Discovering advanced materials for energy applications: theory, high-throughput calculations, and automated experiments
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Anubhav Jain presented on using density functional theory and high-throughput calculations to design advanced materials for energy applications. Key points included: 1) Density functional theory can be used to model materials physics and properties by approximating many-body quantum mechanics. 2) Thermoelectric materials were discussed as an example application, where the goal is to optimize the figure of merit which depends on conductivity, Seebeck coefficient, and thermal conductivity. 3) High-throughput calculations were performed on over 50,000 materials to efficiently screen for promising thermoelectric candidates like TmAgTe2, though experimental validation is still needed due to approximations.
![As mentioned, screening studies for
thermoelectrics use approximate models for zT
20
Electron mobility Thermal conductivity Figure of merit
1. Constant / uniform relaxation
time approximation
2. Semi-empirical models[1]
1. Glassy limit thermal
conductivity models[2]
2. Semi-empirical models[1]
1. Combining previous
models for electron and
thermal conductivity
(optimize for doping, T)
2. Descriptors that implicitly
optimize[1] for doping, T
!=constant
[1] Yan, J.; Gorai, P.; Ortiz, B.; Miller, S.; Barnett, S. A.; Mason, T.; Stevanović, V.; Toberer, E. S. Material
Descriptors for Predicting Thermoelectric Performance. Energy Environ. Sci. 2015, 8 (3), 983–994
[2] D. G. Cahill, S. K. Watson, R. O. Pohl, Phys. Rev. B 1992, 46, 6133-6140 ; D. G. Cahill, R. O. Pohl, Ann. Rev.
Phys. Chem. 1988, 39, 93-121.
3 fitted parameters (A0, B, s)
Band effective mass based on
DOS effective mass and valley
degeneracy
2 fitted parameters (A1, A2)
Later extended to include
coordination number effects](https://image.slidesharecdn.com/jainucd-240209180833-bcb021af/85/Discovering-advanced-materials-for-energy-applications-theory-high-throughput-calculations-and-automated-experiments-20-320.jpg)
Discovering advanced materials for energy applications: theory, high-throughput calculations, and automated experiments
- 1. Discovering advanced materials for energy applications: theory, high-throughput calculations, and automated experiments Anubhav Jain Lawrence Berkeley National Laboratory UC Davis seminar, Feb 2024 Slides (already) posted to hackingmaterials.lbl.gov
- 2. Outline • Introduction to density functional theory • Thermoelectric Materials Design • Fast and accurate methods for electronic transport • Fast and accurate methods for thermal transport • Automated laboratories for synthesis 2
- 3. 3 Today, computer aided design of products is ubiquitous – but what are the governing equations to model materials? ?
- 4. Materials physics is determined by quantum mechanics 4 −!2 2m ∇2 Ψ(r)+V (r)Ψ(r) = EΨ(r) Schrödinger equation describes all the properties of a system through the wavefunction: Time-independent, non-relativistic Schrödinger equation Unfortunately, it cannot be directly solved for all but the simplest systems
- 5. What is density functional theory (DFT)? 5 DFT replaces many-body interactions with a mean field interaction (non- interacting particles) that reproduces the same charge density. In theory, it is exact for the ground state. In practice, accuracy depends on: • the choice of (some) parameters • the type of material and its electronic structure • the property to be studied • whether the simulated system (crystal) is a good approximation of reality. DFT resulted in the 1999 Nobel Prize for chemistry (W. Kohn). It is responsible for 2 of the top 10 cited papers of all time, across all sciences. e– e– e– e– e– e–
- 6. How does one use DFT to design new materials? 6 A. Jain, Y. Shin, and K. A. Persson, Nat. Rev. Mater. 1, 15004 (2016).
- 7. Outline • Introduction to density functional theory • Thermoelectric Materials Design • Fast and accurate methods for electronic transport • Fast and accurate methods for thermal transport • Automated laboratories for synthesis 7
- 8. Thermoelectric materials are an interesting application for DFT modeling 8 • A thermoelectric material generates a voltage based on thermal gradient • Applications • Heat to electricity • Refrigeration • Advantages include: • Reliability • Easy to scale to different sizes (including compact)
- 9. Thermoelectric figure of merit 9 • Many materials properties are important for thermoelectrics • Focus is usually on finding materials that possess a high “figure of merit”, or zT, for high efficiency • Target: zT at least 1, ideally >2 ZT = α2σT/κ power factor >2 mW/mK2 (PbTe=10 mW/mK2) Seebeck coefficient > 100 V/K Band structure + Boltztrap electrical conductivity 103 /(ohm-cm) Band structure + Boltztrap thermal conductivity 1 W/(m*K) • e from Boltztrap • l difficult (phonon-phonon scattering) Very difficult to balance these properties using intuition alone!
