# Thermal fluctuations and carrier localization induced by dynamic disorder in
MAPbI3 described by a first-principles based tight-binding model

David J. Abramovitch  Department of Physics, University of California,
Berkeley  Wissam A. Saidi  Department of Mechanical Engineering and Materials
Science, University of Pittsburgh  Liang Z. Tan  Molecular Foundry, Lawrence
Berkeley National Laboratory

###### Abstract

Halide perovskites are strongly influenced by large amplitude anharmonic
lattice fluctuations at room temperature. We develop a tight binding model for
dynamically disordered MAPbI3 based on density functional theory (DFT)
calculations to calculate electronic structure for finite temperature crystal
structures at the length scale of thermal disorder and carrier localization.
The model predicts individual Hamiltonian matrix elements and band structures
with high accuracy, owing to the inclusion of additional matrix elements and
descriptors for non-Coulombic interactions. We apply this model to electronic
structure at length and time scales inaccessible to first principles methods,
finding an increase in band gap, carrier mass, and the sub-picosecond
fluctuations in these quantities with increasing temperature as well as the
onset of carrier localization in large supercells induced by thermal disorder
at 300 K. We identify the length scale $L^{*}=5$ nm as the onset of
localization in the electronic structure, associated with associated with
decreasing band edge fluctuations, increasing carrier mass, and Rashba
splitting approaching zero.

##  I Introduction

Halide perovskites present an unusual combination of electronic and atomic
structures [1, 2, 3]. The electronic structure in halide perovskites displays
properties of a crystalline semiconductor, including ballistic charge
transport, low carrier masses [4], sharp absorption onsets [4], long carrier
lifetimes [4, 5, 3], and long diffusion lengths [4, 6, 7, 5]. These favorable
electronic properties are employed in optoelectronic applications, including
photovoltaics [4, 8, 5, 9], in which power conversion efficiencies exceed 24%
[9]. However, in contrast to conventional semiconductors, electronic structure
in halide perovskites is strongly coupled to large, polar, and anharmonic
lattice fluctuations [1, 10, 11, 12]. In room temperature MAPbI3, tilting of
the Pb-I octahedra and rotation of the A-site MA cation are both strongly
anharmonic, creating a glass-like phonon structure with a short phonon mean
free path and diffusive phonon transport [10, 2, 13]. Therefore, understanding
the properties of band-like electronic states requires treating their
interaction with local fluctuations in a dynamically disordered atomic lattice
[1, 14]. This combination of electronic and atomic structure has been linked
to a wide variety of observed properties, including high defect tolerance [15,
16, 3], low radiative and non-radiative recombination leading to long carrier
lifetimes and diffusion lengths [7, 17], and large polaron formation [18].

The length and time scales required to simulate dynamic disorder pose a
challenge to traditional electronic structure methods such as conventional
density functional theory (DFT), while anharmonic fluctuations require going
beyond linear electron-phonon coupling [19]. The dynamically disordered
lattice [10] means atomic motion can be viewed in terms of local fluctuations
[11], with fluctuations of the charged lattice and bond displacements
respectively leading to fluctuations in the onsite energy and hopping of
localized electronic orbitals. Several previous studies have modeled halide
perovskite electronic structure with tight binding methods [20, 21, 22, 23,
24, 25, 26, 27], using empirical modifications of equilibrium crystal
structures to calculate the effects of strain [20], spin orbit coupling [20,
26], spin splitting [22], or random fluctuations in onsite energies [26, 27]
and MA orientations [25]. However, accurate prediction of finite-temperature
electronic structure requires accurate prediction of the instantaneous
Hamiltonian matrix elements for a realistic thermally disordered structure.

Herein, we develop an atomic orbital tight binding model for MAPbI3 which can
calculate the electronic structure of systems involving 100,000s of electrons
with moderate computational cost. The model predicts orbital onsite energies,
hopping amplitudes, and spin-orbit couplings from atomic positions, which, in
practice, are often generated from molecular dynamics trajectories. Through a
tight-binding Hamiltonian, these quantities together provide a complete
description of the frontier band electronic structure, which dictates carrier
dynamics under normal operating conditions. This model builds upon earlier
work [21, 28] by adding a new set of physically motivated structural
descriptors that result in increased accuracy, allowing for the investigation
of electronic structure at energy scales relevant to localization and dynamic
disorder in these materials. We find that a detailed description of Coulomb
and short-ranged interactions is essential for an accurate description of the
Hamiltonian, as is the inclusion of a number of non-bonding hopping amplitudes
whose fluctuations play an important role in finite temperature electronic
structure. The interpretations of these descriptors are discussed in Section
II and resulting predictions of physical observables like bandgaps and carrier
masses, in Section IV.

We then apply this model to large-scale simulations at finite temperatures,
showing the onset of carrier localization induced by dynamic disorder. We
study the length scale dependence of bandgap fluctuations, carrier masses, and
spin splittings, finding significant disorder-induced renormalization of
carrier masses and rapid reduction in spin splittings over progressively
larger length scales. These results are collected in Section VI.

