# Giant Huang-Rhys Factor for Electron Capture by the Iodine Intersitial in
Perovskite Solar Cells

Lucy D. Whalley<EMAIL_ADDRESS>Department of Mathematics,
Physics and Electrical Engineering, Northumbria University, Newcastle upon
Tyne, NE1 8QH, UK  Puck van Gerwen  Department of Materials, Imperial College
London, London SW7 2AZ, UK  Jarvist M. Frost  Department of Physics, Imperial
College London, London SW7 2AZ, UK  Sunghyun Kim  Department of Materials,
Imperial College London, London SW7 2AZ, UK  Samantha N. Hood  Department of
Materials, Imperial College London, London SW7 2AZ, UK  Aron Walsh
<EMAIL_ADDRESS>Department of Materials, Imperial College London,
London SW7 2AZ, UK

###### Abstract

Improvement in the optoelectronic performance of halide perovskite
semiconductors requires the identification and suppression of non-radiative
carrier trapping processes. The iodine interstitial has been established as a
deep level defect, and implicated as an active recombination centre. We
analyse the quantum mechanics of carrier trapping. Fast and irreversible
electron capture by the neutral iodine interstitial is found. The effective
Huang-Rhys factor exceeds 300, indicative of the strong electron-phonon
coupling that is possible in soft semiconductors. The accepting phonon mode
has a frequency of $53\text{\,}{\mathrm{cm}}^{-1}$ and has an associated
electron capture coefficient of
$1\text{\times}{10}^{-10}\text{\,}{\mathrm{cm}}^{3}\text{\,}{\mathrm{s}}^{-1}$.
The inverse participation ratio is used to quantify the localisation of phonon
modes associated with the transition. We infer that suppression of octahedral
rotations is an important factor to enhance defect tolerance.

Department of Materials Science and Engineering, Yonsei University, Seoul
03722, Korea

##  1 Introduction

The unusual defect chemistry and physics of lead halide perovskites has
attracted significant attention.1, 2, 3 Slow non-radiative electron-hole
recombination is unusual for solution processed semiconductors, and supports
high voltage and efficient light-to-electricity conversion in a solar cell.4
While significant defect populations are expected based on equilibrium
thermodynamics5 of these soft crystalline materials, and solution processing
introduces additional disorder,6 the native defects don’t appear to contribute
to non-radiative recombination of electrons and holes. This behaviour of
halide perovskites has been broadly termed “defect tolerance”.7, 8, 9 Further
improvement in the performance of halide perovskite devices requires
suppression of non-radiative carrier capture and recombination events.10 In
this report, we perform a quantum mechanical carrier-capture analysis of the
interaction between electrons and the iodine interstitial in CH3NH3PbI3
(MAPI).

Figure 1: Electron capture by a charged defect in a semiconductor. In the
initial configuration (blue), there is a charged defect (D+) and a free
electron (e-) residing in the conduction band. In the final configuration
configuration (pink), the electron is captured to yield a neutral charge state
(D0). The coordinate $Q$ maps out the change in structure between the two
configurations.

Shockley-Read-Hall (SRH) recombination is associated with the successive
capture of an electron and hole following the photo-excitation (nil ¡=¿[hν]
e^- + h^+) of a semiconductor. The kinetics reduce to first-order in a heavily
doped ($n$ or $p$ type) semiconductor; minority carrier capture becomes the
rate-limiting process. A necessary step is the change from delocalized-to-
localized electronic wavefunctions, and multiple phonon ($\hbar\omega$)
emission through the associated structural relaxation,11 as illustrated in
Figure 1. The excess electronic energy of the charge carriers is lost to heat.
Taking the example of a neutral defect (D0), the overall recombination process
is

$\ce{D^{0}<=>[h\nu]D^{0}+e^{-}+h^{+}<=>[-\sum\hbar\omega]D^{-}+h^{+}<=>[-\sum\hbar\omega]D^{0}}$
(1)

###  1.1 Non-radiative carrier capture from first principles

Making use of Fermi’s golden rule, the carrier capture coefficient from an
initial state i to a final state f can be described by

