11institutetext: Maastricht University, Maastricht, Netherlands
22institutetext: Utrecht University, Utrecht, Netherlands 33institutetext:
Studio Europa, Maastricht, Netherlands

# Closing the Loop: Testing ChatGPT to Generate Model Explanations to Improve
Human Labelling of Sponsored Content on Social Media

Thales Bertaglia 11 3 3 0000-0003-0897-4005 Stefan Huber  Catalina Goanta 22
0000-0002-1044-9800 Gerasimos Spanakis 11 0000-0002-0799-0241 Adriana
Iamnitchi 11 0000-0002-2397-8963

###### Abstract

Regulatory bodies worldwide are intensifying their efforts to ensure
transparency in influencer marketing on social media through instruments like
the Unfair Commercial Practices Directive (UCPD) in the European Union, or
Section 5 of the Federal Trade Commission Act. Yet enforcing these obligations
has proven to be highly problematic due to the sheer scale of the influencer
market. The task of automatically detecting sponsored content aims to enable
the monitoring and enforcement of such regulations at scale. Current research
in this field primarily frames this problem as a machine learning task,
focusing on developing models that achieve high classification performance in
detecting ads. These machine learning tasks rely on human data annotation to
provide ground truth information. However, agreement between annotators is
often low, leading to inconsistent labels that hinder the reliability of
models. To improve annotation accuracy and, thus, the detection of sponsored
content, we propose using chatGPT to augment the annotation process with
phrases identified as relevant features and brief explanations. Our
experiments show that this approach consistently improves inter-annotator
agreement and annotation accuracy. Additionally, our survey of user experience
in the annotation task indicates that the explanations improve the annotators’
confidence and streamline the process. Our proposed methods can ultimately
lead to more transparency and alignment with regulatory requirements in
sponsored content detection.

###### Keywords:

sponsored content detection human-AI collaboration legal compliance social
media

##  1 Introduction

The rise of influencers, content creators monetising online content through
native advertising, has drastically changed the landscape of advertising on
social media [8, 13]. This shift has increased concern about hidden
advertising practices that might harm social media users. For decades,
advertising rules have been applied to legacy media in such a way as to
separate commercial communication from other types of content. The primary
rationale behind rules relating to mandated disclosures has been that hidden
advertising leads to consumer deception. Despite the increasing legal
certainty that native advertising, such as influencer marketing, must be
clearly disclosed, monitoring and enforcing compliance remains a significant
challenge [25].

The task of automatically detecting sponsored content aims to enable the
monitoring and enforcement of such regulations at scale. For instance, in the
United Kingdom, the Competition and Markets Authority is one of the
enforcement agencies tasked with monitoring influencer disclosures on social
media, which is done using some automated techniques developed by their
internal data unit111https://www.gov.uk/cma-cases/social-media-endorsements.
In published scholarship, most existing methods frame the problem as a machine
learning task, focusing on developing models with high classification
performance. The success of these models depends on the quality and
consistency of human-annotated data, which often suffer from low inter-
annotator agreement, compromising the reliability and performance of the
models [9, 27]. Moreover, fully-automated approaches are insufficient for
regulatory compliance, where human decision-makers are ultimately responsible
for imposing fines or pursuing further investigations.

To bridge this gap, we propose a novel annotation framework that augments the
annotation process with AI-generated explanations, which, to our knowledge, is
the first attempt in this domain. These explanations, presented as text and
tokens or phrases identified as relevant features, aim to improve annotation
accuracy and inter-annotator agreement. Our experiments show that our proposed
framework consistently increases agreement metrics and annotation accuracy,
thus leading to higher data quality and more reliable and accurate models for
detecting sponsored content. Critically, our work tackles the need for
explainability in AI tools used for regulatory compliance, ensuring that human
decision-makers can better understand and trust the outputs of these models.
This is particularly important for market surveillance activities, which have
not yet caught up with the transparency and accountability issues at the core
of discussions around individual surveillance [19].

##  2 Related Work

Sponsored content detection has primarily been studied as a text
classification problem. Works in this field generally train models in a semi-
supervised setting, using posts disclosed as ads with specific hashtags as
weak labels. Generally, there is a lack of focus on evaluating model
performance with labelled data. Most works collect their own datasets and do
not describe whether (and how) data is annotated. Since social media platforms
typically do not allow data sharing, there are no standardised datasets for
evaluating the task; thus, comparing results is challenging. Furthermore, the
absence of labelled data for evaluation affects the reliability of results, as
models are often not tested on undisclosed ads.

From a technical perspective, previous studies have employed traditional
machine learning models with basic text features [7, 29], neural networks with
text embeddings [32], and multimodal deep learning architectures combining
text, image, and network features [17, 16]. In this paper we experiment with
some of these models in addition to chatGPT and GPT-4 for classification.
Although peer-reviewed research is limited due to chatGPT’s recent release,
some technical reports have found chatGPT to achieve state-of-the-art
performance in several text classification tasks [23, 30, 12].