##  II Descriptors of the electron-lattice interaction

While the complexities of the electron-lattice interaction stem from the large
numbers of electronic and ionic degrees of freedom, there are guiding
principles that simplify the task of parameterizing these interactions. Long-
range interactions arise from Coulombic forces, while short-range interactions
are limited in the number of ions they can involve. In the following, we
identify structural descriptors of the electron-ion interaction, and
investigate the relative extents to which they control the electronic
structure.

Our starting point is the tight binding Hamiltonian

$\hat{H}_{TB}=\sum_{i}\epsilon_{i}(\\{\vec{R}_{a}\\})c^{\dagger}_{i}c_{i}+\sum_{<i,j>}t_{ij}(\\{\vec{R}_{a}\\})c^{\dagger}_{i}c_{j}+h.c.$
(1)

where $\\{\vec{R}_{a}\\}$ are the classical atom positions and $c_{i}$ and
$c^{\dagger}_{i}$ are the electron annihilation and creation operators for
atomic orbital states. Our model consists of Pb $s$\- and $p$-orbitals, and I
$p$-orbitals, as these are the orbitals with dominant contributions to the
conduction and valence bands. We use an atomic orbital basis rather than a
maximally localized Wannier function (MLWF) basis because atomic orbitals are
independent of crystal structure while MLWFs are not. This allows for easier
prediction of the atomic orbital Hamiltonian from atomic positions. In our
model, the orientation of $p$-orbitals is fixed to lie along the
crystallographic axes.

The purpose of the model is to predict the terms
$\epsilon_{i}(\\{\bm{R}_{a}\\})$ and $t_{ij}(\\{\bm{R}_{a}\\})$ (choosing
which $t_{ij}$ are non-zero), allowing for electronic structure calculations
from a set of classical atom positions. In our model,
$\epsilon_{i}(\\{\bm{R}_{a}\\})$ is a linear combination of potentials
calculated from atomic positions and $t_{ij}(\\{\bm{R}_{a}\\})$ is calculated
from the bond geometry between the $i$ and $j$ atoms for bonding and non-
bonding interactions, and selection rules and potentials for spin-orbit
coupling. The numerical parameters for each matrix element are given in
Supplementary Information Section 3.

###  II.1 Onsite Energies

As the main contribution to the onsite energies is expected to be the
electrostatic potentials, previous tight binding models have approximated
onsite energy fluctuations based on the potential at the center of the atom
[21, 27]. However, we find that this is insufficient because this potential
does not distinguish between different $p$-orbitals on the same atom, which
can have significant onsite energy variations. Furthermore, modeling onsite
energies using point-charge Coulomb potentials would neglect several effects,
such as electronic screening, finite orbital and ionic radii, electron-
electron Hartree and exchange-correlation interactions, and hydrogen bonds,
which are believed to be important in MAPbI3 [29].

To address these issues, we include not only the Coulomb potential $V_{C}$ in
the model, but also a short-range potential $V_{SR}$. We model the effective
potential felt by orbitals as $V(\vec{r})=V_{C}(\vec{r})+V_{SR}(\vec{r})$,
where

$V_{C}(\vec{r})=\sum_{A}\sum_{a\in
A}\frac{q_{A}}{4\pi\epsilon_{0}|\vec{R_{a}}-\vec{r}|}$ (2)

$V_{SR}(\vec{r})=\sum_{A}\sum_{a\in A}\frac{\eta_{A}}{8\pi
l_{A}^{3}}\exp(-|\vec{r}-\vec{R}_{a}|/l_{A})=\sum_{A}V_{SR,A}(\vec{r})$ (3)

Here, the potentials are written as a sum over the contribution from atoms of
type $A$ , which belong to the set $\\{Pb,I,N,C,H(-N),H(-C)\\}$. In
anticipation of the fitting done below, we define the contribution to the
short-range potential from atoms of type $A$ to be $V_{SR,A}$. The charges
$q_{A}$ and the ionic radii $l_{A}$ are fixed in our model (Table 1), while
the relative magnitudes of the short-range potentials $\eta_{A}$ are allowed
to vary. The charges on the N, C, and H atoms are chosen based on their
percent ionic character [30] and constraining the MA dipole moment to be
$0.477e\AA$ [31]. The orbital onsite energies are then given by the effective
potential

$\varepsilon_{i}(\\{\vec{R}_{a}\\})=\int
d^{3}r\,V(\vec{r})\,\lvert\psi_{i}(\vec{r})\rvert^{2}$ (4)

where $\lvert\psi_{i}(\vec{r})\rvert^{2}$ is the charge density for orbital
$i$. We simplify the evaluation of Eq. 4 by performing a Taylor series
expansion in the spatial extent of the orbitals in the cartesian directions
$\\{x_{1},x_{2},x_{3}\\}$, obtaining

