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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

fi1−mik

mk2

(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|

ω2i

~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

kBT2exp(~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

Bare 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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