Phonon Engineering: an introduction

Transcription

1 Phonon Engineering: an introduction II. Phonon engineering and heat conduction Dr P.-Olivier Chapuis Institut Catala de Nanotecnolgia (ICN-CIN2) Barcelona, Spain NiPS Summer School on Energy Harvesting at the micro- and nano-scale Avigliano Umbro TR, Italy, 1 st -8 th August 2010

2 The Phononic heat conduction Phononic thermal conductivity Phonon scattering mechanisms Phonons at nanoscale Phonon transmission at interfaces Phonons in novel materials Heat transfer phonons and measurements spectrum! intrinisc solve BTE diffuse? better transport? techniques

3 Phononic thermal conductivity Contributions to the heat conduction Thermal conductivity k has different contributions: k = k phonon + k electron Wiedemann-Franz law for an approximation of electronic contribution in the thermal conductivity 2 kel kb L T 3 e 2 8 W K 2 Silicon (undoped) k Si = 149 Wm -1 K -1 Si= m -1 k Si, e k Si L 0 T e Si Si 1 Graphite k C_graphite = 140 Wm -1 K -1 C_graphite= m -1 k C _ graphite, e k C _ graphite L T 0 e k C _ graphite C _ graphite 0.3%

4 Phononic thermal conductivity The model of the thermal conductivity Solution of a Boltzmann transport equation (Peierls) i f f E i f f v i 1 (, T) / k e B 0 T f t 1 v r f F m coll ).( v r ) ( d 0 ( f0 i, x g, p) E( ) f ( ) vi, x(, p) pol 0 v f f t Bose-Einstein statistics f ( f 0 ) k (Relaxation time approximation) pola. 0 df0 dt k. r T 1 3 ( v ) g( ) vd NB: Isotropic approx. for v,,

5 Phononic thermal conductivity The model of the thermal conductivity Solution of a Boltzmann transport equation (Peierls) f t v r f F m v f f t coll f ( f 0 ) k (Relaxation time approximation) pola df0 dt ( v ) g( ) vd x e 1 f0(, T) / k B T Bose-Einstein statistics e 1 Planck s law for phonons x Wien s law for phonons D=hv s /2.8k B T v s, Si ~ ms -1 x hv s / 1 10 k B T Debye wavelength

6 Phononic thermal conductivity Phonon spectrum MD calculations with bulk Si Calculated phonon density of states (D) in a e=37 nm Si nanowire Balandin Debye k Real dispersion relation peak=2 nm DOS =1 nm Henry and Chen, J. Comp. Theo. Nanosci 5, 1 (2008) Lü, JAP 104, (2008) (Balandin & Wang PRB NIPS (1998)) Summer school, August 2010

7 Phononic thermal conductivity Which phonons? The acoustic phonons are carrying the heat. k 0 pola c 1 3 v( v ) d df0 dt ( v ) k g( 1 3 ) vd c p v s Si (Y. Garcia) Chen, JHT (1998) v g ( opt) k 0 X k NB: Different from the specific heat!

8 Phonon scattering mechnisms Finiteness of the thermal conductivity..? Critical parameter: The phonon relaxation time as without it the propagation would be infinite! In this absence of defects, it is due to the nonlinearity of the force field between atoms NB: k has a 3D meaning FPI (Fermi Pasta Ulam) paradox of the atomic chain k does not always exist when nonlinearity! k~l not always see Lepri etc.

9 Phonon scattering mechnisms Scattering mechanisms that do not conserve the momentum Origin of the different terms in the mean free path Umklapp (Klemens model) U ~ A 1 e - D/bT T n m Origin: Nonlinearity=Anharmonicity!! ħ 1 + ħ 2 = ħ 3 k 1 + k 2 = k 3 but k 3 in the end k 1 k 3 k 3 k 2 G k 2 2 /a k 1 k 3 G k k 2 3 k 1 2 /a (Very schematic!!) k

10 Phonon scattering mechnisms Scattering mechanisms Origin of the different terms in the mean free path Umklapp (Klemens model) U ~ A 1 e - D/bT T 3 2 Origin: Nonlinearity=Anharmonicity!! Boundary scattering of the particle B ~ A 2 v( )/D D k To be taken into account only in crude model if dispersion relation have not been calculated!

