# GlyphNet: Homoglyph domains dataset and detection using attention-based
Convolutional Neural Networks

Akshat Gupta1, Laxman Singh Tomar2, Ridhima Garg3

###### Abstract

Cyber attacks deceive machines into believing something that does not exist in
the first place. However, there are some to which even humans fall prey. One
such famous attack that attackers have used over the years to exploit the
vulnerability of vision is known to be a Homoglyph attack. It employs a
primary yet effective mechanism to create illegitimate domains that are hard
to differentiate from legit ones. Moreover, as the difference is pretty
indistinguishable for a user to notice, they cannot stop themselves from
clicking on these homoglyph domain names. In many cases, that results in
either information theft or malware attack on their systems. Existing
approaches use simple, string-based comparison techniques applied in primary
language-based tasks. Although they are impactful to some extent, they usually
fail because they are not robust to different types of homoglyphs and are
computationally not feasible because of their time requirement proportional to
the string’s length. Similarly, neural network-based approaches are employed
to determine real domain strings from fake ones. Nevertheless, the problem
with both methods is that they require paired sequences of real and fake
domain strings to work with, which is often not the case in the real world, as
the attacker only sends the illegitimate or homoglyph domain to the vulnerable
user. Therefore, existing approaches are not suitable for practical scenarios
in the real world. In our work, we created GlyphNet, an image dataset that
contains 4M domains, both real and homoglyphs. Additionally, we introduce a
baseline method for a homoglyph attack detection system using an attention-
based convolutional Neural Network. We show that our model can reach state-of-
the-art accuracy in detecting homoglyph attacks with a 0.93 AUC on our
dataset.

Keywords: Homoglyph Attacks, Convolutional Neural Networks, Cyber Security,
Phishing

## Introduction

In cyber security, attackers employ different attacks to infiltrate our
systems and networks, with the objective varying from stealing crucial
information to inflicting system damage. One such deceptive attack is the
homoglyph attack (Woodbridge et al. 2018), which involves an attacker trying
to fool humans and computer systems by using characters and symbols that may
appear visually similar to characters used in real domain and process names
but are different. For example, a typical homoglyph attack may involve
changing “d” to “cl”, “o” to “$\theta$”, and “l” to “1”.

Figure 1: example of a real domain and their homoglyphs

Some of the above substitutions can be difficult for the naked eye to detect,
as shown in Figure1, It would mean that users would be easily susceptible to
clicking on the homoglyph links, more so when navigating from one website to
another. The problems arising from such an attack are of two types: a)
Deceiving humans to believe that an illegitimate domain name is real by
fooling the users, resulting in users using fake webpages as if they were the
real ones. b) Create fake academic documents and papers by changing the real
strings with homoglyphs to deceive plagiarism detection tools such as
Grammarly.com

Figure 2: Real text on Grammarly plagiarism detector Figure 3: Homoglyph text
on Grammarly plagiarism detector

Both types of problems are hard to detect and hence require robust methods to
identify an attack before it causes any information breach. Previous
approaches mainly used methods from comparative algorithms such as edit
distance to identify homoglyph attacks from legit strings(Damerau 1964). Any
domain name string that returned an edit distance beyond an acceptable
threshold was considered a homoglyph. Edit distance covers simple approaches
like insertion, deletion, transposition, swapping, and substitution. Due to
this shortcoming, a slight change to the illegitimate domain name can easily
bypass it as quickly as a real one. Now, a slightly better version of it was
proposed, which was called Visual Edit Distance(Ristad and Yianilos 1998). It
proposes to have a particular edit distance for the visual similarity of the
two domain name strings. However, these methods were more relevant in academia
and had negligible prevalence in the real world. A homoglyph attack differs
from a phishing attack because domain names in the former are hardly
distinguishable but can be apparent in the latter.

We have taken the famous poem ”The Road Not Taken” by Robert Frost to
demonstrate this concept. In Figure 2, we have taken the poem text and run
Grammarly’s Plagiarism Detector tool. It reports 100% plagiarism, which is
correct but later when we passed the homoglyphed version of the same text, it
reports the text to be 100% unique, as shown in Figure 3. This proves that
even today’s state-of-the-art systems cannot effectively deal with texts
comprising homoglyphs.

Recently, Microsoft obtained a court order to remove many fraudulent
”homoglyph” domains used to conduct fraud and pose as Office 365 users.(Page
2021) following a customer complaint about a business email compromise attack,
Microsoft conducted an investigation and discovered that the unidentified
criminal organization responsible for the attack also created 17 other
malicious domains, which were combined with the customer credentials that had
been stolen to gain unauthorized access to Office 365 accounts and monitor the
contacts of the customers.

Microsoft stated that the cybercriminals have caused and continue to cause
irreparable injury to Microsoft, its customers, and the general public. The
complaint also stated that the cybercriminals have illegally accessed customer
accounts, monitored customer email traffic, gathered information on pending
financial transactions, and criminally impersonated [Office 365] customers.

According to studies, this attack hit $71\%$ organizations in $2021$. Sixty-
two countries people were the subject of a massive cyberattack last year.

In this research, we aim to create a data set that can help expand research on
homogylph attacks. We propose to apply an attention-based Convolutional Neural
Network (CNN) to detect homoglyphs without the need to provide paired data.
Additionally, our model achieves a significant performance boost compared to
other approaches due to its architectural design. Our method can be applied
directly as an API or web service for an organization to check a domain or
process name before accessing it. For evaluation, we compared our performance
with other baselines and found that our model outperforms them. Moreover, our
approach also addresses the problem of unpaired data setting, which is often
the case in the real world.

The major contributions of our research are as follows:

  1. 1.

Created a benchmark dataset of 4 million real and homoglyph domain images
based on known homoglyph patterns and URLs. It is generated via strings from
single-and dual-character noise sampled using a Gaussian distribution over a
homoglyph character pool.

  2. 2.

A method that uses an image dataset for detecting homoglyphs involving an
attention-based convolutional neural network trained in a supervised fashion
achieves a better AUC score than the other existing baselines.

The paper’s organization starts by introducing the problem faced by existing
approaches to detect homoglyph-based phishing attacks in both academia and the
real world. In Related Work, we discuss the existing approaches which propound
the idea of solving this problem with either string matching or Deep Learning
based methods like Siamese Neural Networks and GANs. We have explained their
major pitfalls in terms of generalizing capabilities and feasibility. In the
Dataset section, a comprehensive description is provided of the generation of
the proposed images dataset. It follows a brief description of our attention-
based CNN baseline implementation. The Experimentation section describes
dataset splitting, metrics used, and other settings. Later, in the Results
section, we examine the results and scores obtained after the experiments
conducted in the last section. Both data and baseline implementation results
are validated and explained with the help of an elegant table within the same
section. The following section, Discussion, presents experiments we tried that
did not work. Finally, the Conclusion Section summarizes the observations and
contributions.