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

Here $V$ is the supercell volume, $g$ is the degeneracy of the final state,
$W_{\textrm{if}}$ is the electron-phonon coupling matrix element, $\Theta_{m}$
is the thermal occupation of the vibration state $m$, $\langle\chi_{im}|\Delta
Q|\chi_{fn}\rangle$ is the overlap of the vibrational wavefunctions $\chi$,
and the Dirac $\delta(\Delta
E+m\hbar\omega_{\textrm{i}}-n\hbar\omega_{\textrm{f}})$ ensures conservation
of energy. Each of these quantities can be derived from density functional
theory (DFT) calculations.12, 13

The meaning of Equation 2 becomes clearer when you realise that $\Delta E$ and
$\Delta Q$ refer to vertical and horizontal offsets in a configuration
coordinate diagram, and that $\omega_{\textrm{i}}$ ($\omega_{\textrm{f}}$) is
the frequency of the effective one-dimensional vibration in the initial
(final) charge state (Figure 1). Figure 2 provides a useful guide to the
sensitivity of the underlying model parameters and the accessible range of
$C$.

##  2 Methods

###  2.1 Electronic structure

A quantum mechanical treatment of electron capture was performed using the
open-source CarrierCapture package.13 The 1D Schrödinger equation for the
potential energy surface was solved using a finite difference method. This
builds on the approach of Alkauskas et al,12 and the implementation has been
applied to a range of semiconductors.14, 15

The underlying electronic structures were calculated using density functional
theory (DFT) as implemented in the GPU port of VASP,16 using a plane wave
basis set with an energy cut-off of $400\text{\,}\mathrm{eV}$. Projection
operators were optimised in real space with an accuracy of
$0.02\text{\,}\mathrm{meV}$ per atom, and a $2\\!\times\\!2\\!\times\\!2$
gamma centred Monkhorst-Pack mesh was used for the Brillouin zone integration.

So that no preference was given to a particular combination of octahedral
tilts the starting point for atomic relaxation was MAPI in the pseudo-cubic
phase. The interstitial was placed in a 192-atom supercell built from an
expansion of the 12-atom unit cell, using the transformation matrix $m_{t}$:

$m_{t}=\begin{bmatrix}2&-2&0\\\ 2&2&0\\\ 0&0&2\\\ \end{bmatrix}$

Ground states geometries were found using PBEsol functional,17 which has been
shown to accurately describe the structures and phonons of these materials.
The internal atomic coordinates were relaxed until the force acting on each
atom was less than
$0.01\text{\,}\mathrm{eV}\text{\,}{\mathrm{\SIUnitSymbolAngstrom}}^{-1}$.
Defect formation energies were converged to within $0.01\text{\,}\mathrm{eV}$
per formula unit between the 192- and 384-atom supercells.

The potential energy surface was calculated using the screened-exchange HSE06
functional18 ($\alpha$ = 0.43) including spin-orbit coupling (SOC), with total
energy converged to within ${10}^{-5}\text{\,}\mathrm{eV}$. The electron-
phonon coupling term is derived from wave functions calculated with the PBEsol
functional and SOC.

###  2.2 Lattice dynamics

The harmonic phonon modes were calculated using a
$2\\!\times\\!2\\!\times\\!2$ supercell expansion (93 atoms including the
iodine interstitial). To evaluate the force-constant matrix the finite
displacement method was used with displacements of
$0.01\text{\,}\mathrm{\SIUnitSymbolAngstrom}$. Forces were computed in VASP
using a plane wave basis set of $700\text{\,}\mathrm{eV}$, a total energy
convergence criterion of ${10}^{-8}\text{\,}\mathrm{eV}$ and the PBEsol
functional. A $2\\!\times\\!2\\!\times\\!2$ gamma centred Monkhorst-Pack mesh
was used for the Brillouin zone integration. To extract the phonon
eigenvectors and frequencies the Phonopy package19 was used. The Inverse
Participation Ratio was calculated using the Julia-Phonons package.20

##  3 Results and Discussion

Common vacancy defects do not introduce levels into the band gap of lead
iodide perovskites, whilst iodine interstitials do.1 Interstitial iodine may
be formed by the incorporation of excess iodine (I(g) ¡=¿ I_i) or through
Frenkel pair formation (I_I ¡=¿ V_I + I_i). There are three accessible charge
states for the iodine interstitial (+/0/$-$). The calculated geometries of the
three charge states are shown in Figure 3.

