# Steric engineering of point defects in lead halide perovskites

Lucy D. Whalley<EMAIL_ADDRESS>[

###### Abstract

Due to their high photovoltaic efficiency and low-cost synthesis, lead halide
perovskites have attracted wide interest for application in new solar cell
technologies. The most stable and efficient ABX_3 perovskite solar cells
employ mixed A-site cations, however the impact of cation mixing on carrier
trapping and recombination—key processes that limit photovoltaic
performance—is not fully understood. Here we analyse non-radiative carrier
trapping in the mixed A-cation hybrid halide perovskite MA_1-xCs_xPbI_3. By
using rigorous first-principles simulations we show that cation mixing leads
to a hole trapping rate at the iodine interstitial that is eight orders of
magnitude greater than in the single cation system. We demonstrate that the
same defect in the same material can display a wide variety of defect
activity—from electrically inactive to recombination centre—and, in doing so,
resolve conflicting reports in the literature. Finally, we propose a new
mechanism in which steric effects can be used to determine the rate of carrier
trapping; this is achieved by controlling the phase and dynamical response of
the lattice through the A-site composition. Our findings elucidate crucial
links between chemical composition, defect activity and optoelectronic
performance, and suggest a general approach that can help to rationalise the
development of new crystalline materials with target defect properties.

Northumbria University] Department of Mathematics, Physics and Electrical
Engineering, Northumbria Universiry, Newcastle Upon Tyne, NE1 8ST, UK

##  1 Introduction

Organic-inorganic lead halide perovskites (OLHPs) have attracted wide interest
for their application in optoelectronics. Single junction halide perovskite
solar cells now exceed 25% power conversion efficiency 1, with the most stable
and efficient devices employing mixed organic (methylammonium, MA and
formamidinium, FA) or inorganic (Cs, Rb) cations on the A-site of the ABX3
perovskite structure. A-site cation engineering is primarily used to improve
thermal and chemical stability 2; the impact of A-site mixing on defect
activity is not fully understood, despite this being crucial for the
development of devices with increased efficiency 3.

The A-site cation indirectly determines various optoelectronic, transport and
defect properties through an influence on the symmetry and dynamic response of
the crystal lattice 3, 4. Although defect formation and activity is sensitive
to the exact system under consideration, the common observation across OLHPs
is that halide ions form abundant, mobile point defects which are active in
carrier trapping and recombination 5, 6. Furthermore, the bonding in OLHPs is
relatively weak, leading to an easily distorted (‘soft’) lattice and large
lattice relaxation after carrier capture at a defect site 7, 8.

Large lattice relaxation is not commonly observed in all-inorganic
optoelectronic materials. It is more akin to what occurs in organic molecular
materials 9, where steric engineering through the incorporation of bulky
ligands is used to restrict the vibrational modes associated with lattice
relaxation after electronic excitation 9, 10. Here we propose that a similar
approach can be used to rationalise the design of defect-tolerant OLHPs. Using
first-principles quantum chemical calculations, combined with methods adapted
from the field of organic semiconductors, we show that cation mixing in
MA_1-xCs_xPbI3 can be used to adjust the hole trapping rate at the iodine
interstitial by eight orders of magnitude. Furthermore, using symmetry mode
analysis we demonstrate that defect activity can be tuned by controlling phase
formation through the steric size of the A-site cations. This provides a new
route for engineering defect properties without altering the metal–halide
chemistry that is beneficial for photovoltaic performance.

##  2 Methods

###  2.1 Electronic structure calculations

The underlying electronic structures were calculated using density functional
theory (DFT) as implemented in VASP 11 using a plane wave basis set with an
energy cutoff of $400\text{\,}\mathrm{eV}$. A $2\\!\times\\!2\\!\times\\!2$
gamma centered Monkhorst–Pack mesh was used for the Brillouin zone
integration. The interstitial was placed in a 192-atom supercell. We
calculated $\Delta Q$ for the iodine interstitial in a 768 atom unit cell and
found it to be converged within
$1\text{\,}\mathrm{a}\mathrm{m}\mathrm{u}^{1/2}\mathrm{\text{Å}}$. A-site
ordering for the mixed cation systems were determined using Special Quasi-
random Structures 12 as implemented in the ICET code 13.

Ground state geometries were found using the PBEsol functional 14 with a force
cutoff of $0.01\text{\,}\mathrm{eV}\text{\,}{\mathrm{\text{Å}}}^{-1}$. The
interpolated geometries were generated using custom code (available in an
online repository 15) and the Atomic Simulation Environment 16. The potential
energy surface was calculated using the screened-exchange HSE06 functional 17
with $\alpha=0.43$ and spin–orbit coupling. The total energy cutoff was
${10}^{-5}\text{\,}\mathrm{eV}$. We used a uniform reduction factor to
evaluate HSE06 energies at the gamma point only. We calculated the neutral
defect formation energy in MAPbI3 using the full $2\\!\times\\!2\\!\times\\!2$
k-point grid and found it to be converged within $0.01\text{\,}\mathrm{eV}$
per formula unit. We employed a delta self-consistent field approach to
constrain the occupation of the defect states near energy level crossings. The
electron–phonon coupling term was derived from wavefunctions calculated at the
same level of theory.

###  2.2 Defect properties

The formation energy of a defect in charge state $q$ is given by

$E_{\mathrm{f}}(q)=E_{\mathrm{d}}(q)-E_{\mathrm{b}}-\sum_{i}\mu_{i}n_{i}+q(\epsilon_{\mathrm{VBM}}+E_{\mathrm{F}}),$

where $E_{\mathrm{d}}(q)$ is the total energy of the defect lattice in charge
state $q$, $E_{\mathrm{b}}$ is the total energy of the pristine lattice,
$\mu_{i}$ is the chemical potential of species $i$ and $n_{i}$ is the number
of atoms that are added or removed. $E_{\mathrm{d}}(q)$, $E_{\mathrm{b}}$,
$\mu_{i}$ and $\epsilon_{\mathrm{VBM}}$ were calculated using DFT, as outlined
in the previous section. $E_{\mathrm{d}}(q)$ includes a correction term for
charged defects, which was calculated using sxdefectalign with a value of
$\bar{\epsilon_{0}}=22.67$ for the static dielectric constant 18, 19. More
details on the methodology as applied to hybrid halide perovskites have
already been published in Reference 8.

A quantum mechanical treatment of electron capture was performed using the
open-source CarrierCapture package 20 which builds on the approach outlined in
Reference 21. In this model the carrier capture coefficient for capture from
an initial state i to a final state f is given by

$C=V\frac{2\pi}{\hbar}gW_{\mathrm{if}}^{2}\sum_{m}\Theta_{m}\sum_{n}|\langle\chi_{\mathrm{im}}|Q-Q_{0}|\chi_{\mathrm{fn}}\rangle|^{2}\times\delta(\Delta
E+m\hbar\omega_{\mathrm{i}}-n\hbar\omega_{\mathrm{f}}),$

where $\Delta E$ is the total change in energy, $V$ is the supercell volume,
$g$ is the energetic degeneracy of the final state, $W_{\textrm{if}}$ is the
electron-phonon coupling matrix element,
$\langle\chi_{\mathrm{im}}|Q-Q_{0}|\chi_{\mathrm{fn}}\rangle$ is the overlap
of the vibrational wavefunctions $\chi$, and the Dirac $\delta$ ensures that
there is conservation of energy. In practice the Dirac $\delta$ term is
replaced by a smearing function; for the calculations in this study this is a
gaussian function of width $0.01\text{\,}\mathrm{eV}$. $\Theta_{m}$ is the
thermal occupation of the vibrational state $m$. The electron-phonon coupling
term was calculated using the Nonrad package 22. Further details of the
methodology can be found in the literature 21.

