\lambda^{\prime}}-n_{\lambda^{\prime\prime}})[\delta(\omega+\omega_{\lambda^{\prime}}-\omega_{\lambda^{\prime\prime}})-\delta(\omega-\omega_{\lambda^{\prime}}+\omega_{\lambda^{\prime\prime}})]\\}$
(21)

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

$\frac{1}{\tau_{\lambda}^{I}(\omega)}=\frac{\pi\omega_{\lambda}^{2}}{2N}\sum_{\lambda^{\prime}}\delta\left(\omega-\omega^{\prime}_{\lambda}\right)\sum_{k}g_{k}|\sum_{\alpha}\textbf{W}_{\alpha}\left(k,\lambda\right)\textbf{W}_{\alpha}^{*}\left(k,\lambda\right)|^{2}$

(22)

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

$g_{k}=\sum_{i}f_{i}\left(1-\frac{m_{ik}}{\overline{m}_{k}}\right)^{2}$ (23)

$f_{i}$ 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.

Figure 5: Experimental and theoretical thermal conductivity ($\kappa$) of GeTe as a function of temperature are presented in a fast change process. Phase change behavior is realized in the studied temperature range: 322 K $\leqslant$ T $\leqslant$ 503 K. Electronic ($\kappa_{el}$) and phonon contributions ($\kappa_{L}$) to the total thermal conductivity ($\kappa_{tot}$) are shown. $\kappa_{L}$ is evaluated using RTA, direct solution of LBTE and Slack model. N, U, I and V define normal, umklapp, isotope and vacancy scattering processes respectively. Table 3: Experimental and theoretical total thermal conductivity ($\kappa$) of the crystalline R3m-GeTe is presented as a function of temperature (T). The lattice contribution of thermal conductivity ($\kappa_{L}$) is taken as an average of $\kappa_{x}$ and $\kappa_{z}$ as $\kappa_{av}$ = (2$\kappa_{x}$ \+ $\kappa_{z}$)/3 (see text). The unit of $\kappa$ is Wm-1K-1. N, U, I and V represent normal, umklapp, isotope and vacancy scattering processes respectively. T (K) | Expt. | Direct (N,U) | Direct (N,U,I) | Slack | RTA(N,U) | RTA(N,U,I) | Direct (N, U, I, V)
---|---|---|---|---|---|---|---
322 | 3.59 | 3.90 | 3.84 | 2.88 | 2.90 | 2.87 | 3.36
372 | 3.09 | 3.53 | 3.48 | 2.69 | 2.66 | 2.64 | 3.06
412 | 2.70 | 3.31 | 3.27 | 2.53 | 2.53 | 2.51 | 2.89
452 | 2.62 | 3.12 | 3.09 | 2.40 | 2.41 | 2.39 | 2.75
503 | 2.43 | 2.92 | 2.90 | 2.30 | 2.28 | 2.27 | 2.59

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

${}\frac{1}{\tau_{V}}=x\left(\frac{\Delta
M}{M}\right)^{2}\frac{\pi}{2}\frac{\omega^{2}g(\omega)}{G^{\prime}}$ (24)

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:

$\kappa_{L}=C\frac{\overline{M}\,\Theta_{D}^{3}\,\delta}{G^{2}\,n_{c}^{2/3}\,T}$
(25)

with:

$C=\frac{2.43\times 10^{-6}}{1-0.514/G+0.228/G^{2}}$ (26)

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.

###  IV.4 Variation of lattice thermal conductivity ($\kappa_{L}$) with phonon
frequency: Contribution of acoustic and optical modes

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)

$\kappa_{L}^{c}=\int_{0}^{\omega}\kappa_{L}(\omega^{\prime})d\omega^{\prime}$
(27)

Figure 6: Cumulative lattice thermal conductivities ($\kappa_{L}$) of
crystalline GeTe are presented as a function of frequencies at four different
temperatures: (a) T = 322 K, (b) T = 372 K, (c) T = 452 K and (d) T = 503 K.
Cumulative $\kappa_{L}$, computed within the RTA framework, along hexagonal c
axis ($\kappa_{z}$), along its perpendicular direction ($\kappa_{x}$) and
their average $\kappa_{av}$ = (2$\kappa_{x}$ \+ $\kappa_{z}$)/3 are shown. The
derivatives of $\kappa_{z}$ and $\kappa_{x}$ with respect to frequencies are
also shown for each temperature.

