Heat Transfer in Nanostructures : From Particles to Waves

1. Heat Transfer in Nanostructures : From Particles to Waves Sebastian Volz Laboratoire d’Energétique Moléculaire et Macroscopique, Combustion, CNRS - Ecole Centrale Paris - France TIENCS08 - Singapour June 4th

2. 2. Phononic crystals 3. Contact resistance and quantum of conductance 1. Quasi-Ballistic Heat Transfer From Scattering to Wave Mechanisms

3. Hendrik Casimir QUASI-BALLISTIC HEAT TRANSFER Joseph Fourier 1824 Ludwig Boltzman Martin Knudsen

4. ELECTRON-PHONON COUPLING in NANOPARTICLES « we have performed the first investigation of the internal electron thermalization dynamics in metal clusters. » PRL, 85, 2200, (2000) Pump-Probe Femtolaser Electron absorbing Relaxing on Phonons in Nanoparticles

5. CORE-SHELL • 2-STEP MODEL Majid Rashidi • BALLISTIC-DIFFUSIVE EQUATIONS G. Chen, PRL, 86, 2297

6. QUASI-BALLISTIC EFFECT Submitted to PRB Journal of Heat Transfer, in press

7. PHONON PARTICLE ? k.x>2 x k k< /a => x > 10a = 5nm

8. PHONON WAVE p(, , pol, ) Hokudai U.

9. CONFINEMENT e ikL=e ika=0 k = n . 2/L L/a~1 BVK: e ik(L+x)=e ikx k = n . 2/L L/a>>>n un ~ expi(kna-wt)+ expi(-kna-wt) ~ cos(kna)e-iwt STATIONNARY WAVE, zero group velocity

10. Phononic Cristals THERMOELECTRIC FACTOR OF MERIT Jean-Numa Gillet In a superlattice: l can be several orders of magnitude smaller than bulk material But ZT > alloy limit because: 2 major drawbacks: Lattice mismatch can occur between layers of a superlattice as in Si/Ge superlattices: formation of defects and dislocations: reduces s and avoids increase of ZT compared to alloy limit. 2) Superlattices only decrease heat conduction in the perpendicular direction to the thin-film surfaces

11. Insulating Materials Recent work on a stack layered WSe2 SL =0.04W/mK at ambiant!! Einstein model (Mean FP=WaveL/2) =0.2W/mK The lowest theoretical limit for a crystal. ELECTRICAL CONDUCTIVITY STRONGLY DECREASED PRESERVE CRYSTAL STRUCTURE? Chiritescu et al., Science 315, 351, 2007.

12. “Macroscopic” phononic crystals Phononic crystals: inspired by remarkable properties of photonic crystals Show band gaps -destructive interferences- of acoustic wave. “Macroscopic” phononic crystals: periodic structure of elastic rods for 2D crystals or beads for 3D crystals within a solid matrix or a fluid. Lattice constant: usually of  1 to  10 mm • Problem: band gap cannot occur at frequencies that are higher than  1 MHz

13. Atomic-scale 3D phononic crystal Supercell with N = 5 and M = 3 Equilibrium positions of the Si (blue) and Ge (red) atoms obtained by conjugated gradient method d = Na = 2.715 nm Ma = 1.629 nm • Band Gaps? • And low thermal conductivity?

14. 3 x 8 x N3 = 3000 dispersion curves Lattice dynamics: to compute the dispersion curves: - Use of the Stillinger-Weber potential - 3000 dispersion curves (3 × 1000 = 3000 d.o.f. in supercell) Use of General Utility Lattice Program (GULP) Quasi ab initio: only potential shape is set Curves are very flat: Phonon confinement results in very low group velocities instead of band gaps Phonon Trap Half of first BZ in [1 0 0]

15. Thermal conductivity model Boltzmann and from Fourier law: Mode MFP Mode heat capacity DOS / V Mode group velocity Angular frequencies wk,m and group velocities obtained by lattice dynamics! Much easier to compute than preceding eqs. based on integration over , because Debye approximation cannot be used for the 3000 - 3 = 2997 optical dispersion curves!