$\varepsilon_{i}(\\{\vec{R}_{a}\\})=\sum_{n=0}\sum_{\mu=1}^{3}a^{(n)}_{\mu}\frac{d^{n}V_{C}}{dx_{\mu}^{n}}+\sum_{A}\sum_{n=0}\sum_{\mu=1}^{3}b^{(n)}_{A,\mu}\frac{d^{n}V_{SR,A}}{dx_{\mu}^{n}}+\bar{\varepsilon}$
(5)

where the $a^{(n)}_{\mu}$ and $b^{(n)}_{A,\mu}$ coefficients are treated as
fitting parameters, and the potentials and their derivatives are evaluated at
the position of orbital $i$. We have included an explicit potential offset
$\bar{\varepsilon}$ to account for the vacuum level. The fitting parameters
thus obtained can be interpreted as encoding the amount of screening of the
potentials $V_{C}$ and $V_{SR}$, as well as the sizes of the orbital charge
density $\lvert\psi_{i}(\vec{r})\rvert^{2}$ (the parameters $\eta_{A}$ are
absorbed into $b^{(n)}_{A,\mu}$). The spatial derivatives of $V_{C}$ and
$V_{SR}$ capture the directional dependence of the $p$-orbital onsite
energies. In our model, we include up to 4th order in $V_{C}$ derivatives and
2nd order in $V_{SR}$ derivatives. Symmetry requirements place restrictions on
the $a^{(n)}_{\mu}$ and $b^{(n)}_{A,\mu}$ coefficients — for an $s$-orbital,
derivatives in all directions carry equal weight, so that
$a^{(n)}_{1}=a^{(n)}_{2}=a^{(n)}_{3}$. For a $p$-orbital oriented along the
$x_{1}$ axis, $a^{(n)}_{1}\neq a^{(n)}_{2}=a^{(n)}_{3}$. Odd derivatives do
not contribute because the orbital electron density is inversion symmetric. In
our implementation, we have used $\nabla^{2}V_{C}=0$ to simplify the numerical
evaluation of the Coulomb terms.

By including higher order potential derivatives, the electronic structure
becomes sensitive to an expanded set of structural distortions. This is of
particular relevance in this material class where the soft lattice can undergo
large amplitude anharmonic fluctuations. For instance, rotations of a single
PbI6 octahedron do not change the potential at the central Pb site, but do
modify the directional potential derivatives. The “head-to-head” A-site modes
predominant at low frequencies [11] are likewise strongly directional in their
potential fluctuations. Because of this, we find that fits lacking either
directional derivatives or short ranged potentials have limited accuracy (see
SI Section 5).

###  II.2 Hopping Parameters

We consider nearest-neighbor hopping parameters $t_{ij}$, which can be
categorized into Pb s - I p $\sigma$-bonds, Pb p - I p $\sigma$-bonds, Pb p -
I p $\pi$-bonds and Pb s - I p, Pb p - I p, and I p - Pb p nonbonding hopping
parameters. The latter three nonbonding parameters are zero in an idealized
perovskite crystal structure, but can take on non-zero values in a dynamically
disordered crystal structure at finite temperature. We model the hopping
parameters $t_{ij}(\\{\vec{R}_{a}\\})$ with a Koster-Slater form based on the
displacement $(x_{1},x_{2},x_{3})=\vec{r}_{j}-\vec{r}_{i}$ between the two
bonded atoms, with hopping parameters decreasing exponentially with bond
displacement. In general, the form of the bond hopping parameters is

$t_{ij}=a\exp\left(-\sum_{\mu}b_{\mu}x_{\mu}-\sum_{\mu}c_{\mu}x_{\mu}^{2}\right)+d$
(6)

and the form of the non-bonding hopping parameters is

$t_{ij}=ax_{\nu}\exp\left(-\sum_{\mu}b_{\mu}x_{\mu}-\sum_{\mu}c_{\mu}x_{\mu}^{2}\right)$
(7)

where $a$, $b_{\mu}$, $c_{\mu}$, and $d$ are fitting parameters and for non-
bonding $\nu$ is the direction of the non-bonded p-orbital. This functional
form captures changes in bond strength with shortening or lengthening of the
bond along the primary bonding direction ($b_{\mu}$), as well as the effects
of atom displacement along perpendicular directions ($c_{\mu}$). As with the
onsite energies, symmetry requirements place constraints on the $b_{\mu}$,
$c_{\mu}$ parameters; additionally, the parameters are used only to the
leading order in each direction. For instance, a Pb s - I p $\sigma$-bond
along the $x_{1}$ direction requires that $b_{2}=b_{3}=0$, and $c_{1}=0$ as it
is a second order effect in the $x_{1}$ direction. The functional forms for
each of the six cases are given in Supplementary Information section 3.

We find that bond hopping parameter fluctuations are among the most important
Hamiltonian matrix elements in finite temperature bandgap, mass, and spin
splitting calculations. Additionally, we find that non-bonding hopping
parameters play an important role in finite temperature fluctuations despite
being zero in the time-averaged tetragonal and cubic structures with all atoms
in high-symmetry positions.

###  II.3 Spin Orbit Coupling

The spin orbit coupling (SOC) has a strong effect on the electronic structure
of the halide perovskites, due to the presence of multiple high-$Z$ elements.
The effects include a strong renormalization of the bandgap, possible Rashba
splitting [32, 33], circular dichroism [34], unusual spin dynamics[35, 36,
37], and modified excitonic fine structure[38]. The Rashba effect for example
involves the combination of SOC with crystal structure distortions that break
inversion symmetry, and has been linked to indirect bandgaps in the halide
perovskites[39]. Spin orbit coupling fluctuations also play an important role
in bandgap fluctuations. It is therefore necessary to include SOC and account
for structural distortions.

