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

Abstract

We study the effect of characteristic size variation on the phonon thermal transport in crystalline GeTe for a wide range of temperatures using the first-principles density-functional method coupled with the kinetic collective model approach. The characteristic size dependence of phonon thermal transport reveals an intriguing collective phonon transport regime, located in between the ballistic and the diffusive transport regimes. Therefore, systematic investigations have been carried out to describe the signatures of phonon hydrodynamics via the competitive effects between grain size and temperature. A characteristic nonlocal length associated with phonon hydrodynamics and a heat wave propagation length has been extracted. The connections between phonon hydrodynamics and these length scales are discussed in terms of the Knudsen number. Further, the scaling relation of thermal conductivity as a function of characteristic size in the intermediate size range emerges as a crucial indicator of the strength of the hydrodynamic behavior. A ratio concerning normal and resistive scattering rates has been employed to understand these different scaling relations, which seems to control the strength and prominent visibility of the collective phonon transport in GeTe. This systematic investigation emphasizes the importance of the competitive effects between temperature and characteristic size on phonon hydrodynamics in GeTe, which can lead to a better understanding of the generic behavior and consequences of the phonon hydrodynamics and its controlling parameters in low-thermal conductivity materials.

Full text

Effect of characteristic size on the collective phonon transport in crystalline GeTe
Kanka Ghosh,∗Andrzej Kusiak, and Jean-Luc Battaglia
University of Bordeaux, CNRS, Arts et Metiers Institute of Technology, Bordeaux INP,
INRAE, I2M Bordeaux, 351 Cours de la lib´eration, F-33400 Talence, France
We study the effect of characteristic size variation on the phonon thermal transport in crystalline
GeTe for a wide range of temperatures using the first-principles density functional method coupled
with the kinetic collective model approach. The characteristic size dependence of phonon thermal
transport reveals an intriguing collective phonon transport regime, located in between the ballistic
and the diffusive transport regimes. Therefore, systematic investigations have been carried out to
describe the signatures of phonon hydrodynamics via the competitive effects between grain size and
temperature. A characteristic non-local length, associated with phonon hydrodynamics and a heat
wave propagation length has been extracted. The connections between phonon hydrodynamics and
these length scales are discussed in terms of the Knudsen number. Further, the scaling relation
of thermal conductivity as a function of characteristic size in the intermediate size range emerges
as a crucial indicator of the strength of the hydrodynamic behavior. A ratio concerning normal
and resistive scattering rates has been employed to understand these different scaling relations,
which seems to control the strength and prominent visibility of the collective phonon transport in
GeTe. This systematic investigation emphasizes the importance of the competitive effects between
temperature and characteristic size on phonon hydrodynamics in GeTe, which can lead to a better
understanding of the generic behavior and consequences of the phonon hydrodynamics and its
controlling parameters in low-thermal conductivity materials.
I. INTRODUCTION
Detecting phonon hydrodynamics and associated col-
lective phonon transport in low-thermal conductivity
materials is a challenging task due to the not-so-
overwhelming differences between the normal and the
resistive phonon scattering rates. This leads to the ex-
ploration of very low cryogenic temperatures to see a vis-
ible effect of collective motion of phonons. On the other
hand, 2D materials draw an appreciable amount of stud-
ies [1–3] concerning phonon hydrodynamics because of
their enhanced normal scattering phenomena. This helps
in realizing phonon hydrodynamics even at higher tem-
peratures and therefore can be understood using exper-
iments. Nevertheless, collective phonon transport holds
fundamental interest in materials as it draws parallel to
the hydrodynamic flow in fluids. Investigating this collec-
tive phonon transport in low-thermal conductivity mate-
rials is crucial to understand the role of different compet-
ing effects that influences phonon hydrodynamics and in-
vokes fundamental question on its generality and validity
in both high and low conductivity materials. The com-
plete understanding of the origin of this phenomena thus
demands a systematic decoupling between various con-
trolling parameters that dictate phonon hydrodynamics
in materials.
Phonon hydrodynamics is a heat transport phenom-
ena where the collective flow of phonons dominate the
heat conduction in materials [1, 4–8]. This is enabled by
significantly higher momentum conserving normal scat-
tering (N) events compared to other dissipative scatter-
ing events [Umklapp (U), isotope (I) and boundary scat-
∗kanka.ghosh@u-bordeaux.fr
tering (B)], favoring damped wave propagation of tem-
perature fluctuations [9, 10]. In their consecutive two
pioneering theoretical works [11, 12] published in 1966,
Guyer and Krumhansl distinguished the phonon hydro-
dynamics for nonmetallic crystals using the comparison
between normal and resistive average scattering rates.
Phonon hydrodynamics have also been realized by the
deviation from Fourier’s law at certain length and time
scales [2, 13]. The concept of the kinetic theory of re-
laxons to characterize phonon hydrodynamics have been
introduced by Cepellotti and Marzari [14]. Very recently,
Sendra et al. [15] introduced a framework to use hydrody-
namic heat equations from phonon Boltzmann equation
to study the hydrodynamic effects in semiconductors.
Experiments and theoretical investigations over the
years suggest that only few and mostly two-dimensional
(2D) materials possess phonon hydrodynamics [2, 3, 5,
16, 17]. Some of these 2D materials like graphene and
boron nitride [1] can even persist phonon hydrodynam-
ics at room temperature due to the presence of strong
normal scattering realized via first-principles simulations.
Recently, the relation between the thickness and ther-
mal conductivity and consequently their connection to
the phonon hydrodynamics was studied for graphite [18].
The presence of second sound, a prominent manifestation
of phonon hydrodynamics, was also observed in graphite
at a temperature higher than 100 K via the experiments
carried out by Huberman et al. [19]. This validates the
predictions of the simulation studies done by Ding et al.
[20] on graphite. Similarly, theoretical evaluations by
Markov et al. [9] confirmed the experimental observation
[21] of hydrodynamic Poiseuille phonon flow in bismuth
(Bi) at low temperature. A faster than T3scaling of the
lattice thermal conductivity was described as a marker
to identify phonon hydrodynamics in bulk black phos-
arXiv:2107.04731v1 [cond-mat.mtrl-sci] 10 Jul 2021

