vast oqect marseille - cpt.univ-mrs.frcrepieux/stock/vast_oqect_marseille_final_w… · workshop «...

Workshop « Open questions on energy transport & conversionin nanoscale quantum systems »N. Vast, https://portail.polytechnique.edu/lsi/fr/recherche/theorie-de-la-science-des-materiaux

Outline :

Collective heat transport in the hydrodynamic regime, not by single phonons

1. Motivations

2. Results: ab initio calculation of the thermal conductivity

3. Conclusions

Keywords: high performance computing, atomic scale, density functional theory, Boltzmann’s transport equation

1

Hydrodynamic heat transport regime in bismuth: a theoretical viewpoint

NathalieVASTLaboratoiredesSolidesIrradiés (LSI),EcolePolytechnique,CEA,CNRS,Palaiseau

LSI:MaximeMARKOV,JelenaSJAKSTE,Giuliana BARBARINO,Giorgia FUGALLOIMPMC,SorbonneUniversité:LorenzoPAULATTO,Michele LAZZERIDipartimento diFisica,Università diRomaLaSapienza,Italy:FrancescoMAURI


• Improvement of ZT through the reduction of κL

High Seebeck coefficientLow thermal conductivity

But low electrical conductivity

Thermal properties of bismuth

Rhombohedral structure

2

• Overall figure of merit ZT = 0.4 at T = 300 K in the trigonal direction

Lee, Esfarjani, Mendoza, Dresselhaus, Gang Chen, Phys. Rev. B (2014) Markov, Sjakste, Fugallo, Paulatto, Lazzeri, Mauri, Vast, Phys. Rev. B 93, 064301 (2016) M. Markov’s PhD (2016)


Observation of second sound in experiment

15

1. Pulse propagation technique 2. Optical technique

T. F. McNelly, S. J. Rogers, D. J. Channin, R. J. Rollefson, W. M. Goubau, G. E. Schmidt, J. A. Krurnhansl, and R. O. Pohl. Phys. Rev. Lett., 24,100 (1970)

D. W. Pohl and V. Irniger Phys. Rev. Lett., 36:480 (1976)

3


Second sound

So far second sound has been observed in a very few materials

3D-materials:

Solid helium ( T ~ 1.8 K) Bismuth (T ~ 1.5 – 3.5 K)NaF (T ~ 10 -18 K)SrTiO3 (recent work)Graphite (recent work)

2D-materials: (recent theoretical predictions)exists even at high T > 50 K

Suspended graphene (MIT, USA)Black phosporous (ESPCI, France)

4

Ackermann, Bertman, Fairbank, Guyer,PRL 16, 789 (1966)

Jackson, Walker, McNelly, PR L25, 26 (1970)


Si

What are the heat transport regimes in bismuth?

De Tomas, Acantarero, Lopeandia, Alvarez, J. Appl. Phys 118, 134305 (2015)

Bi?

Bi is an isotopically pure material : Second sound T ~ ei (k·r−ωt) has been measured between T=1.5 and 3.5 KHydrodynamic (collective) transport of heat, not single-phonon mode (kinetic) regime

Second sound

5

Narayanamurti and Murti, Phys. Rev. Lett. 78, 11461 (1972)

Workshop « Open questions on energy transport & conversionin nanoscale quantum systems »N. Vast, https://portail.polytechnique.edu/lsi/fr/recherche/theorie-de-la-science-des-materiaux 6

Some methods to compute the lattice thermal conductivity kL

The heat flux JL from the hot to the cold pole of the sample reads

Several methods to compute kL impose the temperature gradient or heat flux

!" =1

%&'()< + , + 0 > /,

1

2

• Equilibrium (equ.) molecular dynamics (MD)

• Non-equ. MD !" = − lim()(*→,

lim-→.

< 0 1 >3435

• Approach to equ. MD Lampin, Nguyen, Francioso, Cleri, App. Phys. Let. 100, 131906 (2012)

In the present work we use Boltzmann’s transport equation for phononsTo compute the stationary solution of phonon distribution nq

!" = −%"&'∂)

ph-ph ph-boundary


Phonon-phonon scattering

7

L. Paulatto, F. Mauri, M. Lazzeri Phys. Rev. B 87, 214303 (2013)

D3Q code

ph-ph ph-boundary

(2n+1) theorem within DFPT

Boltzmann’stransport equation


Casimir's model for boundary scattering

The shortest dimension can be• Grains or Wire diameter

• Film thickness

- shortest sample dimension

ph-ph ph-boundary

Boltzmann’stransport equation

8


How to model the heat transport regimes?