## Related Work

The work by (Ginsberg and Yu 2018) used a Siamese Neural Network to detect
homoglyphs using a paired dataset. This dataset included pairs of strings; one
was a real domain name, and the other was a homoglyph domain name. In their
work, they converted this pair of strings into binary images that were later
fed to a Siamese Neural Network(Koch et al. 2015). The Siamese neural network
uses two identical convolutional neural networks(LeCun, Bengio et al. 1995) to
learn the visual difference between a pair of images. They were applied to
domains such as healthcare, finance, and others but have recently gained
popularity in cyber security.

Though(LeCun, Bengio et al. 1995) their work showed significant improvement
from previous baselines but suffered from two major pitfalls:

  1. 1.

In online security systems, it is impossible to provide paired data, without
which these systems will not work.

  2. 2.

It cannot be used in academia due to the inability to find a paired word for
each word present in a scientific article.

Therefore, although this approach performs well, it cannot be employed in
real-world systems.

The traditional solutions to prevent homoglyph attacks were inspired by
genomics(Lu, Mohammed, and Wang 2019), which proposed the idea that homoglyph
domains are in string formats and, therefore, should be compared with
legitimate ones to detect whether they are real or not. Edit Distance(Ristad
and Yianilos 1998) is the measure of the minimum number of operations required
to transform one sequence (domain or process name string in our case) into
another. If the value exceeds an acceptable threshold, it should predict as
homoglyph. This looks effective but not when giving it a second thought. The
reason is that in cases like ”google.com” and ”go0gle.com”, edit distance
would return only ’1’ which does not look so threatening but is a homoglyph
domain name. Furthermore, a paired sequence of strings is required to make
comparisons, which would not be the case if it was a homoglyph of a new domain
name. Finally, in the real world, this approach lacked severely good results.

Phishing attacks(Hong 2012) should not be confused with homoglyph attacks.
Phishing is an attack involving the attacker sending homoglyph/false/fake
links that appear to be coming from a trusted vendor. It leads to information
compromise(Helms, Ettkin, and Morris 2000), data breaches(Cheng, Liu, and Yao
2017), and financial fraud(Reurink 2018). The difference between Phishing and
Homoglyphs is that the former uses tricks such as longer, bad, and misspelled
domain names and URLs(Ma et al. 2009) to fool people. At the same time, the
latter takes advantage of the inability to differ in visually similar but
different domain name strings. Thus, it is required to create better solutions
for homoglyph detection.

### Siamese Neural Networks

Figure 4: Siamese neural network architecture(Woodbridge et al. 2018)

The Siamese neural network architecture is proposed to detect homoglyphs using
a paired data set. This dataset included pairs of strings, one was a real
domain name, and the other was a homoglyph domain name. Each instance was a
tuple that contained a real domain string, a homoglyph domain string, and a
score that denotes whether the second element is a valid homoglyph of the
first or not as part of its elements. In their work, they converted this pair
of strings into binary images, images that were later fed to a Siamese Neural
Network. However, we observed a significant difference while reproducing the
results in our dataset.

### PhishGANs

Figure 5: PhishGAN architecture(Sern, David, and Hao 2020)

Approaches such as Siamese Neural Networks suffered severely in terms of
performance due to lack of data, as they only had close to $91k$ real domain
images. As a remedial solution, we were required to produce comprehensive data
to train our models well. Recently, Lee Joon Sern et al. proposed PhishGANs to
generate synthetic data(Sern, David, and Hao 2020). They discussed creating a
generative adversarial network(Goodfellow et al. 2014) that aimed to create
images similar to real domain names to increase existing data sets.
PhishGANs(Sern, David, and Hao 2020) being a GAN(Goodfellow et al. 2014)
involved a generator and a discriminator, both trained in an adversarial
fashion such that the generator is trained well enough to produce images
similar to those of a real domain which the discriminator cannot detect.
Later, these images were fed to a different network for binary classification
aimed at distinguishing real domain names from homoglyphs. They used
UNet(Ronneberger, Fischer, and Brox 2015) architecture as a generator using a
custom loss function called a dot product loss. The PhishGANs(Sern, David, and
Hao 2020) were trained similarly to how Pix2Pix(Isola et al. 2017) is trained.
Later, for classification purposes, a new architecture was defined and called
homoglyph identifier (HI) using CNN(LeCun, Bengio et al. 1995) as an encoder
using a triplet loss function (Hoffer and Ailon 2015) as input, the positive
domain (google.com), the anchor domain (go0gle.com) and the negative domain
(apple.com). On some popular domains, such as youtube.com and facebook.com, HI
achieved an accuracy of roughly $0.81$ while testing their homoglyphs. On an
unseen domain, HI achieved an accuracy of $0.76$ while feeding it back again
in PhishGANs(Sern, David, and Hao 2020) and generating its homoglyphs, and
later training on them, which helped detect its homoglyphs using $0.86$
accuracy. Although the idea of generating synthetic data using GANs(Goodfellow
et al. 2014) looks promising and intriguing but is not motivating when it
comes to real-world usage. GANs(Goodfellow et al. 2014), in general, is one of
the trickiest architectures in Deep Learning(LeCun, Bengio, and Hinton 2015)
since their advent and are often found to have issues while training in the
real world, which is not the case in the constrained environment of academia.
It is common to encounter issues like problems in convergence, generator
oscillating between generating specific examples in the domain, and multiple
inputs resulting in generating the same output. Also, the performance increase
was not drastic enough to compel us for its usage.

## Dataset

The work by (Woodbridge et al. 2018) proposed their custom paired data set
that comprises $91k$ real domains and $900k$ homoglyphs. Each real domain is
used to generate its respective homoglyphs. Each point in this dataset is a
three-element tuple denoting domain, homoglyph, and score. Here, if the value
of the score is $1.0$, then it is a valid homoglyph of the real domain. The
real-world data limitation to Homoglpyh-based attacks is the lack of publicly
available data sets.

Proposed dataset: GlyphNet
We have proposed a dataset consisting of real and homoglyph domains. To
generate homoglyph domains, real domains are needed. We have obtained domains
from the Domains Project(Turkynewych 2020). This repository is one of the
largest collections of publicly available active domains. The entire
repository comprises 500M domains, and we restricted our work to 2M domains
due to hardware restrictions.