2
phorus [8] and SrTiO3[22, 23]. Koreeda et al. [24] stud-
ied collective phonon transport in KTaO3using low fre-
quency light-scattering and time-domain light-scattering
techniques and phonon hydrodynamics was found to ex-
ist below 30 K . Further second sound was also observed
in solid helium (0.6 - 1 K) [25], NaF (∼15 K) [26] at low
temperatures.
As discussed earlier, the studies of phonon hydrody-
namics for low-thermal conductivity materials are sub-
stantially less compared to its high-thermal conductivity
counterpart. However, a systematic decoupling of vari-
ous controlling parameters can help manipulate phonon
hydrodynamic behavior in the low-thermal conductivity
materials. Torres et al. [27] showed a strong phonon
hydrodynamic behavior in low-lattice thermal conduc-
tivity (κL) materials such as single layer transition metal
dichalcogenides (MoS2, MoSe2, WS2and WSe2). In our
earlier paper [28], we investigated the low temperature
thermal transport in crystalline GeTe, a chalcogenide-
based material of diverse practical interests [29, 30],
which shows even lower lattice thermal conductivity com-
pared to metal dichalcogenides and found that it exhibits
phonon hydrodynamics. However we found that the pres-
ence of hydrodynamic phonon transport in crystalline
GeTe is sensitive to the grain size and vacancies present
in the material. Further, temperature was found to play
an important role in favoring appreciable normal scatter-
ing events to enable collective phonon transport.
For low-thermal conductivity materials like GeTe, the
characteristic size of the material and temperature are
two crucial parameters that influence the existence of
phonon hydrodynamics. Distinguishing the competing
effects of these two factors is important for general under-
standing of collective phonon transport in GeTe. There-
fore, in the current paper, we investigate the effects of
characteristic size (L) on the collective thermal trans-
port in low-thermal conductivity crystalline GeTe for
temperatures ranging from 4 K to around 500 K. We
use first-principles calculations with a kinetic collective
model approach [31] for this paper. We first identify the
L-regimes corresponding to ballistic and complete diffu-
sive regimes. Then we explore the regime of collective
phonon transport that comprises both ballistic and dif-
fusive phonons. Average scattering rates have been used
to identify phonon hydrodynamic regimes both in terms
of temperature and characteristic size. Further, temper-
ature and L-regimes are quantified using the Knudsen
number obtained using two different length scales con-
cerning phonon hydrodynamics. The prominent signa-
ture of phonon hydrodynamics in GeTe is found to de-
pend on the scaling exponent of thermal conductivity
as a function of Lin the intermediate L-regime where
phonon transport shifts from ballistic to complete diffu-
sive. The ratio of normal to resistive scattering rates at
this L-regime seems to dictate the strength of the hydro-
dynamic behavior.
II. COMPUTATIONAL DETAILS
First-principles density functional methods are em-
ployed to optimize the structural parameters of crys-
talline GeTe (space group R3m). The details of the pa-
rameters for GeTe can be found in our earlier paper [32].
The phonon lifetime is calculated using PHONO3PY [33]
software package. The supercell approach with finite dis-
placement of 0.03 ˚
A is employed to obtain the harmonic
(second order) and the anharmonic (third order) force
constants, given via
Φαβ(lκ, l0κ0) = ∂2Φ
∂uα(lκ)∂uβ(l0κ0)(1)
and
Φαβγ (lκ, l0κ0, l00κ00 ) = ∂3Φ
∂uα(lκ)∂uβ(l0κ0)∂uγ(l00κ00 )(2)
respectively. Density functional method is implemented
with QUANTUM-ESPRESSO [34] to calculate the forces
acting on atoms in supercells. The harmonic force con-
stants are approximated as [33]
Φαβ(lκ, l0κ0)' − Fβ[l0κ0;u(lκ)]
uα(lκ)(3)
where F[l0κ0;u(lκ)] is atomic force computed at r(l0κ0)
with an atomic displacement u(lκ) in a supercell. Simi-
larly, third order force constants are calculated using[33]
Φαβγ (lκ, l0κ0, l00κ00 )' −Fγ[l00κ00 ;u(lκ),u(l0κ0)]
uα(lκ)uβ(l0κ0)(4)
where F[l00κ00 ;u(lκ), u(l0κ0)] is the atomic force com-
puted at r(l00 κ00) with a pair of atomic displacements
u(lκ) and u(l0κ0) in a supercell. These two sets of linear
equations are solved using the Moore-Penrose pseudoin-
verse as is implemented in PHONO3PY [33].
Using the 2×2×2 supercell and finite displacement
method, we obtain 228 supercell configurations with dif-
ferent pairs of displaced atoms, for the calculations of the
anharmonic force constants. A larger 3×3×3 supercell is
employed for the harmonic force constants calculation.
For force calculations, the reciprocal space is sampled
with a 3×3×3 k-sampling Monkhorst-Pack (MP) mesh
[35] shifted by a half-grid distances along all three direc-
tions from the Γ- point. For the density functional cal-
culations, the Perdew-Burke-Ernzerhof (PBE) [36] gen-
eralized gradient approximation (GGA) is used as the
exchange-correlation functional. Due to its negligible ef-
fects on the vibrational features of GeTe, as mentioned in
earlier studies [37, 38], the spin-orbit interaction has been
ignored. Electron-ion interactions are represented by
pseudopotentials using the framework of the projector-
augmented-wave (PAW) method [39]. The Kohn-Sham
(KS) orbitals are expanded in a plane-wave (PW) basis
with a kinetic cutoff of 60 Ry and a charge density cut-
off of 240 Ry as specified by the pseudopotentials of Ge

3
and Te. The total energy convergence threshold has been
kept at 10−10 a.u. for supercell calculations. The imag-
inary part of the self-energy has been calculated using
the tetrahedron method from which phonon lifetimes are
obtained.
III. LATTICE DYNAMICS AND KINETIC
COLLECTIVE MODEL (KCM)
In the theory of lattice dynamics, the crystal poten-
tial is expanded with respect to atomic displacements
and the third-order coefficients associated with anhar-
monicity are used to calculate the imaginary part of the
self-energy [33]. Generally, in a harmonic approximation,
phonon lifetimes are infinite whereas, anharmonicity in
a crystal yields a phonon self-energy ∆ωλ+iΓλ. The
phonon lifetime (τph−ph) has been computed from the
imaginary part of the phonon self energy as τλ=1
2Γλ(ωλ)
using PHONO3PY [33, 40] from the following equation
Γλ(ωλ) = 18π
~2X
λ0λ00
∆ (q+q0+q00)|Φ−λλ0λ00 |2{(nλ0+nλ00 +1)δ(ω−ωλ0−ωλ00 )+(nλ0−nλ00 )[δ(ω+ωλ0−ωλ00 )−δ(ω−ωλ0+ωλ00 )]}
(5)
where nλ=1
exp(~ωλ/kBT)−1is the phonon occupation
number at the equilibrium. ∆ (q+q0+q00 ) = 1 if
q+q0+q00 =G, or 0 otherwise. Here Grepresents recip-
rocal lattice vector. Integration over q-point triplets for
the calculation is made separately for normal (G= 0) and
umklapp processes (G6= 0) and therefore phonon umk-
lapp (τU) and phonon normal lifetime (τN) have been
distinguished. Using second-order perturbation theory,
the scattering of phonon modes by randomly distributed
isotopes (τ−1
I) is given by Tamura [41] as
1
τI
λ(ω)=πω2
λ
2NPλ0δ(ω−ω0
λ)Pkgk|PαWα(k, λ)W∗
α(k, λ)|2
(6)
where gkis the mass variance parameter, defined as
gk=X
i
fi1−mik
mk2
(7)
fiis the mole fraction, mik is the relative atomic mass of
ith isotope, mkis the average mass = Pifimik, and Wis
a polarization vector. The database of the natural abun-
dance data for elements [42] is used for the mass variance
parameters. The phonon-boundary scattering has been
implemented using Casimir diffuse boundary scattering
[43] as τB
λ=L
|vλ|, where, vλis the average phonon group
velocity of phonon mode λand Lis the grain size, which
is also called Casimir length, the length phonons travel
before the boundary absorption or re-emission [43].
We use the kinetic collective model (KCM) [31] to ob-
tain the lattice thermal conductivity of GeTe. The KCM
method has emerged as a useful approach to depict heat
transport at all length scales with the computational cost
being substantially less than that of the full solution of
the linearized Boltzmann transport equation. According
to the KCM method, the heat transfer process occurs
via both collective phonon modes, emerges from the nor-
mal scattering events and via independent phonon col-
lisions. Therefore, lattice thermal conductivity can be
expressed as a sum of both kinetic and collective contri-
butions weighed by a switching factor (Σ ∈[0,1]), which
indicates the relative weight of normal and resistive scat-
tering processes [27, 31]. While each mode exhibits indi-
vidual phonon relaxation time in the kinetic contribution,
the collective contribution is designated by an identical
relaxation time for all modes [31, 44]. In the kinetic con-
tribution term, the boundary scattering is included via
the Matthiessen’s rule as
τ−1
k=τ−1
U+τ−1
I+τ−1
B(8)
where τkis the total kinetic phonon relaxation time. On
the contrary, a form factor F, calculated from the sample
geometry, is employed to incorporate boundary scatter-
ing in the collective term [31, 44]. The KCM equations
are:
κL=κk+κc(9)
κk= (1 −Σ) Z~ω∂f
∂T v2τkDdω(10)
κc= (ΣF)Z~ω∂f
∂T v2τcDdω(11)
Σ = 1
1 + hτNi
hτRBi
(12)
where κkand κcare kinetic and collective contribu-
tions to κL, respectively. hτNiand hτRB idesignate aver-
age normal phonon lifetime and average resistive (consid-
ering U,I, and B) phonon lifetimes, respectively. hτNi
and hτRB iare defined in the KCM [31] as integrated
mean-free times,
hτRB i=RC1τkdω
RC1dω (13)