Figure 2: Variation of the carrier capture coefficient $C$ with the parameters
$\Delta E$ and $\Delta Q$ using Equation 2. This data is for the single
frequency approximation and taking frequencies near the extrema of the
physical range: (a) $\hbar\omega$ = 8 meV and (b) $\hbar\omega$ = 50 meV.

The neutral interstitial, rather than remaining as an isolated iodine atom in
an interstitial region, bonds with a lattice iodine to form an I${}_{2}^{-}$
complex. This is referred to as an H-center by the metal halide community and
has a characteristic paramagnetic $S=\frac{1}{2}$ spin configuration.21 Iodine
is well known to form polyiodide chains with bond lengths that are sensitive
to the charge state. The I–I bond length in solid orthorhombic crystalline
iodine is $2.67\text{\,}\mathrm{\SIUnitSymbolAngstrom}$, which lengthens to
$3.23\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ upon formation of
$\mathrm{I}_{2}^{-}$.22 Our bond lengths, calculated using a 193-atom pseudo-
cubic supercell and plane wave basis set cut-off at $400\text{\,}\mathrm{eV}$,
are within $0.05\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ of this value. Owing
to the octahedral tilting pattern, we distinguish between in-plane (IP) and
out-of-plane (OP) configurations for this defect.

Figure 3: Defect geometries of
$\mathrm{I}_{\mathrm{i}}^{+}$,$\mathrm{I}_{\mathrm{i}}^{-}$,$\mathrm{I}_{\mathrm{i}}^{\mathrm{IP}}$
and $\mathrm{I}_{\mathrm{i}}^{\mathrm{OP}}$ in CH3NH3PbI3. IP indicates that
the neutral defect is lying in the $ab$-plane. OP indicates that the defect is
lying along the $c$-axis. All distances are measured in units of Å. The iodine
is coloured in purple and the lead in grey. For clarity, the organic cations
are not show.

.

For the positive charge state, an asymmetric trimer structure is found that is
typical of $\mathrm{I}_{3}^{-}$. For example, the tri-iodide group in CsI_3
has interbond distances of $2.82\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ and
$3.10\text{\,}\mathrm{\SIUnitSymbolAngstrom}$, with the longer bond possessing
the majority of the additional charge.23 In the negative charge state, the
antibonding orbitals are filled, resulting in a split-interstitial
configuration with the longest I–I bond length of all three charge states.

In the dark, the positive charge state is thermodynamically favoured in a
p-type regime (EF close to the valence band) and the negative charge is
favoured in an n-type regime (EF close to the conduction band). The neutral
interstitial is metastable but has been suggested to form under
illumination,24, 25 which is our focus here.

Various first-principles studies have attempted to elucidate the nature of
charge capture at the iodine interstitial site. Fast non-radiative
recombination 25 and fast radiative recombination24 at the neutral iodine
interstitial have each been suggested. This contradiction is despite both
studies evaluating the electronic structure using the same Heyd-Scuseria-
Ernserhof (HSE) hybrid density functional18 and incorporating spin-orbit
coupling (SOC), demonstrating the sensitivity of this system to the exact
calculation parameters and defect geometries. As an extension to static DFT,
ab-initio molecular dynamics (MD) can be used to model the motion at room
temperature. Large fluctuations in the halide vacancy defect energy level (up
to $1\text{\,}\mathrm{eV}$),26 and the formation of hole polarons that
suppress charge recombination27 have both been reported. However, a
comprehensive MD analysis—using a hybrid functional and SOC— remains
computationally prohibitive.26, 28 We find a charge capture rate that is
highly sensitive to the defect geometry; the importance of geometry
relaxation, comparison with other computational reports, and the reference
configuration for defect energetics are discussed in the Supplementary
Information.