16. Relaxation-time model Use of Matthiessen rule: Bulk Si and Ge: High-purity Silicon at T > 100 K: boundary (B) and isotopic-and-defects (l) scattering can be neglected compared to umklapp (u) scattering Phononic nanomaterial: tScatdue to phonon scattering by the Ge QDs Low T: tScat << tB and tI

17. Incoherent scattering cross section M = 3 M = 2 M = 1 HBZ For the smallest Ge QD (M = 1), scattering in HBZ is not efficient For the largest Ge QD (M = 3), scattering in HBZ becomes very efficient with sinc that can be higher than d2/2

18. Bulk Si (exp.) M = 1 M = 2 M = 3 300 K: lmax = 0.37 W/Km, decrease factor: 418 ‘Einstein Model’ for crystal 0.2 W/mK Peak value (48 K): lmax = 0.56 W/Km, peak-to-peak decrease factor: 8535 Thermal conductivity results lmax> lrealis an upper bound becausesinc<sinc+coh

19. max decrease at T = 300 K < Cv > / < Cv >Si < l >max/< l >Si lmax/lSi For small Ge atomic densities x 0.2: < Cv > decrease effect are predominant over < l >max decrease at T= 300 K To be published in Journal o f Heat Transfer.

20. Conclusion / Phononic Cristal At T = 300 K, the thermal conductivity can be reduced by at least a 400 factor in an atomic-scale 3D phononic crIstal to reach a value lower than 0.4 W/Km.This reduction is due to the shrunk MFPs of phonons but also to their very low GROUP VELOCITIES.No Gap effect is noticed.We expect a further decrease when coherent scattering will be introduced.This material may break the Einstein limit and be an interesting candidate for thermoelectric applications.

21. Nanowires and Thermal Contact Resistances Metrology Spectrometry NanoElectronics Molecular Junction

22. Thermal Bath T1 DIFFUSION  Macro-FOURIER Thermal Bath T0

23. WEXLER RESISTANCE  NanoContact  >  D Thermal Bath T1 Thermal Bath T0

24. Multireflections IFD<PHONON WAVELENGTH? S. Volz et al., J.App. Phys. 103, 34306

25. Quantum of Conductance 1D in k-space Wire Thermal Conductance -Conductance of 1 phononBRANCH /NOT/ 1 Quantum -Temperature Dependent -Predominant CONDUCTANCE?

26. Qb Qw CONTACT CONDUCTANCE Diffuse Transmission: Phonons loosememory at interface

27. 2x2cells in section 5x5 10x10 DISPERSION CURVES Lattice dynamics Solution: Stillinger Weber potential

28. Group Velocities

29. WIRE vs CONTACT CONTACT RESISTANCE PREDOMINANT 2 ODM SUBSTRATE HEAT CAPACITY Phys. Rev B in press

30. THERMAL RESERVOIRS (k) ? TL TR 1D DOS ? Excited Modes 1D WIRE 3D SUBSTRATE T

31. CONCLUSION Nanowire and Contact At low temperatures, Heat flux in of SUB-10nm Nanowires is dominated by CONTACT RESISTANCE . Quantum thermal conductance can not be measured in those wires. At low temperatures, Specific Heat of the substrate following a T3 law is smaller than the nanowire specific heat, which is proportional to T. We also infer that at low temperatures: The conductance of a membrane suspended between two 3D substrate follows a T3 law (=Cv of a 3D substrate). The conductance of a wire suspended between two membranes follows a T2 law (=Cv of a 2D subtrate);

32. Collaborators Academic: J.-J. Greffet, EM2C-ECP M. Laroche, EM2C-ECP M. Massot, EM2C-ECP J. Bai, LMSSMAT-ECP B. Palpant, INSP-Paris J.-Y. Duquesne, INSP-Paris L. Jullien, ENS Ulm Paris Charlie Goss, LPN - Paris M. Mortier, ENSCP-Paris G. Tessier, ESPCI-Paris S. Ravaine, ICMCB-Bordeaux S. Dilhaire, CPMOH Bordeaux S. Gomès, CETHIL-Lyon S. Lefèvre, CETHIL-Lyon G. Domingues, LT-Nantes B. Charlot, TIMA-Grenoble C. Bergaud, LAAS-Toulouse Industries/Institutes: D. Rochais, CEA_Le Ripault N. Mingo, CEA-LITEN M. Plissonnier, CEA-LITEN J.-J. Greffet M. Laroche Post-Doc Students J.-N. Gillet E. Rousseau P.-O. Chapuis Ph.D. Students Y. Chalopin C. Bera

33. THANK YOU !

34. THERMAL DIODES Arunava Majumdar ‘With the availability of nonlinear thermal control, phonons should no longer be considered the unwanted by-products of electronics. Phonons, like electrons and photons, are information carriers and should be processed accordingly.’

35. The Mysterious Story of Nanowires Thermal Conductivity APL Li&Majumdar, 83, 2934, 2003 Nature, Roukes 27/04/00 M.R. says: notobtained for long wires