4
and
hτNi=RC0τNdω
RC0dω (14)
where τkis the total kinetic relaxation time and phonon
distribution function in the momentum space, repre-
sented in terms of Ci=0,1(ω), defined in Ref. [31] as
Ci(ω) = v|q|
ω2i
~ω∂f
∂T D(15)
where v(ω) is the phonon mode velocity and |q|is mod-
ulus wave vector. C0represents the specific heat of mode
ω.fstands for Bose-Einstein distribution function, vis
mode velocity and D(ω) is phonon density of states for
each mode. τcdenotes the total collective phonon relax-
ation time and defined as
τc(T) = RC1dω
R(τ−1
I+τ−1
U)C1dω (16)
Σ stands for the switching factor. Fis the form factor
approximated via [44]
F(Lef f ) = L2
eff
2π2l2 s1 + 4π2l2
L2
eff
−1!(17)
where, Leff is the effective length of the sample (in our
system, we use Leff =L, the grain size) and lis the
characteristic non-local scale [11, 44]. This characteristic
non-local length lemerges from the complete hydrody-
namic description of the KCM and is defined as a pa-
rameter that determines the non-local range in phonon
transport. In our earlier paper [28], comparing the results
for thermal conductivity obtained using both direct solu-
tions of linearized Boltzmann transport equation (LBTE)
and KCM for GeTe, we found an excellent agreement be-
tween them at low temperature. At higher temperatures,
a reasonable matching trend is retrieved, with KCM ex-
hibiting slightly lower values than the LBTE solutions.
However, in the low temperature hydrodynamic regime
for GeTe, the solutions of LBTE and KCM collapse sat-
isfactorily. For all KCM [31] calculations of lattice ther-
mal conductivity and associated parameters, KCM.PY
code [31] is implemented with the outputs obtained us-
ing PHONO3PY [33].
IV. RESULTS AND DISCUSSIONS
A. Ballistic and diffusive phonon transport
As a first step to elucidating the complex collective
behavior of phonons as a function of characteristic size
(L), it is imperative to explore the variation of κLwith
Land therefore to identify the effect of Lon the ballistic
and diffusive phonon transport. Figure 1 describes this
FIG. 1. The variation of lattice thermal conductivity (κL) as
a function of characteristic length (L) of the GeTe sample at
different temperatures.
variation of GeTe for a wide temperature range (4 -
500 K). As the Lvaries almost 106orders of magnitude
(from 0.001 µm to 5000 µm), κLundergoes a transition
from a linear variation of Lto a plateau-like regime, and
corresponds to complete ballistic and complete diffusive
transport respectively. As we gradually go from lower to
higher temperatures, the ballistic regime shrinks and the
diffusive regime starts growing. Also, the onset of diffu-
sive transport gradually seems to take place at lower
FIG. 2. The variation of lattice thermal conductivity (κL)
with temperature as a function of Lfor crystalline GeTe.

5
FIG. 3. The variation of (a) Lball and (b) Ldiff are represented as a function of temperature for crystalline GeTe. The insets
of (a) and (b) display the defining procedure of Lball and Ldiff respectively for a representative case of T= 10 K.
values of Las we increase the temperature. It is well
known in the literature [13, 45] that ballistic conduction
of phonons occurs without ph-ph scattering and displays
a linear variation with L, whereas diffusive conduction
of phonons manifests when scattered phonons carry the
heat. The effect of the characteristic size on κLcan also
be represented via the variation of κLwith temperatures
for different L, as shown in Fig 2. At higher tempera-
tures, it is well known [32] that κLdecreases with T, with
1/Tscaling due to the dominant umklapp scattering at
high temperatures. As temperature is lowered, gradually
κLattains a peak following a gradual decrement at very
low temperature. As we go towards higher L, the peaks
of κLas a function of temperature are gradually seen to
be shifted towards lower temperatures (Fig. 2).
The effect of Lon the temperature variation of κLgives
rise to an interesting feature as we increase Labove a
certain limit. It is known that Lplays a crucial role via
phonon-boundary scattering as gradual increment of L
assists in weakening the boundary scattering. This weak-
ening of boundary scattering and strong normal scat-
tering rates (to be discussed later) at low temperatures
transforms the peak of κLinto a cusp-like feature when
L≥1µm and κLis further seen to be increased at very
low temperatures.
To give a more precise account of ballistic and diffu-
sive conduction of phonons in GeTe, we further investi-
gate the characteristic size range of ballistic and diffusive
conduction as a function of temperature. The complete
ballistic length regime (Lball) is defined via the maxi-
mum value of L, until which κLvaries linearly with L.
Similarly, the complete diffusive length regime (Ldif f )
is defined via the minimum length L, above which κL
reaches the thermodynamic limit and therefore reaches
a plateau. In other words, Ldiff represents the longest
mean free path of the heat carriers at a particular tem-
perature [13]. Figures 3.(a) and 3.(b) represent the vari-
ations of Lball and Ldiff respectively, as a function of
temperature. As temperature increases, we see a grad-
ual decrement of both Lball and Ldif f . We note here
that at very high temperatures, we hardly observe any
ballistic conduction of phonons and the Ldiff acquires a
very low value. This is representative of the fact that at
high temperatures, internal phonon-phonon scattering is
so dominant that no ballistic heat conduction is seen to
exist, even for very small grains of the order of 1 nm.
To delve deeper into the origin of length dependent
κLin the ballistic phonon conduction regime of GeTe,
we investigate the contribution of acoustic and opti-
cal modes in the ballistic propagation of heat. Ear-
lier, molecular dynamics simulations and experiments on
suspended single-layer graphene [46, 47] suggested the
ballistic propagation of long-wavelength, low-frequency
acoustic phonon to be solely responsible for the length-
dependent κLin the ballistic regime. Our previous stud-
ies on GeTe [28, 32] suggested that GeTe shows a clear
distinction between acoustic and optical modes in the fre-
quency domain around 2.87 THz. The density of states
goes to zero around a frequency of 2.87 THz [32], distin-
guishing two distinct frequency regimes: acoustic regime
(ω < 2.87 THz) and optical regime (ω > 2.87 THz). We
calculate the cumulative lattice thermal