- 10. Example: Seebeck and e– conductivity tradeoff 10 Heavy band: ü Large DOS (higher Seebeck and more carriers) ✗ Large effective mass (poor mobility) Light band: ü Small effective mass (improved mobility) ✗ Small DOS (lower Seebeck, fewer carriers) Multiple bands, off symmetry: ü Large DOS with small effective mass ✗ Difficult to design! E k
- 11. 11 The experimental community has been steadily finding diverse, high zT thermoelectric materials
- 12. Can new computational approaches help find better thermoelectrics even faster? As proposed as early as 2003 by Blake and Metiu1: 12 “With the cost of computing become relatively inexpensive one can envisage a time where one runs multiple computer test tube reactions like these on large Beowulf clusters - as a means of screening for new TE materials. Certainly it appears that in the future theory may be a very competent dance partner for what has previously been a solo experimental effort in searching for ever better TE materials.” 1. Blake and Metiu. Can theory help in the search for better thermoelectric materials? Chemistry, Physics, and Materials Science of Thermoelectric Materials: Beyond Bismuth Telluride, 2003
- 13. We calculated a large amount of transport data under constant relaxation time approximation 13 ~50,000 crystal structures and band structures from Materials Project are used as a source F. Ricci, et al., An ab initio electronic transport database for inorganic materials, Sci. Data. 4 (2017) 170085. We compute electronic transport properties with BoltzTraP and atomate About 300GB of electronic transport data is generated. All data is available free for download https://contribs.materialsproject.org/projects/carrier_transport/ All data is available free for download via Materials Project or direct download from journal article.
- 14. • Advantage – we can screen *many* materials and be quite comprehensive. ~50,000 materials can be computed and compared. • Some disadvantages • Fixed relaxation time often prioritizes materials with flat bands, which is undesirable in reality • Properties are overestimated at high temperatures and high doping • Thermal conductivity numbers are rough estimates • No modeling of dopability / carrier type in high-throughput • Functionals don’t include vdW interactions • So we can’t take theory numbers at face value! • e.g., look at the band structure (is it flat bands?) • does the material require high temperatures or high doping? If so, less reason to believe we can achieve it in reality • experimental factors taken into account • Goal was to run higher levels of theory, doping, etc. – but never got to develop this well 14 Advantages and disadvantages of approach
- 15. 15 New Materials from screening – TmAgTe2 • Calculations: trigonal p-TmAgTe2 could have power factor up to 8 mW/mK2 • Tetragonal form also interesting • Ag-Te compounds known to have low thermal conductivity • BUT requires 1020/cm3 carriers of doping to hit peak in PF Zhu, H.; Hautier, G.; Aydemir, U.; Gibbs, Z. M.; Li, G.; Bajaj, S.; Pöhls, J.-H.; Broberg, D.; Chen, W.; Jain, A.; White, M. A.; Asta, M.; Snyder, G. J.; Persson, K.; Ceder, G. Computational and experimental investigation of TmAgTe 2 and XYZ 2 compounds, a new group of thermoelectric materials identified by first-principles high-throughput screening, J. Mater. Chem. C, 2015, 3
- 16. TmAgTe2 (experiments) 16 1. Zhu, H.; et al. Computational and experimental investigation of TmAgTe 2 and XYZ 2 compounds, a new group of thermoelectric materials identified by first-principles high-throughput screening, J. Mater. Chem. C, 2015, 3 • Expt: p-zT only 0.35 despite very low thermal conductivity (~0.25 W/mK) • Limitation: carrier concentration (~1017/cm3) • likely limited by TmAg defects, as determined by followup calculations • Later, we achieved zT ~ 0.47 using Zn-doping 2. Pöhls, J.-H., et al. First-principles calculations and experimental studies of XYZ2 thermoelectric compounds: detailed analysis of van der Waals interactions. J. Mater. Chem. A 6, 19502–19519. https://doi.org/10.1039/C8TA06470A
- 17. YCuTe2 – friendlier elements, higher zT (0.75) 17 Aydemir, U.; Pöhls, J.-H.; Zhu, H., Hautier, G.; Bajaj, S.; Gibbs, Z. M.; Chen, W.; Li, G.; Broberg, D.; Kang, S.D.; White, M. A.; Asta, M.; Ceder, G.; Persson, K.; Jain, A.; Snyder, G. J. YCuTe2: A Member of a New Class of Thermoelectric Materials with CuTe4-Based Layered Structure. J. Mat Chem C, 2016 experiment computation • Calculations: p-YCuTe2 could only reach PF of 0.4 mW/mK2 • SOC inhibits PF • if thermal conductivity is low (e.g., 0.4, we get zT ~1) • Expt: zT ~0.75 – not too far from calculation limit • carrier concentration of 1019 • Decent performance, but unlikely to be improved with further optimization
- 18. The data set is available on the Materials Project 18 https://next-gen.materialsproject.org/contribs/projects/carrier_transport
- 19. Outline • Introduction to density functional theory • Thermoelectric Materials Design • Fast and accurate methods for electronic transport • Fast and accurate methods for thermal transport • Automated laboratories for synthesis 19
- 20. As mentioned, screening studies for thermoelectrics use approximate models for zT 20 Electron mobility Thermal conductivity Figure of merit 1. Constant / uniform relaxation time approximation 2. Semi-empirical models[1] 1. Glassy limit thermal conductivity models[2] 2. Semi-empirical models[1] 1. Combining previous models for electron and thermal conductivity (optimize for doping, T) 2. Descriptors that implicitly optimize[1] for doping, T !=constant [1] Yan, J.; Gorai, P.; Ortiz, B.; Miller, S.; Barnett, S. A.; Mason, T.; Stevanović, V.; Toberer, E. S. Material Descriptors for Predicting Thermoelectric Performance. Energy Environ. Sci. 2015, 8 (3), 983–994 [2] D. G. Cahill, S. K. Watson, R. O. Pohl, Phys. Rev. B 1992, 46, 6133-6140 ; D. G. Cahill, R. O. Pohl, Ann. Rev. Phys. Chem. 1988, 39, 93-121. 3 fitted parameters (A0, B, s) Band effective mass based on DOS effective mass and valley degeneracy 2 fitted parameters (A1, A2) Later extended to include coordination number effects
- 21. The Boltzmann transport equation determines carrier transport properties 21 group velocity (easy) lifetime (hard)
- 22. τ ... DFPT AMSET ∝ DOS–1 / semi- empirical constant lifetime A model to explicitly calculate scattering rates while remaining computationally efficient Aim: accuracy comparable to EPW at 1/100th – 1/1000th computational cost AND with rich information content AMSET is a new framework for calculating transport properties including e- lifetimes
- 23. primary input: uniform k-mesh band structure calculation Step 1: Band structure (probably DFT)
- 24. Fourier interpolation of eigenvalues and group velocities (there is some custom resampling to get even more accurate integrations) Step 2: Interpolation of band structure to dense mesh
- 25. lifetimes calculated using scattering equations that depend on first-principles inputs Step 3: Use band structure and scattering equations to determine scattering rates
- 26. calculate mobility, conductivity, Seebeck thermal conductivity Step 4: Transport properties by solving BTE
- 27. Acoustic deformation potential (ad) deformation potential, elastic constant Ionized impurity (ii) dielectric constant Piezoelectric (pi) dielectric constant, piezoelectric coefficient Polar optical phonon (po) dielectric constant, polar phonon frequency Scattering rates determined by DFT inputs *note: scalar equations shown, these are generalized to tensor forms
- 28. • All first principles inputs • Nothing fit or tuned to experimental data • Rich information content • scattering mechanisms • E k-dependent scattering rates • Short runtime • ~60 mins of DFT calculations on 64 cores • Plus ~40 mins of time to run AMSET AMSET predicts switch from impurity to polar phonon scattering in GaN Ganose, A. M.; Park, J.; Faghaninia, A.; Woods-Robinson, R.; Persson, K. A.; Jain, A. Efficient Calculation of Carrier Scattering Rates from First Principles. Nat Commun 2021, 12 (1), 2222.
- 29. • Anisotropy is also captured by AMSET • Allows for analyzing non- cubic systems and getting direction- dependent properties AMSET can also calculate the anisotropic transport properties of realistic materials Crystal structure image from: Pletikosić, Ivo et al. (2017). Band structure of a IV- VI black phosphorus analogue, the thermoelectric SnSe. Physical Review Letters. 120. Ganose, A. M.; Park, J.; Faghaninia, A.; Woods-Robinson, R.; Persson, K. A.; Jain, A. Efficient Calculation of Carrier Scattering Rates from First Principles. Nat Commun 2021, 12 (1), 2222. SnSe
- 30. AMSET shows close agreement to experiment for the mobility and Seebeck coefficient across many materials Ganose, A. M.; Park, J.; Faghaninia, A.; Woods-Robinson, R.; Persson, K. A.; Jain, A. Efficient Calculation of Carrier Scattering Rates from First Principles. Nat Commun 2021, 12 (1), 2222.
- 31. Timing for calculations are very reasonable and within reach of most groups Ganose, A. M.; Park, J.; Faghaninia, A.; Woods-Robinson, R.; Persson, K. A.; Jain, A. Efficient Calculation of Carrier Scattering Rates from First Principles. Nat Commun 2021, 12 (1), 2222. Total calculation times are ~500X faster than DFPT+Wannier The AMSET portion of the calculation scales well with system size
- 32. Docs: https://hackingmaterials.lbl.gov/amset/ Support: https://matsci.org/c/amset Paper: installation pip install amset usage amset run --static-dielectric 10 ... Can be controlled through the command line or python interface, integration with atomate2 for automatic workflows Ganose, A. M.; Park, J.; Faghaninia, A.; Woods-Robinson, R.; Persson, K. A.; Jain, A. Efficient Calculation of Carrier Scattering Rates from First Principles. Nat Commun 2021, 12 (1), 2222. AMSET is an open source python package and has already been used in 200+ downstream studies
- 33. Outline • Introduction to density functional theory • Thermoelectric Materials Design • Fast and accurate methods for electronic transport • Fast and accurate methods for thermal transport • Automated laboratories for synthesis 33
- 34. Goal - add automated, high-throughput thermal properties to Materials Project – particularly 3rd + 4th order MP Annual Meeting 2023 34
- 35. The problem – obtaining force constants can require many DFT calculations 35 To obtain 2nd order IFCs To obtain 3rd order IFCs 2 displacements in a supercell (# of supercells needed: 1000s-10000s) … 1 displacement in a supercell (Usually 5 supercells needed) Finite-displacement method IFCs extracted from HiPhive To obtain any order of IFCs (2nd, 3rd,…) in one shot … displace each atom in a supercell (Only need 5~10 supercells in total!) • Traditionally, one performs systematic displacements, each of which only has a few atom movements and solves only a small portion of the IFC matrix • Primitive cells with reduced symmetry and many atoms can easily require 1000 or more calculations • The scaling goes something like: O(Nn) where N is the number of sites and n is the order of IFC you want. Not scalable!
- 36. The solution – perform non-systematic displacements • Instead of performing systematic displacements, perform non-systematic displacements in which many IFC terms are “mixed up” • Then, perform a best fit procedure to fit the IFC matrix elements to the observed data • Typically undetermined, so regularization is important • This method has been suggested by several groups, for now we focus on the implementation in the HiPhive code (Erhart group, Chalmers University of Technology) • Disadvantage: this method requires careful selection of fit parameters to get correct results 36 IFCs extracted from HiPhive To obtain any order of IFCs (2nd, 3rd,…) in one shot … displace each atom in a supercell (Only need 5~10 supercells in total!) Monte Carlo rattle penalizes displacements that lead to very small interatomic distances Fransson, E.; Eriksson, F.; Erhart, P. Efficient Construction of Linear Models in Materials Modeling and Applications to Force Constant Expansions. npj Comput Mater 2020, 6 (1), 135.
- 37. We have been working on a workflow for lattice dynamics that gives 100 – 1000X speedup and is automatic 37 VASP DFT relaxation of primitive cell VASP SCF on supercells (u = 0.01-0.05 Å) VASP SCF on supercells (u = 0.1-0.5 Å) HiPhive Fit harmonic Φ2 HiPhive Fit anharmonic Φ3 ,Φ4 etc Complete Φ Imaginary modes? Stable Phonon INPUT Bulk modulus ShengBTE/ FourPhonon Boltzmann Transport • Free Energy • Entropy • Heat Capacity • Gruneisen • Thermal Expansion • Lattice Thermal Conductivity No Yes Inner Loop Outer Loop No • Quantum Covariance • Renormalize Φ2 Imaginary modes? Converged free energy? Free Energy Converged free energy? • Expand Lattice at T Yes Yes No • Phase transition • Thermoelectric zT Renormalization at T ≥ 0 K Renormalization at T ≥ 0 K Renormalized Φ • Corrected Free Energy No Yes 100x speedup 1000x speedup
- 38. Results from automated workflow are promising 38 Rhombo-to-Cubic in GeTe (Tc = 650 K) Tetragonal-to-Cubic in ZrO2 (Tc = 2650 K) ZrO2 (cubic, Fm-3m) GeTe (cubic, Fm-3m) GeTe (rhombohedral, R3m) ZrO2 (tetragonal, P42/nmc) a b c d GeTe (cubic, Fm-3m) GeTe (rhombohedral, R3m) c d Zr (BCC, Im-3m) e f Zr (HCP, P63/mmc)
- 39. • Expect to submit a paper on this in Feb 2024 • Workflow will be available in both the “atomate” and “atomate2” codes 39 Next steps
- 40. Outline • Introduction to density functional theory • Thermoelectric Materials Design • Fast and accurate methods for electronic transport • Fast and accurate methods for thermal transport • Automated laboratories for synthesis 40
- 41. Machine learning is now predicting very large numbers of new stable compounds 0 500000 1000000 1500000 2000000 MP stable ICSD PDF M3GNet stable In a short period of time, ML algorithms can generate potentially millions of potentially stable compounds M3GNet data: Chen, C., Ong, S.P. A universal graph deep learning interatomic potential for the periodic table. Nat Comput Sci 2, 718–728 (2022). Note that rate of new experimental deposition into ICSD / PDF is 10K – 20K materials per year 41
- 42. Synthesis recipe 50 mg Li2CO3 80 mg MnO 20 mg TiO2 800 °C (air) 24 hours 50 mg 80 mg Target LiMnTiO4 20 mg 800 °C, 24 hours Final product! There are no well-defined rules for choosing the most effective precursors and conditions Experimental issues like precursor melting, volatility, or reactivity with the container Initial experiments often give zero target yield. What to do next? Making new materials is inherently slow and unpredictable Even when you are successful, it is very time and labor intensive! 42
- 43. The A-lab aims to close the loop on rapid synthesis Robotics Optimization algorithms Machine learning 43
- 44. The A-Lab: three robotic stations work together Precursor preparation: Gravimetric dispenser works with a robot arm to weigh and mix powders Heating station: A second robot arm operates on a rail, transferring samples to and from box furnaces Characterization: A third robot arm extracts the synthesis products and prepares them for X-ray diffraction (XRD) The hardware team 44
- 45. Science use case: Synthesizing unknown (to A-lab) compounds 42,000 stable cmpds 146 final cmpds “Google-stable” Stable in air Not in ICSD or mined literature Of these, we selected 58 cmpds for which precursors were readily available No rare or unsafe elements Objective: target some compounds that are computationally predicted in Materials Project, but not present in that database or several others … And do it in 3 weeks! 45
- 46. Results from the A-Lab syntheses: 41/58 targets made! Making 41 “unknown-to-system” chemical compositions in 3 weeks is a major achievement 71% success per target 37% success per recipe 46 N.J. Szymanski, et al. Nature. 624 (2023).
- 47. Four major reasons for inability to make compounds 47
- 48. What’s next? • We are working to expand automated characterization capabilities of the A-lab, giving greater confidence in synthesized products closer to manual analysis • New capabilities such as synthesis under reducing gases as well as glove box synthesis are being targeted • Things are always a work in progress! 48
- 49. Acknowledgements 49 Thermoelectrics • G. Jeffrey Snyder • Jan Pohls • Umut Aydemir • Mary Anne White Slides (already) posted to hackingmaterials.lbl.gov AMSET thermal • Alex Ganose • Zhuoying Zhu • Junsoo Park • Alireza Faghininia A-lab • Gerbrand Ceder • Yan Zeng • Nate Szymanski • Yuxing Fei • Bernardus Rendy Funding from the U.S. Department of Energy Toyota Research Institute