Hydrodynamic heat transport regime in bismuth: a theoretical viewpoint

Transcription

1 Hydrodynamic heat transport regime in bismuth: a theoretical viewpoint Nathalie VAST Laboratoire des Solides Irradiés (LSI), Ecole Polytechnique, CEA, CNRS, Palaiseau LSI: Maxime MARKOV, Jelena SJAKSTE, Giuliana BARBARINO, Giorgia FUGALLO IMPMC, Sorbonne Université: Lorenzo PAULATTO, Michele LAZZERI Dipartimento di Fisica, Università di Roma La Sapienza, Italy: Francesco MAURI 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

2 Thermal properties of bismuth Overall figure of merit ZT = 0.4 at T = 300 K in the trigonal direction High Seebeck coefficient Low thermal conductivity But low electrical conductivity Rhombohedral structure Improvement of ZT through the reduction of κ L Lee, Esfarjani, Mendoza, Dresselhaus, Gang Chen, Phys. Rev. B (2014) Markov, Sjakste, Fugallo, Paulatto, Lazzeri, Mauri, Vast, Phys. Rev. B 93, (2016) M. Markov s PhD (2016) 2

3 What are the heat transport regimes in bismuth? Bi is an isotopically pure material : Second sound T ~ e i (k r ωt) has been measured between T=1.5 and 3.5 K Hydrodynamic (collective) transport of heat, not single-phonon mode (kinetic) regime Second sound Narayanamurti and Murti, Phys. Rev. Lett. 78, (1972) Si? Bi De Tomas, Acantarero, Lopeandia, Alvarez, J. Appl. Phys 118, (2015) 3

4 Some methods to compute the lattice thermal conductivity k L The heat flux J L from the hot to the cold pole of the sample reads! " = % " &' ) Several methods to compute k L impose the temperature gradient or heat flux! " = 1 1 Equilibrium (equ.) molecular dynamics (MD) < +, + 0 > /, % & ' ( ) 2 < 0 1 > Non-equ. MD! " = lim lim () (*, 35 Approach to equ. MD Lampin, Nguyen, Francioso, Cleri, App. Phys. Let. 100, (2012) In the present work we use Boltzmann s transport equation for phonons To compute the stationary solution of phonon distribution n q ph-ph ph-boundary 4

5 How to model the heat transport regimes? Phonon distribution n q versus inverse scattering rate Γ q -1 n q : exact variational solution Γ q -1 : single mode approximation : c q is the phonon group velocity ħω q is the phonon energy n 0 q is the equilibrium (Bose-Einstein) phonon distribution Δn q = n q - n 0 q is the change in the phonon distribution Important at low T Collective transport Sufficient at room T Kinetic transport 5

6 Hydrodynamic heat transport regime in bismuth: a theoretical viewpoint Outline : Collective heat transport in the hydrodynamic regime Not a heat transport 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 6

7 Phonons scattering by phonon-phonon coupling (anharmonicity) The three acoustic phonons in bismuth The longitudinal LA phonon has a short lifetime (>10 ps) (high scattering probability Forte phonon-phonon interaction) Phonon frequency (cm -1 ) Bi Green, red, blue colors: scattering probability by phonon-phonon interaction (one color per phonon) Maksim MARKOV PhD, Ecole Polytechnique (2016) High symmetry direction in the Brillouin zone 7

8 Lattice thermal conductivity in bismuth Trigonal crystal of bismuth : binary direction (perpendicular to the trigonal axis) Bi Phonon-phonon scattering Black: Our ab initio calculation Green: measurements Issi, Aust. J. Physics (1979) Scattering By sample boundary 100-nm 50-nm Normal + Umklapp Umklapp? 1/T behavior Red: Calculations (USA) Lee, Esfarjani, Mendoza, Dresselhaus, Gang Chen, Phys. Rev. B (2014) Effet of the nanostructuring: Decrease of thermal conductivity by 50% in a 100-nmnanostructure at 100 K Markov, Sjakste, Fugallo, Paulatto, Lazzeri, Mauri, Vast, Phys. Rev. B 93, (2016) 8