###  2.3 Geometry and symmetry analysis

Bond lengths and bond angles were analysed using the Atomic Simulation
Environment 16. Crystal structures were visualised using vesta 23. Isodistort
was used for symmetry mode analysis, with MAPbI3 in the parent cubic phase
$Pm\bar{3}m$ as a reference structure. The rotational motion of the MA
molecule was not considered in this analysis; all A-sites were modelled as
point particles. The phonon mode amplitudes were normalised to the parent cell
volume to allow comparison between different compositions.

##  3 Results

###  3.1 Carrier capture rates with molecular rotations

We focus our analysis on the iodine interstitial defect in the negative
(I${}_{\mathrm{i}}^{-}$) and neutral (I${}_{\mathrm{i}}^{0}$) charge states as
these have been found to be most detrimental to solar cell efficiency 24. The
iodine interstitial is a negative-U defect so it follows that
I${}_{\mathrm{i}}^{0}$ is metastable 25, 26, 5, 8, 27. However
I${}_{\mathrm{i}}^{0}$ can still be formed through electron capture at
I${}_{\mathrm{i}}^{+}$ or hole capture at I${}_{\mathrm{i}}^{-}$. As shown in
Figure 1a, the neutral iodine interstitial I${}_{\mathrm{i}}^{0}$ bonds with a
lattice iodine to produce a molecular I${}_{2}^{-}$ H-centre with a trapped
hole 27. After electron capture or hole release the negative charge state
I${}_{\mathrm{i}}^{-}$ is formed in a split-interstitial configuration. This
is accompanied by tilting and distortions of the inorganic PbI6 octahedra to
accomodate the I-I bond lengthening. The inorganic structural changes are
coupled to MA rotations, similar to the behaviour observed during thermal
phase transitions 28.

To model the kinetics of charge capture at a defect site we must consider the
coupling of electronic and atomic structure. We map the potential energy
surface (PES) between two charge states as a function of a collective
coordinate

$Q=\sqrt{\sum_{i}m_{i}\Delta r_{i}^{2}},$

where the sum is over atoms $i$ with mass $m_{i}$ and a displacement from
equilibrium of $\Delta r_{i}$. For inorganic and hybrid materials the standard
procedure is to conduct a linear interpolation between each atomic position of
the two equilibrium structures 21, 29, 30, and this is the procedure that has
been previously applied to hybrid perovskites 26, 8, 25, 31. Contributions
from the rotation of the molecular cation have previously been ignored as
linear interpolation adjusts the intramolecular bond lengths to give
unphysical energies 8.

We introduce an interpolation method (which we will term ‘Kabsch
interpolation’) most commonly used for molecular materials, but transferred
here for the first time to a hybrid inorganic-organic material. To interpolate
the molecular species we first express atomic positions in the initial and
final geometries as a set of vectors, $\mathbf{a}_{i}$ and $\mathbf{a}_{f}$
respectively. We use the Kabsch algorithm to calculate an optimal rotation
axis $e$ and angle $\theta$ that maps between $\mathbf{a}_{i}$ and
$\mathbf{a}_{f}$ whilst minimising the root mean squared deviation between
each vector pair 32. This allows us to rotate the molecule around $e$ using a
linear interpolation of $\theta$. Finally, we combine this with a linear
interpolation of both the molecular centre of mass and inorganic framework.
This method generates a more accurate energy surface that allows us to
consider molecular translations and rotations. Unphysical energies are avoided
as the molecule is treated as a rigid object that cannot be deformed.

Figure 1:  Carrier capture processes in single and mixed cation perovskites.
Predicted energies are also given in Table S1. (a) Crystal structures for: i)
a pristine (defect-free) lead halide perovskite material; ii) the neutral
iodine interstitial in a H-centre configuration with a localised hole; iii)
the negative iodine interstitial in a split-interstitial configuration. (b)
Configuration coordinate diagram for the neutral and negative iodine
interstitial in MAPbI3. Each scatter point represents a DFT calculated total
energy. The solid grey lines are the potential energy surfaces (PES) generated
using a lower accuracy interpolation method that describes translations only
(no molecular rotations). (c) Nonradiative carrier capture coefficients for
the iodine interstitial in MAPbI3. Calculated for electron capture at a
neutral iodine interstitial (yellow solid line), hole capture at a negative
iodine interstitial (blue solid) and electron capture followed by hole capture
(red dash) (d) Ratio of electron capture rate (from the conduction band) and
hole emission rate (into the valence band) for the neutral iodine interstitial
in MAPbI3. (e) Configuration coordinate for the neutral and negative iodine
interstitial in MA_0.875Cs_0.125PbI3. The grey dot line is the PES of the
negative iodine interstitial in MAPbI3, given for comparison. (f-g) As in
(c-d), but for MA_0.875Cs_0.125PbI3.

Figure 1b shows configuration coordinate diagrams for the
I${}^{0}_{\mathrm{i}}\Leftrightarrow\mathrm{I}^{-}_{\mathrm{i}}$ transitions
in MAPbI3, using standard linear interpolation (in grey) and Kabsch
interpolation (in colour). We use the Heyd-Scurseria-Ernzerhof hybrid
functional 17 alongside spin-orbit coupling to obtain accurate defect
energetics. $Q=0$ corresponds to the equilibrium configuration of
I${}^{0}_{\mathrm{i}}$ and $Q=\Delta Q=$
$36\text{\,}\mathrm{a}\mathrm{m}\mathrm{u}^{1/2}\mathrm{\text{Å}}$ corresponds
to that of I${}^{-}_{\mathrm{i}}$. A nonradiative recombination process beings
at I${}^{0}_{\mathrm{i}}$ with an electron at the conduction band minimum
(CBM) and hole at the valence band maximum (VBM). This is represented with the
orange dash line in Figure 1b. After electron capture there is a transition to
I${}^{-}_{\mathrm{i}}$ (blue dot line); in the semiclassical picture this
requires overcoming the energy barrier $\Delta E_{\mathrm{n}}$. Subsequent
hole capture over the energy barrier $\Delta E_{\mathrm{p}}$ to
I${}^{0}_{\mathrm{i}}$ (green solid line) completes the recombination cycle.
For Kabsch interpolation the energy surface is softened as energy is
dissipated through rotations of the MA cation, resulting in a significant
reduction of $\Delta E_{\mathrm{n}}$ ($0.15\text{\,}\mathrm{eV}$ to
$0.025\text{\,}\mathrm{eV}$) and $\Delta E_{\mathrm{p}}$
($0.92\text{\,}\mathrm{eV}$ to $0.63\text{\,}\mathrm{eV}$). In semiclassical
models the capture rate has an exponential dependence on the ratio of barrier
height to $k_{\mathrm{B}}T$, so changes of $\sim$$100\text{\,}\mathrm{meV}$,
as seen here, can have a significant impact on defect activity.