For two orbitals $\lvert\psi_{i}\rangle$ and $\lvert\psi_{j}\rangle$ in a
spherically symmetric atomic potential, the SOC takes the form:

$\displaystyle t_{ij}=$
$\displaystyle\left\langle\psi_{i}\left\lvert\frac{-e}{2m^{2}c^{2}}\frac{1}{r}\frac{dV(r)}{dr}\hat{\bm{L}}\cdot\hat{\bm{S}}\right\rvert\psi_{i}\right\rangle$
(8)

In our model, we consider SOC between onsite $p$-orbitals on Pb and I only, as
$\hat{\bm{L}}\cdot\hat{\bm{S}}$ vanishes for $s$-orbitals, and the SOC of
orbitals on different sites is negligible. In a dynamically fluctuating
crystalline environment, the potential in Eq. 8 is not fully spherically
symmetric, with finite temperature fluctuations modifying each atom’s central
potential. We therefore modify Eq. 8 to

$\displaystyle t_{ij}(\\{\vec{R}_{a}\\})=$
$\displaystyle\gamma_{ij}(\\{\vec{R}_{a}\\})\left\langle\psi_{i}\left\lvert\hat{\bm{L}}\cdot\hat{\bm{S}}\right\rvert\psi_{j}\right\rangle$
(9)

where the magnitude of the SOC between orbitals on the same atom
$\gamma_{ij}(\\{\vec{R}_{a}\\})$ is expected to be affected by the
electrostatic potential and its derivatives in the vicinity of the atom. We
therefore take it to be a linear combination of the potentials $V_{C}$ and
$V_{SR}$ and their derivatives, as in section II.1 (although with different
coefficients):

$\gamma_{ij}(\\{\vec{R}_{a}\\})=\sum_{n=0}\sum_{\mu=1}^{3}c^{(n)}_{\mu}\frac{d^{n}V_{C}}{dx_{\mu}^{n}}+\sum_{A}\sum_{n=0}\sum_{\mu=1}^{3}d^{(n)}_{A,\mu}\frac{d^{n}V_{SR,A}}{dx_{\mu}^{n}}+\bar{\gamma}$
(10)

with $\bar{\gamma}$ being a constant fitting parameter. The
$\hat{\bm{L}}\cdot\hat{\bm{S}}$ term in Eq. 9 introduces a phase factor that
is either purely real or purely imaginary, with time-reversal symmetry
imposing the following constraints on the SOC matrix elements [21]:

$\displaystyle t_{p_{x}^{\uparrow},p_{y}^{\uparrow}}=\,$ $\displaystyle
t_{p_{x}^{\downarrow},p_{y}^{\downarrow}}^{*}$ (11) $\displaystyle
t_{p_{x}^{\downarrow},p_{z}^{\uparrow}}=\,$ $\displaystyle-
t_{p_{x}^{\uparrow},p_{z}^{\downarrow}}^{*}$ (12) $\displaystyle
t_{p_{y}^{\uparrow},p_{z}^{\downarrow}}=\,$ $\displaystyle-
t_{p_{y}^{\downarrow},p_{z}^{\uparrow}}^{*}$ (13)

##  III Methods

The model outlined in Section II is parameterized from DFT calculations on
crystal structures generated by classical molecular dynamics (MD). We use the
molecular dynamics force field presented by Mattoni et al [40] to generate
atomic trajectories in the NVT ensemble. We use experimental lattice
parameters [41, 42], scaled proportionately to their equilibrium MD values
(calculated in cells of dimension $6\sqrt{2}\times 12\times 6\sqrt{2}$). For
parameterization of the model, we use 40 instantaneous 2 $\times$ 2 $\times$ 2
supercell structures, taken at 1.5 picosecond intervals to minimize
correlations between instances, at each of the following temperatures: 150 K,
220 K, 240 K, 260 K, 280 K, 300 K, 320 K, 340 K. The structure is orthorhombic
at 150 K, tetragonal from 220 K to 320 K, and cubic at 340 K. DFT electronic
structure is calculated along the MD trajectories, with the PBE GGA exchange-
correlation functional[43], norm-conserving non-local pseudopotentials [44], a
4 x 4 x 4 $\Gamma$ centered k-point grid, 50 Rydberg energy cutoff, and fully-
relativistic spin orbit coupling, using the Quantum Espresso[45] software
package. We obtain a tight-binding Hamiltonian by projecting DFT electronic
structures onto the Pb s and p and I p atomic orbitals, using the Wannier90
code[46]. These Hamiltonian matrix elements are used to fit the model
according to Eq. 1. Carrier mass and spin splittings are calculated along the
$\Gamma\rightarrow X$, $\Gamma\rightarrow Y$, and $\Gamma\rightarrow Z$
directions.