9 Phonon calculation in the harmonic approximation Phonon frequency as a function of high symetry directions in the Brillouin zone Bi Indirect phonon gap modulates the strength of the anharmonic interaction between acoustic and optical phonon (AOPI) 9

10 Lattice thermal conductivity in bismuth Trigonal crystal of bismuth : binary direction (perpendicular to the trigonal axis) Bi Phonon-phonon scattering Black: Our ab initio calculation Green: measurements Issi, Aust. J. Physics (1979) Scattering By sample boundary 100-nm 50-nm Normal + Umklapp Umklapp? 1/T behavior Red: Calculations (USA) Lee, Esfarjani, Mendoza, Dresselhaus, Gang Chen, Phys. Rev. B (2014) Effet of the nanostructuring: Decrease of thermal conductivity by 50% in a 100-nmnanostructure at 100 K Markov, Sjakste, Fugallo, Paulatto, Lazzeri, Mauri, Vast, Phys. Rev. B 93, (2016) 10

11 Thermodynamical average of phonon-phonon & ph-boundary scattering rates i av = Trigonal crystal of bismuth : binary direction (perpendicular to the trigonal axis) P C P C C = n 0 (n 0 + 1) (~! ) 2 k B T 2 i P Markov, Sjakste, Barbarino, Fugallo, Paulatto, Lazzeri, Mauri, Phys. Rev. Lett. 120, (2018) 11

12 Hydrodynamic regime of thermal transport in bismuth (1/2) Trigonal crystal of bismuth : binary direction (perpendicular to the trigonal axis) Increase of Δn qj : ph- repopulation by normal processes : j Thermal conductivity κ, Wm -1 K Theory exact Idem, L Cas = 9.72 mm Expt. Issi Theory SMA T<20K, Uher T> 20`K Idem, L Cas = 9.72 mm Temperature, K Markov, Sjakste, Barbarino, Fugallo, Paulatto, Lazzeri, Mauri, Phys. Rev. Lett. 120, (2018) 12

13 Hydrodynamic regime of thermal transport in bismuth (2/2) Heat wave propagation length L h in the binary and trigonal directions κ L measured for various sample sizes L h, µm Single-phonon attenuation measurements We defined L h such as apple(t,l Cas = L h )=apple(t,1)/e, Binary direction Expt. Second sound λ hydro = v /Γ U av λ gas = v /(Γ U av +Γn av ) L h - wires - SMA L h - wires - VAR Temperature, K hydro(1) =v/ U av, Gedanken experiment: the hydrodynamic regime could be deduced from a comparison between : - measurements of lattice thermal conductivity on samples of various sizes -measurement average of «monochromatic» phonon attenuation Average phonon propagation length : vj 2 = gas(1) =v/( U av + n av) P C c j c P j C, Markov, Sjakste, Barbarino, Fugallo, Paulatto, Lazzeri, Mauri, Phys. Rev. Lett. 120, (2018) 13

14 Hydrodynamic heat transport regime in bismuth: a theoretical viewpoint Thermal conductivity of Bi Acoustic-optical phonon interaction plays a crucial role on the magnitude of is predicted in the trigonal direction where it as not been measured experimentally Hydrodynamics heat transport regime is characterized by a heat wave propagation length L h deduced from of samples of different sizes Coupled electron and phonon transports Phonon-drag contribution to the Seebeck coefficient S i Phonon-drag effect depends on nanostructure size and shape Towards the modeling of doping in Bi 2 Te 3 compounds and SiGe alloys 14

15 Hydrodynamic heat transport regime in bismuth: a theoretical viewpoint Nathalie VAST Laboratoire des Solides Irradiés (LSI), Ecole Polytechnique, CEA, CNRS, Palaiseau LSI: Maxime MARKOV, Jelena SJAKSTE, Giuliana BARBARINO, Giorgia FUGALLO IMPMC, Sorbonne Université: Lorenzo PAULATTO, Michele LAZZERI Dipartimento di Fisica, Università di Roma La Sapienza, Italy: Francesco MAURI Maxime Markov Jelena Sjaskte 15