For accurate predictions of capture rates, we use a quantum chemical theory to
calculate coefficients for electron capture ($C_{\mathrm{n}}$) and hole
capture ($C_{\mathrm{p}}$) 21, 29. This moves beyond the semiclassical picture
to include the strength of coupling between the defect state and CBM
($W^{\mathrm{n}}_{\mathrm{if}}$, for electron capture), or the defect state
and VBM ($W^{\mathrm{p}}_{\mathrm{if}}$, for hole capture). It also considers
the vibronic overlap between each PES, thus allowing quantum tunnelling below
the classical barrier. The capture coefficients determine the capture rate $R$
at a defect. To take electron capture at a neutral defect as an example,

$R_{n}=C_{n}N_{0}n,$ (1)

where $N_{0}$ is the neutral defect density and $n$ is the electron density.
We consider electron capture followed by hole capture, the total rate of which
is quantified using

$C_{\mathrm{np}}=\frac{C_{\mathrm{n}}C_{\mathrm{p}}}{C_{\mathrm{n}}+C_{\mathrm{p}}}.$
(2)

We note that this does not correspond to the total rate of non-radiative
recombination at the iodine interstitial, as we do not consider non-radiative
recombination with hole capture as the initial step.

Figure 1c shows that although electron capture is fast at
$300\text{\,}\mathrm{K}$, the non-radiative recombination process is limited
by the slow rate of hole capture. Importantly, competing with the first step
of this process (electron capture) is hole emission from the localised state
associated with I${}^{0}_{\mathrm{i}}$ (green solid line) to a delocalised
state in the valence band (forming I${}^{-}_{\mathrm{i}}$, blue dot line).
This process does not require photoexcitation so can happen ‘in the dark’. The
ratio of the electron capture rate $R_{\mathrm{n}}$ to the hole emission rate
$R^{\mathrm{em}}_{\mathrm{p}}$ is given by:

$\frac{R_{\mathrm{n}}}{R^{\mathrm{em}}_{\mathrm{p}}}=\frac{nC_{n}}{N_{v}C_{p}},$
(3)

where $N_{\mathrm{v}}$ is the density of occupied states in the valence band
and $n$ is the electron concentration 33. Assuming a hole effective mass value
of $0.2\text{\,}\emph{m}_{\mathrm{e}}$ 34 and that the electron concentration
is $1\text{\times}{10}^{15}\text{\,}{\mathrm{cm}}^{-3}$ 35, hole emission at
the neutral iodine interstitial will occur faster than electron capture at
temperatures above $160\text{\,}\mathrm{K}$ (Figure 1d). It is important to
highlight that once the negatively charged iodine interstitial is formed,
whether through electron capture or hole emission, it is limited by the slow
rate of hole capture ($C_{\mathrm{p}}=$
$6.0\text{\times}{10}^{-17}\text{\,}{\mathrm{cm}}^{3}\text{\,}{\mathrm{s}}^{-1}$
at $300\text{\,}\mathrm{K}$) and is electrically inactive.

###  3.2 Carrier capture in mixed A-cation systems

We now investigate the impact of cation mixing on nonradiative trapping and
recombination processes. We consider the mixed cation system MA_1-xCs_xPbI3
and compare this against the control case of MAPbI3. Initially we focus our
analysis on the mixed A-cation system MA_0.875Cs_0.125PbI3, which is close to
the Cs concentration reported to be optimal for device efficiency 36. We find
that a H-centre defect is formed with charge localisation around the iodine
dimer (Figure S1), indicating that the basic defect activity is comparable to
MAPbI3.

Figure 1e shows that the total lattice relaxation $\Delta Q$ is suppressed
through Cs incorporation
($36.8\text{\,}\mathrm{a}\mathrm{m}\mathrm{u}^{1/2}\mathrm{\text{Å}}$ to
$28.9\text{\,}\mathrm{a}\mathrm{m}\mathrm{u}^{1/2}\mathrm{\text{Å}}$). We note
that, despite the reduction, this lattice relaxation is still relatively large
as $\Delta Q$ is typically less than
$10\text{\,}\mathrm{a}\mathrm{m}\mathrm{u}^{1/2}\mathrm{\text{Å}}$ for all-
inorganic materials 30, 29. Analysis of the defect structure for each charge
state shows that the $\Delta Q$ reduction can be primarily attributed to
reduced displacements of Pb and I. In particular, the change in Pb-I-Pb bond
angle after charge capture is reduced, suggesting that octahedral rotations
are suppressed in the mixed cation materials (Table S3).

Figure 1e also shows an increase in the neutral to negative charge transition
level $\epsilon(0/-)$ ($-0.23\text{\,}\mathrm{eV}$ to
$0.06\text{\,}\mathrm{eV}$). The charge transition level
$\epsilon(q/q^{\prime})$ corresponds to the difference in total energy
$E_{\mathrm{d}}$ of defect states $q$ and $q^{\prime}$ evaluated at their
equilibrium configurations $Q$ and $Q^{\prime}$, respectively, and referenced
to the valence band edge $\epsilon_{\mathrm{VBM}}$ of the host material 37:

$\epsilon(q/q^{\prime})=\frac{E_{\mathrm{d}}(q^{\prime})\rvert_{Q^{\prime}}-E_{\mathrm{d}}(q)\rvert_{Q}}{q-q^{\prime}}-\epsilon_{\mathrm{VBM}}.$
(4)

The charge transition level (CTL) is a key parameter for classifying defect
activity in the semi-classical defect model. For example, defects within a few
$K_{\mathrm{B}}T$ of the band edge typically show single carrier trapping and
de-trapping behaviour, whilst ‘deep’ defects towards the middle of a band gap
may successively trap both carrier species and form a site for non-radiative
recombination 33. We note here that the CTL is distinct from the Kohn-Sham
defect eigenvalue as the former accounts for structural relaxation after
charge capture or release. The Kohn-Sham defect eigenvalue, often referred to
simply as ‘defect level’, has been monitored in previous studies which show
that there are large energy fluctuations over $1\text{\,}\mathrm{eV}$38, 39,
40. After cation mixing we find that $\epsilon(0/-)$ CTL increases so that the
two charge states become close to thermodynamic equilibrium. As a result, fast
hole trapping and de-trapping behaviour is expected.

We find that the shape of the PES is largely unchanged after Cs mixing. This
is especially true at small displacements around the equilibrium structures,
where there is an equal softening of the harmonic PES after Cs incorporation
for both the negative ($43\text{\,}{\mathrm{cm}}^{-1}$ to
$35\text{\,}{\mathrm{cm}}^{-1}$) and neutral ($58\text{\,}{\mathrm{cm}}^{-1}$
to $50\text{\,}{\mathrm{cm}}^{-1}$) charge states. For comparison, the _DX_
-Centre in GaAs has an effective harmonic frequency of
$81\text{\,}{\mathrm{cm}}^{-1}$. The increase in $\epsilon(0/-)$, combined
with a relatively rigid displacement of the $I^{-}_{\mathrm{i}}$ and
$I^{0}_{\mathrm{i}}$ PES in $E$-$Q$ space, leads to a reduction in the hole
capture barrier $\Delta E_{\mathrm{p}}$ ($0.63\text{\,}\mathrm{eV}$ to
$0.23\text{\,}\mathrm{eV}$) and a small increase in the electron barrier
height $\Delta E_{\mathrm{n}}$ ($0.025\text{\,}\mathrm{eV}$ to
$0.045\text{\,}\mathrm{eV}$). The reduction in $\Delta E_{\mathrm{p}}$
indicates that there will be an increased rate of hole trapping at the
negatively charged iodine interstitial after cation mixing.

