This document presents a new tight-binding model for simulating the electronic structure of methylammonium lead iodide (MAPbI3) at finite temperatures. The model improves upon previous work by better predicting transverse phonon modes and charge fluctuations. It is parameterized using DFT calculations and Wannier90 projections onto atomic orbitals. Tests show the new model more accurately captures bandgap fluctuations with temperature compared to DFT. The model allows for new simulations of nanostructures, surfaces, and other properties of perovskites.
APS D63.00002 Tight Binding Simulation of Finite Temperature Electronic Structure Dynamics in MAPbI3
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
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
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)
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)
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.
Notas del editor
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.