11 Phonon scattering mechnisms Scattering mechanisms Origin of the different terms in the mean free path Umklapp (Klemens model) U ~ A 1 e - D/bT T 3 2 Origin: Nonlinearity=Anharmonicity!! Boundary scattering of the particle B ~ A 2 v( )/D Rayleigh scattering due to impurities Similar to electromagnetics Mie theory ~ A 3 4 (dpart << ) Majumdar, JAP (2005)

12 Phonon scattering mechnisms Scattering mechanisms Origin of the different terms in the mean free path Umklapp (Klemens model) U ~ A 1 e - D/bT T 3 2 Origin: Nonlinearity=Anharmonicity!! Boundary scattering of the particle B ~ A 2 v( )/D Rayleigh scattering due to impurities Similar to electromagnetics Mie theory ~ A 3 4 (dpart << ) Electron-phonon interaction e-ph ~ T

13 Phonon scattering mechnisms Scattering mechanisms Origin of the different terms in the mean free path Umklapp (Klemens model) U ~ A 1 e - D/bT T 3 2 Origin: Nonlinearity=Anharmonicity!! Boundary scattering of the particle B ~ A 2 v( )/D Rayleigh scattering due to impurities Similar to electromagnetics Mie theory ~ A 3 4 (dpart << ) Electron-phonon interaction e-ph ~ T Usually: Mathiessen rule of the relaxation time i NB: Curious: Same treatment of elastic, inelastic etc. liftetime

14 Phonon scattering mechnisms Scattering mechanisms (2) Leading mean free paths =v g Boundaries Wave! J.Y. Duquesne, INSP, Paris 10nm Si particles in a matrix of Ge

15 Phonon scattering mechnisms Mean free path distribution = v g MD calculations with bulk Si Henry and Chen, J. Comp. Theo. Nanosci 5, 1 (2008)

16 Phonons at nanoscale How to deal with BTE at low D? At small scale (space/time), the Fourier approach breaks down! One needs then or to solve the BTE (long!) Phonon density of states Limitation of the approach: L~ v L~ nm? - Probabilistic: Monte-Carlo method - Approx: Discrete ordinate (Radiation) - Approx.: Ballistic-diffusive equation Dispersion relation wave effect Phonon mean free path Particle transport effect to use a simulation method at the atomic scale - Molecular dynamics - Lattice dynamics - Atomistic Green s function method Grey approximation

17 Phonons at nanoscale NB: Cattaneo-Vernotte 2 1 T T c k 2 t t T Propagation of heat Fourier vs BTE at nanoscale Examples taken from Lacroix, Joulain, PRB (2005) also incomplete Stationary temperature profile between two parallel thermalized media Transverse Longitudinal Temperature jump

18 Phonons at nanoscale Reducing the thermal conductivity Impurities or nanoparticles Useful for the generation of thermoelectricity! Efficiency depends on figure-of-merit ZT Z= S / ( k el + k ph ) Majumdar, PRL (2007) Strategies to decrease k ph (without impact on and S ) Adding impurities or nanoparticles! impacts the high-frequency acoustic phonons ErAs in InGaAs

19 Phonons at nanoscale Reducing the thermal conductivity Boundaries Useful for the generation of thermoelectricity! Efficiency depends on figure-of-merit ZT Z= S / ( k el + k ph ) Strategies to decrease k ph (without impact on and S ) Adding boundaries impacts all phonons Ball-milling Chen and Ren, Science (2008)

20 Phonons at nanoscale Reducing the thermal conductivity Boundaries Useful for the generation of thermoelectricity! Efficiency depends on figure-of-merit ZT Z= S / ( k el + k ph ) Strategies to decrease k ph (without impact on and S ) Adding boundaries impacts all phonons Here in nanowires Majumdar, APL (2003)

21 Phonons at nanoscale Reducing the thermal conductivity Roughness Useful for the generation of thermoelectricity! Efficiency depends on figure-of-merit ZT Z= S / ( k el + k ph ) TEM Strategies to decrease k ph (without impact on and S ) Adding amorphous layers at the boundaries further reduces the thermal conductivity! Majumdar, Nature (2009) See also Heat, same issue

22 Phonon transmission at interfaces Phonon transmission at interfaces? Wave model for the low-frequency phonons Acoustic wave! 1 2 T=4Z 1 Z 2 /(Z 1 +Z 2 ) 2 Z 1 = c 1 Z 2 NB: Terminology issue: Kapitza resistance (fluid-solid) Thermal interface resistance (thick interface) Thermal boundary resistance surface Transistor level Polymer-based layer Heat spreader