To quantify the impact of cation mixing we analyse the same multiphonon
capture and recombination processes as outlined in Section 3.1. We find that
the defect state couples more strongly with the valence band
($W^{\mathrm{p}}_{\mathrm{if}}=$
$0.034\text{\,}\mathrm{eV}\,\mathrm{a}\mathrm{m}\mathrm{u}^{-1/2}\mathrm{\text{Å}}^{-1}$)
compared to the conduction ($W^{\mathrm{n}}_{\mathrm{if}}=$
$0.002\text{\,}\mathrm{eV}\,\mathrm{a}\mathrm{m}\mathrm{u}^{-1/2}\mathrm{\text{Å}}^{-1}$);
this is expected as the VBM has a larger density of electronic states derived
from iodine $p$-orbitals (Figure S2). This, combined with the reduction in
$\Delta E_{\mathrm{p}}$, results in a carrier recombination process that is
more balanced between electron and hole capture, and that shows two clear
regimes. At temperatures below $200\text{\,}\mathrm{K}$ the process is limited
by low vibronic overlap between occupied states of the I${}^{-}_{\mathrm{i}}$
and I${}^{0}_{\mathrm{i}}$ energy surfaces, leading to slow hole capture at
I${}^{-}_{\mathrm{i}}$ and a recombination coefficient $C_{\mathrm{np}}$ that
is strongly temperature dependent. At higher temperatures the small
$W^{\mathrm{n}}_{\mathrm{if}}$ makes electron capture at the
I${}^{0}_{\mathrm{i}}$ the limiting process, and $C_{\mathrm{np}}$ has a
reduced dependence on temperature (Figure 1f).

As in the single cation material, we must also consider hole emission from the
localised state associated with I${}^{0}_{\mathrm{i}}$. We find that that the
rate of hole emission exceeds the rate of electron capture across the whole
temperature range (Figure 1g). Furthermore, in contrast to the single cation
material, hole trapping is also fast with $C_{\mathrm{p}}$=
$8.4\text{\times}{10}^{-9}\text{\,}{\mathrm{cm}}^{3}\text{\,}{\mathrm{s}}^{-1}$
at $300\text{\,}\mathrm{K}$; an increase by over eight orders of magnitude
compared to single cation MAPbI3. We conclude that Ii will form a site for
successive hole trapping and de-trapping in this system.

###  3.3 Steric engineering of point defect properties

Figure 2:  Relationship between defect properties for the MA_1-xCs_xPbI3
series of materials. (a) Schematic showing key defect properties:
reorganisation energy $\lambda$, absorption energy
$E_{\mathrm{abs}}(q/q^{\prime})$ and charge transition level
$\epsilon(q/q^{\prime})$ (b) Relationship between reorganisation energy
$\lambda$ and $\Delta Q$. The purple circle, green triangle, orange cross and
blue square denote $x=0,0.125,0.25$ and 0.5 compositions respectively. Values
for the single cation $Pnma$ phase are taken from Reference 26. The upper dash
line is a first-order polynomial fit to the mixed cation data, the lower dash
line is a fit to the single cation data. The dot line is a fit to all scatter
points. (c) Relationship between charge transition level $\epsilon(0/-)$ and
$\Delta Q$. The dash line is a fit to the single cation data. The dot line is
a fit to all scatter points. The inset shows energies calculated with the
pbesol exchange-correlation functional (empty scatter points) and the hybrid
HSE06 functional with spin-orbit coupling (filled scatter points).

The increase in $\epsilon(0/-)$ after cation mixing can be rationalised using
the concept of reorganisation energy $\lambda$ from electron transfer theory
41, 42. As shown in Figure 2a, $\lambda$ is the free energy change associated
with the relaxation $\Delta Q$ between equilibrium geometries:

$\lambda=E_{\mathrm{d}}(q^{\prime})\rvert_{Q^{\prime}}-E_{\mathrm{d}}(q^{\prime})\rvert_{Q}.$
(5)

Note that by definition $\lambda$ is always negative. $\epsilon(q/q^{\prime})$
and $\lambda$ are related through the expression
$\epsilon(q/q^{\prime})=E_{\mathrm{abs}}(q/q^{\prime})\pm\lambda$ 37, where
$E_{\mathrm{abs}}$ is the Franck-Condon absorption energy for a vertical
transition with no change in lattice geometry:

$E_{\mathrm{abs}}(q/q^{\prime})=\frac{E_{\mathrm{d}}(q^{\prime})\rvert_{Q}-E_{\mathrm{d}}(q)\rvert_{Q}}{q-q^{\prime}}-\epsilon_{\mathrm{VBM}}.$
(6)

For relaxation after electron capture in MAPbI3 $|\lambda|>E_{\mathrm{abs}}$,
so that there is a thermodynamic penalty for re-forming the neutral charge
state. After Cs incorporation there is a reduction in $\Delta Q$ and, as a
result, the magnitude of $\lambda$ is also reduced. This leads to an increase
in $\epsilon(0/-)$ so that the two charge states become close to thermodynamic
equilibrium.

Reorganisation energy is a key parameter that controls charge mobility in
organic electronics and the efficiency of organic photovoltaic devices 43.
Across a wide range of molecular systems there a correlation between the
charge transfer reaction coordinate and $\lambda$; a smaller geometric
relaxation is associated with a smaller reorganisation energy. This
relationship enables charge transport engineering, whereby the size of groups
within a molecule and/or molecular packing density is used to determine the
extent of lattice relaxation and, following this, $\lambda$ 10, 44.

In order to evaluate the extension of steric engineering to lead halide
perovskites we begin by investigating the relationship between $\lambda$ and
$\Delta Q$ for carrier capture at the iodine interstitial in MA_1-xCs_xPbI3
across a range of stoichiometries ($x=0,0.125,0.25$ and 0.5). At higher
stoichiometries ($x>0.5$) we found that the valence band maximum increases so
that there is no localised defect state for hole capture within the band gap,
in agreement with previous reports for CsPbI3 45.

In Figure 2b we show that after Cs incorporation $\Delta Q$ is reduced
relative to the single cation system ($Fm\overline{3}m$ phase) for all
compositions. Note that to allow comparison between different A-site
compositions we scale $\Delta Q$ by $\sqrt{V_{c}/V}$ where $V$ is the volume
of the defect supercell and $V_{c}$ is the volume of the control supercell
(MAPbI3 in the $Fm\bar{3}m$ phase). We find that there is a strong linear
correlation between $\lambda$ and $\Delta Q$ for the mixed A-cation systems
(upper dash line). This analysis includes the point (0,0), as required by the
definition of reorganisation energy.

We combine our results with data from the literature at the same level of
theory for the single cation system 26. This enables us to consider three
lattice relaxation processes in MAPbI3, each corresponding to a different
regime of lattice relaxation: i) a vertical transition (no relaxation, $\Delta
Q=0$); ii) relaxation in the $Pnma$ phase (large relaxation, $\Delta Q\sim
20$); and iii) relaxation in the $Fm\overline{3}m$ phase (giant relaxation,
$\Delta Q\sim 35$). Figure 2b confirms a linear correlation between $\lambda$
and $\Delta Q$ for the single cation systems (lower dash line).

Across all data points (single and mixed cation materials) we find a linear
correlation (dot line). We also note that for all materials $|\lambda|$ is in
the range $1\text{\,}\mathrm{eV}$–$2\text{\,}\mathrm{eV}$, which is
significantly larger than that typically found for rigid organic molecules
($0.1\text{\,}\mathrm{eV}$–$0.2\text{\,}\mathrm{eV}$) 43 or point defects in
2D materials ($<$$1\text{\,}\mathrm{eV}$) 46.