6
FIG. 4. The variation of cumulative lattice thermal conductivity (κc
L) of crystalline GeTe as a function of phonon frequency
(ω) for four different temperatures: (a) 10 K, (b) 30 K, (c) 50 K and (d) 300 K. For each temperature, the variation of κC
Lwith
ωis presented for three different L: 0.001, 0.002 and 0.003 µm. The gray shaded region denotes the acoustic modes regime for
GeTe.
conductivity (κc
L) as a function of phonon frequency de-
fined as [33, 40]
κc
L=Zω
0
κL(ω0)dω0(18)
where κL(ω0) is defined as [33, 40]
κL(ω0)≡1
NV0X
λ
Cλvλ⊗vλτλδ(ω0−ωλ) (19)
with 1
NPλδ(ω0−ωλ) the weighted density of states
(DOS). Figure 4 presents the variation of average κc
L
(= (κC
xx+κC
yy +κC
zz )
3) with phonon frequency. The density
of states goes to zero at a frequency where κc
Lreaches
a plateau defining the separation between acoustic (fre-
quency <2.87 THz) and optical (frequency >2.87 THz)
modes. Except at low temperature (T= 10 K), the con-
tribution from optical modes seem to present at all other
temperatures. As we gradually increase the temperature
[from Fig 4.(b) to Fig 4.(d)], the contributions from op-
tical modes are seen to be enhanced. For example, for L
= 0.003 µm, the contribution of optical modes at T= 10
K, 30 K, 50 K and 300 K are 0 %, 9.9 %, 24.2 %, and
37.7 % respectively. Therefore, contrary to the under-
standing of ballistic propagation for a 2D material like
single-layer graphene, except for very low temperatures,
GeTe also shows a weak contribution from optical modes
in the ballistic phonon propagation regime. However,
the significant contributions come from acoustic modes
in this regime.
To visualize the consequences on the mean-free path of
the phonons at small L, we present the variation of the
effective mean-free path variation with phonon frequency
for different Lat different temperatures in Fig. 5. In the
KCM nomenclature, the kinetic mean free path [lk(ω)]
and the collective mean free path [lc(T)] are defined as
lk(ω) = vτkand lc(T) = vτcrespectively, where vis the
group velocity and
v=Rv~ω∂f
∂T D(ω)dω
R~ω∂f
∂T D(ω)dω (20)
is the mean phonon velocity [31]. As the kinetic MFP

7
FIG. 5. Effective mean free path (MFP) of crystalline GeTe
are presented as a function of frequencies for three different
L: 0.001 µm (blue points), 0.002 µm (green points) and 0.003
µm (red points) at four different temperatures: (a) T= 10
K, (b) T= 30 K, (c) T= 50 K, (d) and T= 300 K. The gray
shaded region denotes the acoustic modes regime for GeTe.
is a function of phonon frequency whereas the collec-
tive MFP is frequency independent and varies only with
temperature, we present an effective MFP as leff (ω) =
(1-Σ)lk(ω)+Σ lc. The separate contributions from col-
lective and kinetic MFPs are described in Supplemental
Fig. S1. Two effects can be observed from this represen-
tation. First, at low temperature, as the Lis increased,
the optical modes at higher frequencies exhibit more scat-
tered mean-free paths. Figure 5.(a) shows that at T=
10 K, at higher frequencies in the optical modes, L=
0.003 µm persists more scattered MFPs compared to the
L= 0.001 µm case. This feature indicates that the bal-
listic conduction is stronger for L= 0.001 µm, where
Lstrongly controls the mean-free path than that of the
L= 0.003 µm case. Second, increasing temperature for
fixed L, also leads to the gradual weakening of the bal-
listic conduction of phonons, as can be seen from Fig.
5. This is evident from the gradual broadening of MFPs
with temperature [follow fixed color points for four dif-
ferent temperatures in Fig. 5.(a), (b), (c) and (d).] due
to the gradually weakening control of Lon dictating the
mean-free paths of the system.
B. Collective phonon transport
After understanding the effect of characteristic size (L)
on the ballistic and diffusive conduction of phonons, we
turn our attention to the effect of Lon the collective
phonon transport of crystalline GeTe. The connection
between ballistic and diffusive phonon transport and the
collective motion of phonons are crucial to determine
the origin of the exotic hydrodynamic phonon trans-
port in materials. In our earlier work [28], unusually,
low-thermal conductivity chalcogenide GeTe emerged as
a possible candidate to feature phonon hydrodynamics
with the characteristic size being a dominant factor.
FIG. 6. Thermodynamic average phonon scattering rates as a function of temperature for GeTe for different characteristic
sizes (L). N,U,Iand Rdenote normal, umklapp, isotope and resistive scattering respectively. Boundary scattering rates for
different Lare also presented. The shaded regions in (a) and (b) correspond to the validation of the Guyer’s condition [12] for
Poiseuille’s flow (Eq. 24) for L= 0.08 µm and L= 0.8 µm respectively.