Important at low TCollective transport

nq : exact variational solution

Sufficient at room TKinetic transport

Γq-1 : single mode approximation :

cq is the phonon group velocityħωq is the phonon energyn0q is the equilibrium (Bose-Einstein) phonon distributionΔnq = nq - n0q is the change in the phonon distribution

Phonon distribution nq versus inverse scattering rate Γq-1

G. Fugallo, M. Lazzeri, L. Paulatto, F. Mauri, Phys. Rev. B 88, 045430 (2013)

9


Outline :Collective heat transport in the hydrodynamic regime

Not a heat transport by single phonons

1. Motivations2. Results: ab initio calculation of the thermal conductivity

3. Conclusions

Keywords: high performance computing, atomic scale, density functional theory, Boltzmann’s transport equation

10



Phonons scattering by phonon-phonon coupling (anharmonicity)Ph

onon

freq

uenc

y(c

m-1

)

High symmetry direction in the Brillouin zone

Forte The three acoustic phonons in bismuth

The longitudinal LA phonon has a short lifetime (>10 ps)(high scattering probability phonon-phonon interaction)

Green, red, blue colors:scattering probabilityby phonon-phonon interaction (one color per phonon)

Maksim MARKOV PhD, Ecole Polytechnique (2016)

Bi


Lattice thermal conductivity in bismuth

Black:Our ab initio calculation

Green: measurementsIssi, Aust. J. Physics (1979)

Red: Calculations (USA)Lee, Esfarjani, Mendoza,Dresselhaus, Gang Chen,Phys. Rev. B (2014) Effet of thenanostructuring:Decrease of thermal conductivity by 50% in a 100-nm-nanostructure at 100 K

Markov, Sjakste, Fugallo, Paulatto, Lazzeri, Mauri, Vast, Phys. Rev. B 93, 064301 (2016)

Bi

Umklapp ? 1/T behavior

Normal + Umklapp


ScatteringBy sample boundary

100-nm50-nm

12

Trigonal crystal of bismuth : binary direction (perpendicular to the trigonal axis)


Phonon calculation in the harmonic approximation

Indirect phonon gap modulates the strength of the an-harmonic interaction between acoustic and optical phonon (AOPI)

Phonon frequency as a function of high symetry directions in the Brillouin zone

Bi

13


Lattice thermal conductivity in bismuth

Black:Our ab initio calculation

Green: measurementsIssi, Aust. J. Physics (1979)

Red: Calculations (USA)Lee, Esfarjani, Mendoza,Dresselhaus, Gang Chen,Phys. Rev. B (2014) Effet of thenanostructuring:Decrease of thermal conductivity by 50% in a 100-nm-nanostructure at 100 K

Markov, Sjakste, Fugallo, Paulatto, Lazzeri, Mauri, Vast, Phys. Rev. B 93, 064301 (2016)

Bi

Umklapp ? 1/T behavior

Normal + Umklapp


ScatteringBy sample boundary

100-nm50-nm

14



Thermodynamical average ofphonon-phonon &ph-boundary scattering rates

15

2

1 2 5 10 50 100 300Temperature, K

100

101

102

103

104

Theory exact

Idem, LCas

= 9.72 mmExpt. IssiTheory SMA

Idem, LCas

= 9.72 mm

Th

erm

al c

on

du

ctiv

ity

κ⊥,

Wm

-1K

-1

FIG. 1: Temperature dependence of the lattice thermal con-ductivity (LTC) in the binary direction for a single crystalwithout (solid lines) or with (dotted lines) millimeter-sizedsample boundaries (MSSB). Black curves : exact variationalcalculation (V-BTE). Red curves : single mode approxima-tion (SMA-BTE). MSSB modeled with the wire geometry andLCas = 9.72 mm [21]. Green dashed lines : LTC extracted byus from expt. of Ref. 22 for T > 20 K; for T < 20 K, LTCfrom a sample having a rectangular cross-section 8.8 ⇥ 8.6mm2 (Ref. 11). We used the T-independent bulk value of6 W(K.m)�1 [23, 24] of the electronic contribution to extractthe LTC from the total thermal conductivity of Bi [11].