To understand how $\Delta Q$ may be used to tune the the charge state and
activity of Ii we investigate the relationship between $\epsilon(0/-)$ and
$\Delta Q$. In Figure 2c we show a negative correlation between
$\epsilon(0/-)$ and $\Delta Q$ (dot line), indicating that $\epsilon(0/-)$ can
be tuned through variation of $\Delta Q$. As in Figure 2a, if we confine our
analysis to a single composition and $E_{\mathrm{abs}}$ value, the trend
becomes stronger (dash line).

The inset in Figure 2c compares $\epsilon(0/-)$ values calculated using the
semilocal pbesol exchange-correlation functional without spin-orbit coupling
(empty scatter points), and the HSE06 exchange correlation functional with
spin-orbit coupling (HSE06-SOC, filled scatter points). Across all materials
we observe a systematic increase in $\epsilon(0/-)$ at the higher HSE06-SOC
level of theory. This is due to a shift in the the predicted electronic band
edge, as has been observed in previous studies 5. Our results suggest that
once the change in $\epsilon(q/q^{\prime})$ for a single cation system is
known, this can be applied as a correction term to lower accuracy predictions
for related mixed cation systems.

Figure 2c provides insight into the significant $0.7\text{\,}\mathrm{eV}$
discrepancy between previously published values for $\epsilon(0/-)$ 8, 26. Our
results identify that the source of this discrepancy is the perovskite phase
used for modelling, and the influence this has on the predicted $\Delta Q$:
whilst Reference 8 uses the high symmetry $Pm\bar{3}m$ pseudo-cubic phase,
Reference 26 uses the lower symmetry orthorhombic phase, which is formed from
condensation of the $M_{3}^{+}$ and $R_{4}^{+}$ phonon modes associated with
octahedral tilting. If a subset of the distortions associated with lattice
relaxation in the cubic phase is not available in the orthorhombic phase, this
will lead to the observed reduction in $\Delta Q$ and increase in
$\epsilon(0/-)$.

Our analysis suggests that a symmetry lowering mechanism might be responsible
for the $\Delta Q$ reduction observed after cation mixing. To test this
hypothesis we use symmetry mode analysis 47. This allows us to decompose the
structural changes after A-cation mixing into the normal phonon modes of the
single cation cubic structure, giving insight into the specific atomic
displacements that are contributing to any lattice distortion. Figure 3a shows
that for all three mixed cation systems the most dominant phonon modes is
$\Gamma_{4}^{-}$. This is a polar zone centre displacement corresponding to an
off-centering of Pb and I. The second most dominant phonon modes is
$M_{3}^{+}$. This is a zone boundary mode and a dominant component of thermal
phase transitions in perovskite materials 48. It corresponds to rigid
rotations of the PbI6 octahedra, and consequently describes motions of I only.
The amplitudes for all phonon modes are given in Figures S3–S5.

Figure 3:  Symmetry mode analysis of MA_1-xCs_xPbI3 with the cubic perovskite
phase $Pm\bar{3}m$ used as a reference (parent) structure. a) Amplitude of the
two most dominant phonon modes $M_{3}^{+}$ and $\Gamma_{4}^{-}$. Green (Pb)
and orange (I) are used to indicate the contribution from each atomic species;
the A-site species is not displaced for either mode. b) Schematic of the
atomic displacements that correspond to each phonon mode. For easier
visualisation of the octahedral tilt patterns the iodine atoms are not shown.

Inspection of the $\Gamma_{4}^{-}$ polarisation vector shows that this mode is
primarily a rigid shift relative to the A-site cation and so does not lead to
a reduction in volume. In contrast, mapping along the $M_{3}^{+}$ mode reduces
the metal-halide-metal bonding angle and the cubo-octahedral volume around the
A-site. This is in agreement with our measured volumes and Pb-I-Pb bond angle
after Cs incorporation (Table S2). We conclude from this that the reduced
steric size of the Cs cation ($1.81\text{\,}\mathrm{\text{Å}}$) compared to MA
($2.70\text{\,}\mathrm{\text{Å}}$) leads to a volume contraction primarily
mediated via condensation of the $M_{3}^{+}$ octahedral tilting mode.

Group theoretical analysis confirms that displacement along $M_{3}^{+}$
reduces the symmetry from cubic to tetragonal ($P4/mbm$), which is in
agreement with the structures reported for FA/Cs and FA/MA/Cs mixed cation
systems 49 50. These experimental observations of mixed A-cation perovskites
in the tetragonal phase suggest two important points. Firstly, that this
symmetry lowering behaviour is common across other systems where cation mixing
leads to a reduction in unit cell volume. Secondly, that any dynamic disorder
leading to an effective pseudo-cubic phase is suppressed; the PbX6 octahedral
rotations are ‘locked-in’ and do not time-average to a higher symmetry
structure. The latter point is supported by molecular dynamics simulations
which show that a low concentration of Cs or Rb in FAPbI3 suppresses
octahedral tilting 51, 52.

##  4 Discussion

Our calculations show that Cs incorporation leads to a significant increase in
the rate of non-radiative trapping and de-trapping at the iodine interstitial.
At first this may appear to contradict the well-established improvement in
performance for mixed cation perovskite materials, which are used in cells
with the highest power conversion efficiencies (PCEs) 3, 53. However for high-
performance PV materials, both high PCE and good stability (chemical, thermal
and mechanical) are a prerequisite. Figure 1 suggests that the high PCEs for
mixed A-cation perovskites do not derive from a reduced rate of non-radiative
trapping or recombination in the bulk (in-grain) material as-synthesised, but
follows from other well-documented factors including increased
(photo-)stability 2, 3, 54, 55 and phase purity 56, 57.

Our results are in agreement the relatively limited experimental
characterisation of mixed MA/Cs perovskite materials. Microwave
photoconductivity decay measurements show that Cs-Br incorporation reduces
carrier lifetime in MA1-xCsxPbI3-xBrx ($x=$ 0.05, 0.1 and 0.15) perovskites
films immediately after deposition 58. The films were then kept in ambient
conditions and the carrier lifetime was measured hourly for nine hours; whilst
the carrier lifetime of MAPbI3 films decreased over this time period, the
carrier lifetime of the Cs/MA compounds increased. In a separate study, powder
x-ray diffraction shows that incorporation of Cs in MA1-xCsxPbBr leads to a
reduction in unit cell volume and increased tilting of the PbBr3 inorganic
cage 59. Time-resolved photoluminescence measurements confirm that there is a
decrease in the non-radiative recombination lifetime for the MA-rich compounds
($x=$0.2, 0.4) compared to the pure MA compound. For increasing Cs content
($x=$0.6, 0.8) the non-radiative lifetime is restored, suggesting that the
dominant defect-meditated recombination channel has been removed. This
supports our calculations showing that there is no active hole trapping state
at higher Cs proportion ($x>0.5$).

Figure 4:  Schematic illustration outlining the impact of steric engineering
and crystal phase on Ii defect activity. $\epsilon(0/-)$ is the neutral to
negative charge transition level, $\Delta Q$ is a measure of lattice
relaxation between the equilibrium geometries of I${}_{\mathrm{i}}^{0}$ and
I${}_{\mathrm{i}}^{-}$. The control case of single cation MAPbI3 in the
pseudo-cubic phase shows the largest lattice relaxation after carrier capture,
leading to an electrically inactive defect (orange illustrations). A-site
mixing with a smaller cation reduces lattice relaxation ($\Delta Q^{\prime}$)
and the hole capture barrier ($\Delta E_{\mathrm{p}}^{\prime}$), leading to
fast hole trapping and de-trapping (blue). Full symmetry lowering to the
orthorhombic phase leads to further reductions in $\Delta Q^{\prime\prime}$
and $\Delta E_{\mathrm{p}}^{\prime\prime}$, and a charge transition level that
is in the electronic band gap (green).