8
FIG. 7. The spectral representation of lattice thermal conductivity (κL) as a function of phonon frequency at T= 10 K for
four different characteristic size or grain sizes (L): (a) 0.2 µm, (b) 0.5 µm, (c) 1 µm and (d) 5 µm. The kinetic contribution
(κkinetic) is defined using light violet and the collective contribution (κcollective) is defined using light brown color.
In this context, we start by investigating the relative
strengths of the average phonon scattering rates, which
is defined as
hτ−1
iiave =PλCλτ−1
λi
PλCλ
(21)
Here, λdenotes phonon modes (q,j) comprising wave
vector qand branch j. Index idenotes normal, umklapp,
isotope, and boundary scattering processes used, marked
by N, U, I, and B respectively. Cλis the modal heat
capacity, given by the following equation
Cλ=kB~ωλ
kBT2exp(~ωλ/kBT)
[exp(~ωλ/kBT)−1]2(22)
where, Tdenotes temperature, ~is the reduced Planck’s
constant, and kBis the Boltzmann constant. In one of
the earliest works on phonon hydrodynamics, Guyer and
Krumhansl [12] found that the hydrodynamic regime ex-
ists if
hτ−1
Uiave  hτ−1
Niave (23)
Further, Guyer’s condition [12] for the presence of second
sound and Poiseuille’s flow reads:
hτ−1
Uiave <hτ−1
Biave <hτ−1
Niave (24)
In Fig 6, we explore the Lwindow that satisfies
the aforementioned Guyer and Krumhansl condition of
phonon hydrodynamics in crystalline GeTe. Figure 6
presents the average scattering rates due to normal (N),
resistive (R) [comprised of umklapp (U) and isotope scat-
tering (I)] and the phonon-boundary scattering as a func-
tion of temperature for GeTe. We observe a substantial
width of L, that persists phonon hydrodynamic condi-
tions, as the boundary scattering rates decrease gradu-
ally on increasing L. This is shown via the gray shaded
regions in Figs. 6.(a) and 6.(b) for two representative

9
grain sizes: L= 0.08 µm and L= 0.8 µm, respec-
tively. In the scattering rate approach, we also iden-
tified the ballistic conduction region, mentioned earlier
through the linear dependence of κLwith L, as the re-
gion where hτ−1
Biave  hτ−1
ph−phiave . Similarly, the purely
diffusive conduction region, mentioned earlier as the L-
regime where κLis independent of L, as the region where
hτ−1
Biave  hτ−1
ph−phiave . At this point, we go back to Fig.
2 to explain the cusp-like behavior of κLas a function of
temperature. This cusp-like pattern of κLis found to
present for L > 1µm, as we gradually decrease the tem-
perature. In Fig. 6.(b), this Lregime is identified as L
values for which normal scattering overpowers boundary
scattering rates. At low temperatures, umklapp scatter-
ing is rare and boundary scattering acts as the dominant
resistive phonon scattering. So, the effect of boundary
scattering tries to reduce the κLwhile the momentum
conserving normal scattering tries to increase κL. Over-
powering normal scattering compared to boundary scat-
tering for L > 1µm forces κLto increase after an appar-
ent shallow dip or a plateau and gives rise to the cusp-like
pattern in Fig. 2.
Once the Guyer and Krumhansl conditions are satisfied
and a prominent Lwindow is observed to feature phonon
hydrodynamics, we next investigate the spectral repre-
sentation of lattice thermal conductivity (κL) in this L
window. In Fig. 7, using the KCM approach, we present
a spectral representation of κL, distinguished by its ki-
netic (κkinetic) and collective contributions (κcollective ),
as a function of phonon frequency at T= 10K for four
different L. We choose T= 10 K as a representative tem-
perature to feature collective transport of phonons. The
four different Lvalues have been chosen such that it cov-
ers a wide range that traverses from ballistic transport to
the hydrodynamic regime at T= 10 K. As we gradually
increase the L[from Figs. 7(a) to 7(d)), a gradual in-
crement of the contributions coming from the collective
transport is observed (shown via the red shaded regions
inside the curve). The spectral κLgoes to zero before
2.87 THz, indicating the sole contribution of acoustic
phonons in thermal transport at 10 K, as was realized
earlier in Fig. 4(a).
To quantify the collective motion as a function of tem-
perature for different L, we investigate the variation of
characteristic non-local length (l) in GeTe at different
temperatures and grain sizes. In a complete hydrody-
namic description of thermal transport, the extension of
the Guyer and Krumhansl equation [11] done in the KCM
framework [44], namely, the hydrodynamic KCM equa-
tion, yields
τdQ
dt +Q=−κ∇T+l2∇2Q+ 2∇∇ · Q(25)
where τis the total phonon relaxation time, Qis the
heat flux, κis phonon thermal conductivity, and lis the
non-local length, that determines the non-local range in
FIG. 8. The variation of Knudsen number (Kn) with temperature for different Lvalues of crystalline GeTe. The shaded region
satisfies 0.1 ≤Kn ≤10 while the rectangular boxes define phonon hydrodynamic regimes calculated from average scattering
rates. Blue dashed lines to guide the eye for T= 6 K, 10 K and 20 K.

10
phonon transport. If we employ the steady state,
strong geometric effects, and neglect the term 2∇∇ · Q,
then the equation can be visualized as analogous to
Navier-Stokes equation with lresembling heat viscos-
ity. The Knudsen number (Kn) can be obtained from
Kn =l/L to study the collective motion quantitatively.
Lower values of Kn define the recovery of Fourier’s law
whereas the hydrodynamic behavior becomes prominent
when Kn gets higher values [7, 44]. Figure 8 presents the
variation of Kn as a function of temperature for different
L. As temperature is lowered, a gradual increment of Kn
is observed, concomitant with the gradual prominence of
non-local behavior. Kn has earlier been described [7, 9]
to indicate a phonon hydrodynamic regime when 0.1 ≤
Kn ≤10, bearing similarities with fluid hydrodynamics.
We denote this region via a shaded region in Fig. 8. In
Fig. 8, we also superpose the hydrodynamic L-window,
identified using average scattering rates following Guyer
and Krumhansl conditions for three representative tem-
peratures: T= 6 K, 10 K and 20 K. We observe that both
definitions match well and the hydrodynamic L-window
obtained by scattering rate analysis falls within the Kn
range for hydrodynamics.
Knudsen number calculation also reveals the Ziman
hydrodynamic regime for GeTe. Looking at the verti-
cal dashed lines corresponding to T= 6 K and T= 10
K in Fig. 8, a small L-region is observed which does
not fall into the rectangles, defined to indicate a hydro-
dynamic regime using scattering rate hierarchy. How-
ever, they fall inside the regime of 0.1 <Kn <10, es-
pecially in the regime where Kn is close to 0.1. This
corresponds to the Ziman hydrodynamic regime which
corresponds to a regime where N scattering dominates
but dissipates mostly by R scattering contrary to the
Poiseuille hydrodynamic regime where N scattering dissi-
pates mostly by the boundary scattering of the phonons.
On the other hand, looking at Lvalues that lie inside
0.1 <Kn <10 but with values close to 10, also some-
times do not lie inside the rectangular region (see the
case of L= 0.04 and 0.1 µm at T= 10 K in Fig 8).
Recalling Fig 6.(a), we observe that L= 0.04 µm at
T= 10 K designates a scattering rate hierarchy, where
hτ−1
Biave >hτ−1
Niave >hτ−1
Riave, but hτ−1
Biave is not 
hτ−1
ph−phiave . Therefore, though it follows the prescribed
hierarchy for hydrodynamics, the Lvalues do not enable
a complete ballistic propagation and a competition be-
tween ballistic and diffusive phonons operates. This com-
petition makes it difficult to distinguish sharp boundaries
between different regimes. We will discuss more about
this competition later. To characterize the repopulation
of phonons in a different way, following Markov et al.
[9], we extract a length scale related to the propagation
of heat wave before being dissipated, called the heat wave
propagation length (Lhwpl), defined as a length at which
the completely diffusive thermal conductivity decays 1/e
times:
κL(T, L)|L=Lhwpl =κL(T, L > Ldif f )/e (26)
FIG. 9. Heat wave propagation length (Lhwpl) as a function
of temperature for crystalline GeTe. The temperature varia-
tion of phonon propagation lengths, correspond to the damp-
ing due to resistive scattering (λhydro) and both resistive and
normal scattering (λgas) along aand hexagonal c-axis are also
presented.
where Ldiff is the minimum length L, above which κL
reaches the thermodynamic limit, as mentioned earlier
in Fig. 3. Lhwpl is connected to second sound, a typi-
cal characteristic for hydrodynamic heat transport phe-
nomenon, which demonstrates the heat propagation as
damped waves in a crystal [1, 12, 48] as a result of coher-
ent collective motion of phonons due to the domination
of N scattering. In this context, drift velocity of phonons
(v) and phonon propagation length (λph) are defined as
v2
j=PαCαvg
αj ·vg
αj
PαCα
(27)
and
λph =v/hτ−1iave (28)
where, Cαis heat capacity of mode α,vg
αj is phonon
group velocity of mode αand jcan be either the compo-
nent along the aaxis (x) or the hexagonal caxis (z). Heat
transfer of GeTe is anisotropic, as can be recalled from
our earlier studies [28, 32], featuring different group ve-
locities along the hexagonal caxis and its perpendicular
(aaxis) direction and therefore yields different drift ve-
locities and different phonon propagation lengths along
xand z. Figure 9 presents the variation of heat wave
propagation length (Lhwpl) with temperature along with
the variation of phonon propagation lengths along xand
z. Phonon propagation lengths are distinguished [9] as
λhydro and λgas via
λhydro =v/hτ−1
Riave (29)
λgas =v/ hτ−1
Riave +hτ−1
Niave(30)