heat carried by single (uncorrelated) phonons with finitelifetimes, is valid in the kinetic regime only. Thus, asignificant di↵erence between the LTC obtained by a so-lution of V-BTE and the one obtained in the single modeapproximation (SMA-BTE) is the indication that the hy-drodynamic regime is achieved. Second, we compare thethermodynamic averages of the phonon-scattering ratesfor normal and resistive processes �n and �U , and thehydrodynamic regime occurs when [20]

�U

av

⌧ �n

av

. (1)

Then, we address the question of the occurrence ofPoiseuille’s flow inside the hydrodynamic regime. Here aswell, various methods are employed, which now accountfor the additional scattering rate by sample boundaries�b. We first use Guyer’s conditions [20],

�U

av

< �b

av

< �n

av

(2)

and find the temperature interval in which second-soundis calculated to be observable. In the second method, weextract the heat wave propagation length directly fromthe LTC calculated with V-BTE and compare it to thesample size. The sample size comparable to the HWPLsets the threshold for the second sound observability, asabove this threshold, the heat-wave is damped beforereaching the sample boundary.

The thermodynamic averages of phonon scatteringrates for normal, Umklapp and boundary collisional pro-cesses that condition the transport regime read

�i

av

=

P⌫

C⌫

�i

⌫P⌫

C⌫

(3)

where C⌫

is the specific heat (see below) of the phononmode ⌫ = {qj} and the index i = n, U, b stands for nor-mal, Umklapp and extrinsic (boundary) scattering re-spectively. Besides the scattering rate (or inverse relax-ation time), the quantities characterizing heat transportare the drift velocity v of the heat carriers defined below,and the phonon propagation length � = v ��1

av

which isthe characteristic distance that heat carrying phononscover before damping. As a source of damping, we con-sider, in infinite samples, either Umklapp processes only

�hydro

(1) = v/�U

av

, (4)

or their combination with normal processes throughMatthiessen’s rule

�gas

(1) = v/(�U

av

+ �n

av

) . (5)

When scattering by sample boundaries is accounted for,the phonon propagation length reads �(LCas) instead of�(1) in eqs. 4�5, where Casimir’s length LCas representsthe smallest dimension of the sample or nanostructure.

In bismuth the transport is anisotropic and has compo-nents along the trigonal axis (k) and perpendicular (?)to it, i.e. along the binary and bisectrix directions. Thedrift velocity in these directions reads [9]

v2j

=

P⌫

C⌫

c⌫ j

· c⌫ jP

⌫

C⌫

, (6)

where j stands for k or ? direction and c⌫

is thephonon group velocity. In the thermodynamic aver-ages the specific heat of a phonon mode is calculated as

C⌫

= n0⌫

(n0⌫

+ 1) (~!⌫)2

kBT

2 , where n0 stands for the temper-ature (T) dependent Bose-Einstein phonon occupationnumber and !

⌫

is the phonon frequency.The lattice thermal conductivity, third-order anhar-

monic constants of the normal and Umklapp phonon in-teractions, and thermodynamical averages have been cal-culated on a 28x28x28 q-point grid in the Brillouin zone,but for the drift velocity below 2 K, which required a40⇥40⇥40 grid. Details of the calculation are given inthe supplemental material. We have used the wire geom-etry for the boundary scattering with Casimir’s model,

�b = 2n0⌫(n

0⌫+1)|cb

⌫ |L

Cas , where cb⌫

is the group-velocity in thedirection of the smallest dimension. The factor of twoaccounts for surface roughness [23, 24].Remarkably, our calculated lattice thermal conductiv-

ity (LTC) shows the same evolution as the experimentalone over three orders of magnitude (Fig. 1, resp. black

2

1 2 5 10 50 100 300Temperature, K

100

101

102

103

104

Theory exact

Idem, LCas


Idem, LCas

= 9.72 mm

Therm

al

conducti

vit

y κ⊥, W

m-1

K-1



�U

av

⌧ �n

av

. (1)


�U

av

< �b

av

< �n

av

(2)



