Patrick Hopkins University of Virginia Department of Mechanical and Aerospace Engineering

1. Phonon Scattering Processes Affecting Thermal Conductance at Solid-solid Interfaces in Nanomaterial Systems Patrick Hopkins University of Virginia Department of Mechanical and Aerospace Engineering March 10, 2008

2. Moore’s Law Rocket nozzle 107 W/m2 Nuclear reactor 106 W/m2 Hot plate 105 W/m2 Equivalent power density [W/m2] 45 nm 100 nm 500 nm Transistor size

3. Heat generated Rejected heat Field effect transistors Thermal management is highly dependent on the boundary between materials

4. Thermoelectrics ZT = figure of merit S = Seebeck coefficient σ = electrical conductivity k = thermal conductivity T = temperature

5. Thermal boundary resistance • Thermal boundary resistance creates a temperature drop, DT, across an interface between two different materials when a heat flux is applied. • First observed by Kapitza for a solid and liquid helium interface in 1941. A typical resistance of 10-9-10-7 m2K/W is equivalent to ~ 0.15-15 mm Si ~ 1-100 nm SiO2 T Mismatch in materials causes a resistance to heat flow across an interface.

6. B Voids, imperfect contact Tcold Thot A Thot DT Tcold Distance Two types of interface resistance Thermal Boundary Resistance Thermal Contact Resistance • Due to difference in the acoustic properties: Phonon reflection at the interface • Electron-phonon interaction • Present even in the case of perfect contact with no roughness • Microscopic quantity • Important for bulk surfaces • Macroscopic quantity • Due to imperfect contact or voids in microstructure Tcold Thot B A hBD= thermal boundary conductance 1/hBD = thermal boundary resistance Thot DT Tcold Distance

7. Major research objectives • the role of interface disorder on interfacial heat transfer • the effects of different phonon scattering mechanisms on interfacial heat transfer

8. Outline of presentation • Theory of phonon interfacial transport • Measurement of interfacial transport with the transient thermoreflectance (TTR) technique • Influence of atomic mixing on interfacial phonon transport • Influence of high temperatures on interfacial phonon transport • Summary

9. T Thermal conduction in bulk materials Thermal conduction Microscopic picture L Z l = mean free path [m] phonon-phonon scattering length in homogeneous material k = thermal conductivity [Wm-1K-1] = thermal flux [Wm-2] What happens if l is on the order of L?

10. T Z Thermal conduction in nanomaterials Microscopic picture of nanocomposite T Z < ln Ln q=hBDDT keffective of nanocomposite does not depend on phonon scattering in the individual materials but on phonon scattering at the interfaces hBD = thermal boundary conductance [Wm-2K-1] Change in material properties gives rise to hBD

11. Theory of hBD Phonon flux transmitted across interface 2 1 I q Phonon distribution Projects phonon transport perpendicular to interface Phonon speed [m s-1] Phonon energy [J] Phonon interfacial transmission Spectral phonon density of states [s m-3]

12. Diffuse scattering • Scattering completely diffuse • Elastically isotropic materials • Single phonon elastic scattering Diffuse Mismatch Model (DMM) E. T. Swartz and R. O. Pohl, 1989, "Thermal boundary resistance,” Reviews of Modern Physics, 61, 605-668. diffuse scattering – phonon “looses memory” when scattered T > 50 K and realistic interfaces Averaged properties in different crystallographic directions Is this assumption valid?

13. Maximum hBDwith elastic scattering Phonon radiation limit (PRL) Same assumptions as DMM DOS side 1 (softer) in DMM DOS side 2 (harder) in PRL DMM PRL

14. DMM and PRL calculations

15. Outline of presentation • Theory of phonon interfacial transport • Measurement of hBD with the TTR technique • Influence of atomic mixing on hBD • Influence of high temperatures (T > D) on hBD • Summary

16. Verdi V10 = 532 nm 10 W RegA 9000 tp ~ 190 fs single shot - 250 kHz 4 mJ/pulse Mira 900 tp ~ 190 fs @ 76 MHz l = 720-880 nm 16 nJ/pulse Verdi V5 = 532 nm 5 W Transient ThermoReflectance (TTR) Probe Beam Delay ~ 1500 ps l/2 plate Beam Splitter Sample Dovetail Prism Lenses Detector Polarizer Pump Beam Variable ND Filter Acousto-Optic Modulator Lock-in Amplifier Automated Data Acquisition System

17. Transient ThermoReflectance (TTR) Free Electrons Absorb Laser Radiation Ballistic Electron Transport PROBE HEATING “PUMP” Electron-Phonon Coupling (~2 ps) Electrons Transfer Energy to the Lattice Thermal Diffusion by Hot Electrons Thermal Equilibrium Thermal Diffusion within Thin Film Thermal Diffusion (~100 ps) FILM Thermal Diffusion Thermal Conductance Across the Film/Substrate Interface Thermal Boundary Conductance (~2 ns) SUBSTRATE Substrate Thermal Diffusion (~100 ps – 100 ns) Thermal Diffusion within Substrate Focus of current analysis