Homoglyph Creation Algorithm
Homoglyph Generation is an important task, as one needs to ensure enough
randomness to make it appear real and keep the process simple enough to fool
the target. Publicly available tools like dnstwist(Ulikowski 2015) replace
every character in the real input domain with their respective glyphs. It
generates poor homoglyphs for the large part because it relies on paired data
which is not fit to serve the purpose practically. We created our novel
algorithm for the generation of homoglyph domains to ensure that real
homoglyphs are generated with randomness and closeness. To achieve this, we
sample homoglyph noise characters using Gaussian sampling(Boor, Overmars, and
Van Der Stappen 1999) from the glyph pool. We used 1M real domains to generate
$2M$ homoglyphs with a single glyph character and introduce diversity in our
dataset; we reran this algorithm on the remaining 1M real domains to generate
homoglyph domains with two character glyphs. Finally, we have the 4M real and
homoglyph domains.

Image Generation
Homoglyph attacks exploit the weakness of human vision to differentiate real
from homoglyph domain names. From a visual perspective, we are interested in
learning the visual characteristics of real and homoglyph domain names. To do
so, we rendered images from the real and homoglyph strings generated via our
algorithm. We have used ARIAL Typeface as our chosen font, a $28$ font size,
on a black background with white text from the middle left of the image; the
image size is $150\times 150$.

Dataset Name | Real | Homoglyph | Total
---|---|---|---
Domain and Process Strings(Woodbridge et al. 2018) | $90k$ | $900k$ | $990k$
Similar and Dissimilar Pairs(Majumder et al. 2020) | $2257$ | $2257$ | $4514$
GlyphNET (Ours) | $2000k$ | $2000k$ | $4000k$
Table 1: Dataset comparison Figure 6: Rendered images from the dataset, $0$;
homoglyph domain and, $1$; real domain

## Methodology

This section presents our approach to building an end-to-end homoglyph
detection system. We build on attention-based(Bahdanau, Cho, and Bengio
2014)(Vaswani et al. 2017) convolutional neural network(LeCun, Bengio et al.
1995) that aims to exploit the visual dissimilarity between real and homoglyph
domain names. The architecture of our model is shown in Figure 7 and Figure 8.

The rendered images are then used as input to the CNN to learn the desired
visual feature information. The model consists of four Conv2D layers to learn
visual information such as edges, curves, and strokes. Each convolutional
layer is paired with a max-pooling layer to perform dimensionality reduction
on the learned features. This model is developed in keras(Chollet et al.
2015). Each convolution block is followed by a convolutional block attention
module (CBAM), as described in the following.

Figure 7: Our neural network architecture Figure 8: Zoom in view of conv-
attention module

Attention processes boost the strength of representation by focusing on
essential traits and suppressing unneeded ones. It uses the feed-forward
convolutional neural network’s CBAM, a specific and efficient attention
module. Given a preliminary feature map, the module successively infers
attention maps along the channel and spatial dimensions. It then multiplies
the attention maps by the preliminary feature map to achieve adaptive feature
refinement. The overall attention process is summarized as follows:

$F^{\prime}=M_{c}(F)\otimes F,$

$F^{\prime\prime}=M_{s}(F^{\prime})\otimes F^{\prime}$,

  1. 1.

Given an intermediate feature map $F\in\mathcal{R}^{C\times H\times W}$ as
input. $C$ represents a number of channels, $H$ and $W$ represent the height
and width of the feature map $F$ respectively.

  2. 2.

CBAM sequentially infers a 1D channel attention map
$M_{c}\in\mathcal{R}^{C\times 1\times 1}$

  3. 3.

And a 2D spatial attention map $M_{s}\in\mathcal{R}^{1\times H\times W}$

  4. 4.

$\otimes$ element-wise multiplication

For the sake of non-linearity, the RELU activation function is used.

Architecture | Accuracy | Precision | Recall | F1-score | AUC
---|---|---|---|---|---
Siamese CNN(Woodbridge et al. 2018) | $0.79$ | $0.78$ | $0.71$ | $0.74$ | $0.78$
Ensemble CNN(Majumder et al. 2020) | $0.83$ | $0.82$ | $0.79$ | $0.80$ | $0.83$
PhishGAN(Sern, David, and Hao 2020) | $0.71$ | $0.74$ | $0.65$ | $0.69$ | $0.71$
Attention CNN (Ours) | $0.93$ | $0.93$ | $0.93$ | $0.93$ | $0.93$
Table 2: Model performance comparison on our dataset

## Experimentation

### Dataset and Metrics

We have split our dataset into three parts, train, validation, and test, with
a ratio of $70:20:10$, respectively which amounts to $2.8M$, $0.8M$, and
$0.4M$ images in train, validation, and test sets. Each image size is
$150\times 150$.

We use accuracy for measuring the performance of the classification task.
Since accuracy can sometimes be misleading in a binary classification task,
especially for unbalanced data sets, we consider precision, recall, and F1
score as our evaluation metrics, even though our dataset is balanced. We have
also used the AUC score to compare our solution with some other works.

### Experimental Settings

For the training part, we used binary cross-entropy as a Loss Function. We
have used RMSProp optimizer to optimize the loss obtained from the binary
cross-entropy loss function, with a learning rate of $10e^{-4}$, and the
network is trained for $30$ epochs with early stopping. We trained with a
batch size of $256$. We evaluated the performance of our model in terms of
accuracy vs. epochs and loss vs. epochs plots.

## Results

We evaluated our model on two unpaired data sets for domain names. We took an
input string from the dataset we created in the previous section, converted it
into an image, and fed it to the model to generate a binary label. The results
for the domain names are tabulated in Table 2. Out of the $400k$ test images,
our model correctly categorized $372k$ images resulting in $0.93$ accuracy.
Our model achieved an f1-score of $0.93$, $13$ points higher than the previous
models. Our model outperforms other baselines and comparable works on the
other metrics, including accuracy, precision, recall, and AUC. The performance
of other models on our dataset was also below par compared with the proposed
datasets in their works, signifying our dataset’s variations, difficulty, and
importance.

Our dataset, code, and models are publicly available under MIT LICENSE and can
be accessed from our project’s GitHub
repository111https://github.com/Akshat4112/Glyphnet

## Discussion

We now discuss some interesting observations and experiments which did not
work and possible explanations regarding them.

