Ultra-low Thermal Conductivity of Layered Crystals

1. Ultra-low Thermal Conductivity of Layered Crystals Pawel Keblinski, Department of Materials Science and Engineering, Rensselaer Polytechnic Institute, Troy, NY

2. People Arun Bodapati - graduate student Lin Hu - graduate student Sergei Shenogin - postdoctoral researcher David Cahill - University of Illinois at Urbana

3. Intro to Thermal Conduction - Materials Perspective

4. Heat conduction range Thermal conductivity W/m-K 1 10 100 1000 aSi cSi YSZ Alumina diamond Isotropic polymers copper Phonons Phononselectrons Phonons Most of the variation comes from the mean free path of the carrier  = 1/3 c  v Thermal conductivity Phonon velocity Mean free path Heat capacity

5. Mean free path and minimum thermal conductivity Phonon-phonon scattering • Silicon at 300K • Mean free path L ~100nm • Polysilicon • L ~ few times grain size • Amorphous Si • ~ atom size Einstein limit Minimum thermal conductivity  ~ 100 W/m-K Phonon-grain boundary No real propagating phonons  ~ 1 W/m-K

6. Thermal conductivity temperature dependence Amorphous and nanocrystals Crystal • At low temperatures tc increases with increasing heat capacity • At higher temperature phonon-phonon scattering lowers mean free • Newly excited high frequency phonons doe not contribute much to tc • Phonon mean free path ~ atom size or grain size • Thermal conductivity increases with temperature

7. Ultra-low Thermal Conductivity of LayeredMaterials

8. Below the amorphous material limit Thermal conductivity below the Einstein limit Why it is so low? Is it really below Einstein limit? C. Chiritescu, D. G. Cahill, et al., Science. 315, 353(2007).

9. How to go below the amorphous material limit • Find a crystal with low thermal conductivity to start with • Create nanostructures possibly reducing already low thermal conductivity by 1-2 orders of magnitude • By creating “poly/nanocrystals” • By creating superlattices

10. A B A B A B A B A B Model structures Perfect crystal Stacking disorder Grain boundaries Mass superlattice A C A B C B A B A B

11. Model interactions Harmonic springs Lennard-Jones potential Two sets of  and  parameters are used • Within WSe2 sheet:  = 0.455 eV and  = 2.31 Å • Between layers:  = 0.04 eV and  = 3.4 Å

12. Molecular dynamics results Perfect crystal and stacking disordered structures show strong film thickness dependence not observed in experiment Structures with grain boundaries show lower conductivity and weaker size dependence

13. Phonon Localization and Polarization Perfect crystal Low frequency phonons are delocalized and polarized Higher frequency phonons are delocalized but not well polarized

14. Phonon Localization and Polarization Stacking disorder Low frequency phonons are delocalized and polarized, same way as in perfect crystal Higher frequency phonons can be localized or delocalized but they do not matter much for thermal transport

15. Phonon Localization and Polarization Grain boundary Low frequency phonons are weakly localized and weakly polarized Conductivity reduction and weak size dependence Higher frequency phonons can be localized or not and are not polarized

16. Phonon Localization and Polarization Mass disorder Modes exhibit features of superlattice periodicity and are delocalized Thermal conductivity of a dense solid below that of still air Perhaps the reduction of tc is akin to that observed in semiconductor superlattices

17. A Simple Interfacial Model Nano-structuring eliminates propagating phonons Therefore, one can consider the structure as a series of independent interfaces characterized by interfacial their thermal resistance/conductance Each layer is a 2 d crystal with phonons propagating in-plane Heat flow between planes is proportional to number of in-plane phonons with frequency below a threshold determined by strength of cross plane interactions. Under assumptions 1-4 a simple analysis yields thermal conductance

18. Phonon Perspective WSe2 cross-plane acoustic branches Only cross-plane longitudinal acoustic (LA) branch has a significant group velocity Transverse acoustic (TA) branch has very low group velocity

19. Phonon Perspective WSe2 cross-plane and in-plane branches Cross-plane optical branches have essentially zero group velocities

20. Minimum Thermal Conductivity If you assume that there is the same group velocity for all branches c= cLA, where cLA is the cross plane speed of sound you overestimate by • A factor of 3 since cLA >>cTA • A factor ~ cLA/ cLA(in plane) due to elastic anisotropy Thus for WSe2 the overestimate will be about factor of 5-4 Correct application of the minimum thermal conductivity model is in agreement with the experiment