11
Figure 9 shows the variation of heat wave propagation
length (Lhwpl), superimposed with phonon propagation
lengths with temperature along both aand caxis direc-
tions of GeTe. We observe that Lhwpl follows well the
trend of λhydro as a function of temperature in the whole
temperature range studied. λgas, the phonon propaga-
tion length corresponds to the uncorrelated phonon gas
where both N and R scattering processes contribute to
the damping of heat wave, on the other hand, seems to
diverge from Lhwpl as the temperature is lowered. This
feature is an indication of gradual prominence of hydro-
dynamic behavior of phonons as the temperature is low-
ered. Similarly, the reasonable match between Lhwpl and
λhydro predicts that heat wave propagation length is well
captured by phonon flow with resistive damping caused
by umklapp and isotope scattering. At very low temper-
ature (T= 4 K), a slight deviation is observed between
Lhwpl and λhydro which can be attributed to the impor-
tance of boundary scattering as a significant damping
process at very low temperature.
Therefore, Lhwpl can lead to the identification of the
length scale at different temperatures at which phonon
hydrodynamics can exist and therefore Poiseuille’s flow
and second sound phenomena can be observed. Inter-
estingly, comparing Lhw pl and characteristic size (L) of
the sample, we can define Knudsen number in another
approach as [9] Kn = Lhwpl/L. The variation of Kn
obtained using Lhwpl, is presented as a function of tem-
perature in the Supplemental Material (Fig. S2). The
variation of Kn with Tis found similar to our earlier
evaluation of Kn using nonlocal length (Fig. 8).
The blurry regions of transitions between ballistic, hy-
drodynamic, and diffusive transport are intriguing to un-
derstand the competition between different phonons with
a wide range of mean free paths. Ideally, phonons with
a wide spectrum of mean-free paths can be distinguished
as either ballistic (MFP > L) or diffusive (MFP < L)
phonons. However, the relative strength between ballis-
tic and diffusive phonons are crucial to realize the com-
petition between these two kind of phonons which even-
tually plays a decisive role in dictating the visible hydro-
dynamic phenomena. The phonon Knudsen minimum
is such an indicator for the transition between ballis-
tic and hydrodynamic phonon propagation regimes and
had been used for several materials including graphene
[16], graphite [20], SrTiO3[23], black phosphorus [18] to
detect phonon hydrodynamics. Figures 10.(a) and (b)
present the the variation of normalized thermal conduc-
tivity (κ∗
L=κL/L), a quantity that is similar to dimen-
sionless κL, as a function of inverse Knudsen number,
calculated using nonlocal length and heat wave propa-
gation lengths respectively. Figure 10.(b) shows a wider
range of 1/Kn as the Kn obtained using heat wave prop-
agation length reaches higher values at low temperatures
compared to that of the non-local length calculation from
hydrodynamic KCM method. However, we observe al-
most similar trends of κ∗
Lwith the variation of 1/Kn
coming out of the two different approaches in obtain-
ing the Knudsen number. At T= 300 K, a steep lin-
ear decreasing trend is observed which is associated with
the diffusive phonon scattering events as phonons behave
as uncorrelated gas particles and resistive scattering is
prominent and dominating at this temperature.
FIG. 10. (a) The variation of normalized thermal conductivity (κL/L) as a function of inverse Knudsen number, calculated
using characteristic non-local length for different temperatures. (b) The variation of normalized thermal conductivity (κL/L)
as a function of inverse Knudsen number, calculated using heat wave propagation length for different temperatures.