�i

av

=

P⌫

C⌫

�i

⌫P⌫

C⌫

(3)

where C⌫


av


�hydro

(1) = v/�U

av

, (4)


�gas

(1) = v/(�U

av

+ �n

av

) . (5)



v2j

=

P⌫

C⌫

c⌫ j

· c⌫ jP

⌫

C⌫

, (6)



C⌫

= n0⌫

(n0⌫

+ 1) (~!⌫)2

kBT


⌫



�b = 2n0⌫(n

0⌫+1)|cb

⌫ |L

Cas , where cb⌫


ity (LTC) shows the same evolution as the experimentalone over three orders of magnitude (Fig. 1, resp. blackMarkov, Sjakste, Barbarino, Fugallo, Paulatto, Lazzeri, Mauri, Phys. Rev. Lett. 120, 075901 (2018)


Hydrodynamic regime of thermal transport in bismuth (1/2)

16

1 2 5 10 50 100 300Temperature, K

100

101

102

103

104

Theory exact

Idem, LCas


Idem, LCas

= 9.72 mmT

her

mal

conduct

ivit

y κ⊥, W

m-1

K-1


Increase of Δnqj :ph- repopulationby normal processes : j

T<20K, Uher T> 20`K

Markov, Sjakste, Barbarino, Fugallo, Paulatto, Lazzeri, Mauri, Phys. Rev. Lett. 120, 075901 (2018)


Definition oftheclassical PoiseuilleorKnudsen flow

17

PIPE boundaries Flow velocity =0

Flow velocity =0

Flow velocity is maximal

Driving force : pressure gradient Resistive viscous force : along both boundaries

Fluid

Fluid in a pipe

Science dictionary, online, 2017-04-11

The flow of low-pressure gas through a tube whose diameter is much smaller than the mean free path of the molecules of the gas.

Knuden number : ratio of the molecule mean free path over smallest dimension of the pipeHere: We define the “mean free-path” as the “heat wave propagation length”

The macroscopic concept of viscosity needs to be modified since the resistance to motion is due primarily to molecular collisions with the passage walls.


Hydrodynamic regime of thermal transport in bismuth (2/2)

Heat wave propagation length Lh in the binary and trigonal directions


18

1 2 5 10 50 100 300 500Temperature, K

10-2

10-1

100

101

102

103

104

105

Lh, µ

mExpt.

λhydro

= v⊥/Γ

U

av

λgas

= v⊥/(Γ

U

av+Γ

n

av)

Lh - wires - SMA

Lh - wires - VAR

Binary directionSecond sound

2

1 2 5 10 50 100 300Temperature, K

100

101

102

103

104

Theory exact

Idem, LCas


Idem, LCas

= 9.72 mm

Ther

mal

conduct

ivit

y κ⊥, W

m-1

K-1



�U

av

⌧ �n

av

. (1)


�U

av

< �b

av

< �n

av

(2)



�i

av

=

P⌫

C⌫

�i

⌫P⌫

C⌫

(3)

where C⌫


av


�hydro

(1) = v/�U

av

, (4)


�gas

(1) = v/(�U

av

+ �n

av

) . (5)



v2j

=

P⌫

C⌫

c⌫ j

· c⌫ jP

⌫

C⌫

, (6)



C⌫

= n0⌫

(n0⌫

+ 1) (~!⌫)2

kBT


⌫



�b = 2n0⌫(n

0⌫+1)|cb

⌫ |L

Cas , where cb⌫



2

1 2 5 10 50 100 300Temperature, K

100

101

102

103

104

Theory exact

Idem, LCas


Idem, LCas

= 9.72 mm

Th

erm

al c

on

du

ctiv

ity κ⊥,

Wm

-1K

-1



�U

av

⌧ �n

av

. (1)


�U

av

< �b

av

< �n

av

(2)



�i

av

=

P⌫

C⌫

�i

⌫P⌫

C⌫

(3)

where C⌫


av


�hydro

(1) = v/�U

av

, (4)


�gas

(1) = v/(�U

av

+ �n

av

) . (5)



v2j

=

P⌫

C⌫

c⌫ j

· c⌫ jP

⌫

C⌫

, (6)



