Introduction to phonopy

Transcription

1 Introduction to phonopy

2 Installation Download phonopy Install Python libraries that phonopy relies Install phonopy % sudo apt-get install python-dev python-numpy \ python-matplotlib python-tk python-lxml python-yaml % tar xvfz phonopy tar.gz % cd phonopy % python setup.py install home=. Add the phonopy library directory (e.g., ~/phonopy-0.9.3/lib/python) to PYTHONPATH, maybe in.bashrc,.zshenv, etc. export PYTHONPATH=$PYTHONPATH:~/phonopy-0.9.3/lib/python

3 Run examples Suppose phonopy's /bin is in the list of the execution path. Examples are found at % cd example/si % phonopy -p % cd example/nacl % phonopy -p -nac

4 Phonopy is A phonon calculation toolbox and toolkit Easily installed on Ubuntu (or recent distributions) Currently Windows and Mac OS X are out of consideration. Windows and Mac users are encouraged to boot Ubuntu on a virtual machine (e.g. VMware, Virtualbox) Phonopy requires force calculators, e.g., it work togather with first-principles calculations, or any calculation that can calculate forces on atoms. Processes of phonopy and force calculations are completely separated. Therefore phonopy can be applied to any force calculator with small effort. Written mainly in Python Phonopy Python module is prepared.

5 Calculation steps A supercell with a displacement 1. Prepare a unit cell 2. Relax the structure 3. Build a set of supercells with displacements 4. Calculate forces on atoms of the set of supercells 5. Collect sets of forces Forces Displacement 6. Calculate phonon frequencies

6 Work flow of phonopy Prepare a crystal structure at equilibrium in VASP POSCAR format Create supercells with displacements Force calculation by a force calculator, e.g. VASP Force collection Phonon analysis Collect sets of forces on atoms of the supercells Density of states, Band structure, Thermal properties, etc. Further analysis from a set of phonon calculations Thermal expansion, etc.

7 Pre-process & Force collection Pre-process Force collection Input structure (POSCAR) DISP and vasprun.xml's Supercell size (--dim) Supercell size (--dim) % phonopy -d -dim= % phonopy -dim= \ -f ALL_vapsrun.xml_FILES (supercell size = 2x2x2) Output files - Supercell with displacements (POSCAR-*) - Displacement directions (DISP) Output file Sets of forces (FORCES) for a post-process input.

8 POSCAR NDIM tag Phonopy process SPOSCAR, POSCAR-* DISP VASP calculations VASP DFPT calculations

9 Post process A file containing sets of forces and atomic displacements (FORCES) is transformed to supercell force constants. Dynamical matrices at arbitrary q-point are built from the supercell force constants. Dynamical matrices are solved, and phonon frequency and polarization vectors are obtained. DOS, PDOS, band structure, and thermal properties at constant volume are obtained following a setting file (INPHON) and command-line options. % cat INPHON NDIM = MP = % phonopy -p -t Thermal properties are plotted.

10 Phonopy post-process POSCAR NDIM or MATDIM tag or --dim option band.yaml FORCES or FORCE_CONSTANTS PRIMITIVE_AXIS tag mesh.yaml thermal_properties.yaml total_dos.dat partial_dos.dat

11 Computing system PC Plot Ask force calculation Cluster Ask phonon analysis (via X forwarding) Work station Recent PCs are enough strong that usually PC can be Work station.

12 Tips of force calculation in VASP INCAR of phonon calculation PREC = Accurate LREAL =.FALSE. ADDGRID =.TRUE. EDIFF = 1.0e-08 Geometry optimization of unit cell Set EDIFFG = -1.0e-08 and relax as much as possible watching the recidual forces Recidual forces of ~1e-4 ev/å may be acceptable, but depends on systems. Maybe IBRION = 2; ISIF = 3 Force calculations of supercells Set IBRION = -1 i.e. no relaxation is allowed.

13 An example set of INCARs Geometry optimization PREC ENCUT IBRION ISIF NSW NELMIN EDIFF EDIFFG IALGO ISMEAR LREAL ADDGRID LWAVE LCHARG = = = = = = = = = = = = = = Accurate e e ; SIGMA = 0.1.FALSE..TRUE..FALSE..FALSE. You may need to relax several times by mv CONTCAR POSCAR. Force calculation PREC ENCUT IBRION NELMIN EDIFF IALGO ISMEAR LREAL ADDGRID LWAVE LCHARG = = = = = = = = = = = Accurate e ; SIGMA = 0.1.FALSE..TRUE..FALSE..FALSE.

