Gang Zhang Institute of High Performance Computing, Singapore zhangg@ihpc.a-star.sg

1. Nanoscale Heat Energy Transport: A Computational Study Gang Zhang Institute of High Performance Computing, Singapore zhangg@ihpc.a-star.edu.sg

2. Contributions to the heat conduction Wiedemann-Franz law: contribution from electron to thermal conductivity

3. Outline Lattice Vibration and Phonons Phonon Band Structure Thermal Conductivity Calculations

4. Section 1: Lattice Vibration and Phonons Atomic Vibration 1D single atom chain 1D diatomic chain

5. X=A sin ωt X The physics of oscillations and waves Harmonic oscillator in classical mechanics: Equation of motion: Hooke’s law Solution x

6. Traveling plane waves: Y 0 X X=0: t=0:

7. Transverse wave Longitudinal wave

8. Atom vibration Consider two neighboring atoms that share a chemical bond The bond is like a spring with an energy relationship

9. Pair Potential Lennard-Jones (LJ) potential is a widely used pair potential characteristic energy characteristic length unit the interatomic distance r

10. the equilibrium distance in LJ potential the well depth

11. Parameters in Lennard-Jones potential for different systems

12. Many-body potential Stillinger-Weber (SW) potential It consists of both two-body and three-body terms defined as The reduced pair potential f2 is where a is the cut-off distance beyond which interaction vanishes

13. Many-body potential Tersoff potential: a new approach to model covalent interaction Bond order: the strength of a bond between two atoms depends on the local environment. fR and fA describe the repulsive and attractive forces, respectively, and fC is the cut-off function. The bond order term bij is defined as

14. Near the minimum, the energy is well approximated by a parabola Spring constant

15. Section 1: Lattice Vibration and Phonons Atomic Vibration 1D single atom chain 1D diatomic chain

16. n (n-2) (n-1) (n+2) (n+1) Lattice Dynamics Now consider a one-dimensional atomic chain a One-dimensional Lattice model

17. n n Total force driving atom n back to equilibrium equation of motion Solution of continuous wave equation Boundary conditions Born-Karman: assume that the ends of the chain are connected

18. Then the boundary conditions become Where n is an integer λ is the vibration wavelength Wave vector

19. Solution to the Equations of Motion Substitute exponential solution into equation of motion The dispersion relation for acoustic phonons relates phonon frequency (energy) to wave vector (wavelength)

20. Note: here pictures of transversal waves although calculation for the longitudinal case k Continuum limit of acoustic waves:

21. k Region is called first Brillouin zone , here h=1

22. Section 1: Lattice Vibration and Phonons Atomic Vibration 1D single atom chain 1D diatomic chain

23. a D u2n u2n+1 u2n-2 u2n-1 u2n+2 Vibrational Spectrum for structures with 2 or more atoms/unit cell Linear diatomic chain: 2n 2n-2 2n-1 2n+1 2n+2 2a Equation of motion for atoms on even positions: Equation of motion for atoms on odd positions: Solution with: and

24. 2 2

25. Acoustic wave, K~0： Optical wave, K~0：

26. Optic Mode Atomic Displacement Acoustic Mode Atomic Displacement

27. vibrational modes quantized phonons Phonons are quantized lattice vibrations

28. 1 longitudinal 2 transverse acoustic Dispersion curves of 3D crystals • 3D crystal: clear separation into longitudinal and transverse mode only possible in • particular symmetry directions sound waves of elastic theory • Every crystal has 3 acoustic branches further 3 optical branches • Every additional atom of the unit cell again 2 transverse 1 longitudinal n atoms per unit cell: 3 acoustic branches + 3(n-1) optical branches = 3n branches 1LA +2TA (n-1)LO +2(n-1)TO

29. Transversal Optical degenerated Transversal Acoustic degenerated (0,0,0) Part of the phonon dispersion relation of diamond Longitudinal Optical Longitudinal Acoustic diamond lattice: fcc lattice with basis n=2 2x3=6 branches expected

30. Section 1: Lattice Vibration and Phonons Atomic Vibration 1D single atom chain 1D diatomic chain QUESTIONS?

31. Outline Lattice Vibration and Phonons Phonon Band Structure Thermal Conductivity Calculations

32. Section II: Phonon Band Structure What we can get? How to calculate?

33. Phonon dispersion of single-layer graphene Phonon group velocity Phonon Density of States specific heat

34. Density of Phonon States Consider a 1D chain of total length L carrying M atoms at a separation a Born-von Karman: assume that the ends of the chain are connected

35. The phonon density of states gives the number of modes per unit frequency per unit volume of real space

36. How to calculate? VASP (Vienna Ab Initio Simulation Package) – A Plane Wave Density Functional Code

37. Brief History of Density Functional Theory (DFT) • First suggested by Fermi in 1929 that the total Energy of an electronic system can be determined by the electron density • In 1960’s the exact relationship between electron density and energy. Hohenberg-Kohn (1964); Kohn-Sham (1965) • The ground state energy of a system is a unique functional of the electron density. • The ground state energy can be obtained variationally: the density that minimizes the ground state energy is the exact ground state density. • Kohn was awarded the 1998 Nobel Prize in chemistry for DFT • DFT software, such as VASP, became commercially available (~1996).

38. What is VASP? One of the software packages that uses DFT to solve the quantum problem for materials – Uses periodic boundary conditions – Uses pseudopotential method with a plane waves basis set – Can model systems with maximum no. of atoms in the range of 100-200

39. VASP input files • INCAR – User specified parameters that define the calculation • Global break condition, Energy cutoff, Smearing, Ionic and geometric relaxation parameters • POSCAR – Specifies the periodic simulation cell – Information regarding geometry of the system • POTCAR – Pseudopotential (PP) file – Information on PP and XC functional • KPOINTS – Defines k-point mesh

40. INCAR file INCAR is the central input file of VASP. It determines what to do and how to do it The INCAR file for a Copper surface calculation

41. INCAR file

42. POSCAR file This file contains the lattice geometry and the atom positions, also starting velocities for molecular dynamics calculations lattice constant: scale all lattice vectors and all atomic coordinates the three lattice vectors defining the unit cell of the system the number of atoms per atomic species Positions of atoms

43. POTCAR file The POTCAR file contains the pseudopotential for each atomic species used in the calculation For all elements VASP provides PP files in different flavors – US-PP, PAW-LDA, PAW-GGA, PAW-PBE • If multiple species of atoms are in the system – Concatenate the same type of POTCAR files to make a single POTCAR file – The order of the POTCAR types should correspond to the ordering of atoms in POSCAR

44. KPOINTS file The file KPOINTS must contain the k-point coordinates and weights the number of k-points whether the coordinates are given in cartesian or reciprocal coordinates coordinates and the (symmetry degeneration) weight for each k-points

45. OUTPUT files • CONTCAR – Contains position of the system after the calculation has completed • OSZICAR – Contains data of electronic steps and ionic steps • OUTCAR – Complete output of run including input file data. • CHGCAR – Charge density of system after run

46. Other DFT packages – Quantum espresso (www.quantum-espresso.org) – Abinit( www.abinit.org) – Siesta (http://icmab.cat/leem/siesta/)

47. Section II: Phonon Band Structure What we can get? How to calculate? QUESTIONS?

48. Outline Lattice Vibration and Phonons Phonon Band Structure Thermal Conductivity Calculations

49. Section III: Thermal Conductivity Calculations Phonon Boltzmann Equation Molecular Dynamics NEGF

50. Phonon Boltzmann equation The equilibrium phonon distribution is given by the Bose–Einstein distribution