C⌫

= n0⌫

(n0⌫

+ 1) (~!⌫)2

kBT


⌫



�b = 2n0⌫(n

0⌫+1)|cb

⌫ |L

Cas , where cb⌫



2

1 2 5 10 50 100 300Temperature, K

100

101

102

103

104

Theory exact

Idem, LCas


Idem, LCas

= 9.72 mm

Th

erm

al c

on

du

ctiv

ity

κ⊥,

Wm

-1K

-1



�U

av

⌧ �n

av

. (1)


�U

av

< �b

av

< �n

av

(2)



�i

av

=

P⌫

C⌫

�i

⌫P⌫

C⌫

(3)

where C⌫


av


�hydro

(1) = v/�U

av

, (4)


�gas

(1) = v/(�U

av

+ �n

av

) . (5)



v2j

=

P⌫

C⌫

c⌫ j

· c⌫ jP

⌫

C⌫

, (6)



C⌫

= n0⌫

(n0⌫

+ 1) (~!⌫)2

kBT


⌫



�b = 2n0⌫(n

0⌫+1)|cb

⌫ |L

Cas , where cb⌫



4

aries becomes significant (Fig. 2).

However, the average scattering rates discussed so fardo not contain any information about repopulation mech-anisms [27]. To account for them, we extract a heat wavepropagation length L

h

that we define by the criterion :

(T, LCas = Lh

) = (T,1)/e, (7)

where (T,1) denotes the LTC obtained for an infinitesample at a given temperature, and (T, LCas) denotesthe LTC obtained for a sample of finite dimension. Theextracted HWPL L

h

is the cylindrical wire diameter LCas

needed to reduce (T,1) by e (Fig. 3, filled disks, andsupplemental material for the trigonal direction).

Remarkably, at low temperatures, Lh

is found to beclose to the phonon propagation length computed withUmklapp processes only (eq. 4). These resistive processesdamp the heat wave, thus defining the wave travelingdistance between the instant of heat wave generation tocomplete di↵usion. A strong presence of normal pro-cesses, in turn, favors heat conduction and second soundbehavior. With the increase of temperature, L

h

becomesclose to the phonon propagation length accounting forboth Umklapp and normal processes of eq. 5, i.e. of anuncorrelated phonon gas (empty circles). We see that thebehavior of the HWPL L

h

extracted from the LTC as afunction of temperature is the fingerprint of the tran-sition from the hydrodynamic to kinetic regime. Thetemperature range and sample dimension in which ob-servations of second sound are available in the binarydirection (red line segment) are in extremely satisfactoryagreement with the calculations, which support the oc-currence of second sound at 3.0 K for a 9.72 mm wire.Fig. 3 enables us also to predict the occurrence of sec-ond sound at other temperatures and sample sizes, forinstance at 4.1 K in a wire with a 1 mm size.

We emphasize that in principle, Lh

is a measurablequantity, provided that LTC can be measured in samplesof many di↵erent sizes, including very large ones. In thatsense, the results presented in Fig. 3 can be viewed as aGedanken experiment in which : (i) First, one need to de-termine the heat wave propagation length from the ther-mal conductivity measured in samples of di↵erent sizes,as described with eq. 7. (ii) Secondly, its combinationwith a measurement of the average phonon mean freepath in a bulk sample, given by eq. 5, as done, for exam-ple, in attenuation measurement experiments [28], could,in principle, lead to the identification of the temperatureand sample size ranges in which Poiseuille’s flow occurs.

We turn to the characterization of Poiseuille’s flow,defined in the previous paragraph as the range of tem-peratures and propagation lengths where L

h

and �hydro

are close to each other. For this purpose we use commonhydrodynamic quantities : Knudsen number and driftvelocity. The former is defined as the ratio between the

(a)

(b)

FIG. 4: Heat flow characteristics in Bi as a function of thetemperature. Panel (a) : drift velocity v in the binary (?)and trigonal (k) directions (resp. black solid and red dashedlines). Symbol : saturated second-sound velocity measuredat 3 K [10]. Panel (b) : Knudsen number L

h

/LCas for a wireof Casimir length LCas = 9.72 mm (black solid line). Theratio of the phonon propagation length in the hydrodynamic(resp. gas) regime over LCas is also given (resp. black dashedand dotted lines). The shaded region 3.0 K < T < 3.48 Kcorresponds to the temperature interval in which a secondsound peak has been reported in the binary direction [10].