### Using only Grayscale Images

During the image rendering phase, where we generated images from the data set
containing real and homoglyph domains, we experimented with generating colored
images instead of grayscale ones. We used ($73$, $109$, $137$) as the
background color while ($255$,$255$,$0$) as the color of the text to be
written. However, the network trained from these colored images was always
found to be underperforming the network trained on grayscale images. One
possible reason might be that the grayscale involves black and white as two
colors, which are two extremes. Hence, it preserves the difference in
adjoining pixels at the periphery of the letter and background pixels.
Meanwhile, the colors though appearing to us as distinctly different, suffered
to preserve the difference when later passed through resizing operations. We
perform data augmentation on our data and later train our network using the
data generated, but it leads to a downfall in accuracy. One possible reason
might be that data augmentation(Shorten and Khoshgoftaar 2019) is used in
those scenarios where we expect distinctive image features to exist, but they
do not exist in the actual data set. It can be understood from a Cat vs. Dogs
example. Usually, data sets contain cats and dogs in limited positions in the
pictures, so our model fails to recognize some of the real images. The reason
is that in the real world, either a cat or a dog might turn their heads and
might be sitting in different postures, which makes it difficult for our model
to locate distinctive features like whiskers and pointy ears in cats and
tongues in case of dogs in the absence of large amounts of data catering these
considerations. Therefore, to mimic such behavior, Data Augmentation is used,
which helps to create all these different types of images. However, in our
case, using it leads to flipped characters, and rotated images lead to anchor
and tilde signs over letters going in different directions, which is not the
case with real-world strings. Therefore, data augmentation was, in fact,
counterproductive for our use case.

We rendered images of sizes $256\times 256$ during the image generation phase.
Apart from the image size $256\times 256$, at which we observed the best
results, we tried experimenting with the following image sizes: $128\times
128$, $150\times 150$, $224\times 224$, and $512\times 512$. The smaller the
image size, the more performance degradation there is relative to it. An
increase in size did not lead to any significant improvement but increased the
training time of the model. Hence, we use $256\times 256$ image size.

### Building Model without Transfer Learning

We train a base network on a base dataset and task and then reuse the learned
features or transfer them to a second target network to be trained on a target
dataset. This process will tend to work if the features are general, meaning
suitable for both base and target tasks rather than specific to the base task.
We performed experiments with transfer learning(Pan and Yang 2009) by
incorporating networks like VGG16(Simonyan and Zisserman 2014), Resnet18(He et
al. 2016), Resnet34, Resnet50, Wide ResNet-101-2, ResNeXt-50-32x4d and
ResNeXt-101-32x8d which were trained on ImageNet(Deng et al. 2009) dataset.
Our experiments did not obtain good accuracy using these architectures, either
pre-trained or from scratch.

There are two possible reasons:

1) Large number of hidden layers: These architectures have many hidden layers
ranging from $16$ up to $100$. The deeper the network, the more it tries to
aggregate the learned features to create high-level features. It works well in
images of real-world entities, but in our context, it does not help as these
are just images generated from strings. Going further deep into the network
makes the network lose all the subtle features from parts of strings like
tilde and apostrophes. It has learned to differentiate real from homoglyph
strings.

2) Pre-trained in a data set of different domains: Another reason is that
these networks were pre-trained on the ImageNet dataset, which contains images
from real-world entities, but does not have images similar to our problem.
Hence, using a pre-trained network having weights learned from such images
instead of our domain problem did not help. We obtained an accuracy of $63\%$
to $67\%$ using the above architecture.

## Conclusion

In this work, we created a first-of-its-kind large-scale homoglyph phishing
image dataset comprising 4M images of real and homoglyph domains. Later, we
presented a baseline that relied on learning features from an attention-based
convolutional neural network using our constructed data set to differentiate
real domain names from homoglyph domain names to avoid homoglyph attacks. Our
dataset and approach are robust because we can generalize on unseen homoglyphs
as compared to other approaches which are data-dependent for every single
inference, which leads it to outperform existing approaches. We believe this
work is significant and provides an important benchmark to propel work in this
area, and its applications would serve as a safeguard against phishing attacks
in the real world.

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††thanks: Current Address: ISM, Univ. Bordeaux, CNRS, Talence, France

# Nonlocal screening dictates the radiative lifetimes of excitations
in lead halide perovskites

Yoonjae Park  Department of Chemistry, University of California, Berkeley
Amael Obliger  Department of Chemistry, University of California, Berkeley
David T. Limmer<EMAIL_ADDRESS>Department of Chemistry, University of
California, Berkeley  Materials Science Division, Lawrence Berkeley National
Laboratory  Chemical Science Division, Lawrence Berkeley National Laboratory
Kavli Energy NanoScience Institute, Berkeley, California, Berkeley

###### Abstract

We use path integral molecular dynamics simulations and theory to elucidate
the interactions between charge carriers, as mediated by a lead halide
perovskite lattice. We find that the charge-lattice coupling of MAPbI3 results
in a repulsive interaction between electrons and holes at intermediate
distances. The effective interaction is understood using a Gaussian field
theory, whereby the underlying soft, polar lattice contributes a nonlocal
screening between quasiparticles. Path integral calculations of this nonlocal
screening model are used to rationalize the small exciton binding energy and
low radiative recombination rate observed experimentally and are compared to
traditional Wannier-Mott and Fröhlich models, which fail to do so. These
results clarify the origin of the high power conversion efficiencies in lead
halide perovskites. Emergent repulsive electron-hole interactions provide a
design principle for optimizing soft, polar semiconductors.

Lead halide perovskites are a class of materials that have unique
photophysical properties resulting from their soft, polar lattices. They have
vanishingly small exciton binding energies and despite modest mobilities, have
large free carrier diffusion lengths resulting from exceptionally long carrier
lifetimes.1; 2; 3 These properties make lead halide perovskites ideal
materials for photovoltaic devices.4; 5; 6 Many of their optoelectronic
properties have been thought to arise from electron-phonon coupling, as the
largely ionic bonding of the lead halides admit strong Coulomb interactions
between free charges and the lattice.7 It has been conjectured that polaronic
effects in particular8; 9; 10 act to protect free charges from recombination
and screen their interactions, reducing exciton binding energies.9 However,
the significant anharmonicity of the perovskite lattice has made uncovering
the molecular origin of these properties difficult.11; 12; 13; 14

Here, we apply path integral molecular dynamics15; 16 to study an atomistic
model of quasiparticles embedded in a MAPbI3 lattice, in order to understand
how a fluctuating lattice affects its electronic properties. Much recent
effort has gone into understanding the effects of the lattice on the excitonic
properties of perovskites computationally 17; 18; 19; 20 and analytically.21;
22 However, unlike traditional polar semiconductors where lattice fluctuations
can be described by a harmonic approximation, the tilting and rocking motions
of the inorganic octahedra14 and nearly free motions of the A-site cations12;
13 render the lattice highly anharmonic. This complicates the simplification
to traditional model Hamiltonians like the Fröhlich model or its
generalizations.20; 23 Attempts to include lattice effects into _ab initio_
based approaches have been developed but these are difficult to extend to the
time and length scales necessary to explain the nature of how photogenerated
electrons and holes bind, dissociate and recombine.24 Using an explicit
atomistic representation of the lattice surrounding the quasiparticles allows
us to go beyond simplified models. Employing path integral calculations allows
us to consider finite temperature effects directly on diffusive time and
length scales. These simulations motivate a field theory to describe the
effective electron-hole interactions that dictate the emergent optical
properties of the perovskites. With these simulations and theory, we are able
to elucidate the origin of low exciton binding energies and recombination
rates as a consequence of a nonlocal screening from the lattice.

