![Symmetry constrains
[1] Kennedy et al., PRB 59, 4023 (1999) (experiment)
T, K
Volume,Å^3
200 1600600 1200
6972
Ortho-I
Ortho-II
Tetrag.
Cubic
Pm-3m
I4/mcm
Cmcm
Pbnm
0
−40
−177
−241
Cubic
Tetrag.
Ortho-II
Ortho-I
Calc. ∆Etot,
meV
Phase
[1] [2]
[2] Evarestov et al., Phys.Stat.Sol.(b) 242, R11 (2005) (calculation)
SrZrO3](https://image.slidesharecdn.com/2abinitiotemperaturephononsgrouptheory-130902001937-phpapp02/85/Ab-initio-temperature-phonons-group-theory-4-320.jpg)
![Symmetry constrains
[1] Kennedy et al., PRB 59, 4023 (1999) (experiment)
T, K
Volume,Å^3
200 1600600 1200
6972
Ortho-I
Ortho-II
Tetrag.
Cubic
Pm-3m
I4/mcm
Cmcm
Pbnm
0
−40
−177
−241
Cubic
Tetrag.
Ortho-II
Ortho-I
Calc. ∆Etot,
meV
Phase
[1] [2]
[2] Evarestov et al., Phys.Stat.Sol.(b) 242, R11 (2005) (calculation)
SrZrO3](https://image.slidesharecdn.com/2abinitiotemperaturephononsgrouptheory-130902001937-phpapp02/85/Ab-initio-temperature-phonons-group-theory-16-320.jpg)
Ab initio temperature phonons group theory
- 1. Evgeny Blokhin Chelyabinsk SUSU’2013 summer workshop Max-Planck Institute for Solid State Research Stuttgart, Germany Theory and practice of ab initio materials modeling Part II
- 2. Outlook 1. Considering temperature from ab initio 2. Atomic vibrations and phase transitions 3. Example of four perovskites 4. And example of strontium titanate surface
- 3. How to deal with temperature at 0°K? 1. Symmetry constrains 2. Elastic properties and equation of state, e.g. 3. Configurational disorder entropy 4. Vibrational entropy (phonons)
- 4. Symmetry constrains [1] Kennedy et al., PRB 59, 4023 (1999) (experiment) T, K Volume,Å^3 200 1600600 1200 6972 Ortho-I Ortho-II Tetrag. Cubic Pm-3m I4/mcm Cmcm Pbnm 0 −40 −177 −241 Cubic Tetrag. Ortho-II Ortho-I Calc. ∆Etot, meV Phase [1] [2] [2] Evarestov et al., Phys.Stat.Sol.(b) 242, R11 (2005) (calculation) SrZrO3
- 5. Calculating phonons Brillouin zone of cubic SrTiO3 Cubic SrTiO3 primitive cell ( ) ( ) 0 ( ) ! n ni i i n E E R n δ δ ∞ = ∂ = −∑
- 6. Calculating phonons Brillouin zone of cubic SrTiO3 Cubic SrTiO3 2x2x2 supercell ( ) ( ) 0 ( ) ! n ni i i n E E R n δ δ ∞ = ∂ = −∑
- 7. Cubic SrTiO3 phonon dispersion over the BZ
- 8. T, K 37 1050 “Tetragonal FE” Tetragonal AFD Cubic R4+ Г4- Sr O Ti Soft-mode driven phase transitions in perfect SrTiO3
- 9. 9 ...
- 11. triperiodic (3D) diperiodic (2D), LG61
- 12. Soft-mode driven phase transitions in perfect SrTiO3 surface z Γ Γ M 3-dimensional Brillouin zone 2-dimensional Brillouin zone
- 13. 548517B1g 447439B1g460, 447454Eg 394421Eg 175180Eg235, 229157B2g 162155Eg 147153A1g 129129A1g143, 144142Eg 4848Eg48, 5285A1g 37133iEg15, 4017Eg Expt.TheoryIrrepExpt.TheoryIrrep Surface AFD STOBulk AFD STO Soft-mode driven phase transitions in perfect SrTiO3 surface
- 14. SrTiO3 (2 phases): T > 105 K Pm-3m (221), T < 105 K I4/mcm (140) SrZrO3 (4 phases): T > 1343 K Pm-3m (221), T < 1343 K I4/mcm (140), T < 1113 K Cmcm (63), T < 1000 K Pbnm (62) 14 BaTiO3 (4 phases): T > 403 K Pm-3m (221), T < 403 K P4mm (99), T < 278 K Amm2 (38), T < 183 K R3m (160) BaZrO3 (1 phase): Pm-3m (221) Example of four perovskites
- 15. Image by Amisi et al., PRB85, 064112 (2012) SrZrO3 ab initio predicted phase transitions } } } cubic tetragonal orthorhombic
- 16. Symmetry constrains [1] Kennedy et al., PRB 59, 4023 (1999) (experiment) T, K Volume,Å^3 200 1600600 1200 6972 Ortho-I Ortho-II Tetrag. Cubic Pm-3m I4/mcm Cmcm Pbnm 0 −40 −177 −241 Cubic Tetrag. Ortho-II Ortho-I Calc. ∆Etot, meV Phase [1] [2] [2] Evarestov et al., Phys.Stat.Sol.(b) 242, R11 (2005) (calculation) SrZrO3
- 17. Summary 1. Group-theory analysis allows finding: (a) symmetry of phonons for the high symmetry phase, (b) the set of low symmetry subgroups and possible symmetry of phonons responsible for 2nd order phase transitions 2. The phonon calculations for the high symmetry phase give the soft modes symmetry and define the possible low symmetry phases 3. The phonon calculations for the found low symmetry phases allows one to define the most stable phase