Robert J. Stevens Department of Mechanical Engineering Rochester Institute of Technology

1. Modeling Thermal Transport at Single Interfaces and in Nanostructured Materials Using Non-equilibrium Molecular Dynamics Techniques Robert J. Stevens Department of Mechanical Engineering Rochester Institute of Technology RIT Research Computing Tech Group April 19, 2007

2. Outline • Motivation • NEMD Approach • Single Interfaces • Size effects • Comparison with theoretical models • Defects, temperature • Nanostructures (Si-Ge) • Summary and Future Plans

3. Macroscale  Nanoscale Nanoscale thermal transport is important when either the individual energy carriers must be considered and/or when continuum models break down. Bulk Nanostructure

4. Thermal Boundary Resistance • Mismatch in materials causes a resistance to heat flow across an interface. 10-9-10-7 m2K/W ~ 0.15-15 mm Si ~ 1-100 nm Si2O

5. Thermoelectrics *kagakukan.toshiba.co.jp Superlattice 1-100 nm * Berkeley Nano Engineering Research Program

6. Thermoelectric Performance

7. Other Applications • Vertical Cavity Surface Emitting Lasers • Optical storage • Micro-bolometers • Nanocomposites and nanostructures *Cahill, et al., 2002 *Li, et al., 2003 *Yang and Chen, 2004

8. Motivation Rq = Ratio of Film to Substrate Debye temperatures

9. Motivation

10. Computational Molecular Dynamics Approach • Numerically solve equation of motion for a system of interacting particles. • Rules are set up about how atoms interact with one another. • Precise knowledge and control over interface types. • Ability to vary one parameter to study its impact on interfacial thermal conductance. • Anharmonic potentials. • Probing at high spatial resolution is not possible by traditional experiments.

11. MD Approach

12. Molecular Dynamics Issues • Choice of interatomic potential (approximates). • Classical model, does not account for electron transport nor quantum effect of phonons. • Limit on system size (< 1mm), atomic spacing ~1-2 Å, system sizes 104 – 107. • Limit to short time scales (<100 ns), timesteps of ~1 fs.

13. y x z NEMD: Non-Equilibrium Molecular Dynamics • Build crystal, periodic in x-y direction • Either fixed or periodic in z direction • Apply heating and cooling to bath atoms by velocity scaling, constant heat flux, or Gaussian thermostat method. • System is allowed to come to steady state conditions and data collected over next ~5·106 time steps. • Temperatures, energy flux, and pressures are monitored.

14. Fundamental Interface Study • Lennard-Jones potential with cutoff at 2.5s. • Interfaces are oriented on the FCC (100) plane.

15. NEMD Approach

16. MDS Temperature

17. Errors, BC, and Size Effects • Statistical errors due to system noise for 6 million time step simulations was ~6-8%. • Crystal sizes of 5x5x40 were typically used, so size effect errors were less than statistical error. • Negligible differences between different BC and temperature regulation methods.

18. Transient MDS Experiment Results are similar to NEMD approach, but there is difficulty in defining thermal mass.

19. Results for Perfect Interface with Mass Mismatch

20. Fitting DMM to MDS Data

21. Comparison with DMM

22. Lattice Mismatch

23. Mixed Interface

24. Temperature Dependence

25. Experimental Evidence of Temperature Dependence

26. Modeling Real Structures (Si-Ge) 4,000 atoms LJ Potential Pair-wise potential 100,000 atoms SW Potential 3 body potential

27. LAMMPS • Large-scale Atomic/Molecular Massively Parallel Simulator • Developed in the mid 1990’s at Sandia National Laboratories as an open source C++ code, funded by DOE. http://lammps.sandia.gov/ • Distributed-memory message-passing parallelism (MPI). • Spatial decomposition of simulation domain using “ghost” atoms. • Has been used to model atomic, biological, metallic, and granular systems based on classical molecular dynamics. • Uses neighbor lists to reduce computational effort. • Velocity-Verlet integrator, with constant NVE, NVT, or NPT.

28. Modeling Real Structures (Si-Ge) • RIT Cluster • 47 IBM Xseries 330 Servers 2-1.4GHZ Pentium 3 Xeon Processors • 1 IBM Server for the Head Node 2-2.0GHZ Pentium 4 Xeon Processors

29. Modeling Real Structures (Si-Ge) Stillinger and Weber, Phys. Rev. B, 1985 Ding and Andersen, Phys. Rev. B., 1986 • Stillinger-Weber potential • Mixing rules Ethier and Lewis, J. Matr. Res., 1992

30. Crystal Size Impact

31. Si-Ge Superlattices

32. Si-Ge Superlattices

33. Si-Ge Superlattices

34. Summary and Future Plans • NEMD is one means of exploring thermal transport at interfaces and nanostructured materials. • For LJ interfaces with defects when compared to DMM, partially captures the trend seen in real interfaces. • Thermal boundary conductance is linearly dependent with temperature in the classical limit, indicating potential role for inelastic scattering mechanisms for thermal transport at LJ interfaces. • Stillinger-Weber potential with NEMD predicts bulk conductivity of Si well. Still need to confirm for Ge material using naturally occurring isotope breakdown. • Reduced effective thermal conductivity for SiGe superlattice as period size is reduced. Results compare with existing experimental data on SiGe SL. • Did not observe reduction in thermal conductivity when increasing superlattice period above 10 nm, as observed experimentally. • Need to explore size impact on SiGe SL results. • Expand simulations to examine nanocomposite materials. • Temperature dependence in SL.

35. Acknowledgements • Leonid Zhigilei, University of Virginia • Patrick Hopkins, University of Virginia • Rick Bohn and Gurcharan Khanna, RIT • New Faculty Development Funds, RIT • Steve Plimpton, Sandia National Lab, LAMMPS

36. DMM Calculations for Debye Approximation

37. Transmission Coefficient-DMM Interface scattering with no memory: Principle of detailed balance: = Debye Model: