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Abstract: D63.00002 : Improved Accuracy Tight Binding Model for Finite Temperature Electronic Structure Dynamics in Methyl Ammonium Lead Iodide (MAPbI3)

Presenter:

David Abramovitch

(Department of Physics, University of California, Berkeley)

Authors:

David Abramovitch

(Department of Physics, University of California, Berkeley)

Liang Tan

(Molecular Foundry, Lawrence Berkeley National Lab)

Halide perovskites are promising photovoltaic and optoelectronic materials. However, computing electronic properties and dynamics at finite temperature is challenging due to nonlinear lattice dynamics and prohibitive computational costs for ab initio methods. Tight binding models decrease computational costs, but current models lack the ability to accurately model instantaneous atom displacement and reduced symmetry at finite temperature. We present a parameterized tight binding model for MAPbI3 capable of predicting instantaneous electronic structures for large systems based on atomic positions extracted from classical molecular dynamics. Our tight binding Hamiltonian predicts instantaneous atomic orbital onsite energies and hopping parameters accurate to 0.1 to 0.01 eV compared to DFT across the orthorhombic, tetragonal, and cubic phases, including effects of temperature, reduced symmetry, and spin orbit coupling. This model allows for efficient calculation of instantaneous and dynamical electronic structure at the length and time scales required to address coupled electronic and ionic dynamics, as required for predicting temperature dependence of carrier mass, band structure, free carrier scattering, and polaron transport and recombination.

Presenter:

David Abramovitch

(Department of Physics, University of California, Berkeley)

Authors:

David Abramovitch

(Department of Physics, University of California, Berkeley)

Liang Tan

(Molecular Foundry, Lawrence Berkeley National Lab)

Halide perovskites are promising photovoltaic and optoelectronic materials. However, computing electronic properties and dynamics at finite temperature is challenging due to nonlinear lattice dynamics and prohibitive computational costs for ab initio methods. Tight binding models decrease computational costs, but current models lack the ability to accurately model instantaneous atom displacement and reduced symmetry at finite temperature. We present a parameterized tight binding model for MAPbI3 capable of predicting instantaneous electronic structures for large systems based on atomic positions extracted from classical molecular dynamics. Our tight binding Hamiltonian predicts instantaneous atomic orbital onsite energies and hopping parameters accurate to 0.1 to 0.01 eV compared to DFT across the orthorhombic, tetragonal, and cubic phases, including effects of temperature, reduced symmetry, and spin orbit coupling. This model allows for efficient calculation of instantaneous and dynamical electronic structure at the length and time scales required to address coupled electronic and ionic dynamics, as required for predicting temperature dependence of carrier mass, band structure, free carrier scattering, and polaron transport and recombination.

- 1. Tight Binding Simulation of Finite Temperature Electronic Structure in Methylammonium Lead Iodide (MAPbI3) David Abramovitch1 and Liang Tan2 1Department of Physics, University of California Berkeley and 2Molecular Foundry, Lawrence Berkeley National Lab
- 2. Motivation Halide Perovskite electronic structure is connected to nonlinear lattice dynamics on length and time scales too large for DFT A. Poglitsch and D. Weber J. Chem. Phys. 87, 6373 (1987) Atomic orbital Tight Binding Models: Previous: Boyer-Richard et al J. Phys. Chem. Lett. 2016, 7, 19, 3833-3840, Zheng et al Energy Environ. Sci.,2019,12,1219 Our model modifies original finite temperature model by Mayers et al Nano Lett. 2018, 18, 12, 8041- 8046 → Better prediction of transverse modes and charge fluctuations → Thorough benchmarking against DFT shows improvements
- 3. Methods Orbitals: Wannier90 projections of Pb s, Pb p, and I p orbitals Atom positions from classical MD ↓ DFT electronic structure and Wannier90 projection onto atomic orbitals ↓ Fit onsite and hopping parameters from distances, potentials, etc. ↓ Model Hamiltonian ↓ Tight Binding Electronic Structure
- 4. Onsite Energies Onsite Energies: Fit: E = E0 + aV(0) + b d2V/dx2 Symmetry based Spin Orbit Coupling between p orbitals on same atom as in Mayers et al Nano Lett. 2018, 18, 12, 8041-8046
- 5. Hopping Parameters
- 6. Tests - Bandgap and Fluctuations The new model predicts fluctuations in the gap gap much more accurately, across a range of temperatures and phases For 300 K: DFT Bandgap = 0.44 eV TB Bandgap = 0.54 eV
- 7. Tests - Band Edge
- 8. Urbach Tail Density of states calculations show a small (5 - 20 meV) temperature dependent Urbach tail. 30 8x8x8 structures with 10x10x10 kpoint grid and 2 meV Gaussian smoothing
- 9. Low Temperature Carrier Mobility Collaboration with McClintock et al (J. Phys. Chem. Lett 2020, 11, 3) found dramatically increased transport at low temperatures Our TB model indicates temperature effects on free carrier mass and scattering are too small, supporting excitonic explanation
- 10. Conclusion and Outlook Quantitatively accurate tight binding allows for new calculations Applicable to nanostructures, surfaces, etc. General procedure is transferable to other perovskites and likely beyond Mayers et al Nano Lett. 2018, 18, 12, 8041-8046
- 11. Acknowledgements This research was supported by the Computational Materials Sciences Program funded by the US Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division. This research used resources of the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy under Contract No. DE- AC02-05CH11231. Travel costs were supported in part by the National Society of Physics Students
- 12. References [1] M. Z. Mayers, L. Z. Tan, D. A. Egger, A. M.Rappe, and D. R. Reichman, How lattice and charge fluctuations control carrier dynamics in halide perovskites, Nano Letters 18, 8041 (2018), pMID: 30387614,https://doi.org/10.1021/acs.nanolett.8b04276. [2] A. Mattoni, A. Filippetti, M. I. Saba, and P. Delugas,Methylammonium rotational dynamics in lead halide perovskite by classical molecular dynamics: The role of temperature, The Journal of Physical Chemistry C119, 17421(2015), https://doi.org/10.1021/acs.jpcc.5b04283. [3] M. T. Weller, O. J. Weber, P. F. Henry, A. M. Di Pumpo,and T. C. Hansen, Complete structure and cation orientation in the perovskite photovoltaic methylammonium lead iodide between 100 and 352 k, Chem. Commun.51, 4180(2015). [4] T. Baikie, Y. Fang, J. M. Kadro, M. Schreyer, F. Wei,S. G. Mhaisalkar, M. Graetzel, and T. J. White, Synthesis and crystal chemistry of the hybrid perovskite(ch3nh3)pbi3 for solid-state sensitised solar cell applications, J. Mater. Chem. A1, 5628 (2013). [5] S. Boyer-Richard, C. Katan, B. Traor ́e, R. Scholz, J.-M. Jancu, and J. Even, Symmetry-based tight binding modeling of halide perovskite semiconductors, The Journal of Physical Chemistry Letters 7, 3833 (2016), pMID:27623678, https://doi.org/10.1021/acs.jpclett.6b01749. [6] L. McClintock, R. Xiao, Y. Hou, C. Gibson, H. C.Travaglini, D. Abramovitch, L. Z. Tan, R. T. Senger,Y. Fu, S. Jin, and D. Yu, Temperature and gate dependence of carrier diffusion in single crystal methylammonium lead iodide perovskite microstructures, The Journal of Physical Chemistry Letters 11, 1000 (2020), pMID:31958953, https://doi.org/10.1021/acs.jpclett.9b03643. [7]A.Poglitsch and D.Weber, Dynamic disorder in methylammonium trihalogenoplumbates (ii) observed by millimeter - wave spectroscopy ,The Journal of Chemical Physics 87,6373 (1987),https://doi.org/10.1063/1.453467. [8] F. Zheng and L.-w. Wang, Large polaron formation and its effect on electron transport in hybrid perovskites, EnergyEnviron. Sci.12, 1219 (2019)
- 13. Supporting Slides
- 14. Other Hopping Parameters Next nearest neighbors: Spin orbit coupling occurs between p orbitals on the same atom, as follows: With 𝛾 = 0.31 eV for I and 𝛾 = 0.56 eV for Pb, as in Mayers et al. Nano Letters 2018 18 (12), 8041-8046 Mean (eV) 0.139 0.160 Standard Deviation (eV) 0.0209 0.0349 Best Fit Error (eV) -- 0.0114
- 15. Methods Atom positions from classical MD ↓ Parameterize model by computing DFT electronic structure and using Wannier90 to project to atomic orbitals. ↓ Fit onsite and neighbor hopping parameters based on structure, potentials, etc. Calculation Details: MD: Force field from Mattoni et al2, use experimental lattice ratios3,4 and equilibrate volume on 6x6x6 orthorhombic supercell. Ran NVT trajectories on 2x2x2 stoichiometric (MAPbI3) supercells and stoichiometric 8x8x8 supercells. 2x2x2 supercells are used for parameterizing tight binding. DFT: Quantum Espresso, PBE GGA functional, OPIUM norm-conserving non-local pseudopotentials, 4x4x4 𝚪 centered k-point grid, 50 Rydberg energy cutoff, non-collinear spin orbit.
- 16. Atomic Orbitals Orbitals: Wannier90 projections of Pb s, Pb p, and I p orbitals Atomic Orbital Wannier Hamiltonian shows very good prediction of band structure
- 17. Onsite Energies Orbital Mean (eV) Standard Deviation (eV) Fit Error (eV) Pb s -2.45 0.119 0.0583 Pb p 5.93 0.129 0.0598 I p 2.51 0.204 0.0942 Onsite energies are fitted linearly using coulomb potential, second derivative, and unit cell volume. Onsite Energies: Fit: E = E0 + aV(0) + b d2V/dx2
- 18. Hopping - Sigma Bonds Mean 1.029 eV (geometry determines sign) 1.663 eV Standard Deviation 0.192 eV 0.178 eV Best Fit Error 0.0146 eV 0.0351 eV
- 19. Hopping - Pi Bonds and Junctions Mean -0.373 eV 0.0 eV (sign from displacement and geometry) Standard Deviation 0.0852 eV 0.227 eV Best Fit Error 0.0239 eV 0.0247 eV
- 20. Tests - Bandgap and Fluctuations The new model predicts fluctuations in the gap gap much more accurately, across a range of temperatures and phases Tempe rature (K) DFT Bandgap mean (SD) (eV) New Model Bandgap mean (SD) (eV) Model 8x8x8 Bandgap mean (SD) (eV) 150 0.574 (0.036) 0.737 (0.034) 0.869 (0.0059) 220 0.422 (0.050) 0.546 (0.061) 0.725 (0.014) 300 0.440 (0.049) 0.535 (0.060) 0.643 (0.0072)
- 21. Tests - Original Model Band Structure
- 22. Tests - New Model Band Structure
- 23. Thing Explained Better Computer Way to Understand Tight Waves and Excited Stuff in Warm Sun Catching Rock These new sun catching rocks are good, but hard to understand even on a computer when warm and moving. Using tight waves is easier for the computer, but it is hard to guess how the waves touch when they move because they are warm. We have a new set of tight pieces which can be turned into waves if you know where the pieces are. It is pretty good, close to using a computer to get the full waves, even with pieces out of order. Since it is much easier than finding full waves, the computer can understand the waves in big, warm pieces of rock and how they change and move. This allows us to understand how to move light power through the rock.

