1. Computational Solid State Physics 計算物性学特論 第２回 2.Interaction between atoms and the lattice properties of crystals

2. Atomic interaction Lennard-Jones potential: for inert gas atoms: He, Ne, Ar, Kr, Xe Stillinger- Weber potential: for covalent bonding atoms: C, Si, Ge

3. Lennard-Jones potential (1) r / σ V LJ /ε V LJ (r) minimum at r: inter-atomic distance repulsive force attractive force

4. 1 st term: repulsive interaction caused by Pauli ’ s principle 2 nd term: Van der Waals interaction (attractive) E 1 : electric field generated by a temporal dipole moment p 1 p1p1 r p 2 (r) temporal dipole moment induced dipole moment Lennard-Jones potential (2)

5. 1-dimensional crystal ax a: lattice constant Energy per atom: cohesive energy ε c =1.04ε E minimum at a=1.12σ

6. Bulk modulus B : Bulk modulus N : the number of atoms in a crystal a : lattice constant

7. Lattice vibration a xnxn displacement x Na: length of a crystal The first derivative of the inter-atomic potential vanishes because atoms are located at the equilibrium positions. The second derivative of the inter-atomic potential gives the spring constant κ between atoms. assume: neglect the 2nd neighbor interaction

8. Equation of motion for atoms m: mass of an atom a x n-1 x n x n+1 Force on the n-th atom: Equation of motion for atoms:

9. Solution for equation of motion Periodic boundary condition: Assume: 1 st Brillouin zone N modes k: wave vector

10. Dispersion relation of lattice vibration ka ω(k)/ω 0 sound velocity: phase velocity at k=0 acoustic mode v becomes larger for larger κ and smaller m.

11. Phonon Energy quantization of lattice vibration l=0,1,2,3 Bose distribution function for phonon number: for :zero point oscillation

12. Role of the acoustic phonon in semiconductors at a room temperature Main electron scattering mechanism in crystals Determine the lattice heat capacity Determine the thermal conductivity

13. Lattice heat capacity: Debye model (1) Density of states of acoustic phonos for 1 polarization Debye temperature θ N: number of unit cell N k : Allowed number of k points in a sphere with a radius k phonon dispersion relation

14. Thermal energy U and lattice heat capacity C V : Debye model (2) 3 polarizations for acoustic modes

15. ・ Low temperature T<<θ ・ High temperature T>>θ Equipartition law: energy per 1 freedom is k B T/2 Debye model (3)

16. Heat capacity C V of the Debye approximation: Debye model (4) k B =1.38x10 -23 JK -1 k B mol=7.70JK -1 3k B mol=23.1JK -1

17. Heat capacity of Si, Ge and solid Ar: Debye model (5) cal/mol K=4.185J/mol K 3k B mol=5.52cal K -1 Si and Ge Solid Ar

18. Thermal conductivity (1) T: temperature c: heat capacity per particle n: average number of phonons v: group velocity of phonon τ: scattering time Diffusive energy flux x 3k B T(x) vxτvxτ c v x τdT/dx Energy Energy emission

19. Thermal conductivity (2) K is largest for diamond because of the high sound velocity! C: heat capacity per unit volume, l=vτ: phonon mean free path v: sound velocity of acoustic phonon Thermal conductivity coefficient

20. Molecular dynamics simulation for atoms Equation of motion for atoms: r: position of an atom v: velocity a: acceleration F: force t: time m: mass of an atom

21. (1) velocity Verlet ’ s method Time evolution for small time interval :

22. Proof of (1)

23. (2) Verlet method Time evolution for small time interval

24. Temperature Equipartition theorem Temperature is determined from the average kinetic energy.

25. Periodic boundary condition 2-dimensional system

26. Trajectories of 20 atoms interacting via Lennard-Jones potential

27. Setting of energy and temperature triangular crystal melting

28. formation of triangular crystal Time-lapse snapshots of interacting particles (1)

29. melting Time-lapse snapshots with increasing Temperatures (2)

30. Problems 2-1 Calculate two branches of the dispersion relation of the lattice vibration for a diatomic linear lattice using a simple spring model, and describe the characteristics of each branch. Calculate the dispersion relation for a graphen sheet using a simple spring model between nearest neighbor atoms. Study the role of the optical phonon in semiconductor physics.

31. Problems 2-2 Find the most stable 2-dimensional crystal structure, using the Lennard Jones potential. Find the most stable 3-dimensional crystal structure, using the Lennard Jones potential. Write a computer simulation program to study the motion of 3 atoms interacting with Lennard-Jones potential. Assume the space of motion to be within a 2-dimensional square region.

32. Problems 2-3 Study experimental methods to observe the dispersion relation of phonons. Study the phonon dispersion relations for Si and Ge crystals and discuss about the similarity and the difference between them. Study the phonon dispersion relations for Ge and GaAs crystals and discuss about the similarity and the difference between them.