We consider a system of an electron-hole pair, interacting with a MAPbI3
perovskite employing a fully atomistic description of the lattice. The full
system Hamiltonian, $\mathcal{H}$, consists of electronic, lattice, and
interaction pieces,
$\mathcal{H}=\mathcal{H}_{\mathrm{el}}+\mathcal{H}_{\mathrm{l}}+\mathcal{H}_{\mathrm{int}}$.
The highly dispersive bands of MAPbI3 allow us to make an effective mass
approximation, so that the electronic Hamiltonian is defined as

$\displaystyle\mathcal{H}_{\mathrm{el}}=\frac{\hat{\mathbf{p}}_{e}^{2}}{2m_{e}}+\frac{\hat{\mathbf{p}}_{h}^{2}}{2m_{h}}-\frac{e^{2}}{4\pi\varepsilon_{0}\varepsilon_{\infty}|\hat{\mathbf{r}}_{e}-\hat{\mathbf{r}}_{h}|}$
(1)

where the subscripts $e$ and $h$ indicate electron and hole,
$\hat{\mathbf{p}}$ and $\hat{\mathbf{r}}$ are the momentum and position
operators, $m_{e}/m=m_{h}/m=0.2$ are the band masses of the quasiparticles
taken from recent GW calculations in units of the bare electron mass $m$,25
$\varepsilon_{0}$ is the vacuum permittivity while $\varepsilon_{\infty}$ is
the optical dielectric constant for charge $e$. For the lattice, we use an
atomistic model developed by Mattoni et al., that has been demonstrated to
reproduce the structural and dielectric properties of lead halide
perovskites.26; 27 Its Hamiltonian is decomposable as

$\displaystyle\mathcal{H}_{\mathrm{l}}=\sum_{i=1}^{N}\frac{\hat{\mathbf{p}}_{i}^{2}}{2m_{i}}+U_{\mathrm{l}}(\hat{\mathbf{r}}^{N})$
(2)

where $\hat{\mathbf{p}}_{i}$, $\hat{\mathbf{r}}_{i}$ and $m_{i}$ are the
momentum, position, and mass of $i^{th}$ atom, $N$ is the total number of
atoms in the lattice, and $U_{\mathrm{l}}(\hat{\mathbf{r}}^{N})$ is the pair-
wise interaction potential between atoms with configuration
$\hat{\mathbf{r}}^{N}=\\{\hat{\mathbf{r}}_{1},\hat{\mathbf{r}}_{2},\dots,\hat{\mathbf{r}}_{N}\\}$.
The potential includes electrostatic and excluded volume interactions. The
charge-lattice interaction term is given by
$\mathcal{H}_{\mathrm{int}}=U_{e,\mathrm{l}}(\hat{\mathbf{r}}_{e},\hat{\mathbf{r}}^{N})+U_{h,\mathrm{l}}(\hat{\mathbf{r}}_{h},\hat{\mathbf{r}}^{N})$
where $U_{e,\mathrm{l}}$ and $U_{h,\mathrm{l}}$ denote sums of
pseudopotentials. Consistent with the largely ionic nature of MAPbI3, we
employ pseudopotentials of the form of short-ranged truncated Coulomb
potentials, with a cut-off radii chosen as the ionic radii of each species.28;
29; 30

As the atoms are heavy and we are largely interested in room temperature
behavior, we adopt a classical description of the MAPbI3 lattice. However, we
use a path integral description for the two quasiparticles. Such a
quasiparticle path integral approach has been employed previously to study
polaronic effects in the lead halides and trapping in other semiconductors.31;
32; 33; 34 The partition function, $\mathcal{Z}$, for the composite system can
be written as

$\displaystyle\mathcal{Z}=\int\mathcal{D}[{\mathbf{r}}_{e},{\mathbf{r}}_{h},\mathbf{r}^{N}]\,e^{-\mathcal{S}[{\mathbf{r}}_{e},{\mathbf{r}}_{h},\mathbf{r}^{N}]/\hbar}$
(3)

with the action
$\mathcal{S}[{\mathbf{r}}_{e},{\mathbf{r}}_{h},\mathbf{r}^{N}]=\mathcal{S}_{\mathrm{el}}+\mathcal{S}_{\mathrm{l}}+\mathcal{S}_{\mathrm{int}}$.
The corresponding imaginary time path action for the electronic part becomes

$\displaystyle\mathcal{S}_{\mathrm{el}}=\int_{\tau}\,\frac{m_{e}\dot{\mathbf{r}}_{e,\tau}^{2}}{2}+\frac{m_{h}\dot{\mathbf{r}}_{h,\tau}^{2}}{2}-\frac{e^{2}}{4\pi\varepsilon_{0}\varepsilon_{\infty}|\mathbf{r}_{e,\tau}-\mathbf{r}_{h,\tau}|}$
(4)

where the imaginary time $\tau$ is defined over the interval 0 to
$\beta\hbar$, $\beta^{-1}=k_{\textrm{B}}T$, $T$ is temperature,
$k_{\textrm{B}}$ is Boltzmann’s constant, and $\hbar$ is Planck’s constant.
The velocity and position of electron/hole are denoted
$\dot{\mathbf{r}}_{e/h,\tau}$ and $\mathbf{r}_{e/h,\tau}$. Under the
assumption of a classical lattice, the contributions to the path action from
MAPbI3 and its interaction with the quasiparticles become
$\mathcal{S}_{\mathrm{l}}=\beta\hbar\mathcal{H}_{\mathrm{l}}$ and

$\displaystyle\mathcal{S}_{\mathrm{int}}=\int_{\tau}\,U_{e,\mathrm{l}}(\mathbf{r}_{e,\tau},\mathbf{r}^{N})+U_{h,\mathrm{l}}(\mathbf{r}_{h,\tau},\mathbf{r}^{N})$
(5)

an integral over the pseudopotentials. By discretizing the path action into a
finite number of imaginary time slices, the classical counterpart of each
quantum particle becomes a ring polymer consisting of beads connected by
harmonic springs.35

We perform molecular dynamics (MD) simulations of two ring polymers with 1000
beads representing the electron and hole and an MAPbI3 lattice with
40$\times$15$\times$15 unit cells at 300K. The large system size is necessary
in order to ensure that self-interaction errors between the quasiparticles are
minimized. This atomistic description allows us to capture all orders of
interaction between the quasiparticles and the MAPbI3 lattice, free of low
temperature harmonic approximations. The simulation details including pseudo-
potentials and the force field of the lattice can be found in Supporting
Information (SI).