We now consider the potential energy surface (PES) associated with sequential
electron and hole capture, Figure 4.

$\ce{I_{i}<=>[h\nu]I_{i}+e^{-}+h^{+}<=>[-\sum\hbar\omega]I_{i}^{-}+h^{+}<=>[-\sum\hbar\omega]I_{i}}$
(3)

The coordinate $Q$ is defined as $\sqrt{\sum_{i}m_{i}\Delta r_{i}^{2}}$, where
the sum is over all inorganic atoms $i$ with mass $m_{i}$ and a displacement
from equilibrium of $\Delta r_{i}$. After electron capture at I_i^OP, not only
the I dimer but the Pb atoms and the surrounding I atoms that form the
octahedra relax significantly, as shown in Figure 3. Analysis of the Pb-I-Pb
angles in each charge state demonstrates that charge capture is associated
with rotations of the inorganic octahedral cage (Figure S6). As the large
displacements of the heavy Pb and I atoms are involved, $\Delta Q$ is large
($\Delta Q=$
$36\text{\,}\mathrm{a}\mathrm{m}\mathrm{u}^{1/2}\mathrm{\SIUnitSymbolAngstrom}$).
This value is double that typically found for non-radiative recombination
centres in kesterites,15 and demonstrates that there is strong coupling
between the electronic charge state of the defect and the lattice distortion.

The definition of configuration coordinate $Q$ is not unique. For example, we
might define $\Delta Q$ in the MAPI:Ii system as the root mean squared
displacement of the two bonding iodine. For this definition of $\Delta Q$ we
find that electron trapping at the neutral iodine interstitial proceeds with a
small geometrical rearrangement, $\Delta
Q=$$0.073\text{\,}\mathrm{\SIUnitSymbolAngstrom}$, resulting in fast radiative
electron capture. This small distortion of the iodine dimer has been reported
previously.24 However, this model excludes the large relaxation of the
surrounding perovskite structure, leading to a significant underestimation of
$\Delta Q$ and $S$.

There are two classes of phonon mode associated with non-radiative
transitions. Promoting modes couple the initial and final electronic states by
producing a sizeable electron-phonon coupling matrix element
$W_{\textrm{if}}$. Accepting modes take up the excess electronic energy after
charge capture, resulting in a change of mean displacement. Rather than
consider each phonon mode in turn (which would be computationally
prohibitive), we use the effective mode $Q$ to consider the accepting modes
only.29

$W_{\textrm{if}}$, a pre-factor in Equation 2, is proportional to: i) the
change in overlap between the initial (delocalised) and final (localised)
single particle electron wavefunctions, as a function of $Q$; and ii) the
change in energy between these two states. For electron capture at the neutral
iodine interstitial, $W_{\textrm{if}}$ is
$0.0036\text{\,}\mathrm{eV}\mathrm{a}\mathrm{m}\mathrm{u}^{-1/2}\mathrm{\SIUnitSymbolAngstrom}^{-1}$.
This is comparable to an estimate for the maximum possible value,
$W_{\textrm{if}}^{\textrm{max}}=$
$0.0048\text{\,}\mathrm{eV}\mathrm{a}\mathrm{m}\mathrm{u}^{-1/2}\mathrm{\SIUnitSymbolAngstrom}^{-1}$,
demonstrating that $Q$ has both accepting and promoting character, and
justifying the use of the configuration coordinate. Further details of this
comparison can be found in the Supplementary Information.

