1 / 54

Predicting Phonon Properties from Molecular Dynamics Simulations Alan McGaughey Department of Mechanical Engineering Car

Predicting Phonon Properties from Molecular Dynamics Simulations Alan McGaughey Department of Mechanical Engineering Carnegie Mellon University, Pittsburgh PA 2 nd International Conference on Phononics and Thermal Energy Science Tongji University, Shanghai, China May 27, 2014.

wood
Download Presentation

Predicting Phonon Properties from Molecular Dynamics Simulations Alan McGaughey Department of Mechanical Engineering Car

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Predicting Phonon Propertiesfrom Molecular Dynamics Simulations Alan McGaughey Department of Mechanical Engineering Carnegie Mellon University, Pittsburgh PA 2nd International Conference on Phononics and Thermal Energy Science Tongji University, Shanghai, China May 27, 2014

  2. Acknowledgements • Collaborators • Cristina Amon (Toronto) • Massoud Kaviany (Michigan) • Funding • Department of Energy (U. Michigan) • Air Force Office of Scientific Research YIP • Harrington Fellowship at UT Austin • National Science Foundation (Landry, Thomas fellowships) Students • Dr. Eric Landry (→ UTRC) • Dr. John Thomas (→ JHU APL) • Dr. Joe Turney (w/ C. Amon) (→ UTRC) • Dr. Dan Sellan (Toronto → UT Austin) • Dr. Jason Larkin (→ SpiralGen) • Alexandre Massicotte (CMU M.S.) • Kevin Parrish (CMU Ph.D.) • Sam Huberman (Toronto→ MIT) • Ryan Iutzi (Waterloo → MIT) Colleagues • Bo Qiu and Xiulin Ruan (Purdue) • Junichiro Shiomi (Tokyo)

  3. Outline • Phonons and Thermal Conductivity • Harmonic Lattice Dynamics Calculations • Phonon Lifetimes from MD • Lifetimes from Anharmonic Lattice Dynamics • Summary

  4. Review of Monday Lecture • MD simulation • Use Newton’s laws of motion to generate atomic trajectories • Limited by classical statistics, lack of good potentials • Can access huge systems, naturally include disorder, full anharmonicity • Thermal conductivity prediction • Green-Kubo method (equilibrium) • Direct Method (non-equilibrium) • Today: Predicting phonon properties from the atomic trajectories • Need harmonic lattice dynamics calculations (GULP)

  5. Model System: LJ Argon Computationally fast, so good for methodology development Will consider crystal, alloy, and amorphous phases as a case study. Melting temperature ~ 90 K. http://www.cmbi.ru.nl/redock/

  6. What is a Phonon? • Quantized lattice vibration with energyħω • Wave number (=2p/wavelength), frequency, and polarization → define the phonon mode • Phonons are the primary carriers of thermal energy in semiconductors and dielectrics (e.g., Si, GaN, graphene, quartz). • There is a spectrum of phonons in every material.

  7. Phonon Formula for Thermal Conductivity Boltzmann transport equation + Fourier law

  8. Specific Heat • Phonons are bosons, described by Bose-Einstein statistics • Energy of phonon mode i is • Specific heat is Classical, Harmonic Limit

  9. Quantum vs. Classical Statistics MD simulations are classical • High temperature limit of Bose-Einstein, x = ħw/kBT→ 0 • Equipartition of energy in a harmonic system

  10. Dispersion and Group Velocity • With wave vectors and frequencies, build dispersion curves. • Gradient gives the group velocity: Wang et al., Eur. Phys. J. B62, 381 (2008) LJ Crystal McGaughey,Ph.D. thesis, U. Michigan (2004).