Figure 1: Quasiparticle path integral molecular dynamics simulations.
Representative snapshots of the simulation of electron(red) and hole(blue)
with the MAPbI3 lattice where electron and hole are (a) close to and (b) far
from each other. Zoomed in structure in (a) represents the MAPbI3 lattice
where gray, yellow, and blue atoms represent Pb2+, I-, and MA+, respectively.
(c) Free energy between electron and hole as a function of the distance
between quasiparticle centroids from molecular dynamics simulation (black
circles) and an effective exciton interaction from Eq.13 (red solid line) with
parameters as $\varepsilon^{*}=5$, $l_{s}=2.47$nm, and $l_{c}=1.26$nm.

To analyze the emergent exciton interaction in MAPbI3, we compute the free
energy between electron and hole using Umbrella sampling with the Weighted
Histogram Analysis Method.36 We compute the reversible work to move two charge
centers relative to each other

$\beta
F(R)=-\ln\langle\delta\left(R-|{\mathbf{r}}^{\mathrm{c}}_{e}-{\mathbf{r}}^{\mathrm{c}}_{h}|\right)\rangle$
(6)

where $R$ is the distance between the electron and hole centroid
${\mathbf{r}}^{\mathrm{c}}_{e/h}$, $\delta(x)$ is Dirac’s delta function, and
$\langle..\rangle$ represents an ensemble average. Simulation snapshots are
shown in Figs. 1a and 1b, where spatially delocalized charges extend with a
radius of gyration between 1.5 - 3 nm. Figure 1c shows $F(R)$, which is
nonmonotonic. The free energy exhibits a minima at $R=0$ reflecting the
binding of the electron-hole pair into an exciton, a plateau at large $R$, and
a barrier at intermediate $R\approx 8$ nm. The binding energy is large due to
the neglect of polarizability in this description of the lattice. Considering
the bare Coulomb potential is a monotonic function, the repulsive interaction
found in Fig. 1c at intermediate electron-hole distances must arise from the
lattice. An effective electron-hole repulsion has been speculated in lead
halide perovskites previously,37; 38; 39 but had defied direct observation or
theoretical validation.

In order to understand the emergent lattice effects on the electron-hole
interaction and surprising intermediate repulsion, we assume that the
fluctuations of the lattice are well described by a Gaussian field. Such
Gaussian field theories underpin a number of standard effective interactions
including dielectric continuum theory and the Casimir effect.40; 41; 42 A
Gaussian approximation in this context is analogous to assuming that while the
lattice is anharmonic, it responds linearly.43; 44 We consider approximating
the lattice by an effective polar displacement field,
$\mathbf{u}_{\mathbf{k},\tau}$. Within the Gaussian field approximation, the
path action for the lattice becomes

$\displaystyle\mathcal{S}_{\mathrm{l}}$
$\displaystyle\approx\frac{1}{2}\int_{\tau}\int_{\tau^{\prime}}\,\int_{\mathbf{k}}\mathbf{u}_{\mathbf{k},\tau}\chi_{\mathbf{k},\tau-\tau^{\prime}}^{-1}\mathbf{u}_{\mathbf{-k},\tau^{\prime}}$
(7)

where
$\chi_{\mathbf{k},\tau-\tau^{\prime}}=\langle\mathbf{u}_{\mathbf{k},\tau}\mathbf{u}_{-\mathbf{k},\tau^{\prime}}\rangle$
is the susceptibility at wave vector $\mathbf{k}$ and imaginary time
displacement $\tau-\tau^{\prime}$. The susceptibility is determined by a
phonon dispersion relationship only in the limit of zero temperature, and
generally reflects the correlations within the effective polar displacement
field.45 Consistent with the Coulombic pseudopotentials used in the MD
simulations, we take the coupling between the charges and the lattice to be
linear

$\displaystyle\mathcal{S}_{\mathrm{int}}\approx\int_{\tau}\,\int_{\mathbf{k}}\,\mathbf{u}_{\mathbf{k},\tau}\,\lambda\frac{e^{i\mathbf{k}\cdot\mathbf{r}_{e,\tau}}-e^{i\mathbf{k}\cdot\mathbf{r}_{h,\tau}}}{k}$
(8)

and described by a Fröhlich-like interaction,46 where $\lambda$ is a Fröhlich
coupling constant. The lattice variables can be integrated out, leaving a
Gaussian approximation to the partition function, $\mathcal{Z}_{\mathrm{G}}$,

$\displaystyle\mathcal{Z}_{\mathrm{G}}$
$\displaystyle=\int\mathcal{D}[\mathbf{r}_{e},\mathbf{r}_{h},\mathbf{u}_{\mathbf{k}}]\,e^{-\mathcal{S}_{\mathrm{all}}[{\mathbf{r}}_{e},{\mathbf{r}}_{h},\mathbf{u}_{\mathbf{k}}]/\hbar}$
(9)
$\displaystyle=Z_{\mathrm{l}}\int\mathcal{D}[\mathbf{r}_{e},\mathbf{r}_{h}]\,e^{-\mathcal{S}_{\mathrm{el}}/\hbar}\,e^{-\mathcal{S}_{\mathrm{eff}}[{\mathbf{r}}_{e},{\mathbf{r}}_{h}]/\hbar}$

where $Z_{\mathrm{l}}$ is the partition function for a displacement field
without couplings to the charges. This integration results in an effective
path action of the form (see SI for details)

$\displaystyle\mathcal{S}_{\mathrm{eff}}=-\sum_{i,j}\int_{\tau}\int_{\tau^{\prime}}\int_{\mathbf{k}}\Gamma_{ij}\chi_{\mathbf{k},\tau,\tau^{\prime}}\frac{|\lambda|^{2}}{2k^{2}}e^{i\mathbf{k}\cdot|\mathbf{r}_{i,\tau}-\mathbf{r}_{j,\tau^{\prime}}|}$
(10)

where $i,j\in\\{e,h\\}$ and $\Gamma_{ij}$ takes the value of $\Gamma_{ij}=1$
if $i=j$ and $\Gamma_{ij}=-1$ if $i\neq j$.