HWPL and the characteristic dimension of transport,

Kn =Lh

LCas. (8)

Interestingly, the transition between the hydrodynamicand kinetic regime is found for a calculated Knudsennumber Kn ⇡ 0.58 at T = 3.5 K in agreement with thecriteria of phonon hydrodynamics 0.1 . Kn . 10 [5](Fig. 4, bottom panel, black solid line). Our drift veloc-ity calculated with eq. 6 in the binary direction shows amaximum of v? = 770 m/s at 3.0 K whose value matcheswell with the second sound velocity v = 780 m/s mea-sured in Ref. 10. At variance with the experiment [10], wefind however a dependence on the propagation direction(top panel, black solid and red dashed lines).In conclusion, repopulation of phonon states by nor-

mal processes turns out to be particularly strong at lowtemperatures and leads to the occurrence of the hydro-dynamic regime in bismuth. We have shown that this

We defined Lh such as

Single-phonon attenuationmeasurements

κL measuredfor varioussample sizes

Gedanken experiment:the hydrodynamic regime could be deduced from a comparison between :- measurements of lattice thermal conductivity on samples of various sizes - average measurementof « monochromatic » phonon attenuation

Average phonon propagation length :

See POSTER


Hydrodynamic regime ofthermaltransport inbismuth(2/2)

Heat wave propagation length Lh in the binary and trigonal directions


19

1 2 5 10 50 100 300 500Temperature, K

10-2

10-1

100

101

102

103

104

105

Lh, µ

m

Expt.

λhydro

= v⊥/Γ

U

av

λgas

= v⊥/(Γ

U

av+Γ

n

av)

Lh - wires - SMA

Lh - wires - VAR

Binary direction

1 2 5 10 50 100 300 500Temperature, K

10-2

10-1

100

101

102

103

104

105

Lh, µ

m

Expt.

λhydro

= v||/Γ

U

av

λgas

= v||/(Γ

U

av+Γ

n

av)

Lh - wires - SMA

Lh - wires - VAR

Trigonal directionSecond sound Second sound

Average phonon propagation length :

2

1 2 5 10 50 100 300Temperature, K

100

101

102

103

104

Theory exact

Idem, LCas


Idem, LCas

= 9.72 mm

Ther

mal

conduct

ivit

y κ⊥, W

m-1

K-1



�U

av

⌧ �n

av

. (1)


�U

av

< �b

av

< �n

av

(2)



�i

av

=

P⌫

C⌫

�i

⌫P⌫

C⌫

(3)

where C⌫


av


�hydro

(1) = v/�U

av

, (4)


�gas

(1) = v/(�U

av

+ �n

av

) . (5)



v2j

=

P⌫

C⌫

c⌫ j

· c⌫ jP

⌫

C⌫

, (6)



C⌫

= n0⌫

(n0⌫

+ 1) (~!⌫)2

kBT


⌫



�b = 2n0⌫(n

0⌫+1)|cb

⌫ |L

Cas , where cb⌫



2

1 2 5 10 50 100 300Temperature, K

100

101

102

103

104

Theory exact

Idem, LCas


Idem, LCas

= 9.72 mm

Th

erm

al c

on

du

ctiv

ity κ⊥,

Wm

-1K

-1



�U

av

⌧ �n

av

. (1)


�U

av

< �b

av

< �n

av

(2)



�i

av

=

P⌫

C⌫

�i

⌫P⌫

C⌫

(3)

where C⌫


av


�hydro

(1) = v/�U

av

, (4)


�gas

(1) = v/(�U

av

+ �n

av

) . (5)



v2j

=

P⌫

C⌫

c⌫ j

· c⌫ jP

⌫

C⌫

, (6)



C⌫

= n0⌫

(n0⌫

+ 1) (~!⌫)2

kBT


⌫



�b = 2n0⌫(n

0⌫+1)|cb

⌫ |L

Cas , where cb⌫



2

1 2 5 10 50 100 300Temperature, K

100

101

102

103

104

Theory exact

Idem, LCas


Idem, LCas

= 9.72 mm

Th

erm

al c

on

du

ctiv

ity

κ⊥,

Wm

-1K

-1



�U

av

⌧ �n

av

. (1)