##  IV Thermal fluctuations of Hamiltonian matrix elements, band masses, and
Rashba effects

To understand the mechanisms by which thermal fluctuations affect the
electronic structure, we compute finite-temperature carrier properties using
the descriptors introduced in section II, and compare them to ab-initio
predictions. Scatter plots and statistics for selected onsite energy, hopping
parameter, and spin-orbit coupling fits are shown in Fig. 1, showing that
model predictions of individual Hamiltonian matrix elements closely reproduce
DFT values. Plots for the full set of matrix elements are available in SI
section 4. This testifies to the high fidelity of the physically-motivated
features and fits employed in our tight binding model. Furthermore, we have
found that using more complex models for the tight binding parameters or using
non-linear optimization improves the hopping and spin-orbit terms in the
tight-binding Hamiltonian, however, these were found not to appreciably affect
the accuracy of band structure predictions.

We find that fluctuations of almost all the matrix elements along the
molecular dynamics trajectories greatly exceed the thermal energy scale at
room temperature, with standard deviations of 0.1-0.2 eV for the onsite
energies and $\sim$0.2 eV for the bonding and non-bonding hopping amplitudes,
implying that a thermally-averaged picture of the electronic structure would
not be valid. Thus, off-diagonal disorder of the hopping amplitudes, including
the nominally non-bonding matrix elements, cannot be neglected in this system.
Even though spin-orbit coupling has the smallest fluctuations among all the
matrix elements, the Pb spin-orbit matrix elements have fluctuations that are
still larger than the thermal energy scale.

We find that the model predictions of matrix elements have RMS errors in the
range of 0.01-0.04 eV, depending on the matrix element. These errors are a
factor of 3-10 times smaller than the fluctuations in the matrix elements
arising from lattice dynamics. The I spin-orbit coupling is the sole
exception, with an exceptionally small fluctuation scale of 0.005 eV, and an
RMS error of a similar magnitude, which is likely limited by accuracy of the
DFT calculation. Overall, these results show that the potential and bond
geometry descriptors introduced above provide a faithful description of the
important electronic structure fluctuations that are present at room
temperature.

This picture is supported by calculations of the tight binding $\Gamma$ point
energy gap, mass, and spin splitting from the modeled matrix elements, and by
comparing to predictions from the same set of matrix elements obtained from
DFT (hereafter, referred to as ab initio TB). Collectively, these 3 properties
describe the band structure near the band edge. We consider the band edge
energy dispersion of

$E_{\pm}(\vec{k})=\frac{\hbar^{2}k_{i}^{2}}{2m_{i}}\pm\alpha_{i}\cdot k_{i}$
(14)

which is valid for materials with time-reversal symmetry but not necessarily
inversion symmetry. The Rashba splitting term $\pm\vec{\alpha}\cdot\vec{k}$ is
a spin-dependent energy splitting present when inversion symmetry is broken.
We obtain the mass and Rashba $\vec{\alpha}$ from fitting
$\frac{E_{+}+E_{-}}{2}=\frac{\hbar^{2}k^{2}}{2m}$ and
$\frac{E_{+}-E_{-}}{2}=\vec{\alpha}\cdot\vec{k}$, respectively, for k-points
near the band edge.

Since the transformation from Hamiltonian matrix elements to the band edge
properties is highly nonlinear, errors in the matrix element model could in
principle become magnified or give biased predictions of band edge electronic
structure. For the model given above, we find that these issues are largely
absent, with the predicted $\Gamma$-point bandgap, band masses, and Rashba
parameters in good agreement with those calculated from ab initio TB on a
$2\times 2\times 2$ supercell (Fig. 3). Likewise, the band structure over the
entire Brillouin zone is well reproduced by the model, as shown in Figure 2
for a typical timestep along a 300 K MD trajectory. We note that the potential
and geometry descriptors are not explicitly dependent on temperature, as they
are functions of atomic positions only. As such, they perform well across the
tested temperature range of the tetragonal and cubic phases, which they have
been parameterized on. Comparisons to the full DFT band structure are given in
SI section 8, where it is seen that fluctuations are generally well predicted.

Next, we investigate the relative importance of the various types of
fluctuations on the electronic structure, by considering a series of
simplified models. By setting each onsite energy to its average value, we
neglect potential fluctuations in a manner akin to a mean-field approximation.
This approximation results in significantly overestimated bandgaps and band
masses compared to ab initio tight-binding values on a 2 $\times$ 2 $\times$ 2
supercell (Figure 4), implying that bandgaps and masses are strongly
renormalized by the presence of potential fluctuations. This can be understood
as the result of potential fluctuations increasing the bandwidth of valence
conduction bands, leading to a reduced bandgap and band masses.

Instead, if non-bonding hopping amplitudes are neglected (or equivalently,
assumed to not fluctuate), we find that bandgaps and band masses are both
underestimated. This suggests that non-bonding fluctuations tend to decrease
the bandwidths of conduction and valence bands, leading to increases in the
bandgap and band masses. In contrast, bond hopping fluctuations play an
important role in bandgap and mass fluctuations but do not significantly
affect the average in the manner of onsite and non-bonding fluctuations
(Figure 4). Furthermore, the calculation of Rashba splitting is inaccurate if
either bonding or non-bonding fluctuations are neglected, as these hopping
amplitudes are crucial for describing the form of local inversion symmetry
breaking [22].