Interdisciplinary research combining computational methods with fields such as
communication and media studies and law has focused on identifying
influencers, describing their characteristics, and mapping the prevalence of
their disclosures [21, 2, 4]. In the context of using explanations to improve
data labelling or decision-making, research has explored AI-human
collaboration and investigated the optimal integration of explanations for
human interaction [18, 22, 6, 28]. To the best of our knowledge, our paper is
the first to propose using AI-generated explanations to improve the detection
of sponsored content, bridging the gap between explainable AI and regulatory
compliance in the context of sponsored content on social media.

##  3 Experimental Setup

This section describes the dataset we use, how we selected the model for
sponsored content detection, generated explanations to augment the annotation
process, and designed the annotation task and the user-experience survey.

###  3.1 Data Collection

We collected and curated our own dataset of Instagram posts for this study. We
manually selected 100 influencers based in the United States using the
influencer discovery platform Heepsy222https://heepsy.com. We selected 50
micro-influencers (between $100k$ and $600k$ followers) and 50 mega-
influencers (over $600k$ followers). Then, we collected all available data and
metadata from all posts for each account using
CrowdTangle333https://www.crowdtangle.com/, the Meta platform that provides
access to social media data for (among others) academic purposes. Our dataset
includes $294.6k$ posts, 66.1% from mega-influencers and 33.9% from micro-
influencers. CrowdTangle’s Terms of Service do not allow (re)sharing datasets
that include user-generated content; thus, we cannot share the full dataset.
However, the list of the ids of accounts and posts is publicly available on
https://github.com/thalesbertaglia/chatgpt-explanations-sponsored-content/

###  3.2 Detecting Sponsored Content

In the first step of our experimental setup, we aim to select the most
suitable sponsored content classifier for generating explanations. We evaluate
three previously proposed models: (1) a logistic regression classifier with
term frequency inverse-document frequency (TF-IDF) features, analogous to the
approach used by [7, 29], (2) a pre-trained BERT model fine-tuned for our
task, comparable to [17, 32], and (3) OpenAI’s chatGPT (GPT-3.5-turbo as of
March 2022), which achieves state-of-the-art results in various text
classification tasks [12, 23]. We generate GPT predictions using OpenAI’s API.

To evaluate the models’ performance, we select a sample from our original
dataset and split the data into training and test sets by year, using 2022 for
testing and all prior posts for training. This division simulates a real-world
scenario where a model is deployed and used to classify unseen data for
regulatory compliance. By ensuring no temporal overlap between the sets, we
prevent the model from learning features correlated with a specific period.
Given the high imbalance in the data (only 1.72% of posts are disclosed as
sponsored), we apply the random undersampling approach proposed by Zarei et
al. (2020) [32] to balance the data. We include all disclosed posts ($n$) and
randomly sample ($2*n$) posts without disclosures as negative examples. We
allocate 90% of the balanced data before 2022 to training and the remaining
10% to validation. We use all data in 2022 as the test set.

Additionally, we labelled a sample of the test set to evaluate the model’s
performance in detecting undisclosed ads. Four annotators labelled 1283 posts
in total, with a sample of 50 posts labelled by all annotators for calculating
agreement metrics. The inter-annotator agreement was 52% in absolute agreement
and 53.37 in $\alpha$, indicating moderate agreement. 654 posts were labelled
as sponsored (50.97%) and 629 as non-sponsored (49.03%). 91.59% of the
sponsored posts did not have disclosures – i.e., they were identified as
undisclosed ads.

We employ a semi-supervised approach to train the models, treating disclosed
sponsored posts as positive labels for the sponsored class. We consider #ad,
#advertisement, #spons, and #sponsored as ad disclosures. We then remove
disclosures from the posts to prevent models from learning a direct mapping
between disclosure and sponsorship. We train the logistic regression model
using TF-IDF features extracted from word-level n-grams from the captions
(unigrams, bigrams, and trigrams). For the BERT-based model, we use the bert-
base-multilingual-uncased pre-trained model weights from HuggingFace [31]. We
fine-tuned the BERT-based model for three epochs using the default
hyperparameters (specified in Devlin et al. (2019) [5]).

We apply various prompt-engineering techniques to enhance GPT’s predictions.
As we use the same methodology for generating explanations, we provide a
detailed description in the following subsection. We evaluate all models using
F1 for the positive and negative classes, Macro F1 (the simple average of both
classes) and Accuracy in detecting undisclosed ads – a critical metric for
determining the models’ effectiveness in detecting sponsored posts without
explicit disclosures, which is ultimately our goal. Table 1 presents the
classification metrics for the three models, calculated based on the labelled
test set.

Table 1: Performance of the different models on the labelled test set. Acc represents the models’ accuracy in detecting undisclosed ads. Model | Pos F1 | Neg F1 | Macro F1 | Acc
---|---|---|---|---
Log Reg | 45.33 | 66.50 | 55.92 | 28.71
BERT | 29.30 | 68.84 | 49.07 | 10.85
GPT-3.5 | 76.09 | 63.93 | 70.01 | 88.98

GPT-3.5 outperforms the other models in Macro F1 and accuracy in detecting
undisclosed ads. Logistic regression (Log Reg) and BERT achieve significantly
low accuracy, suggesting their inability to identify undisclosed sponsored
posts effectively. The difference in Macro F1 is smaller, highlighting that
relying solely on this metric for evaluating models may not accurately reflect
their actual performance. Therefore, having high-quality labelled data,
including undisclosed ads, is crucial for proper evaluation.