�U

av

< �b

av

< �n

av

(2)



�i

av

=

P⌫

C⌫

�i

⌫P⌫

C⌫

(3)

where C⌫


av


�hydro

(1) = v/�U

av

, (4)


�gas

(1) = v/(�U

av

+ �n

av

) . (5)



v2j

=

P⌫

C⌫

c⌫ j

· c⌫ jP

⌫

C⌫

, (6)



C⌫

= n0⌫

(n0⌫

+ 1) (~!⌫)2

kBT


⌫



�b = 2n0⌫(n

0⌫+1)|cb

⌫ |L

Cas , where cb⌫



4

aries becomes significant (Fig. 2).

However, the average scattering rates discussed so fardo not contain any information about repopulation mech-anisms [27]. To account for them, we extract a heat wavepropagation length L

h

that we define by the criterion :

(T, LCas = Lh

) = (T,1)/e, (7)

where (T,1) denotes the LTC obtained for an infinitesample at a given temperature, and (T, LCas) denotesthe LTC obtained for a sample of finite dimension. Theextracted HWPL L

h

is the cylindrical wire diameter LCas

needed to reduce (T,1) by e (Fig. 3, filled disks, andsupplemental material for the trigonal direction).

Remarkably, at low temperatures, Lh

is found to beclose to the phonon propagation length computed withUmklapp processes only (eq. 4). These resistive processesdamp the heat wave, thus defining the wave travelingdistance between the instant of heat wave generation tocomplete di↵usion. A strong presence of normal pro-cesses, in turn, favors heat conduction and second soundbehavior. With the increase of temperature, L

h

becomesclose to the phonon propagation length accounting forboth Umklapp and normal processes of eq. 5, i.e. of anuncorrelated phonon gas (empty circles). We see that thebehavior of the HWPL L

h

extracted from the LTC as afunction of temperature is the fingerprint of the tran-sition from the hydrodynamic to kinetic regime. Thetemperature range and sample dimension in which ob-servations of second sound are available in the binarydirection (red line segment) are in extremely satisfactoryagreement with the calculations, which support the oc-currence of second sound at 3.0 K for a 9.72 mm wire.Fig. 3 enables us also to predict the occurrence of sec-ond sound at other temperatures and sample sizes, forinstance at 4.1 K in a wire with a 1 mm size.

We emphasize that in principle, Lh

is a measurablequantity, provided that LTC can be measured in samplesof many di↵erent sizes, including very large ones. In thatsense, the results presented in Fig. 3 can be viewed as aGedanken experiment in which : (i) First, one need to de-termine the heat wave propagation length from the ther-mal conductivity measured in samples of di↵erent sizes,as described with eq. 7. (ii) Secondly, its combinationwith a measurement of the average phonon mean freepath in a bulk sample, given by eq. 5, as done, for exam-ple, in attenuation measurement experiments [28], could,in principle, lead to the identification of the temperatureand sample size ranges in which Poiseuille’s flow occurs.

We turn to the characterization of Poiseuille’s flow,defined in the previous paragraph as the range of tem-peratures and propagation lengths where L

h

and �hydro

are close to each other. For this purpose we use commonhydrodynamic quantities : Knudsen number and driftvelocity. The former is defined as the ratio between the

(a)

(b)

FIG. 4: Heat flow characteristics in Bi as a function of thetemperature. Panel (a) : drift velocity v in the binary (?)and trigonal (k) directions (resp. black solid and red dashedlines). Symbol : saturated second-sound velocity measuredat 3 K [10]. Panel (b) : Knudsen number L

h

/LCas for a wireof Casimir length LCas = 9.72 mm (black solid line). Theratio of the phonon propagation length in the hydrodynamic(resp. gas) regime over LCas is also given (resp. black dashedand dotted lines). The shaded region 3.0 K < T < 3.48 Kcorresponds to the temperature interval in which a secondsound peak has been reported in the binary direction [10].