In contrast, the approximation of neglecting the fluctuations in the spin-
orbit couplings (see SI section 9) does not severely affect the accuracy of
Rashba parameter predictions, as these are to first order determined by the
average values of the spin-orbit couplings. However, the predictions of the
bandgap fluctuations are improved by the inclusion of the spin-orbit coupling
fluctuations, as can be seen by comparing the quality of fit with the full
model. The spin-orbit fluctuations are therefore not responsible for
renormalization of the time-averaged bandgap, but do affect transient
electronic structure at the sub-picosecond time scale. Overall, these results
demonstrate that fluctuations in potentials, bonding, and spin-orbit coupling
are mutually competing factors that renormalize band edge electronic
structure, and should all be included for an accurate prediction of electronic
structure.

Lastly, we investigate the temperature dependence of bandgap and mass averages
and fluctuations in 300 ps trajectories sampled every 150 fs for thermal
averages and every 30 fs for fluctuation spectral analysis, a timescale
accessible through tight binding. These results are shown in Fig. 5. Previous
work has highlighted the importance of thermal expansion and anharmonic
electron-lattice interactions in predictions of the temperature dependence of
the bandgap [47, 19, 48, 21]. Our calculations show an increase in bandgap of
approximately 17 meV over a 100 K range, consistent with or somewhat below
experimental temperature dependence [49, 50, 47] depending on experimental
method. A possible reason for this underestimate is the somewhat
underestimated thermal expansion coefficient of our MD calculations compared
to experiment ($\alpha_{v}=9.5*10^{-5}/\text{K}$ between 220 K to 320 K in MD
compared to $1.57*10^{-4}/\text{K}$ in Ref. [51]). We also find average
carrier mass increases and becomes less anisotropic with increasing
temperature which we attribute to a combination of disorder and thermal
expansion. The bandgap standard deviation shows only a weak temperature
dependence, however, higher frequency fluctuations ($\gtrsim$ 1 THz) show a
significant increase with temperature. We rationalize this result by noting
that all temperatures show some degree of dynamic disorder which fluctuates at
longer timescales ($\gtrsim$ 1 ps), while increasing temperature increases the
energy of short timescale fluctuations.

##  V Effect of Dynamic Disorder on Carrier Masses

At scales smaller than the onset of localization, two major factors increase
charge carrier mass at finite temperature compared to the ideal perovskite
structure without thermal fluctuations. The first is the overall increase in
average bond length due to thermal fluctuations which decrease the hopping
between orbitals. The second is scattering caused by spatial disorder in the
finite temperature lattice, with every unit cell having different bond
lengths. To quantify the relative sizes of these effects, we compare the mass
calculated with tight binding under 3 conditions: 1) No fluctuations, using a
Hamiltonian with hopping parameters calculated from the fully symmetric ideal
tetragonal perovskite structure at 300 K and averaged onsite and spin orbit
coupling matrix elements, 2) spatially uniform fluctuations, obtained by
averaging 300 K thermally disordered Hamiltonian matrix elements over all unit
cells so that fluctuations enter only into the average of the Hamiltonian
matrix elements, and 3) 300 K thermal disorder, using the fully thermally
disordered 300 K structures. For these calculations, we use 100 structures
taken from 300 K trajectories, at a simulation size of 8 $\times$ 8 $\times$ 8
unit cells (a scale before any significant localization occurs as shown in
Sec. VI). Results are shown in Table II. In comparing the case of no
fluctuations with the case of spatially uniform fluctuations, which do include
the effects of octahedral rotations at an average level, we find that the
increased bond length of the latter case leads to an approximately 21%
increase in electron and hole mass above the ideal tetragonal perovskite
structure. Comparing to the case of full thermal disorder instead leads to an
approximately 66 % increase for holes and 71 % increase for electrons. We
therefore identify disorder as the larger factor increasing mass at finite
temperature. The large effect of thermal disorder on room temperature carrier
masses is consistent with previous calculations using anharmonic electron-
phonon interactions [48].

##  VI Localization induced by dynamic disorder

Because of the prevalence of potential and hopping amplitude fluctuations,
some amount of localization can be expected of electronic states in this
system. Localization in halide perovskites has been predicted to modify finite
temperature carrier properties, depending on the strength of the disorder [26,
27]. We proceed to quantify the extent of localization using tight binding
electronic structure obtained from the DFT-based model introduced above. We
apply our tight binding model to classical MD trajectories in large
supercells, calculating electronic structure on the scale of dynamic disorder,
for supercells of size N = 4, 6, 8, 10, 12, 14, and 16, at temperature T=300K.