The susceptibility $\chi_{\mathbf{k},\tau}$ is proportional to a dielectric
function evaluated in the absence of the quasiparticles. Different functional
forms of its imaginary time and wavevector dependence imply different ways in
which the lattice can screen the quasiparticles. In the classical limit,
$\chi_{\mathbf{k},\tau}=\chi_{\mathbf{k}}\delta(\tau)$, Eqs. 4, 10, and 12,
imply an effective interaction between the electron and hole,

$\hat{V}_{\mathrm{eff}}(\mathbf{k})=-\frac{1}{k^{2}}\left[\frac{e^{2}}{\varepsilon_{0}\varepsilon_{\infty}}-\chi_{\mathbf{k}}|\lambda|^{2}\right]$
(11)

which is a sum of the bare interaction, here screened by
$\varepsilon_{\infty}=4.5$,47 and the contribution from the lattice
proportional to $\chi_{\mathbf{k}}$. In the zero wavevector limit this is a
constant, and if taken as
$\chi_{\mathbf{k}}|\lambda|^{2}=e^{2}/\varepsilon_{0}(1/\varepsilon_{\infty}-1/\varepsilon_{r})$
we recover the Wannier-Mott model of a exciton. With an effective dielectric
constant $\varepsilon_{r}=6.1$,25 this local, _static_ screening is manifestly
insufficient to produce the repulsive interaction observed from the free
energy calculations. Rather an explicit $\mathbf{k}$ dependence to
$\chi_{\mathbf{k}}$ is required.

Using explicit MD simulations of the bulk classical MAPbI3 lattice, we find
$\chi_{\mathbf{k}}$ is well approximated by

$\displaystyle\chi_{\mathbf{k}}\approx\frac{\chi_{0}}{1-l_{s}^{2}\mathbf{k}^{2}+l_{s}^{2}l_{c}^{2}\mathbf{k}^{4}}$
(12)

characterized by three positive real parameters, $\chi_{0}$, $l_{s}$, and
$l_{c}$. This functional form includes a single resonant peak and is assumed
isotropic on length scales greater than the lattice spacing. The resonant peak
results from the negative second order coefficient and manifests the double
well potential of the optical mode.18 Performing the inverse Fourier transform
gives an effective potential

$\displaystyle V_{\mathrm{eff}}(r)=$ (13)
$\displaystyle-\frac{e^{2}}{4\pi\varepsilon_{0}r}\left[\frac{1}{\varepsilon_{\infty}}+\frac{1}{\varepsilon^{*}}+\frac{\gamma}{4\delta\varepsilon^{*}}e^{-r\delta}\sin[r\gamma-\theta]\right]$

where $r$ is the distance between two charges. Details of the derivation with
analytic expressions for $1/\varepsilon^{*}$,
$\gamma^{-1}\approx\sqrt{2}\ell_{c}$, $\delta^{-1}\approx
2\ell_{c}/\sqrt{1-\ell_{s}/2\ell_{c}}$, and $\theta=\arctan[2\delta/\gamma]$
are shown in the SI. This form is plotted in Fig. 1c and provides an excellent
fit at large $r$ to the free energy from the MD simulation. We refer to this
effective electron-hole interaction arising from spatially dependent screening
from the MAPbI3 lattice as nonlocal screening.48 The theory clarifies that the
non-monotonic interaction potential results from deformations generated within
the lattice due to the charges. At specific characteristic distances these
deformations are sufficiently unfavorable that the electron and hole are
effectively repelled from each other.

This effective interaction in Eq. 13 is distinct from what has been considered
previously by Pollman and Buttner,49 in which coupling to a single
dispersionless optical phonon results in a excitonic polaron that screens the
bare Coulomb potential. In their approximation, the polar displacement field
is treated quantum mechanically by including a $\tau$ dependence of the
susceptibility

$\displaystyle\chi_{\mathbf{k},\tau}=\frac{1}{2\omega}e^{-\omega|\tau|}$ (14)

where the phonon mode is characterized by a single longitudinal optical
frequency $\omega$. Plugging Eq. 14 into Eq. 10, we obtain the effective path
action

$\displaystyle\mathcal{S}_{\mathrm{eff}}$
$\displaystyle=-\sum_{i,j}\Gamma_{ij}\frac{\alpha\omega^{2}\hbar^{3/2}}{\sqrt{8m_{ij}\omega}}\int_{\tau}\int_{\tau^{\prime}}\,\frac{e^{-\omega|\tau-\tau^{\prime}|}}{|\mathbf{r}_{i,\tau}-\mathbf{r}_{j,\tau^{\prime}}|}$
(15)

where we have written $|\lambda|^{2}$ in the traditional Fröhlich form,
introducing $\alpha$ as the dimensionless coupling constant. Pollman and
Buttner further approximate this _dynamic_ screening approach in order to
obtain a closed form effective potential. Their potential is an exponentially
screened Coulomb potential in the classical limit with a screening length
given by the polaron radus.49 While for certain parameters it is possible that
the Pollman-Buttner potential is repulsive,37 employing known values for
$\alpha=1.72$ and $\omega=40$cm-1 for MAPbI3,47 the resultant effective
potential is monotonic and inconsistent with our MD result. Analogous
approaches including sums of two or three dispersionless phonons similarly
fail to describe a repulsive interaction.

Figure 2: Implications of different screening models from explicit path
integral simulations with electron and hole quasiparticles. (a) Free energy as
a function of the distance between quasiparticle centroids at 300K under
static (blue), dynamic (green), and nonlocal (red) screening models. (b)
Radial probability distribution for electron and hole, with the same color
scheme as in (a).

To investigate the implication of a nonlocal screening on the observable
properties of MAPbI3, we simulated an electron and hole pair using our
quasiparticle path integral approach under (i) static, (ii) dynamic, and (iii)
nonlocal screenings. In each case, we employed known experimental parameters
for the dielectric constants, optical frequencies, and effective masses and
thus expect our results to be quantitatively accurate. To our knowledge the
$\mathbf{k}$-dependent dielectric susceptibility has not been reported for
MAPbI3, so we parameterized the nonlocal screening interaction using our MD
results. For each case, we extract the exciton binding energies and
bimolecular recombination rates as both have been difficult to reconcile
theoretically.50

The exciton binding energy, $\Delta E_{\mathrm{B}}$ is definable within our
path integral framework as

$\Delta E_{\mathrm{B}}=\lim_{T\rightarrow 0}\min_{R}\Delta F(R)$ (16)

and to evaluate it we computed the free energy $\Delta F(R)=F(R)-F(\infty)$ at
a variety of temperatures ranging from 200K to 400K and extrapolate its value
to 0K (SI). Representative free energies at $T=300$K are shown in Fig. 2a. As
anticipated from the theory, we find that both dynamic and nonlocal screening
reduce the effective attraction between electron and hole but only the
nonlocal screening results in a barrier to recombination.