HWPL and the characteristic dimension of transport,

Kn =Lh

LCas. (8)

Interestingly, the transition between the hydrodynamicand kinetic regime is found for a calculated Knudsennumber Kn ⇡ 0.58 at T = 3.5 K in agreement with thecriteria of phonon hydrodynamics 0.1 . Kn . 10 [5](Fig. 4, bottom panel, black solid line). Our drift veloc-ity calculated with eq. 6 in the binary direction shows amaximum of v? = 770 m/s at 3.0 K whose value matcheswell with the second sound velocity v = 780 m/s mea-sured in Ref. 10. At variance with the experiment [10], wefind however a dependence on the propagation direction(top panel, black solid and red dashed lines).In conclusion, repopulation of phonon states by nor-

mal processes turns out to be particularly strong at lowtemperatures and leads to the occurrence of the hydro-dynamic regime in bismuth. We have shown that this

We defined Lh such as

κL

Single-phonon Single-phonon

κL

See POSTER See POSTER


Characterization oftheheat transportPoiseuille flow


20



Conclusions?

?

No clear kinetic regime found in the calculations!

Transition from ballistic s hydrodynamic

Predicted Poiseuille regime at good temperature region1.5 K < T < 3.5 K in agreement with experiment.

BiBi

Tran

sitio

n re

gim

e

21


Thermodynamical average ofphonon-phonon &ph-boundary scattering rates

22

2

1 2 5 10 50 100 300Temperature, K

100

101

102

103

104

Theory exact

Idem, LCas


Idem, LCas

= 9.72 mm

Th

erm

al c

on

du

ctiv

ity

κ⊥,

Wm

-1K

-1



�U

av

⌧ �n

av

. (1)


�U

av

< �b

av

< �n

av

(2)



�i

av

=

P⌫

C⌫

�i

⌫P⌫

C⌫

(3)

where C⌫


av


�hydro

(1) = v/�U

av

, (4)


�gas

(1) = v/(�U

av

+ �n

av

) . (5)



v2j

=

P⌫

C⌫

c⌫ j

· c⌫ jP

⌫

C⌫

, (6)



C⌫

= n0⌫

(n0⌫

+ 1) (~!⌫)2

kBT


⌫



�b = 2n0⌫(n

0⌫+1)|cb

⌫ |L

Cas , where cb⌫



2

1 2 5 10 50 100 300Temperature, K

100

101

102

103

104

Theory exact

Idem, LCas


Idem, LCas

= 9.72 mm

Therm

al

conducti

vit

y κ⊥, W

m-1

K-1



�U

av

⌧ �n

av

. (1)


�U

av

< �b

av

< �n

av

(2)



�i

av

=

P⌫

C⌫

�i

⌫P⌫

C⌫

(3)

where C⌫


av


�hydro

(1) = v/�U

av

, (4)


�gas

(1) = v/(�U

av

+ �n

av

) . (5)



v2j

=

P⌫

C⌫

c⌫ j

· c⌫ jP

⌫

C⌫

, (6)



C⌫

= n0⌫

(n0⌫

+ 1) (~!⌫)2

kBT


⌫



�b = 2n0⌫(n

0⌫+1)|cb

⌫ |L

Cas , where cb⌫


ity (LTC) shows the same evolution as the experimentalone over three orders of magnitude (Fig. 1, resp. blackMarkov, Sjakste, Barbarino, Fugallo, Paulatto, Lazzeri, Mauri, Phys. Rev. Lett. 120, 075901 (2018)



Towards the modeling of doping in Bi2Te3 compounds and SiGe alloys

Thermal conductivity of BiAcoustic-optical phonon interactionplays a crucial role on the magnitude of

is predicted in the trigonal direction where it as not been measured experimentallyHydrodynamics heat transport regime ischaracterized by a heat wave propagation length Lhdeduced from of samples of different sizes

S i

Coupled electron and phonon transports

Phonon-drag contribution to the Seebeck coefficient

Phonon-drag effect depends on nanostructuresize and shape


NathalieVASTLaboratoiredesSolidesIrradiés (LSI),EcolePolytechnique,CEA,CNRS,Palaiseau

LSI:MaximeMARKOV,JelenaSJAKSTE,Giuliana BARBARINO,Giorgia FUGALLO

IMPMC,SorbonneUniversité:LorenzoPAULATTO,Michele LAZZERI

Dipartimento diFisica,Università diRomaLaSapienza,Italy:FrancescoMAURI

24

Maxime Markov Jelena Sjaskte


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