The Huang-Rhys factor, $S=\frac{\Delta E}{\hbar\omega}$, is the number of
phonons emitted after carrier capture. $S$ in the strong coupling regime
($S>>1$) is typically associated with non-radiative carrier capture
accompanied by multi-phonon emission.29 The harmonic PES of the iodine
interstitial is soft, with an effective frequency of
$38\text{\,}{\mathrm{cm}}^{-1}$ ($4.7\text{\,}\mathrm{meV}$) and
$53\text{\,}{\mathrm{cm}}^{-1}$ ($6.6\text{\,}\mathrm{meV}$) for the neutral
and negative states, respectively. The low frequency of the negative charge
state, and large $\Delta E$, gives a ‘giant’ Huang-Rhys factor of 350. For
comparison, a substitutional Si atom in GaAs (also known as the DX-center),
the archetypal defect exhibiting the large lattice relaxation, has a Huang-
Rhys factor of $S=75$ with $\omega=$$81\text{\,}{\mathrm{cm}}^{-1}$ and
$\Delta
Q=$$9\text{\,}\mathrm{a}\mathrm{m}\mathrm{u}^{1/2}\mathrm{\SIUnitSymbolAngstrom}$.30,
14

Figure 4: Configuration coordinate diagram for carrier capture by the iodine
interstitial. The DFT energies (solid circles) were calculated using the
hybrid HSE06 functional. The coordinate $Q$, which corresponds to a linear
combination of phonon modes that map between the two charge states, is defined
in the main text. (a) To model electron trapping at the neutral iodine
interstitial, the excited state of the system corresponds to the neutral
defect with a photo-generated electron in the conduction band and hole in the
valence band. The ground state corresponds to the negatively charged defect
with a hole in the valence band. (b) A comparison between the harmonic (dashed
line) and anharmonic (solid line) PES. The curvatures are determined from
second and fourth order spline fits to the DFT calculated energies.

To identify the normal modes that are associated with lattice relaxation we
analyse the phonon dispersion of the negatively charged iodine interstitial.
For each phonon eigenvector ($e_{i}$) at the gamma point of the Brillouin
zone, we calculate the inverse participation ratio (IPR):

$\mathrm{IPR}=\frac{\sum_{i=1}^{N}\left(|e_{i}|\right)^{2}}{\left(\sum_{i=1}^{N}|e_{i}|\right)^{2}},$
(4)

where $N$ is the number of phonon modes.31 A fully localised phonon mode has
an IPR of 1. For a 96-atom supercell, a fully delocalised mode has an IPR of
$\frac{1}{96}=0.0104$. We find that the lowest energy resonant mode (with an
$\textrm{IPR}=0.046$) has a frequency of $53\text{\,}{\mathrm{cm}}^{-1}$,
equal to the frequency of the negative charge state PES in the harmonic
approximation. The agreement indicates that this resonant mode has strong
accepting carrier and will be active in the uptake of excess electronic energy
after the charge transition. Additional analysis of the phonon modes
associated with the defective crystal can be found in Supplementary
Information.

Finally, we consider the capture coefficients associated with sequential
electron and hole capture, as outlined in Equation 3. The electron capture
coefficient $C_{n}$ determines the rate of electron capture $R_{n}$ at a
neutral iodine interstitial

$R_{n}=C_{n}N_{\mathrm{t}}n$ (5)

where $N_{\mathrm{t}}$ is the density of neutral defect traps and $n$ is the
electron density.

It is evident from Figure 4 that an anharmonic description is necessary, as
parabolic functions poorly describe the PES away from the equilibrium
structures. For capture processes where the atomic displacement is smaller, a
harmonic oscillator model can be used to predict non-radiative capture
coefficients using bulk material parameters (e.g. carrier effective mass,
dielectric constant) and the defect energetics ($\Delta E$ and the charge
transition level).32 Unfortunately this model, whilst computationally economic
and ideal for high-throughput type studies, is not valid for the iodine
interstitial due to the anharmonicity of the PES.

In the harmonic picture, the iodine interstitial has an electron capture
coefficient of
$8\text{\times}{10}^{-17}\text{\,}{\mathrm{cm}}^{3}\text{\,}{\mathrm{s}}^{-1}$.
Analysis of the vibrational wavefunctions for each PES shows that quantum
tunnelling between the initial and final charge states is significant at
energies below the classical barrier (Figure S5). Despite tunnelling, the
capture coefficient is low due to the large electron capture barrier
($600\text{\,}\mathrm{meV}$). Fast radiative recombination, which is of the
order
$1\text{\times}{10}^{-10}\text{\,}{\mathrm{cm}}^{3}\text{\,}{\mathrm{s}}^{-1}$
in MAPI,33 dominates in the harmonic approximation.

