Crystal Relaxation, Elasticity, and Lattice Dynamics

Transcription

1 Crystal Relaxation, Elasticity, and Lattice Dynamics Pasquale Pavone Humboldt-Universität zu Berlin

2 PART I: Structure Optimization Pasquale Pavone Humboldt-Universität zu Berlin

3 Outline Part I Structure optimization Cell optimization Internal degrees of freedom Relaxing molecules HoW exciting! 2016

4 Structure Optimization (a): Cell shape stress (b): (Relative) atomic positions force

5 Energy Minimization parabola paraboloid

6 Energy Minimization

7 Lattice (Cell) Optimization E = E(a, b, c, α, β, γ) E = E(V, b, c, α, β, γ)

8 Equation of State (EOS) E = E(V) Murnaghan EOS X Birch-Murnaghan EOS Vinet EOS Polynomial EOS

9 Equation of State of Silver

10 Lattice Optimization in Tool: OPTIMIZE-lattice.sh Example E = E V, c STEP1: opt. V at fixed c: get V 1 STEP2: opt. c at fixed V 1 : get c 2 STEP3: opt. V at fixed c 2 : get V 3...

11 Energy Minimization: Relaxation Internal degree of freedom: atomic positions

12 Relaxation methods in newton harmonic bfgs

13 newton

14 newton

15 harmonic A parabola has a constant 2nd derivative

16 bfgs

17 bfgs Extension to N-degrees of freedom: Similar to harmonic Hessian matrix vs. 2nd derivative Very efficient if close to minimum Default in exciting

18 input.xml <input> <structure /> <groundstate /> <relax method= bfgs > <relax/> </input>

19 Relaxation of Pyridine

20 for Molecules water carbon dioxide pyridine sexithiophene

21 for Molecules In exciting always 3D periodicity:

22 for Molecules Isolated 3D periodical molecules:

23 input.xml <input> <structure cartesian= true > </structure> <groundstate ngridk= > </groundstate> <relax/> </input>

24 Relaxation of Pyridine

25 Visualization

26 Molecular Orbitals of CO 2

27 Molecular Orbitals of CO 2 HOMO

28 Molecular Orbitals of CO 2 LUMO

29 Tutorials

30 PART II: Lattice Dynamics Pasquale Pavone Humboldt-Universität zu Berlin

31 Phonons: Atoms Move Together A phonon in a crystal is a coherent collection of atomic displacements.

32 Phonons: Atoms Move Together

33 Phonons Oscillations Oscillation in space: q = wavevector Oscillation in time: ω = frequency ( energy)

34 Why Are Phonons Interesting

35 Atomic Displacements R u(r) R u(r )

36 Energy, Forces, and Phonons {u} = set of atomic displacements E({u}) E u 2 E u u' E F Φ

37 Force Constants & Dynamical Matrix Φ(R R ) = 2 E u(r) u(r ) D q = Fourier transform of Φ(R) det D q ω 2 q I = 0

38 Phonon Periodicity Equilibrium q = 0 q 0

39 Phonon Periodicity Equilibrium q = 0 q 0

40 Phonon at Γ in Diamond Structure

41 Phonon Periodicity Equilibrium q = 0 q 0

42 Phonon at X in Diamond Structure

43 Phonon at X in Diamond Structure X

44 input.xml <input> <phonons ngridq= > </phonons> </input>

45 Reciprocal and Real-Space Cells Brillouin zone sampling (Super)cell

46 Reciprocal and Real-Space Cells Brillouin zone sampling (Super)cell

47 Reciprocal and Real-Space Cells Brillouin zone sampling (Super)cell

48 input.xml <input> <phonons ngridq= > <phonondos > </phonondos> </phonons> </input>

49 input.xml <input> <phonons ngridq= > <phonondos > </phonondos> <phonondispplot > /> </phonons> </input>

50 Phonon Dispersion: Diamond small ngridk + small ngridq

51 Phonon Dispersion: Diamond larger ngridk + larger ngridq

52 PART III: Elasticity Pasquale Pavone Humboldt-Universität zu Berlin

53 What Is Elasticity? Description of distorsions of rigid bodies and of the energy, forces, and fluctuations arising from these distorsions. Describes mechanics of extended bodies from the macroscopic to the microscopic. Generalizes simple mechanical concepts Force Stress Displacement Strain

54 Strain: State of deformation Equilibrium: Zero strain Zero forces Zero stress Zero displacements Shear strain Uniaxial strain

55 Homogeneous strain r = r s = unstrained position strained position r s = F r = 1 + ε r F = Deformation Matrix ε = Physical Strain Matrix

56 Voigt notation Voigt indices: i, j = xx yy zz yz or zy xz or zx xy or yx α = Representative vector: ε = (ε 1, ε 2, ε 3, ε 4, ε 5, ε 6 )

57 Strain definitions Physical strain: r r s r s = 1 + ε r Lagrangian strain: Δ η = ε ε ε Δ s Δ s 2 Δ 2 = Δ 2η Δ

58 Linear elastic response Low pressure expansion in terms of Lagrangian strain η : E η = E 0 + V 0 2! η C 2 η + E 0, V 0 = Reference (equilibrium) energy and volume Linear elastic constant (2nd order): C (2) = 1 V 0 2 E(η) η η η=0 Diamond C 11, C 12, C 44

59 Numerical derivatives Fitting a polynomial to the calculated points

60 Tutorials

61 Tutorials

62 Tutorials

63 Animated Phonon Modes

64 The Last Slide: We are so Excited!... and let s listen now to Andris!