We quantify the localization of energy eigenstates at the band edge
$\ket{\psi}$ with the inverse participation ratio, given in terms of the
projection of $\ket{\psi}$ onto atomic orbitals $\ket{\alpha}$:

$IPR=\left(\sum_{\alpha}\lvert\bra{\alpha}\ket{\psi}\rvert^{4}\right)^{-1}$
(15)

A completely localized state, i.e. one with limited spatial extent that does
not change with N, will have a small IPR independent of system size. On the
other hand, a completely delocalized state will have an IPR proportional to
the number of atomic orbitals. Plotting the dependence of the $IPR/N^{3}$ on
$N$ (Fig. 6), we find that the band edge wavefunctions of halide perovskites
occupy a regime intermediate between complete localization and complete
delocalization. The average $IPR/N^{3}$ progressively decreases with $N$,
indicating some amount of localization. However, this decrease is slower than
the expected $\frac{1}{N^{3}}$ dependence for complete localization. This
trend is observed for the entire range of supercells considered here, showing
that the system does not reach complete localization by N = 16. Instead, the
band edge wavefunctions show some spatial inhomogeneity at these length
scales, but contain finite density throughout the supercells of these sizes,
as illustrated in Fig. 7. The beginnings of localization behavior observed
here is consistent with previous models that predict the onset of localization
when disorder strengths reach a similar order of magnitude [26, 27], although
our predictions of the disorder strengths at 300K (potential fluctuations of
100-200 meV, Fig. 1) are numerically smaller than the values used in these
previous models (500 meV in Ref. [27]).

The increasing trend of carrier masses with supercell size further supports
this picture of nascent partial localization. We find that electron and hole
masses increase in all directions due to the presence of dynamic disorder, but
remain finite at these supercell sizes, indicating that these states are still
dispersive, and that ballistic transport is still possible at the length
scales of these simulations. These results show that dynamic disorder plays a
significant role in renormalizing carrier masses, with the results at $N=16$
indicating a 20% mass increase, which is likely to increase even further at
larger length scales. We find that carrier masses only start increasing
significantly beyond $N=8$. This nonlinearity of the carrier mass trends
suggests the presence of a length scale, which we identify as $L^{*}=5nm$,
corresponding to $N=8$, which marks the turning point beyond which carrier
masses start to increase rapidly with supercell size.

While $L^{*}$ is connected to the onset of localization, the wavefunctions
cannot be understood as being strictly confined to regions of size $L^{*}$,
due to the observed IPR scaling behavior. To further understand this length
scale, we investigate the scaling of the temporal fluctuations of the bandgap
with system size (Fig. 6). We find that bandgap standard deviation (SD),
calculated over MD trajectories, decreases with system size at length scales
less than $L^{*}$, but remains approximately constant for system sizes larger
than $L^{*}$. For a system close to complete delocalization, which has almost
constant wavefunction density in all unit cells, an increase in the system
size can be expected to decrease the temporal fluctuations of the bandgap, due
to spatial averaging across the entirety of the system. In contrast, for band
edge wavefunctions that are mostly sensitive to fluctuations within a spatial
region of fixed size, increasing the system size would not change the temporal
bandgap fluctuations coming from dynamic disorder within that region. We
therefore interpret $L^{*}$ as a local length scale over which band edge
wavefunctions are sensitive to dynamic disorder. Even though the system
contains long-range interactions (Section II), the bandgap SD is controlled by
the local fluctuations within $L^{*}$, while the long-range fluctuations
affect its average value instead of its SD. The results of these simulations
show that halide perovskite charge carriers exhibit locality in the sense of
electron-lattice interactions, while remaining dispersive and having
appreciable carrier density beyond this length scale.

Rashba splitting may play an important role in halide perovskites, caused by
inversion symmetry breaking due to surfaces, defects, or thermal fluctuations
[52, 53, 33, 54, 55, 56, 39, 57]. Bulk MAPbI3 is believed to be globally
centrosymmetric, forbidding a static Rashba effect[52]. However, lattice
fluctuations which break inversion symmetry have been hypothesized to create a
dynamical Rashba effect at a local scale [56]. We calculate spin splitting as
a function of supercell size, from the scale of local lattice fluctuations, up
to 16 $\times$ 16 $\times$ 16 (10.0 nm $\times$ 10.1 nm $\times$ 10.0 nm),
where the effects of average dynamic disorder rather than one specific local
configuration become important. We find that spin splitting, measured by the
average Rashba parameter $|\bm{\alpha}|$, decreases approaching zero (Fig 6d).
This can be understood through the fact that the correlation length of the
disorder is much smaller than $L^{*}$ [58], so that the effect of the disorder
averages to be inversion symmetric. However, Rashba splitting may still play
an important role near surfaces or crystalline inhomogeneities, important to
thin film applications.

##  VII Conclusions

Using a physically motivated atomic orbital tight binding model, we have
calculated the finite temperature electronic structure of MAPbI3 on the length
and time scales of thermal disorder. With this model, we find an increase in
band gap and carrier masses with increasing temperature as well as an increase
in sub-picosecond fluctuations in these observables. Investigating
localization, we demonstrate that partially localized states at the band edge
begin to emerge on the scale of 5-10 nm, providing a length scale for
localized effects of disorder. The electronic properties of these partially
localized states are controlled by dynamic disorder at the scale of $L^{*}=5$
nm, even though their spatial extent is larger. Our calculations show that
finite temperature effects lead to a significant increase in charge carrier
mass, mostly due to disorder and also in part to increased average bond
length. Thus dynamic disorder should be considered in combination with
polaronic effects when considering the effects of the lattice on charge
carrier mobility. We demonstrate that spin splitting goes to zero on the scale
of localization, as the correlation length of inversion symmetry breaking
disorder is much less than the scale of localization.