18. TTR data Free Electrons Absorb Laser Radiation Thermal Boundary Conductance (~1-10 ns) Thermal Conductance across the Film/Substrate Interface 50 nm Cr/Si

19. Stevens, Smith, Norris, JHT, 2005 Lyeo, Cahill, PRB, 2006 Stoner, Maris, PRB, 1993 New data DMM compared to experimental data Goal: investigate the over- and under-predictive trends of the DMM based on the single phonon elastic scattering assumption

20. Outline of presentation • Theory of phonon interfacial transport • Measurement of hBD with the TTR technique • Influence of atomic mixing on hBD • Influence of high temperatures (T > qD) on hBD • Summary

21. DMM assumptions Slight changes in deposition conditions can give rise to different elemental compositions around solid interfaces DMM Assumption Realistic Interface

22. AES depth profiles Cr/Si mixing layer 9.5 nm Cr-1: no backsputter Si change 9.7 %/nm Elemental Fraction Cr/Si mixing layer 14.8 nm Cr-2: backsputter Si change 16.4 %/nm Depth under Surface [nm]

23. Results from AES data

24. TTR testing P. E. Hopkins and P. M. Norris, Applied Physics Letters 89, 131909 (2006).

25. Decreasing hBD with increasing mixing layer thickness hBD results DMM predicts a constant hBD = 855 MWm-2K-1 Hopkins, Norris, Stevens, Beechem, and Graham, to appear in the Journal of Heat Transfer, 2008

26. Virtual crystal DMM The disordered region is replaced by a homogenized virtual crystal of thickness Dint having effective properties based on the disordered medium with MFP= lint. multiple scattering events from interatomic mixing T. E. Beechem, S. Graham, P. E. Hopkins, and P. M. Norris, Applied Physics Letters 90, 054104 (2007)

27. Virtual crystal DMM multiple scattering events from interatomic mixing In well-matched material systems such as Cr on Si, Rppis very small and on the same order as Rep, so this additional resistance must be considered and added in parallel with Rpp. G = electron-phonon coupling factor Majumdar and Reddy, APL, 2006

28. Virtual crystal DMM multiple scattering events from interatomic mixing Majumdar and Reddy, APL, 2006

29. VCDMM DMM predicts hBD that is almost 8 times larger than that measured on the samples and no dependence on mixing layer thickness or composition. The VCDMM calculations are within 18% of the measured values and show the same trend with mixing layer thickness as the measurements. Hopkins, Norris, Stevens, Beechem, and Graham, to appear in the Journal of Heat Transfer, 2008

30. Summary Investigate the role of interface disorder on interfacial heat transfer • Examined the effects of interfacial properties on hBDin the acoustically matched Cr/Si system with TTR • DMM predicts hBD855 MWm-2K-1 at room temperature • Measured data varies from 100-200 MWm-2K-1, depending on deposition conditions • Multiple phonon elastic scattering could cause this over- prediction of the DMM • DMM only takes into account single scattering events • DMM assumes a perfect interface, but interface disorder will increase the scattering thus decreasing the hBD • VCDMM is introduced and predicts same values and trends for Cr/Si at room temperature as experimental data

31. Stevens, Smith, Norris, JHT, 2005 Lyeo, Cahill, PRB, 2006 Stoner, Maris, PRB, 1993 New data Summary The presence of an interfacial mixing region causing multiple elastic scattering events which are not accounted for and may be the cause of the overestimation of the DMM in well matched material systems with Debye temperature ratios close to one. Goal: investigate the over- and under-predictive trends of the DMM based on the single phonon elastic scattering assumption

32. Outline of presentation • Theory of phonon interfacial transport • Measurement of hBD with the TTR technique • Influence of atomic mixing on hBD • Influence of high temperatures (T > qD) on hBD • Summary

33. Single phonon elastic scattering hBD from DMM limited by f1 f=T/qD f Linear in classical regime (T>qD) *Kittel, 1996, Fig. 5-1

34. Single phonon elastic scattering Elastic scattering – hBD is a function of f/T in lower qD material qDAl=428 K DMM Predictions f qDPb=105 K T/D

35. Molecular dynamics simulations Lennard-Jones Potential with Different Atomic Sizes Kr/Ar Superlattice Nanowire Stevens, Zhigilei, and Norris, IJHMT, 2007 Chen, Li, Yang, Wu, Lukes and Majumdar, Physica B, 2004 Computational results indicate a linear increase in conductance (decrease in resistance) with temperature.