Our results demonstrate that CTLs are highly sensitive to $\Delta Q$, and that
the same defect in the same material can show a wide variety of defect
behaviour. As shown schematically in Figure 4, the iodine interstitial can
vary from being electrically inactive (with $\epsilon(0/-)$ in the valence
band) through being a site for non-radiative hole trapping and de-trapping
(with $\epsilon(0/-)<k_{\textrm{B}}T$), to being a site for non-radiative
recombination (with $\epsilon(0/-)$ towards the middle of the band gap). This
conclusion is supported by time resolved photolouminescence measurements of
MAPbI3 which show that regions with greater compressive strain are associated
with increased non-radiative decay 60. In addition, first-principles molecular
dynamics studies demonstrate that the Kohn-Sham eigenvalues associated with
defect states in hybrid halide perovskites can oscillate by as much as
$1\text{\,}\mathrm{eV}$, reinforcing our finding that defects in this system
have an unusually high level of sensitivity to lattice distortions 38, 39, 40.

Our results also reveal the possibility of tuning defect properties through
control of $\Delta Q$, which is a new approach to defect engineering for
hybrid and inorganic materials. Heterovalent doping to adjust the Fermi level
is firmly established 61, as is adjusting the chemical potentials of reactants
during growth to increase the formation energy (and thus decrease the
concentration) of harmful defects 62. Instead, we propose isovalent doping (in
this case, at the perovskite A-site) to determine the available lattice
relaxation pathways ($\Delta Q$), which in turn adjusts the CTL and carrier
capture barriers. This approach is enabled by a material system which shows
unusually large lattice relaxation and associated re-organisation energies. In
this study we find that the doping-induced volume contraction leads to a
symmetry-lowering distortion which ‘locks in’ the accepting phonon modes that
are active in non-radiative carrier recombination. However there are other
mechanisms beyond this which could used to engineer the available lattice
relaxation pathways; for example, applied hydrostatic pressure and epitaxial
growth have been shown to induce phase transformations in perovskite materials
63, 64.

##  5 Conclusions

We have analysed non-radiative carrier trapping in single and mixed A-cation
perovskite systems. We have introduced an interpolation method for hybrid
materials that describes the coupling between electronic states and molecular
rotations, and applied this to MA_1-xCs_xPbI3 for accurate predictions of
defect activity. We find that cation mixing leads to a significant increase in
the rate of hole trapping and de-trapping compared to the single cation system
in the pseudo-cubic phase. Importantly, we find a linear relationship between
$\epsilon(0/-)$ and $\Delta Q$, demonstrating that the same defect in the same
material can display a wide range of defect activity depending on e.g. film
morphology. The reduction in $\Delta Q$ is associated with a symmetry-lowering
distortion; in the case of mixed A-site cations this is induced through the
decreased steric size of Cs compared to MA. Furthermore, our results suggest
that $\Delta Q$ can be used as an additional parameter to tune carrier
trapping activity in crystalline materials. This is a general approach for
materials that display large lattice relaxation after carrier capture, and may
be extended to other hybrid organic-inorganic materials including metal-
organic frameworks.

The author thanks Marc Etherington, Séan Kavanagh, Jarvist Frost and Aron
Walsh for scientific discussions. This work used the Oswald High Performance
Computing facility operated by Northumbria University (UK). Via our membership
of the UK’s HEC Materials Chemistry Consortium, which is funded by EPSRC
(EP/R029431), this work used the ARCHER2 UK National Supercomputing Service
(http://archer2.ac.uk).

##  6 Data Availability Statement

The custom analysis code and data used to generate the figures and tables in
this work are openly available at https://dx.doi.org/10.5281/zenodo.7689415.
Input and output files for the the HSE06-SoC total energy calculations are
openly available at https://dx.doi.org/10.17172/NOMAD/2023.02.27-2.

Supporting Information: DFT calculated energies, charge density distribution,
geometric analysis data, electronic density of states, and symmetry mode
analysis (PDF)