The relevant physics for designing a quantitatively accurate model also
provides insight on the important factors determining finite-temperature
electronic structure. We show that onsite energy fluctuations, bond
fluctuations, and non-bonding fluctuations all play a large role in finite
temperature electronic structure. The importance of non-bonding hopping
parameters and onsite fluctuations driven by effects other than the Coulomb
potential (including differentiating orbitals on the same atom) are especially
interesting, as to our knowledge they have not been included in previous tight
binding studies of dynamic disorder in halide perovskites. [21, 20, 27]. The
tight binding methods developed here are likely transferable to other
materials as well as halide perovskite systems with nanoscale structure. For
instance, defects [15, 16], interfaces, layered structures, nanocrystals, and
compounds mixing multiple halides [7, 59] or cations [60] are all of great
interest in the halide perovskite family.

##  VIII Acknowledgments

D. A. was supported by the U.S. Department of Energy, Office of Science,
Office of Workforce Development for Teachers and Scientists (WDTS) under the
Science Undergraduate Laboratory Internships (SULI) program. L. Z. T. was
supported by the Molecular Foundry, 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. 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.

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Atom $A$ | Charge $q_{A}$ (e) | Radius $l_{A}$ (Å)
---|---|---
Pb | 2 | 1.33
I | -1 | 2.06
N | 0.2509 | 0.67
C | 0.17352 | 0.56
H (- N) | 0.1617 | 0.53
H (- C) | 0.03016 | 0.53
Table 1: Charges used to compute coulomb potentials, and atomic radii used to compute local potentials. H (- N) refers to hydrogen bonded to the nitrogen in the methylammonium ion, while H (- C) refers to hydrogen bonded to carbon. Calculation | Hole Mass ($m_{e}$) | Electron Mass ($m_{e}$)
---|---|---
No fluctuations | (0.069, 0.074, 0.069 ) | (0.072, 0.074, 0.072)
Spatially averaged fluctuations | (0.084, 0.090, 0.084) | (0.088, 0.089, 0.087)
300 K Thermal Disorder | (0.114 , 0.126, 0.115) | (0.122, 0.128, 0.123)
Table 2: Calculations of charge carrier mass demonstrating the effect of
increased bond length and disorder at finite temperature.  Figure 1: Scatter
plots and statistics for select Hamiltonian matrix element fits in the tight
binding model. Predictions of $\bm{a}$ I onsite energies, $\bm{b}$ Pbp - Ip
$\sigma$ bond hopping matrix elements, $\bm{c}$ Pbp - Ip Nonbonding hopping
matrix elements, and $\bm{d}$ Pb p spin orbit coupling strength show good
accuracy compared against the DFT atomic orbital Hamiltonian matrix elements.
Figure 2: Band structure for an example 2 $\times$ 2 $\times$ 2 structure (the
first timestep at 300 K) is shown for model TB and ab initio TB, showing that
modeled tight binding matrix elements accurately reproduce results of ab
initio matrix elements. The Brillouin zone vectors shown are for the 2 x 2 x 2
supercell. Figure 3: Tight binding band structure predictions at finite
temperature, compared to test data (tight binding performed on DFT matrix
elements). a) $\Gamma$ point energy gap, b) hole mass, c) electron mass, d)
hole Rashba parameter $\alpha_{i}$ , and e) electron Rashba parameter
$\alpha$.  Figure 4: Effects of Hamiltonian matrix element fluctuations on
band structure: (a) $\Gamma$ point energy gap, (b) hole mass, (c) electron
mass, (d) hole Rashba parameter $\alpha_{i}$, and (e) electron Rashba
parameter. Compared to the full model, illustrated in (f), models with (g)
constant onsite energies , (h) constant bond hopping, or (i) no nonbonding
hopping show less accurate predictions as discussed in the text.  Figure 5:
Temperature dependence of observables and their fluctuations. (a) and (b)
Average bandgap and carrier mass as a function of temperature, calculated from
a 300 ps trajectory sampled every 150 fs. Error bars show standard error. (c)
and (d) Fourier transform of bandgap and carrier mass (y component)
fluctuations calculated from a 300 ps trajectory sampled every 30 fs. Gaussian
broaden is used with a width of 0.06 THz. Figure 6:  (a) The average IPR / N3
decreases with supercell size but the IPR does not reach a constant,
indicating only partial localization at this length scale. (b) average charge
carrier mass and (c) bandgap standard deviation show effects of localization
starting around n = 8, above which the mass increases and bandgap fluctuations
remain relatively constant. (d) Rashba parameter magnitude as a function of
supercell size, showing a quenching of spin splitting across all supercell
sizes. This corresponds to a more inversion symmetric average structure.
Statistics collected across 100 structures at a temperature of 300 K. Error
bars for a,b,d show the standard error of the mean, while error bars in c show
the standard error of the bandgap SD. Figure 7: Plot of electron density of
the (a) valence band maximum and (b) conduction band minimum of a 16 $\times$
16 $\times$ 16 structure, simulated at a temperature of 300 K. Radius of
circles is proportional to electron density of a given orbital. Red indicates
Pb s, blue indicates Pb p, and green indicates I p.