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

$\kappa_{L}(\omega^{\prime})\equiv\frac{1}{NV_{0}}\sum_{\lambda}C_{\lambda}\textbf{v}_{\lambda}\otimes\textbf{v}_{\lambda}\tau_{\lambda}\delta(\omega^{\prime}-\omega_{\lambda})$
(28)

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

Table 4: Relative contributions of acoustic and optical modes to the total lattice thermal conductivity ($\kappa_{L}$) for different temperatures. The unit of $\kappa_{L}$ is W/mK. T (K) |  $\kappa_{L}$ (RTA) | Contribution of | Contribution of
---|---|---|---
|  | acoustic modes ($\%$) | optical modes ($\%$)
322 | 2.14 | 77.1 | 22.9
372 | 1.85 | 77.3 | 22.7
452 | 1.53 | 77.1 | 22.9
503 | 1.37 | 77.4 | 22.6

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.

###  IV.5 Anisotropy of lattice thermal conductivity ($\kappa_{L}$) of
crystalline $GeTe$

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

Figure 7: Lattice thermal conductivity ($\kappa_{L}$) of GeTe along $z$
direction ($\kappa_{L}(z)$, along the c axis in the hexagonal notation) and
along $x-y$ direction (($\kappa_{L}(x)$, along the a axis) are presented as a
function of temperature. N, U, I denote normal, umklapp and isotope scattering
effects respectively.

$xy$ 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

$W^{c}(\omega)\equiv\frac{1}{NV_{0}}\sum_{\lambda}\textbf{v}_{\lambda}\otimes\textbf{v}_{\lambda}\delta(\omega-\omega_{\lambda})$
(29)

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.

Figure 8: (a) Ratios of $x$ and $z$ components of cumulative lattice thermal
conductivities, $\kappa^{c}_{L(x)}(\omega)$/$\kappa^{c}_{L(z)}(\omega)$ of
crystalline GeTe as a function of phonon frequencies are presented for four
different temperatures. (b) Ratios of $x$ and $z$ components of cumulative
direct vector products of group velocities,
$W^{c}_{x}$($\omega$)/$W^{c}_{z}$($\omega$) of crystalline GeTe as a function
of phonon frequencies for four different temperatures.

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.

###  IV.6 Variation of lattice thermal conductivity ($\kappa_{L}$) with
temperature: Phonon lifetime and phonon mean free path

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 (Cλ) as a function
of phonon frequencies for different temperatures. Indeed, Cλ 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 Cλ 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.

Figure 9: Phonon lifetimes of crystalline GeTe are shown as a function of
phonon frequencies for four different temperatures: (a) T = 322 K, (b) T = 372
K, (c) T = 452 K and (d) T = 503 K. Phonon lifetimes due to transverse
acoustic (TA), Longitudinal acoustic (LA) and optical modes are presented in
blue, red and green points respectively. Figure 10: Phonon mean free paths of
crystalline GeTe are shown as a function of phonon frequencies for four
different temperatures: (a) T = 322 K, (b) T = 372 K, (c) T = 452 K and (d) T
= 503 K. Phonon mean free paths along a-axis and c-axis are represented in red
and gray dots respectively.

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

$\Lambda_{\lambda}=\textbf{v}_{\lambda}\tau_{\lambda}$ (30)

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

Figure 11: Variation of cumulative lattice thermal conductivity ($\kappa_{L}$)
of crystalline GeTe with phonon mean free path is shown for four different
temperatures: (a) T = 322 K, (b) T = 372 K, (c) T = 452 K and (d) T = 503 K.
The contributions from the mean free paths $\leqslant$ 60 Å to the
$\kappa_{L}$ are mentioned for two extremes of temperatures.

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 $\tau_{\lambda}$.

###  IV.7 Contribution of transverse and longitudinal acoustic phonons

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

Figure 12: Lattice thermal conductivities, contributed solely from acoustic
modes are shown as a function of temperature. ‘Experimental’ values of
acoustic $\kappa_{L}$ are obtained by subtracting $\kappa_{el}$ and optical
mode contributed $\kappa_{L}$, from the experimental values of $\kappa$. RTA
calculated $\kappa_{L}$ for transverse acoustic branch (TA), longitudinal
acoustic branch (LA) and the total acoustic contribution (LA+TA) are also
shown. Figure 13: Phonon lifetimes of GeTe corresponding to umklapp
scattering, along with the umklapp fitted parameter $A$ are shown as a
function of phonon frequencies for four different temperatures: (a) T = 322 K,
(b) T = 372 K, (c) T = 452 K and (d) T = 503 K.