## References

  * Jiang et al. 2022 Jiang, Q. et al. Surface reaction for efficient and stable inverted perovskite solar cells. _Nature_ 2022,
  * Ceratti et al. 2021 Ceratti, D. R. et al. The pursuit of stability in halide perovskites: the monovalent cation and the key for surface and bulk self-healing. _Mater. Horiz._ 2021, _8_ , 1570–1586
  * Lee et al. 2022 Lee, J.-W.; Tan, S.; Seok, S. I.; Yang, Y.; Park, N.-G. Rethinking the A cation in halide perovskites. _Science_ 2022, _375_ , eabj1186
  * Egger et al. 2018 Egger, D. A.; Bera, A.; Cahen, D.; Hodes, G.; Kirchartz, T.; Kronik, L.; Lovrincic, R.; Rappe, A. M.; Reichman, D. R.; Yaffe, O. What Remains Unexplained about the Properties of Halide Perovskites? _Advanced Materials_ 2018, _30_ , 1800691
  * Du 2015 Du, M.-H. Density Functional Calculations of Native Defects in CH3NH3PbI3: Effects of Spin–Orbit Coupling and Self-Interaction Error. _The Journal of Physical Chemistry Letters_ 2015, _6_ , 1461–1466
  * Senocrate and Maier 2019 Senocrate, A.; Maier, J. Solid-State Ionics of Hybrid Halide Perovskites. _Journal of the American Chemical Society_ 2019, _141_ , 8382–8396
  * Frost and Walsh 2016 Frost, J. M.; Walsh, A. What Is Moving in Hybrid Halide Perovskite Solar Cells? _Accounts of Chemical Research_ 2016, _49_ , 528–535
  * Whalley et al. 2021 Whalley, L. D.; van Gerwen, P.; Frost, J. M.; Kim, S.; Hood, S. N.; Walsh, A. Giant Huang–Rhys Factor for Electron Capture by the Iodine Intersitial in Perovskite Solar Cells. _Journal of the American Chemical Society_ 2021, _143_ , 9123–9128
  * Fratini et al. 2020 Fratini, S.; Nikolka, M.; Salleo, A.; Schweicher, G.; Sirringhaus, H. Charge transport in high-mobility conjugated polymers and molecular semiconductors. _Nature Materials_ 2020, _19_ , 491–502
  * Crespo-Otero et al. 2019 Crespo-Otero, R.; Li, Q.; Blancafort, L. Exploring Potential Energy Surfaces for Aggregation-Induced Emission—From Solution to Crystal. _Chemistry – An Asian Journal_ 2019, _14_ , 700–714
  * Kresse and Furthmüller 1996 Kresse, G.; Furthmüller, J. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. _Phys. Rev. B_ 1996, _54_ , 11169
  * Zunger et al. 1990 Zunger, A.; Wei, S.-H.; Ferreira, L. G.; Bernard, J. E. Special quasirandom structures. _Phys. Rev. Lett._ 1990, _65_ , 353–356
  * Ångqvist et al. 2019 Ångqvist, M.; Muñoz, W. A.; Rahm, J. M.; Fransson, E.; Durniak, C.; Rozyczko, P.; Rod, T. H.; Erhart, P. ICET – A Python Library for Constructing and Sampling Alloy Cluster Expansions. _Advanced Theory and Simulations_ 2019, _2_ , 1900015
  * Perdew et al. 2008 Perdew, J. P.; Ruzsinszky, A.; Csonka, G. I.; Vydrov, O. A.; Scuseria, G. E.; Constantin, L. A.; Zhou, X.; Burke, K. Restoring the Density-Gradient Expansion for Exchange in Solids and Surfaces. _Phys. Rev. Lett._ 2008, _100_ , 136406
  * Whalley 2023 Whalley, L. D. Zenodo repository: NU-CEM/Kabsch_interpolation. 2023,
  * Larsen et al. 2017 Larsen, A. H. et al. The atomic simulation environment—a Python library for working with atoms. _Journal of Physics: Condensed Matter_ 2017, _29_ , 273002
  * Heyd et al. 2003 Heyd, J.; Scuseria, G. E.; Ernzerhof, M. Hybrid functionals based on a screened Coulomb potential. _The Journal of Chemical Physics_ 2003, _118_ , 8207–8215
  * 18 Sphinx project, Welcome to the SPHInX repository. https://sxrepo.mpie.de/, Accessed: 2019-08-02
  * Brivio et al. 2013 Brivio, F.; Walker, A. B.; Walsh, A. Structural and electronic properties of hybrid perovskites for high-efficiency thin-film photovoltaics from first-principles. _APL Mater._ 2013, _1_ , 042111
  * Kim et al. 2020 Kim, S.; Hood, S. N.; van Gerwen, P.; Whalley, L. D.; Walsh, A. CarrierCapture.jl: Anharmonic Carrier Capture. _J. Open Source Softw._ 2020, _5_ , 2102
  * Alkauskas et al. 2014 Alkauskas, A.; Yan, Q.; Van de Walle, C. G. First-principles theory of nonradiative carrier capture via multiphonon emission. _Phys. Rev. B_ 2014, _90_ , 075202
  * Turiansky et al. 2021 Turiansky, M. E.; Alkauskas, A.; Engel, M.; Kresse, G.; Wickramaratne, D.; Shen, J.-X.; Dreyer, C. E.; Van de Walle, C. G. Nonrad: Computing nonradiative capture coefficients from first principles. _Comput. Phys. Commun._ 2021, _267_ , 108056
  * Momma and Izumi 2011 Momma, K.; Izumi, F. VESTA3 for three-dimensional visualization of crystal, volumetric and morphology data. _Journal of Applied Crystallography_ 2011, _44_ , 1272–1276
  * Ni et al. 2022 Ni, Z.; Jiao, H.; Fei, C.; Gu, H.; Xu, S.; Yu, Z.; Yang, G.; Deng, Y.; Jiang, Q.; Liu, Y.; Yan, Y.; Huang, J. Evolution of defects during the degradation of metal halide perovskite solar cells under reverse bias and illumination. _Nature Energy_ 2022, _7_ , 65–73
  * Meggiolaro et al. 2018 Meggiolaro, D.; Motti, S. G.; Mosconi, E.; Barker, A. J.; Ball, J.; Andrea Riccardo Perini, C.; Deschler, F.; Petrozza, A.; De Angelis, F. Iodine chemistry determines the defect tolerance of lead-halide perovskites. _Energy Environ. Sci._ 2018, _11_ , 702–713
  * Zhang et al. 2020 Zhang, X.; Turiansky, M. E.; Shen, J.-X.; Van de Walle, C. G. Iodine interstitials as a cause of nonradiative recombination in hybrid perovskites. _Phys. Rev. B_ 2020, _101_ , 140101
  * Whalley et al. 2017 Whalley, L. D.; Crespo-Otero, R.; Walsh, A. H-Center and V-Center Defects in Hybrid Halide Perovskites. _ACS Energy Letters_ 2017, _2_ , 2713–2714
  * Herz 2018 Herz, L. M. How Lattice Dynamics Moderate the Electronic Properties of Metal-Halide Perovskites. _The Journal of Physical Chemistry Letters_ 2018, _9_ , 6853–6863
  * Kim et al. 2019 Kim, S.; Hood, S. N.; Walsh, A. Anharmonic lattice relaxation during nonradiative carrier capture. _Phys. Rev. B_ 2019, _100_ , 041202
  * Kavanagh et al. 2021 Kavanagh, S. R.; Walsh, A.; Scanlon, D. O. Rapid Recombination by Cadmium Vacancies in CdTe. _ACS Energy Letters_ 2021, _6_ , 1392–1398
  * Zhang and Wei 2022 Zhang, X.; Wei, S.-H. Origin of Efficiency Enhancement by Lattice Expansion in Hybrid-Perovskite Solar Cells. _Phys. Rev. Lett._ 2022, _128_ , 136401
  * Kabsch 1976 Kabsch, W. A solution for the best rotation to relate two sets of vectors. _Acta Crystallographica Section A_ 1976, _32_ , 922–923
  * Das et al. 2020 Das, B.; Aguilera, I.; Rau, U.; Kirchartz, T. What is a deep defect? Combining Shockley-Read-Hall statistics with multiphonon recombination theory. _Phys. Rev. Materials_ 2020, _4_ , 024602
  * Whalley et al. 2019 Whalley, L. D.; Frost, J. M.; Morgan, B. J.; Walsh, A. Impact of nonparabolic electronic band structure on the optical and transport properties of photovoltaic materials. _Phys. Rev. B_ 2019, _99_ , 085207
  * Wheeler et al. 2017 Wheeler, S.; Bryant, D.; Troughton, J.; Kirchartz, T.; Watson, T.; Nelson, J.; Durrant, J. R. Transient Optoelectronic Analysis of the Impact of Material Energetics and Recombination Kinetics on the Open-Circuit Voltage of Hybrid Perovskite Solar Cells. _The Journal of Physical Chemistry C_ 2017, _121_ , 13496–13506