14 Tips of k-point sampling mesh in electronic structure calculations Real space Reciprocal space Unit cell 2 2 supercell ½ ½

15 Brief introduction to phonon theory

16 Harmonic oscillator Harmonic potential well Spring constant Mass k m x x Equation of motion F=ma=-kx A solution is frequency V

17 1D-lattice connected with N.N. Nearest neighbor a Mass m n-1 n n+1 n+2 un-1 un un+1 un+2 Let displacement be superposition of traveling waves

18 Dispersion relation Solving equation of motion, frequency is

19 Transverse wave q Atomic modulation is orthogonal to wave vector.

20 Longitudinal wave q Atomic modulation is parallel to wave vector.

21 Interaction among atoms We don't know how far it reaches and how strong it is. Furthermore interaction is not only in pairs.

22 Potential energy expansion Hamiltonian = kinetic + potential Potential energy is expaned with respect to atomic displacements (U). (Monatomic unit cell) : Force constants (FCs) i, j, k, : Cartesian components M, N, P, : Lattice points

23 Phonon from classical mechanics Omit higher terms than 3nd order Harmonic approximation Phonon is found to diagonalize harmonic Hamiltonian: Diagonalization is needed. Multiple atoms in a unit cell are considered. Indices μνπ... are used for internal atomic labeling. This is reduced to eigenvalue problem of dynamical matrix. * M: mass, N0: number of atoms, R: positon vector Phonon frequencies are obtained as square roots of eigenvalues of the dynamical matrix.

24 Diagonalization using computer D: dynamical matrix, w: eigenvector, s: band index It is convenient for solving eigenvalue problem using computer to construct a matrix of in the form like (two atom in unit cell): eigvals, eigvecs = numpy.linalg.eigh( dynmat )

25 Eigenvector w Orthonormality Completeness *Here indices i, j run over the Catesian components and atom indices. w(qs) are eigenvectors in the output files of phonopy.

26 Imaginary mode Imaginary frequency appears when crystal structure is dynamically unstable through the imaginary mode. Sometimes it relates to phase transition, or may be used to check if virtual crystal structure is stable or not. *In phonopy output, imaginary frequency is given as negative frequency. (a>0, b<0) Normal mode coordinate

27 Supercell approach Assume interaction range is confined in supercell size. E.g., 2x2 supercell is used to calculate force constans.

28 Finite difference method Displace one atom, and measure forces on all atoms Force on an atom (Fi) Atomic displacement (Δrj)

29 Thermal properties Thermal properties are calculated from frequencies. Quantum mechanics is necessary to derive them. Helmholtz free energy Entropy Heat capacity at constant volume

30 Crystal symmetry

31 Crystal symmetry Crystal structure has to be correctly symmetrized according to the space group type. Crystal symmetry gives high quality results and reduces computational demands. CIF database may have small number of decimals e.g. 1/ VASP geometry optimization may result in breaking symmetry in hexagonal case, etc. Detailed crystal symmetry is checked by % phonopy -symmetry --tolerance=1e-8

32 Centrings of crystals In space group type names in international table, starting with F, I, C, A, and B means that crystal has centring. F: face centre I: body centre C, A, B: base centre In these cases, the conventional unit cell is 1-, 2-, or 4-fold larger than the primitive cell. Lattice parameters of the conventional and primitive cells are related by transformation matrix. (ITA2002 Sec. 5.1) *ITA: International table volume A

33 Fm3m (e.g. Silicon) Conventional unit cell to primitive cell PRIMITIVE_AXIS =

34 Brillouin zones for space groups Go to click

35 Brillouin zone of Fm3m Coordinates wrt. reciprocal primitive lattice vectors This is to be written in QI and QF tags.

36 More on phonopy

37 Running modes of phonopy Band structure Sample q-points along specified paths Mesh sampling Sample q-points on a uniform mesh Calculations of DOS, PDOS, thermal propertiesblong to this mode. List of q-points Sample q-points listed in QPOINTS file

38 Band structure mode q-point sampling paths: QI = (1) (2) QF = (3) Number of sampling paths: ND = 3 Number of sampling points in a path: NPOINTS = 51 *End points are counted. The output file is band.yaml.