  11. Phonon-Phonon Scattering Two phonons can combine to form a third (and vice-versa) Reciprocal lattice vector • Energy conservation: • Translational invariance of the lattice: • Interactions with > 3 phonons possible, rates increase with temperature

  12. Atomistic Approaches How are atomic trajectories & dynamics related to phonons and their properties? • Phonon → Abstraction • Atom → Real

  13. Outline • Phonons and Thermal Conductivity • Harmonic Lattice Dynamics Calculations • Phonon Lifetimes from MD • Lifetimes from Anharmonic Lattice Dynamics • Summary

  14. MD is Like a Mass-Spring System • 1. Equations of Motion: • 2. Convert to1storder equations: • 3. Solve numerically

  15. Alternative Perspective Find the natural frequencies and mode shapes for m = k = 1. • Equations of motion: • Assume harmonic solution: • Eigenvalue problem: Solution 1: • Solution 2:

  16. Normal Modes • Solution of form: • (coordinate transform) • Total potential energy = • Coordinates decoupled!

  17. Lattice Dynamics on a Crystal Lattice • Building from the mass-spring system, include: • Motion in three-dimensions • More than nearest-neighbor interactions • Non-linear interactions • Periodic boundary conditions • Expand the system potential energy as a Taylor series: 2nd Order Force Constant 3rd Order Force Constant

  18. Harmonic Lattice Dynamics Phonon Coordinate Polarization Vector A transformation that decouples the coordinates into independent harmonic oscillators. Dove,Introduction to Lattice Dynamics(Cambridge, 1993).

  19. Finding k, w, and e • Wave vectors from crystal structure • Harmonic equations of motion for all atoms for a given k • Eigenvalue problem (computationally straightforward) • Dynamical matrix, D, contains the 2nd order (harmonic) force constants • Can now determine specific heat and group velocity

  20. Outline • Phonons and Thermal Conductivity • Harmonic Lattice Dynamics Calculations • Phonon Lifetimes from MD • Lifetimes from Anharmonic Lattice Dynamics • Summary

  21. Normal Mode Decomposition 1. Run an MD simulation, extract atomic velocities. 2. Project velocities onto normal modes from harmonic lattice dynamics. Number of phonon modes depends on size of MD system. • Ladd et al., PRB34 (1986) 5058; • McGaughey & Kaviany, PRB69 (2004) 094303; • Larkin et al., JCTN11 (2014) 257.

  22. Normal Mode Decomposition (cont.) • 3. Normal mode kinetic energy power spectrum • 4. Fit to Lorentzian • 5. t = 1/(2G)

  23. LJ Crystal Lifetimes, T =10 K McGaughey & Larkin, to appear in Annual Review of Heat Transfer, Volume 17.

  24. Size Effects McGaughey & Larkin, to appear in Annual Review of Heat Transfer, Volume 17.

  25. Comparison to Green-Kubo and Direct Method • Deviations above 40 K due to projection onto harmonic modes. Turney et al., PRB79, 075316(2009)

  26. Choose Any Unit Cell McGaughey & Larkin, to appear in Annual Review of Heat Transfer, Volume 17.

  27. Application to Alloys • Can choose any unit cell -> explicit inclusion of disorder Larkin and McGaughey,JAP 114 (2013) 023507.

  28. Application to Amorphous Solids Larkin and McGaughey,PRB 89 (2014) 144303.

  29. Work by Others • Silicon: mean free path spectrum • Henry and Chen, JCTN5 (2008) 1-12 • PbTe, Bi2Te3, GaAs, CNT, zeolites, nanowires • Polyethylene: divergent thermal conductivity? • Henry and Chen, PRL101 (2008) 235502 • Henry and Chen, PRB79 (2009) 144305

  30. Outline • Phonons and Thermal Conductivity • Harmonic Lattice Dynamics Calculations • Phonon Lifetimes from MD • Lifetimes from Anharmonic Lattice Dynamics • Summary

  31. Anharmonic Lattice Dynamics • Force constants from potentials or DFT • Naturally include quantum statistics • Iteratively solve BTE to go beyond relaxation time approximation • Limited to low/medium temperatures • Cannot include disorder explicitly • Computational challenges for large unit cells Extensive work by David Broido and co-workers.