  * Choi et al. 2014 Choi, H.; Jeong, J.; Kim, H.-B.; Kim, S.; Walker, B.; Kim, G.-H.; Kim, J. Y. Cesium-doped methylammonium lead iodide perovskite light absorber for hybrid solar cells. _Nano Energy_ 2014, _7_ , 80–85
  * Gallino et al. 2010 Gallino, F.; Pacchioni, G.; Di Valentin, C. Transition levels of defect centers in ZnO by hybrid functionals and localized basis set approach. _The Journal of Chemical Physics_ 2010, _133_ , 144512
  * Cohen et al. 2019 Cohen, A. V.; Egger, D. A.; Rappe, A. M.; Kronik, L. Breakdown of the Static Picture of Defect Energetics in Halide Perovskites: The Case of the Br Vacancy in CsPbBr3. _The Journal of Physical Chemistry Letters_ 2019, _10_ , 4490–4498
  * Wang et al. 2022 Wang, B.; Chu, W.; Wu, Y.; Casanova, D.; Saidi, W. A.; Prezhdo, O. V. Electron-Volt Fluctuation of Defect Levels in Metal Halide Perovskites on a 100 ps Time Scale. _The Journal of Physical Chemistry Letters_ 2022, _13_ , 5946–5952
  * Chu et al. 2020 Chu, W.; Saidi, W. A.; Zhao, J.; Prezhdo, O. V. Soft Lattice and Defect Covalency Rationalize Tolerance of $\beta$-CsPbI3 Perovskite Solar Cells to Native Defects. _Angewandte Chemie International Edition_ 2020, _59_ , 6435–6441
  * Marcus 1956 Marcus, R. A. On the Theory of Oxidation‐Reduction Reactions Involving Electron Transfer. _The Journal of Chemical Physics_ 1956, _24_ , 966–978
  * Brédas et al. 2002 Brédas, J. L.; Calbert, J. P.; da Silva Filho, D. A.; Cornil, J. Organic semiconductors: A theoretical characterization of the basic parameters governing charge transport. _Proceedings of the National Academy of Sciences_ 2002, _99_ , 5804–5809
  * Shluger and Grutter 2018 Shluger, A. L.; Grutter, P. Reorganization takes energy. _Nature Nanotechnology_ 2018, _13_ , 360–361
  * Ørnsø et al. 2015 Ørnsø, K. B.; Jónsson, E. O.; Jacobsen, K. W.; Thygesen, K. S. Importance of the Reorganization Energy Barrier in Computational Design of Porphyrin-Based Solar Cells with Cobalt-Based Redox Mediators. _The Journal of Physical Chemistry C_ 2015, _119_ , 12792–12800
  * Jin et al. 2020 Jin, H.; Debroye, E.; Keshavarz, M.; Scheblykin, I. G.; Roeffaers, M. B. J.; Hofkens, J.; Steele, J. A. It’s a trap! On the nature of localised states and charge trapping in lead halide perovskites. _Mater. Horiz._ 2020, _7_ , 397–410
  * Bertoldo et al. 2022 Bertoldo, F.; Ali, S.; Manti, S.; Thygesen, K. S. Quantum point defects in 2D materials - the QPOD database. _npj Computational Materials_ 2022, _8_ , 56
  * Stokes et al. 2006 Stokes, H. T.; Hatch, D. M.; Campbell, B. J.; Tanner, D. E. ISODISPLACE: a web-based tool for exploring structural distortions. _Journal of Applied Crystallography_ 2006, _39_ , 607–614
  * Woodward 1997 Woodward, P. M. Octahedral Tilting in Perovskites. II. Structure Stabilizing Forces. _Acta Crystallographica Section B_ 1997, _53_ , 44–66
  * 49 Orri, J. F.; Doherty, T. A.; Johnstone, D.; Collins, S. M.; Simons, H.; Midgley, P. A.; Ducati, C.; Stranks, S. D. Unveiling the interaction mechanisms of electron and X-ray radiation with halide perovskite semiconductors using scanning nano-probe diffraction. _Advanced Materials_ _34_ , 2200383
  * Prasanna et al. 2017 Prasanna, R.; Gold-Parker, A.; Leijtens, T.; Conings, B.; Babayigit, A.; Boyen, H.-G.; Toney, M. F.; McGehee, M. D. Band Gap Tuning via Lattice Contraction and Octahedral Tilting in Perovskite Materials for Photovoltaics. _Journal of the American Chemical Society_ 2017, _139_ , 11117–11124
  * Ghosh et al. 2018 Ghosh, D.; Smith, A. R.; Walker, A. B.; Islam, M. S. Mixed A-Cation Perovskites for Solar Cells: Atomic-Scale Insights Into Structural Distortion, Hydrogen Bonding, and Electronic Properties. _Chemistry of Materials_ 2018, _30_ , 5194–5204
  * Ghosh et al. 2017 Ghosh, D.; Walsh Atkins, P.; Islam, M. S.; Walker, A. B.; Eames, C. Good Vibrations: Locking of Octahedral Tilting in Mixed-Cation Iodide Perovskites for Solar Cells. _ACS Energy Letters_ 2017, _2_ , 2424–2429
  * Hu et al. 2018 Hu, Y. et al. Understanding the Role of Cesium and Rubidium Additives in Perovskite Solar Cells: Trap States, Charge Transport, and Recombination. _Advanced Energy Materials_ 2018, _8_ , 1703057
  * Cho et al. 2018 Cho, H.; Kim, J. S.; Wolf, C.; Kim, Y.-H.; Yun, H. J.; Jeong, S.-H.; Sadhanala, A.; Venugopalan, V.; Choi, J. W.; Lee, C.-L.; Friend, R. H.; Lee, T.-W. High-Efficiency Polycrystalline Perovskite Light-Emitting Diodes Based on Mixed Cations. _ACS Nano_ 2018, _12_ , 2883–2892, PMID: 29494128
  * Niu et al. 2017 Niu, G.; Li, W.; Li, J.; Liang, X.; Wang, L. Enhancement of thermal stability for perovskite solar cells through cesium doping. _RSC Adv._ 2017, _7_ , 17473–17479
  * Saliba et al. 2016 Saliba, M.; Matsui, T.; Seo, J.-Y.; Domanski, K.; Correa-Baena, J.-P.; Nazeeruddin, M. K.; Zakeeruddin, S. M.; Tress, W.; Abate, A.; Hagfeldt, A.; Grätzel, M. Cesium-containing triple cation perovskite solar cells: improved stability, reproducibility and high efficiency. _Energy Environ. Sci._ 2016, _9_ , 1989–1997
  * Jena et al. 2019 Jena, A. K.; Kulkarni, A.; Miyasaka, T. Halide Perovskite Photovoltaics: Background, Status, and Future Prospects. _Chemical Reviews_ 2019, _119_ , 3036–3103
  * Singh et al. 2019 Singh, A.; Chouhan, A. S.; Avasthi, S. Effect of methylamine vapor exposure and ambient ageing on CsxMA1-xPbI3-xBrx perovskites for improved carrier collection. _Materials Research Express_ 2019, _6_ , 085519
  * Premkumar et al. 2019 Premkumar, S.; Kundu, K.; Umapathy, S. Impact of cesium in methylammonium lead bromide perovskites: insights into the microstructures, stability and photophysical properties. _Nanoscale_ 2019, _11_ , 10292–10305
  * Jones et al. 2019 Jones, T. W. et al. Lattice strain causes non-radiative losses in halide perovskites. _Energy Environ. Sci._ 2019, _12_ , 596–606
  * Grotz et al. 2012 Grotz, B.; Hauf, M. V.; Dankerl, M.; Naydenov, B.; Pezzagna, S.; Meijer, J.; Jelezko, F.; Wrachtrup, J.; Stutzmann, M.; Reinhard, F.; Garrido, J. A. Charge state manipulation of qubits in diamond. _Nature Communications_ 2012, _3_ , 729
  * Walsh and Zunger 2017 Walsh, A.; Zunger, A. Instilling defect tolerance in new compounds. _Nature Materials_ 2017, _16_ , 964–967
  * Jiao et al. 2021 Jiao, Y.; Yi, S.; Wang, H.; Li, B.; Hao, W.; Pan, L.; Shi, Y.; Li, X.; Liu, P.; Zhang, H.; Gao, C.; Zhao, J.; Lu, J. Strain Engineering of Metal Halide Perovskites on Coupling Anisotropic Behaviors. _Advanced Functional Materials_ 2021, _31_ , 2006243
  * Prakash et al. 2021 Prakash, A.; Wang, T.; Bucsek, A.; Truttmann, T. K.; Fali, A.; Cotrufo, M.; Yun, H.; Kim, J.-W.; Ryan, P. J.; Mkhoyan, K. A.; Alù, A.; Abate, Y.; James, R. D.; Jalan, B. Self-Assembled Periodic Nanostructures Using Martensitic Phase Transformations. _Nano Letters_ 2021, _21_ , 1246–1252