12
FIG. 11. The variation of lattice thermal conductivity (κL) as a function of characteristic length (L) in log-log scale for different
temperatures: (a) T= 4 K, (b) T= 6 K and (c) T= 10 K. The inset of Fig 11.(b) refers to the zoomed in view around linear
to superlinear scaling at T= 6 K. The intermediate regimes, located in between the ballistic and diffusive propagation regimes
are shown via gray shades.
Starting from T= 20 K, a gradual onset of a horizon-
tal regime is visible before the linearly decreasing trend
of κ∗
Las the temperature is lowered. At T= 4 K, sur-
prisingly, a cusp-like trend, resembling that of a shallow
minimum followed by a prominent maximum is observed
before a linearly decreasing κ∗
Lat higher 1/Kn. The cusp-
like shallow minimum at T= 4 K indicates the phonon
Knudsen minimum and predicts the presence of promi-
nent transition from ballistic to hydrodynamic regime.
Further, a prominent maximum in κ∗
Lhas only been ob-
served at T= 4 K, which designates the strong presence
of hydrodynamic phonon transport in GeTe. Similar ob-
servation can be found by Li et al. [16] for suspended
graphene, where the increasing trend of κL, normalized
by sample width, was attributed to the strong presence
of hydrodynamic phonon transport.
The behavior of phonon Knudsen minimum of GeTe
convinces us to understand the competition between bal-
listic and diffusive phonons in the quasi-ballistic regimes
of phonon transport. We specifically turn our attention
toward the reason behind the strong presence of hydro-
dynamics at T= 4 K visible through Knudsen minimum
in Fig 10. We recall that even T= 6 K, T= 8 K persist
in phonon hydrodynamics, realized via the average scat-
tering rate analysis and Knudsen number variation with
temperature. To perceive the reason behind the differ-
ence between strong and weak phonon hydrodynamics,
we investigate the scaling relation between κLand Lin
the intermediate regime of transport, where the transport
is neither fully ballistic nor fully diffusive.
Figure 11 describes the variation of κLwith Lat T=
4 K, 6 K, and 10 K. Three phonon propagation regimes
have been identified. At lower values of L, ballistic
phonons dominate the transport and therefore a linear
dependency of κLon Lis observed. At very high L, the
phonon transport is completely diffusive and a plateau-
like regime is observed, denoting an independence of κL
over L. The intermediate regime where the phonon prop-
agation shifts from complete ballistic to complete diffu-
sive, plays a crucial role in determining the strong or
weak presence of hydrodynamic propagation of phonons.
Figure 11.(c) indicates a sublinear variation in the inter-
mediate regime at T= 10 K. At T= 6 K [Fig. 11(b)],
a minute superlinear behavior is observed while at T=
4 K [Fig. 11(a)], an enhanced superlinear behavior is
perceived in the intermediate regime.
In the intermediate quasi-ballistic regime of phonon
propagation, where both ballistic and diffusive phonons
FIG. 12. Variation of the scaling exponent αas a function of
Lfor different temperatures. The black dashed line denotes
the α= 1 line.

13
operate and compete with each other, seems to be a
marker to designate sample sizes (L) with strong hy-
drodynamic phonon transport characteristics. To fur-
ther quantify the intermediate nonliearity (both sub
and superlinearity), we evaluate and present the scaling
exponent[20] α=∂log(κL)/∂log(L) as a function of L
for different temperatures in Fig. 12.
α= 0 indicates the size-independent behavior of κL
and therefore describes the completely diffusive phonon
propagation regime. On the other hand, α= 1 reveals
the linear size dependency and henceforth the complete
ballistic phonon conduction regime. The superlinear de-
pendence of κLon Lin the intermediate regime can be
captured by the the condition α > 1. From Fig. 12 we
observe that at low L, for low temperatures, αgoes to 1.
for higher temperatures, as expected almost no ballistic
regime is observed with α < 1. As we increase L, in the
intermediate regime, a gradual departure from α= 1 is
observed. For T= 4 K and T= 6 K, this departure leads
to a regime with α > 1, while for T= 8K and 10 K this
deviation leads to sublinear or α < 1 scaling. At high L
values gradually all phonons become diffusive and αgoes
to zero.
There are several striking features to point out from
Fig. 12. First, prominent contribution of drifting
phonons at 4 K leads to an enhanced superlinear scal-
ing with α > 1, representing the signature of phonon
Poiseuille flow [20] and therefore prominent phonon hy-
drodynamics which assists in featuring the Knudsen min-
imum seen in Fig. 10. Here we mention that even for T
= 4 K, the exponent αgradually starts from 1, reaches a
maximum value around L= 0.8 µm, and goes sublinear
with α < 1 thereafter before going to zero at very high
Lvalues. Therefore, sublinear scaling is universal in the
intermediate regime. For T= 4 K, however, the sub-
linear scaling precedes a superlinear behavior displaying
strong hydrodynamic feature. Second, a minute super-
linear scaling, observed in Fig. 11(b) inset for T= 6 K,
can be realized in a better way by observing the small
L-window for which α > 1 for T= 6 K. At T= 8K and
10 K, though sublinear scaling is observed in the inter-
mediate regime, it decays to zero in different rates. After
L= 10 µm, the decay rate seems faster than that of the
cases below L= 10 µm.
We understand that although average scattering rate
and Knudsen number variation with temperature indi-
cates phonon hydrodynamics to present in GeTe for sev-
eral temperature and characteristic size window, low-κL
material GeTe needs several factors to manifest a strong
hydrodynamic response by phonons. In this context, su-
perlinear scaling of κLon Lplays a crucial role in the
transition from complete ballistic to complete diffusive
propagation regimes. To understand and investigate the
reason behind superlinear and sublinear scaling at the
intermediate quasi-ballistic regime of phonon transport,
we calculate the ratio γas a function of Lfor three tem-
peratures: T= 4 K, 6 K and 10 K. We define γas
γ=τ−1
N
τ−1
R+τ−1
B(31)
where τ−1
N,τ−1
R, and τ−1
Bare average scattering
FIG. 13. (a) Ldependence of γfor three representative temperatures: T= 4 K, 6 K and 10 K. The saturation values of γ
(γdiff ) in the plateau regimes attained at higher Lvalues for different temperatures are denoted via blue dashed lines. The
red dotted line represents γ= 1 and the differences between γ= 1 and γdiff are shown via black double headed arrows. (b)
The variation of ∂log(γ)/∂log(L) as a function of Lfor three representative temperatures: T= 4 K, 6 K and 10 K.