- 20 sec

- (Finish 1:30) Halide perovskites are good semiconductors, but they are known for nonlinear lattice dynamics, defect tolerance, and other factors influence electronic structure which are difficult for ab initio simulations.

An atomic orbital tight binding model reduces the number of states by around 3000.

There have been several previous tight binding models for MAPbI3, and this model is based heavily on a previous model which is specifically designed for finite temperature effects. We have improved its ability to discern between transverse and longitudinal bond motion and the effects of charge fluctuations, and benchmarking against DFT shows improvements. - (Finish 3:00) So, the general procedure is going to be to run classical molecular dynamics to get atom positions as a function of time, and then, in the process of parameterizing the model we run DFT on those structures and use wannier90 to project onto atomic orbitals to get a Hamiltonian in an atomic orbital basis.

Then, we fit those hamiltonian matrix elements based on the atomic structure, with things like bond lengths, coulomb potentials, and so on. From there, we have a model Hamiltonian as a function of the atomic structure and we can use that to get a tight binding electronic structure for much larger structures without ever having to go through DFT.

The atomic orbitals we’re using are the s and p valence orbitals on Pb and the p orbitals on I. Initially, I was concerned that these orbitals might not be enough, but all the testing we’ve done shows that they contain almost all the information about the band structure near the band edges. Here, we’ve plotted DFT bandgap relative to its average against the wannier90 hamiltonian bandgap relative to its average, and we can see that the atomic orbitals predict the bandgap fluctuations very accurately. - (Finish 4:00) Many tight binding models treat onsites as constant.

Original model modified it using the coulomb potential at the site of the atom. We have expanded this to include a second derivative, helping to predict the effect of charge fluctuations and allows for different energies for orbitals on the same atom.

This plot shows the standard deviation of the parameter with the lines and the error of the fit with the boxes. The fluctuations are small relative to the distance between the orbitals, and fairly well predicted by the model.

We also have spin orbit coupling between p orbitals of different spins on the same atom, which is carried over from the original model. - (Finish 5:00) Hopping parameters between adjacent atoms are probably the most important part of the model, and we are able to describe the fluctuations quite well.

We have sigma bonds between Pb s and I p and Pb p and I p orbitals and pi bonds between p orbitals. These are based on simple geometric fits representing the orbital overlap exponentially decaying. The original model used only the overall bond length, but we have found that separating it into components along each axis leads to a more accurate fit.

We have also added in “junction” hopping between perpendicular p orbitals, which is 0 on average for a cubic structure but becomes non zero at finite temperature or with tilting at low temperatures.

Put together, these improvements not only improve the overall accuracy, but also conceptually allow for discerning different kinds of motion (longitudinal, transverse, different directions). - (Finish 6:00) Since we’re interested on finite temperature effects on the band structure, one of the metrics we’ve looked at is the prediction of bandgap fluctuations. Here, we’ve run tight binding and DFT on the same structures, and plotted on the x axis the DFT bandgap relative to its average and on the y axis the tight binding bandgap relative to its average.

The new model predicts fluctuations of the bandgap quite well, across a range of temperatures.

- (Finish 7:00) We’ve also looked at the prediction of other states along the bandedge. Here, we’re plotting, for each structure and each kpoint, the DFT energy of the valence and conduction bands at that kpoint relative to the fermi energy against that energy as predicted by tight binding.

We can see that on the left the Wannier90 hamiltonian is very accurate, as expected. Comparing the new model to the original, not only do the points, tend to lie closer the the line, meaning they’re more accurate in the absolute sense, but the clusters, which represent different kpoints, run parallel to the line, meaning the new model is also predicting the fluctuations more accurately. (each cluster is a bit like a miniature version of the graph of the previous slide for that k-point). - (Finish 8:00) Given the model’s strength at predicting fluctuations, nature applications involve finite temperature effects on the band structure, and one effect we have looked at is the Urbach tail. This is an exponentially decaying tail of states into the bandgap due to structural fluctuations. Halide perovskites are known for having a small urbach tail, despite large structural fluctuations, and our simulations are agreeing with this, with just a small exponential tail of states with a length of 5 - 20 meV. This would have been difficult to calculate using DFT because of the large supercells required.
- (Finish 9:00)Another application where this model has had success was as part of a collaboration with Luke McClintock and Dong Yu and others, and they found that the carrier mobility increases dramatically below around 150K, by around a factor of 80 or 100.

We used our tight binding model to simulate temperature dependent effects on free carriers, such as the mass and scatteriing, and we found that while there were some changes with temperature, they’re far too small to account for the changes they observed, which supports the theory that the effect is due to the formation of excitions. - (Finish 10:00) So, overall, I think we’ve had built a fairly quantitatively accurate tight binding model for MAPbI3 and found some useful applications for it relating to finite temperature effects which require larger length and timescales. This means that it is particularly useful for nanostructures, surfaces, interfaces, defects, etc.

I also think that this general procedure and the programs we have written should be transferable. Nothing in this model is specific to MAPbI3, and I think we could probably parameterize it successfully for other halide perovskites and even other materials, with some changes to the geometry. Up to now, there’s been half a dozen people working for around 2 years to get to this point, but I think with this framework we could hopefully parameterize a model for a new material in a month or two. And I think this will allow us to look at a lot more interesting finite temperature electronic structure physics. - 40 sec

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