23 Phonon transmission at interfaces Phonon transmission at interfaces? (2) More difficulty for the high frequency acoustic phonons 1 2 Diffuse mismatch model = limit of strong diffuse scattering

24 Phonon transmission at interfaces Acoustic mismatch and diffuse mismatch models DMM: All correlations between ingoing and outgoing phonons are ignored t ( )= r ( )= 1-t ( ) t c2 c 1 c (With asssumption on the DOS) Swartz and Pohl, RMP (1987) In bulk systems, the resistances with DMM and AMM are similar (30%)

25 Phonon transmission at interfaces Metal dielectric interface Measured values higher than prediction 1 ph1-ph2 2 ph2 - e - 2 R 2 R Thermal surface resistance = 3 2 /R 1 ph1-ph2 2 ph2 - e - 2 Chapuis Maxwell-Garnett approximation (Phonon particule)

26 Phonons in novel materials Thermal conductivity of new materials Bera, PRL (2010) Porous materials to harvest energy Other types of low-thermal conductivity materials (beating the Einstein limit of amorphous materials) Chiritescu, Science (2007) Goodson, Science (2007) Disordered layered crystal k air (300 K)=0.025 Wm -1 K -1

27 Phonons in novel materials Thermal conductivity of novel materials Carbon nanotubes k=3000 Wm -1 K -1 MWCNT: Kim et al, PRL(2001) Li Shi, Science (2010) Balandin, Nano Letters (2008) Graphene

28 Phonons in novel materials Other types of engineering Rectification? Carbon nanotubes loaded with gradient of molecule density Chang,..,Majumdar, Zettl, Science 2006 Phonon-based motor? For the moment only due to the thermal gradient Bachtold, Science (2008)

29 Heat transfer phonons and measurements Usual methods for heat transport characterisation 3 method (Cahill, RSI, 1989) Based on R=R 0 (1+ T) and T P=R [ I 0 cos t ] 2 ICN and VTT U 3 = /2 R 0 I 0 T 2 Suspended microresistors (Shi and Majumdar) Shi and Majumdar Ultrafast pump-probe spectroscopy S. Dilhaire (Bordeaux)

30 Heat transfer phonons and measurements THE 3 METHOD R(T) = R 0 (1 + T) Resistance depends on temperature I = I 0 cos( t) P(t) = R I(t) 2 = ½ R (1 + cos(2 t)) Joule heating of an electric T(t) = T 0 + T DC + T 2 cos(2 t+ 2 ) wire U = RI = R 0 I 0 [1 + T DC + T 2 cos(2 t+ 2 ) ] cos( t) = R 0 I 0 [(1+ T DC ) cos( t) + ½ T 2 cos( t- 2 ) + ½ T 2 cos(3 t 2 ) ] = U + ½ R 0 I 0 T 2 cos(3 t+ 2 ) Temperature of the wire = f(heat flux to the sample)

31 Conclusions Wave behaviour superimposed to the quasiparticle behaviour Research driven bythermoeletric community and the quest for better insulator [lower k] or by microelectronics for better conductors [higher k] Still plenty of room - Demonstration of the Boltzmann transport equation for phonons? - Phonon relaxation time/mean free path - Degree of diffusivity at the interface - Filters and interference effects - Localization etc. - Amorphous materials [not tackled here!]

32 Useful references - Books - G. Chen, Nanoscale energy transport and conversion - S. Volz (ed), Microscale and Nanoscale Heat Transfer - S. Volz (ed), Thermal Nanosystems and Nanomaterials - Z.M. Zhang, Nano/Microscale heat transfer - Reviews or interesting articles - A. Balandin, Phonon Engineering, J. Nanosc. & Nanotech 5, 1015 (2005) - D. Cahill et al., Nanoscale thermal transport, J. Appl. Phys. 93, 793 (2003) - A. Henry and G. Chen, J. Comp. Theo. Nanosci. 5, 1 (2008) -

33 P2N group (June 2010) Prof Clivia Sotomayor Dr Francesc Alsina Dr Vincent Reboud Dr Nikolaos Kehagias Dr Timothy Kehoe Dr Damian Dudek Dr Olivier Chapuis Dr Yamila Garcia Dr Lars Schneider Ms Noemi Baruch Mr John Cuffe Mr Emigdio Chavez