http://exciting-code.org Crystal Relaxation, Elasticity, and Lattice Dynamics Pasquale Pavone Humboldt-Universität zu Berlin
http://exciting-code.org PART I: Structure Optimization Pasquale Pavone Humboldt-Universität zu Berlin
Outline Part I Structure optimization Cell optimization Internal degrees of freedom Relaxing molecules HoW exciting! 2016
Structure Optimization (a): Cell shape stress (b): (Relative) atomic positions force
Energy Minimization parabola paraboloid
Energy Minimization
Lattice (Cell) Optimization E = E(a, b, c, α, β, γ) E = E(V, b, c, α, β, γ)
Equation of State (EOS) E = E(V) Murnaghan EOS X Birch-Murnaghan EOS Vinet EOS Polynomial EOS
Equation of State of Silver
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...
Energy Minimization: Relaxation Internal degree of freedom: atomic positions
Relaxation methods in newton harmonic bfgs
newton
newton
harmonic A parabola has a constant 2nd derivative
bfgs
bfgs Extension to N-degrees of freedom: Similar to harmonic Hessian matrix vs. 2nd derivative Very efficient if close to minimum Default in exciting
input.xml <input> <structure /> <groundstate /> <relax method= bfgs > <relax/> </input>
Relaxation of Pyridine
for Molecules water carbon dioxide pyridine sexithiophene
for Molecules In exciting always 3D periodicity:
for Molecules Isolated 3D periodical molecules:
input.xml <input> <structure cartesian= true > </structure> <groundstate ngridk= 1 1 1 > </groundstate> <relax/> </input>
Relaxation of Pyridine
Visualization
Molecular Orbitals of CO 2
Molecular Orbitals of CO 2 HOMO
Molecular Orbitals of CO 2 LUMO
Tutorials
http://exciting-code.org PART II: Lattice Dynamics Pasquale Pavone Humboldt-Universität zu Berlin
Phonons: Atoms Move Together A phonon in a crystal is a coherent collection of atomic displacements.
Phonons: Atoms Move Together
Phonons Oscillations Oscillation in space: q = wavevector Oscillation in time: ω = frequency ( energy)
Why Are Phonons Interesting
Atomic Displacements R u(r) R u(r )
Energy, Forces, and Phonons {u} = set of atomic displacements E({u}) E u 2 E u u' E F Φ
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
Phonon Periodicity Equilibrium q = 0 q 0
Phonon Periodicity Equilibrium q = 0 q 0
Phonon at Γ in Diamond Structure
Phonon Periodicity Equilibrium q = 0 q 0
Phonon at X in Diamond Structure
Phonon at X in Diamond Structure X
input.xml <input> <phonons ngridq= 2 2 2 > </phonons> </input>
Reciprocal and Real-Space Cells Brillouin zone sampling (Super)cell
Reciprocal and Real-Space Cells Brillouin zone sampling (Super)cell
Reciprocal and Real-Space Cells Brillouin zone sampling (Super)cell
input.xml <input> <phonons ngridq= 2 2 2 > <phonondos > </phonondos> </phonons> </input>
input.xml <input> <phonons ngridq= 2 2 2 > <phonondos > </phonondos> <phonondispplot > /> </phonons> </input>
Phonon Dispersion: Diamond small ngridk + small ngridq
Phonon Dispersion: Diamond larger ngridk + larger ngridq
http://exciting-code.org PART III: Elasticity Pasquale Pavone Humboldt-Universität zu Berlin
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
Strain: State of deformation Equilibrium: Zero strain Zero forces Zero stress Zero displacements Shear strain Uniaxial strain
Homogeneous strain r = r s = unstrained position strained position r s = F r = 1 + ε r F = Deformation Matrix ε = Physical Strain Matrix
Voigt notation Voigt indices: i, j = xx yy zz yz or zy xz or zx xy or yx α = 1 2 3 4 5 6 Representative vector: ε = (ε 1, ε 2, ε 3, ε 4, ε 5, ε 6 )
Strain definitions Physical strain: r r s r s = 1 + ε r Lagrangian strain: Δ η = ε + 1 2 ε ε Δ s Δ s 2 Δ 2 = Δ 2η Δ
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
Numerical derivatives Fitting a polynomial to the calculated points
Tutorials
Tutorials
Tutorials
Animated Phonon Modes
The Last Slide: We are so Excited!... and let s listen now to Andris!