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)

$\tau_{U}^{-1}=AT\omega^{2}exp\left(-B\frac{\Theta_{D}}{T}\right)$ (31)

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.

##  V Summary and Conclusions

We have carried out a systematic experimental and theoretical study on the
thermal conductivity variation of GeTe at high temperatures. The study
involves fast and reversible phase change between amorphous and crystalline
phases of GeTe. Modulated photothermal radiometry (MPTR) as well as the
Lavenberg-Marquardt (LM) technique are employed to determine thermal
conductivities of GeTe in both amorphous and crystalline phases as a function
of temperature. Thermal boundary resistances, coming from both Pt-GeTe and
GeTe-SiO2 interfaces, have been accurately taken into account for measuring
$\kappa$ experimentally. Van der Pauw technique as well as Boltzmann transport
equations are solved for electrons to estimate electronic thermal conductivity
within the constant relaxation time approximation (CRTA) framework.

To compute lattice thermal conductivity ($\kappa_{L}$), first-principles
density functional theory (DFT) is used and the solution to the linearized
Boltzmann transport equation (LBTE) has been realized via both direct method
and relaxation time approach (RTA). Normal, umklapp and isotope effects are
included in computing the phonon relaxation time in these approaches. While
the direct method is found to capture the trend of $\kappa(T)$ quite well, the
values are a bit overestimated compared to the experimental data. The hole
concentration of 6.24$\times$1019 cm-3, obtained using first principles
calculations and BTE for electrons, necessitates the incorporation of phonon-
vacancy scattering to estimate $\kappa_{L}$. Following a recent work Campi
_et al._ (2017) on crystalline GeTe with almost same hole concentration,
vacancy contribution is incorporated to estimate more realistic values of
$\kappa$. Indeed, the estimate of $\kappa$ using direct method and adding the
temperature independent phonon-vacancy scattering contribution, an excellent
agreement is obtained between experimental and theoretical values of $\kappa$.

$\kappa$ computed from RTA, is also found to retrieve the trend of
experimental $\kappa(T)$ quite well, especially at higher temperatures.
However, the over-resistive nature of RTA due to the treatment of umklapp and
normal scattering in equal footing, causes an underestimation of $\kappa$
compared to the experimental values. Nevertheless, the trend $\kappa(T)$
agrees well at higher temperatures. Cumulative lattice thermal conductivity is
presented as a function of phonon frequencies for different temperatures. The
density of heat carrying phonons or rather the phonon density of states plays
a crucial role in determining $\kappa_{L}$. Acoustic phonons emerge as the
dominant ($\sim$ 77$\%$) contributor to the lattice thermal conductivity while
transverse acoustic modes contribute almost 75$\%$ to the acoustic phonon
transport. The anisotropy of lattice thermal conductivities of crystalline
GeTe between in-plane and out of plane components are understood using the
variation of cumulative outer product of phonon group velocities. Further, the
anisotropy present in phonon mean free path along a-axis and c-axis are found
to control the anisotropy in the heat transfer along these two axes. Phonon
group velocities and mode heat capacities are observed to remain almost
independent of temperature. Therefore, the temperature variation of $\kappa$
is attributed to the variation of phonon mean free path and consequently the
variation of phonon lifetime with temperature.

Phenomenological models are also found to be quite effective in describing and
identifying the underlying physical mechanism of thermal transport. The
experimental values of $\kappa$ in amorphous phase of GeTe are described using
the minimal thermal conductivity model, proposed by Cahill et al. Cahill  _et
al._ (1992). For crystalline phase data, Slack model Morelli and Slack (2006)
is also employed and it has been found to be strikingly similar with RTA based
solutions. Both RTA based solutions as well as expression from
phenomenological Slack model for GeTe indicate that the umklapp phonon phonon
scattering is significant for the temperature regime studied in this work.
Umklapp phonon relaxation time is found to obey $\omega^{-2}$ dependence at
higher frequencies and therefore umklapp scattering parameter has been
obtained, which remains almost constant for the whole temperature range
studied. This complete experimental and theoretical exercise to elucidate the
thermal conductivity of GeTe at high temperatures can further assist in
improving the thermal management for other Te based phase change materials by
understanding heat dissipation, localization and transport with more clarity.

###### Acknowledgements.

This project has received funding from the European Union’s Horizon 2020
research and innovation program under Grant Agreement No. 824957
(“BeforeHand:” Boosting Performance of Phase Change Devices by Hetero- and
Nanostructure Material Design).

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