36. Mismatched samples at low temperatures Lyeo and Cahill, PRB, 2006 Stoner and Maris, PRB, 1993

37. hBD results at temperatures above qD of the softer material Pt/AlN Pt/Al2O3 P. E. Hopkins, R. J. Stevens, and P. M. Norris, To appear in the Journal of Heat Transfer, HT (2008).

38. Analysis • Linear trend in MDS in classical regime (T>>qD) • MDS calculates hBD making no assumption of elastic scattering in interfacial phonon transport • Several samples show linear hBD trends around classical regime DMM f/T JOINT FREQUENCY DMM Substrate (diamond) Film (Pb) P. E. Hopkins and P. M. Norris, Nanoscale and Microscale Thermophysical Engineering 11, 247 (2007)

39. DMM vs. JFDMM DMM JFDMM

40. Summary Investigate the effects of different phonon scattering mechanisms on interfacial heat transfer • Measured hBD at different metal-dielectric interfaces with a range of acoustic similarity • Observed linear trend in hBD around qD • Evidence of inelastic scattering – not predicted with DMM • JFDMM takes into account substrate phonons – and provides better agreement with experimental data

41. Stevens, Smith, Norris, JHT, 2005 Lyeo, Cahill, PRB, 2006 Stoner, Maris, PRB, 1993 New data Summary The presence of inelastic scattering events, which add an additional channel of interfacial energy transport may be the cause of the underestimation of the DMM in mismatched material systems with distinctly different Debye temperatures. Goal: investigate the over- and under-predictive trends of the DMM based on the single phonon elastic scattering assumption

42. Outline of presentation • Theory of phonon interfacial transport • Measurement of hBD with the TTR technique • Influence of atomic mixing on hBD • Influence of high temperatures (T > qD) on hBD • Summary

43. Conclusions Investigate the role of interface disorder on interfacial heat transfer • Determined that interfacial mixing can play a role in phonon transport by inducing multiple phonon scattering events • Accurately described with VCDMM taking into account e-p resistance Investigate the effects of different phonon scattering mechanisms on interfacial heat transfer • Inelastic scattering contributes to hBDat temperatures close to qD of the softer material where substrate phonon population is still quantum mechanically increasing • Developed JFDMM to take into account some portion of inelastic scattering

44. Impact How does the knowledge of phonon scattering affect nanoapplications?

45. Acknowledgments • Pamela Norris, my doctoral advisor and head of the Microscale Heat Transfer laboratory at UVA • Funding from the National Science Foundation (NSF) Graduate Research Fellowship Program (GRFP) • Funding from the Virginia Space Grant Consortium (VSGC) • Collaborators: Leslie Phinney (Sandia), Robert Stevens (RIT), Samuel Graham (GaTech), Thomas Beechem (GaTech) Rob Kelly (UVA), Avik Ghosh (UVA), Mikiyas Tsegaye (UVA), David Cahill (UIUC), John Hostetler (Trumpf Photonics), Mike Klopf (Jefferson Lab), Vickie Connors (NASA Langley) • Microscale Heat Transfer Crew – Rich Salaway, Jennifer Simmons, John Duda, Justin Smoyer

46. PROBE HEATING “PUMP” FILM Thermal Diffusion SUBSTRATE Transient ThermoReflectance (TTR) Free Electrons Absorb Laser Radiation Focus of current analysis Ballistic Electron Transport Electron-Phonon Coupling (~2 ps) Electrons Transfer Energy to the Lattice Thermal Diffusion by Hot Electrons Thermal Equilibrium Thermal Diffusion within Thin Film Thermal Diffusion (~100 ps) Thermal Conductance across the Film/Substrate Interface Thermal Boundary Conductance (~2 ns) Substrate Thermal Diffusion (~100 ps – 100 ns) Thermal Diffusion within Substrate Focus of previous analysis

47. Energy transferred from e- system to l system Energy conducted through e- system Energy stored in e- system Energy deposited into e- system Electron-phonon coupling factor Energy gained by l system from e- system Energy stored in l system Electron-phonon (e-p) nonequlibrium PARABOLIC TWO-STEP MODEL (PTS) *Anisimov, 1974 time z

48. Relate temperature to reflectance DR/R = aDTe + bDTl – only valid for DTe < 150 K • Test at fluences up to 15 J m-2 • DTe in Au of up to ~ 4000 K • ITT 2.4 eV > 1.55 eV TTR energy Intraband reflectance model Valid for all electron temperatures Christensen, PRB, 1976 Smith and Norris, APL, 2001

49. Measure G in Au with TTR 20 nm Au/glass 20 nm Au/glass Different e-p equilibration curves for different fluences But G should be a material property???? Hopkins and Norris, App. Surf. Sci., 2007

50. Single phonon elastic scattering Simplifies transmission coefficient