  32. LJ Crystal Comparison Turney et al., PRB79, 075316(2009)

  33. Silicon Thermal Conductivity from DFT+ALD Esfarjani et al., PRB84, 085204 (2011).

  34. GaN Thermal Conductivity from DFT+ALD Lindsay et al.,PRL 109 (2012) 095901.

  35. Outline • Phonons and Thermal Conductivity • Harmonic Lattice Dynamics Calculations • Phonon Lifetimes from MD • Lifetimes from Anharmonic Lattice Dynamics • Summary

  36. What We Didn’t Discuss Molecular dynamics simulations can be used to study: • Thermal transport across interfaces and thin films • Energy/momentum accommodation between gases and solids • Phase change • Fluid flow • Mechanics • … Anything meaningful aboutanharmonic lattice dynamics

  37. Outlook MD simulation is an extremely powerful tool for studying thermal transport by phonons: • No assumptions about nature of thermal transport • Can access huge systems not possible with lattice dynamics/DFT • Naturally include disorder (not just a perturbation) • Full anharmonicity -> high temperatures • High fidelity potentials are emerging High quality, open source code: LAMMPS and GULP Don’t limit yourself to MD simulation – develop a toolbox. Lattice dynamics techniques are extremely powerful

  38. Suggested Reading • Dove, Introduction to Lattice Dynamics (Cambridge, 1993) • A. J. H. McGaughey and J. M. Larkin, “Predicting phonon properties from equilibrium molecular dynamics simulations." To appear in Annual Reviews of Heat Transfer, Volume 17. Available at ntpl.me.cmu.edu • Ladd et al., “Lattice thermal conductivity: A comparison of molecular dynamics and anharmonic lattice dynamics.” Physical Review B 34 (1986) 5058.

  39. Alan McGaughey Department of Mechanical EngineeringCarnegie Mellon University, Pittsburgh PA mcgaughey@cmu.edu ntpl.me.cmu.edu

  40. Gray approximation • Assume that all phonons have the same properties: • Silicon from DFT • Esfarjani et al., PRB84, 085204 (2011)

  41. More SW Silicon Performance Phonon dispersion curves are related to the atomic vibrations. From Eric Landry’s PhD thesis (CMU, 2009).

  42. Density Functional Theory Atomic Scale Force Constants Lattice Dynamics Calculations Phonon Frequencies & Mean Free Paths Meso-Scale Boundary Scattering Model Boltzmann Transport Equation Thermal Conductivity

  43. Challenge #2: Brillouin zone resolution • Need enough phonon modes to represent bulk • All materials seem to behave differently (minimal effect for silicon) (8,8) CNT, lowest k=0 optical mode LJ argon: Turney et al.,PRB 79 (2009) 064301

  44. Stillinger-Weber Potential for Silicon • F. H. Stillinger and T. A. Weber, “Computer simulation of local order in condensed phases of silicon,” PRB31, 5262 (1985). • Cited more than 2500 times. • Diamond as stable allotrope at low pressure, lattice constant, atomization energy • Melting point and liquid structure in reasonable agreement with experiment

  45. Form of Stillinger-Weber Potential

  46. Performance of SW Silicon From Eric Landry’s PhD thesis (CMU, 2009). Gruneisenparameters are related to thermal expansion. From Porter et al., JAP81 (1997) 96-106.

  47. Silicon Thermal Conductivity from Potentials Broido et al., Physical Review B72, 014308(2005).

  48. Density Functional Theory • Approximate numerical solution to the many-body Schrödinger equation(bodies = electrons and ions) • Theoretically and computationally complex (large number of degrees of freedom) • Expertise is needed to set up and run the calculations. • Graduate students can write a molecular dynamics or harmonic lattice dynamics code in a one-semester class. (not possible for DFT)

  49. Anharmonic Lattice Dynamics Perturb the harmonic solution using 3rd- and 4th-order force constants • Allows phonon modes to interact • Get the frequency shift, D, and linewidth, G = 1/2t • Quantum effects naturally included Wallace,Thermodynamics of Crystals (Wiley, 1972).

  50. Challenge: Satisfy Conservation Rules Finite number of phonon modes in computation Principal value: • is mode dependent • Found using an iterative procedure • Related to relaxation times Delta function: Turney et al.,PRB 79 (2009) 064301. • Challenging for large phonon band gap, range of frequencies. • Other approaches: Chernatynskiy and Phillpot, PRB82, 134301 (2010).

More Related