14
rates for normal, resistive, and boundary scattering re-
spectively. Figure 13.(a) shows that γincreases gradually
and reaches a plateau as we increase L. In the ballistic
phonon conduction regime, we observe γ(T= 4K)<
γ(T= 6K)< γ(T= 10K). This is due to the effect of
strong boundary scattering at low temperature and low
L. However, in the regime of complete diffusive propaga-
tion of phonons a reverse trend is observed: γ(T= 4K)
> γ(T= 6K)> γ(T= 10K) as in this regime, γis
independent of size. We define these saturation values
as γdif f . Again, the crucial crossover is observed in the
intermediate L-regime. We also mark the difference be-
tween γ= 1 and γdiff in Fig. 13(a) via double headed ar-
rows. This difference characterizes the relative strength
of normal scattering compared to the dissipative resis-
tive scattering and therefore indicates the strength for
persisting coherent phonon flow.
However, we tend to understand the reason behind the
nonlinear behavior of κLat the intermediate L-regime.
For this purpose, we present the variation of the expo-
nent of γby calculating ∂log(γ)/∂log(L) as a function of
Lin Fig. 13(b). We observe that the exponent for T= 4
K is higher and for both T= 4 K and 6 K, it stays around
1 (γbeing linearly increasing with L) in the intermediate
regime. However, for T= 10 K, ∂log(γ)/∂log(L) drops
up to several orders (at L= 10 µm, it drops around 10
times) compared to the T= 4 K and T= 6 K cases.
Therefore, the higher the exponent ∂log(γ)/∂log(L) in
the intermediate regime and closer to 1, the higher the
chances of the collective phonon flow due to strong nor-
mal scattering. This eventually can lead to the strong
appearance of phonon hydrodynamics with superlinear
Ldependence of κLand prominent Knudsen minimum
apart from other signatures born out of the assessment of
Knudsen number and average scattering rate as a func-
tion of temperature.
V. SUMMARY AND CONCLUSIONS
We employ KCM in conjunction with first-principles
density functional calculations to investigate the effect
of characteristic size (L) on collective phonon transport
in low-thermal conductivity material GeTe. We observe
phonon hydrodynamics in crystalline GeTe and identify
the competitive effects of both temperature and Lon the
collective phonon transport. As a first step, we distin-
guish heat transport regimes correspond to ballistic and
completely diffusive phonon transport. These regimes
have been identified as a function of both temperature
and L. In the ballistic regime, the frequency dependence
of phonon propagation is understood. Temperature is
found to dominate over Lin deciding the excitation of
acoustic and optical phonons. Even for very small L
values, correspond to ballistic transport regime, we ob-
serve a small contribution coming from optical modes of
GeTe if temperature is raised to 30 K. However, at low
temperature (T= 10 K), only acoustic modes excite to
enable ballistic propagation. The variation of mean free
paths as a function of frequencies also represents this
dependence. At low temperatures, increasing Lgradu-
ally weakens ballistic conduction. On the other hand, for
the same Lvalue, increasing temperature also gradually
weakens the ballistic conduction.
After looking at ballistic and diffusive phonon conduc-
tion regimes, we turn our attention toward the intrigu-
ing intermediate L-regime where both ballistic and dif-
fusive phonons are present. The average scattering rates
seem to follow the Guyer and Krumhansl hierarchy at low
temperatures, indicating the presence of phonon hydro-
dynamics at certain temperatures and Lwindow. KCM
method allows us to distinguish the variation of collec-
tive contribution as functions of both temperature and L.
Therefore, the phonon hydrodynamic regimes in terms of
both temperature and Lhave been realized using non-
local length and Knudsen number (Kn) evaluation which
draws a parallel between fluid hydrodynamics and the
collective flow of phonons. The hydrodynamic regimes
identified using scattering rates are found to satisfy the
condition 0.1 <Kn <10, which is the condition for hy-
drodynamic flow in terms of Kn. Further, exploiting the
variation of lattice thermal conductivity as a function of
L, a heat wave propagation length has been extracted
for different temperatures. Comparing this characteris-
tic length scale with phonon propagation lengths reveals
that the heat wave propagation length is well captured
by phonon propagation with only resistive damping. the
Knudsen number can also be associated with this length
scale which shows almost similar behavior as that of the
Kn obtained using a nonlocal length. For both of these
definitions of Kn, the variation of normalized thermal
conductivity (κ∗
L=κL/L) with 1/Kn shows a Knudsen
minimum like feature only at very low temperature (T=
4 K). Though Kn can capture the hydrodynamic regimes
well in terms of both temperature and L, some of the
prominent features of phonon hydrodynamics, like Knud-
sen minimum, can be weakly present or may be absent
in low-thermal conductivity materials. We have found
that the intermediate L-regime and the scaling of ther-
mal conductivity with Lin this regime works as a marker
to determine the existence of the Knudsen minimumlike
prominent hydrodynamic feature. A superlinear scaling
in this intermediate L-regime seems to assist a Knudsen
minimum and therefore prominent phonon hydrodynam-
ics. On the other hand, sublinear scaling does not lead
to a Knudsen like minimum, featuring weak phonon hy-
drodynamics at those temperatures. A ratio of average
normal and resistive scattering rates have been found to
control the strength and prominent visibility of the col-
lective phonon transport in GeTe.
In summary, this paper reveals crucial details about
how and when the prominent signatures of phonon hydro-
dynamics can be observed in low-thermal conductivity
materials. In this context, it demonstrates and systemat-
ically analyzes the consequences of the competitive effects
between temperature and characteristic size on phonon

15
hydrodynamics in GeTe. The outcome of this study can
lead to a better understanding of the generic behavior
and consequences of the phonon hydrodynamics and its
controlling parameters in any other low-thermal conduc-
tivity materials. The accurate description of phonon hy-
drodynamics in low-κmaterials can also lead to better
theoretical predictions of experimentally observed ther-
mal conductivity at low temperatures for these materials.
ACKNOWLEDGMENTS
This project has received funding from the Euro-
pean Union’s Horizon 2020 research and innovation pro-
gram under Grant Agreement No. 824957 (“Before-
Hand:” Boosting Performance of Phase Change Devices
by Hetero- and Nanostructure Material Design).
VI. SUPPLEMENTARY MATERIAL
A. Collective and Kinetic mean free path
Figure S1 presents the variation of effective mean free
paths of GeTe as a function of phonon frequency for two
different Lvalues at T= 10 K. In KCM approach, mean
free paths can be distinguished as kinetic and collective
mean free paths. As described in the main text, the
kinetic mean free paths (lk(ω)) are different for differ-
ent modes but the collective mean free paths (lc(T)) are
same for all modes and it is only a function of tempera-
ture. The effective mean free path has been realized using
lef f (ω) = (1-Σ)lk(ω)+Σ lc. In Fig S1.(a), we observe that
at T= 10 K, the dominating contribution comes from
the kinetic mean free path for L= 0.003 µm. We recall
from the main text that this Lcorresponds to ballistic
phonon conduction regime for GeTe at 10 K. However, L
= 0.8 µm satisfies the Guyer and Krumhansl condition
for phonon hydrodynamics and consistently the collec-
tive mean free path is seen to dominate over the kinetic
mean free path (Fig S1.(b)).
FIG. S1. Effective mean free path (MFP) of crystalline GeTe, along with collective and kinetic contributions are presented as
a function of frequencies for two different Lat T= 10 K: (a) L= 0.003 µm and (b) L= 0.8 µm. Gray shaded regions denote
acoustic mode frequency regime for GeTe.

16
B. Variation of Knudsen number with temperature
for different Lusing heat wave propagation length
Figure S2 presents the variation of Knudsen number
(Kn) as a function of temperature for different Lval-
ues. Kn has been realized via the heat wave propagation
length (Lhwpl) as Lhw pl/L, where Lhwpl is obtained as a
characteristic length at which the lattice thermal conduc-
tivity in the completely diffusive limit correspond to bulk
sample reduces to 1/e times. The hydrodynamic regime
follows 0.1 <Kn <10 and therefore has been marked
with the shaded region. The hydrodynamic L-regimes
obtained using average scattering rates are also shown
via rectangular boxes for T= 6 K, 10 K and 20 K. Except
the ballistic hydrodynamic boundary regimes for T= 6
K, the regimes evaluated by Kn and average scattering
rates are found to be consistent. The transition between
ballistic and hydrodynamic regimes are often found to
be blurry and without sharp demarcation in low-thermal
conductivity materials. This has been discussed in the
main text.
FIG. S2. The variation of Knudsen number (Kn) with temperature for different Lvalues of crystalline GeTe. The shaded region
satisfies 0.1 ≤Kn ≤10 while the rectangular boxes define phonon hydrodynamic regimes calculated from average scattering
rates. Blue dashed lines to guide the eye for T= 6 K, 10 K and 20 K.
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