In the more accurate anharmonic picture the electron capture coefficient is
$1\text{\times}{10}^{-10}\text{\,}{\mathrm{cm}}^{3}\text{\,}{\mathrm{s}}^{-1}$,
suggesting that non-radiative electron capture will compete with radiative
capture. The electron capture barrier is $148\text{\,}\mathrm{meV}$, much
smaller than in the harmonic case, yielding a larger capture coefficient. As
in the harmonic picture, quantum tunnelling is significant (Figure S5).
Measuring the electron capture rate at the iodine interstitial is difficult as
the trap density is hard to quantify and varies considerably with the material
processing protocol. Nevertheless, a tentative approximation of the capture
rate from a rate constant of
$2\text{\times}{10}^{7}\text{\,}{\mathrm{s}}^{-1}$34 and assuming a trap
density of $1\text{\times}{10}^{16}\text{\,}{\mathrm{cm}}^{-3}$,35 yields a
capture coefficient of
$2\text{\times}{10}^{-9}\text{\,}{\mathrm{cm}}^{3}\text{\,}{\mathrm{s}}^{-1}$.
This estimate of the capture coefficient is within one order of magnitude of
our calculated value for the anharmonic PES.

We observe a striking asymmetry between the electron capture barrier
($148\text{\,}\mathrm{meV}$) and the hole capture barrier
($924\text{\,}\mathrm{meV}$). This is due to the large lattice relaxation
associated with iodine rearrangement upon electron capture. The asymmetry
provides an explanation for the low trap-assisted recombination rate. Although
electron capture at the neutral interstitial is fast, the subsequent capture
of a hole at the negative iodine interstitial is energetically inaccessible
for thermal electrons.

##  4 Conclusions

In conclusion, the soft nature of halide perovskites results in strong
electron-phonon coupling and a large displacement of the surrounding inorganic
octahedra following electron capture. This relaxation process leads to a giant
Huang-Rhys factor and facilitates fast non-radiative electron capture.

We expect to find similarly large, anharmonic lattice relaxations in other
perovskites where dynamic octahedral tilting is evident (e.g. HC(NH2)2PbI3,
FAPI36) and, more generally, in mechanically soft semiconductors that are
prone to structural disorder (e.g. metal-organic frameworks). Defect tolerant
perovskites (materials with low rates of non-radiative recombination) may be
engineered by suppressing octahedral rotations in response to the changes in
defect charge state. This might be achieved through elemental substitutions
that result in local strain fields37 or through defect engineering.38

On the basis of our results, the electron capture process at the neutral
iodine interstitial in MAPI is irreversible; it is energetically unfavourable
for the electron to be released into the band or annihilated by a hole. An
extension of this procedure to cover all native defects, while being
computationally demanding, would help to understanding the nature of non-
radiative losses in halide perovskites solar cells, as well as avenues to
further enhance efficiency towards the radiative limit.

Calculations were performed on the Piz Daint supercomputer at the Swiss
National Supercomputing Centre (CSCS) via the Partnership for Advanced
Computing in Europe (PRACE) project pr51. Via our membership of the UK’s HPC
Materials Chemistry Consortium, which is funded by EPSRC (EP/L000202,
EP/R029431), this work also used the ARCHER Supercomputing Service
(http://www.archer.ac.uk). This work was supported by a National Research
Foundation of Korea (NRF) grant funded by the Korean government (MSIT) (No.
2018R1C1B6008728) and the H2020 Programme under the project STARCELL
(H2020‐NMBP‐03‐2016‐720907). J.M.F. is supported by a Royal Society University
Research Fellowship (URF-R1-191292).

Data access, calculation procedure for defect properties, additional analysis
of the carrier capture rate.

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