39 bandplot Re-plot band structure from band.yaml by % bandplot Command options can be shown by % bandplot -h To print out frequencies in gnuplot data format % bandplot --gnuplot If you want to cut below 0, % bandplot --fmin=0

40 Mesh sampling mode Reciprocal lattice are sampled by a uniform mesh. MP = In phonopy, a mesh point samples the Γ-point when the number is odd. Optionally, the sampling mesh can be shifted with respect to grid space. MP_shift = When MP_SHIFT is 0 or 0.5, the symmetrization of the grid points is fast, which may be important when the mesh is quite dense. The output file is mesh.yaml.

41 Mesh sampling/dos n: number of calculated q-points, g: function used for broadening DOS is obtained by % phonopy --dos or % phonopy -p (without plot) (with plot) The broadening function is the Gaussian function. (Lorentzian is in the code, but no phonopy user interface is implemented. ) Smearing width is controlled by % phonopy sigma=0.1 The output file is total_dos.dat.

42 Mesh sampling/pdos Partial DOS is obtained by PDOS = 1 2, and % phonopy As well as total DOS, smearing is controlled by --sigma. With -p option, PDOS is plotted. The output files is partial_dos.dat, and reploted by % pdosplot -i , 3 6 Where numbers for -i option correspond to 1x, 1y, 1z, 2x, 2y, 2z, (numbers are atom indices.)

43 DOS/Thermal properties Helmholtz free energy, entropy, heat capacity at constant volume are calculated from phonon frequencies by % phonopy -t With -p option, the resutls are plotted. Options --tmax,--tmin,--tstep may be used together. The output file is thermal_properties.yaml. Thermal properties are re-plotted by % propplot cv (or -fe, --entropy) (--gnuplot option is dummy...)

44 Non-analytical term correction Dynamical matrix at q 0 are corrected by nonanalytical term correction. R. M. Pick et al., PRB 1, 910, (1970) *Z: Born effective charge tensor, ε: dielectric tensor At general q-points, force constants are corrected by interpolation scheme of Wang et al. (J. Phys.: Condens. Matter. 22, (2010)) % phonopy --nac Born effective charge and dielectric tensors are necessary to use this option and those are summarized in BORN file.

45 BORN file Unit conversion factor ε Z *Only Z for independent atoms have to be written. Independent atoms are found atom_mapping of output of % phonopy --symmetry These values may be obtained by VASP 5 by setting LEPSILON =.TRUE. VASP results are found in OUTCAR around: BORN EFFECTIVE CHARGES (in e, cummulative output) MACROSCOPIC STATIC DIELECTRIC TENSOR (including local field effects in DFT)

46 Quasi-harmonic approximation (QHA) QHA: Frequencies at volumes % phonopy-qha -p -s v-e.dat \ thermal_properties_yaml_of_volume1 \ thermal_properties_yaml_of_volume2 \ thermal_properties_yaml_of_volume3... The file v-e.dat contains volumes and electronic total energies U(V) of unit cells V U

47 Thermal expansion of Silicon by QHA Origin of thermal expansion is volume dependence of frequency.

48 Volume dependence of frequency of Silicon This point

49 Animation for v_sim and else For v_sim ANIME = q-point For gdis, etc, at only Γ-point Number of pictures Amplitude ANIME = Shift Band index (from the bottom)

50 Eigenvector to displacement Displaced structure along an eigenvector is created by ANIME_TYPE = POSCAR ANIME = POSCAR-001 or POSCAR-003 is the structure that we want.

51 Finite displacements Amplitude Default value is 0.01 Å. Too small value enhances error of forces. Too large value induces anharmonic contribution. Plus-minus displacement Take plus-minus as default This often also compensates residual forces. Only plus when symmetric (automatically searched from crystal symmetry) Combination of these defaults and high enough energy convergence criteria in force calculation is expected to give uniform results.

52 Books and a reference Introduction to Lattice Dynamics (Martin T. Dove) Physical Properties of Crystals (J. F. Nye) Thermodynamics of Crystals (Duane C. Wallace) Thermodynamics and an Introduction to Thermostatistics (Herbert B. Callen) Electrons and Phonons (J. M. Ziman) Principles of Quantum Mechanics (R. Shankar) From ultrasoft pseudopotentials to the projector augmented-wave method, G. Kresse, D. Joubert, PRB 59, 1758 (1999)