The extrapolated binding energies are summarized in Table 1. Within the static
screening approach, the exciton is hydrogenic, and the binding energy is given
by $\Delta
E_{\mathrm{B}}^{s}=\mu{\color[rgb]{0,0,0}e}^{4}/2(4\pi\epsilon_{0}\epsilon_{r})^{2}\hbar^{2}$
where $\mu$ is a reduced mass of the electron and hole. The large decrease in
binding energy under dynamic screening reflects the polaronic effect. Since
the experimentally derived value of $\alpha$ is relatively small,47 we find
the change to the binding energy is well approximated by first order
perturbation theory, yielding the known Fröhlich result, $\Delta
E_{\mathrm{B}}^{d}=\Delta E_{\mathrm{B}}^{s}-2\alpha\hbar\omega$.51 This
reduction in the binding energy is consistent with recent Bethe-Salpeter
calculations with perturbative electron-phonon interactions,24 but higher than
experimental estimates.52 The reduction in the binding energy from the
nonlocal screening is 12 meV, which is close to a prediction assuming
hydrogenic 1s orbits, 17 meV. In the zero temperature limit, our simplified
classical limit for the nonlocal screening model is no longer valid. Including
a constant optical frequency as in the dynamical approximation yields a
binding energy of 39 meV in very good agreement with experiment.52; 53

The bimolecular recombination rate, $k_{r}$ is defined as the rate of change
of the concentration of free charges,

$\frac{d\rho_{e}}{dt}=-k_{r}\rho_{e}\rho_{h}$ (17)

through the reaction $e^{-}+h^{+}\rightarrow\hbar\nu$, where $\rho_{e/h}$ is
the concentration of free electrons/holes. At typical working excitation
densities for MAPbI3 based photovoltaics, radiative recombination is the
limiting factor determining the charge carrier lifetime.50 We can evaluate
$k_{r}$ using Fermi’s golden rule for spontaneous emission, with an effective
mass approximation.54; 55; 56; 57 Within our quasiparticle path integral
framework, the rate is given by a constant times a ratio of path partition
functions,58

$k_{\mathrm{r}}=\frac{e^{2}\sqrt{\varepsilon_{\infty}}E_{\mathrm{gap}}^{2}}{2\pi\varepsilon_{0}\hbar^{2}c^{3}\mu}\frac{{\mathcal{Z}}_{\mathrm{c}}}{\mathcal{Z}}$
(18)

where $E_{\mathrm{gap}}=1.64e$V is the band gap energy for MAPbI3, and $c$ is
the speed of light. The subscript $\mathrm{c}$ on $\mathcal{Z}_{\mathrm{c}}$
stands for combined path integral in which the two separate imaginary time
paths are placed together to form a single, radiating path by linking same
imaginary time slices. The ratio of path partition functions can be evaluated
as,

$\frac{\mathcal{Z}_{\mathrm{c}}}{\mathcal{Z}_{\mathrm{}}}=4\pi\int
dR\,R^{2}\left\langle e^{\Delta S/\hbar}\right\rangle_{R}e^{-\beta\Delta
F(R)}$ (19)

where we replace the ratio of partition functions by an exponential average at
fixed $R$ of the difference in path action $\Delta S=S-S_{\mathrm{c}}$ (see
SI).

Screenings |  Static |  Dynamic |  Nonlocal
---|---|---|---
$\Delta\rm{E}_{B}\,(\rm{meV})$ |  65.65 |  47.67 |  54.12
$\tau_{\mathrm{r}}\,(\rm{ns})$ |  13.5 |  35.5 |  78.1
Table 1: Exciton binding energy $\Delta E_{\mathrm{B}}$, and carrier lifetime
$\tau_{\mathrm{r}}$ estimated using different screening models.

The change in action is a reporter on the overlap between the electron and
hole wavefunctions. The electron and hole radial probability distribution is
described in Fig. 2b, illustrating that the nonlocal screening effect
increases the average distance between electron and hole. This is distinct
from the effect of the dynamic screening, which consistent with the small
value of $\alpha$, leaves the electronic distribution largely unaltered from
the simple Wannier, static screening model. The decrease in electron hole
overlap results in a nearly order of magnitude decrease in $k_{\mathrm{r}}$
using the nonlocal screening theory relative to the static screening theory.
For the nonlocal screening theory, we find $k_{\mathrm{r}}=1.3\times
10^{-10}$cm${}^{3}/$s, in excellent agreement with photoluminescence lifetime
measurements.50

Assuming the only loss mechanism is due to bimolecular recombination, the
lifetime of an electron hole pair is computable from
$\tau_{\mathrm{r}}=1/k_{\mathrm{r}}\rho_{e}$. Summarized in Table 1 for
$\rho_{e}=10^{17}/$cm3 are lifetime estimates using the different screening
models. Both standard polaronic effects incorporated into the dynamic
screening model as well as the nonlocal screening model increase the lifetimes
of free charge carriers, however the contribution of a nonlocal screening
obtained from the MAPbI3 lattice is much more significant. The details on the
bimolecular recombination rate calculations are in the SI.

By comparing these different simplified models that account for charge-lattice
interactions, we find that the ability of the MAPbI3 lattice to nonlocally
screen quasiparticles is sufficient to explain the particularly low exciton
binding energy and recombination rate. Only this screening kernel in our
unified Gaussian field theory formalism can suppress the electron-hole overlap
enough to explain the anomalously long free carrier lifetime with weak lattice
coupling strength. The particular nonlocal screening adopted here was deduced
directly from explicit atomistic molecular dynamics simulations using a
quasiparticle path integral framework. This framework is uniquely able to
study the thermodynamics of this quasiparticle-lattice system at finite
temperature.

The adoption of a spatially dependent screening is consistent with a growing
literature pointing to the importance of dynamic disorder in lead halide
perovskites.59; 60; 61 The barrier to bringing electrons and holes together we
have discovered here undoubtedly has implications apart from offering an
explanation of the particular high power conversion efficiencies of MAPbI3.
For example, this repulsion may help explain observations of anti-binding of
biexcitons.38; 39 The identification of a repulsive electron-hole interaction
generated from the soft, polar modes of the perovskite lattice offers a key
new design principle for photovoltaic materials. Searching for other systems
that admit this type of interaction represents a promising new direction for
materials discovery.

Acknowledgments. This work was supported by the U.S. Department of Energy,
Office of Science, Office of Basic Energy Sciences, Materials Sciences and
Engineering Division under Contract No. DE-AC02-05-CH11231 within the Physical
Chemistry of Inorganic Nanostructures Program (No. KC3103). This research used
resources of the National Energy Research Scientific Computing Center (NERSC),
a U.S. Department of Energy Office of Science User Facility. Y. P. also
acknowledges Kwanjeong Educational Foundation. D. T. L. acknowledges the
Alfred P. Sloan Foundation.

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