BERT’s inferior performance compared to Log Reg could be due to a few factors.
Being pre-trained on longer texts, BERT might struggle to extract sufficient
contextual information from short Instagram captions. In addition, Log Reg,
when combined with TF-IDF features, effectively captures word-level n-grams
that may be more effective at identifying sponsored content patterns. In
contrast, BERT uses subword tokenisation, which could result in less efficient
pattern recognition. Given GPT-3.5’s superior performance, particularly in
detecting undisclosed sponsored posts, we selected it as the model for
generating explanations to augment the annotation task.

###  3.3 Generating Explanations with GPT

We investigated various prompts for all publicly accessible models from the
GPT-3 series and GPT-4. We observed that even the smallest GPT-3 model, Ada
(text-ada-001), performed well in sponsored content detection and identifying
relevant words. Nevertheless, we noted significant performance improvements
for larger models especially when employing chain-of-thought reasoning [30]
and generating explanations – particularly for more ambiguous posts.
Consequently, we focused on GPT-3.5-turbo (the default ChatGPT version as of
March 2022) and GPT-4.

We found a conservative bias for both models, with a strong preference for
predicting the not sponsored class or other negative labels over positive
ones. This phenomenon appeared consistent across all Davinci- and Curie-based
models, with the inverse being true for smaller Babbage and Ada-based models.
We employed several prompt engineering techniques to mitigate this bias and
calibrate the labels. First, we instructed the model to highlight relevant
words and generate explanations before classifying a post. This chain-of-
thought prompting approach, inspired by [30], significantly reduced bias and
improved prediction interpretability. Second, we used few-shot learning to
refine explanation calibration, address known failure modes, and further
alleviate bias [3]. Third, we experimented with different label phrasings,
such as “Likely (not) sponsored”, to enhance the model’s ability to make less
confident predictions. Finally, we directly instructed the model to favour
positive labels in cases of uncertainty, aiming to identify a higher
proportion of undisclosed ads. The final prompt is available on the project’s
GitHub repository 444https://github.com/thalesbertaglia/chatgpt-explanations-
sponsored-content/.

Upon qualitative evaluation, we found that GPT-4 outperformed GPT-3.5-turbo in
explanation quality and classification accuracy, especially for ambiguous
posts. However, for this study, we chose GPT-3.5-turbo (hereafter referred to
as “GPT”) due to its advantages in speed, cost, and public accessibility.
Following this approach, we obtained the most important words in a post and
generated explanations for why a post may or may not be sponsored to assist
annotators. The following is an illustrative example of such an explanation;
we omitted the actual brand name to ensure the post’s anonymity:

    Key indicators: ’@BRAND’, ’LTK’.
    The post promotes a fashion brand and features a discount code,
    indicating a partnership. Additionally, it features a @shop.LTK
    link, a platform for paid partnerships.

###  3.4 Annotation Task

We conducted a user study to evaluate how explanations can help detect
sponsored content. The study consisted of an annotation task in which
participants labelled 200 Instagram posts from our dataset as Sponsored or
Non-Sponsored. Our objective with the task was two-fold: i) Analyse
explanations as a tool for improving annotation as a resource for ML tasks –
i.e., to measure their impact on data quality, which, in turn, allows for the
development of better models and evaluation methods. ii) Simulate regulatory
compliance with sponsored content disclosure regulations – i.e., how a
decision-maker would flag posts as sponsored.

We framed the annotation as a text classification task in which annotators had
to determine whether an Instagram post was sponsored based on its caption.
Generally, we followed the data annotation pipeline proposed by Hovy and Lavid
(2010) [15]. We instructed annotators to consider a post as sponsored if the
influencer who posted it was, directly or indirectly, promoting products or
services for which they received any form of benefits in return. These
benefits included direct financial compensation and non-monetary benefits,
such as free products or services. Self-promotion was an exception: we
considered posts promoting the influencer’s content (e.g. YouTube channel or
podcast) non-sponsored. However, posts advertising merchandise with their
brand or directly selling other goods still fall under sponsored content. We
explained these guidelines to each annotator and provided examples of
sponsored and non-sponsored posts to help reinforce the definitions.

Eleven volunteer annotators with varying levels of expertise participated in
the study. All were between 20 and 30 years old, active social media users,
and familiar with influencer marketing practices on Instagram. Additionally,
all annotators had or were working towards a high-education degree in a
European university. Demographically, the participants came from various
countries. We did not specifically collect country-level information, but at a
continent level, participants were from Asia, Europe, and South America. While
all participants were fluent in English, none were native speakers.

We split annotators into three groups according to their level of expertise in
annotating sponsored content on social media. The first group, with three
people, consisted of participants with no prior experience in data annotation.
The second group included four participants who previously participated in
annotation tasks but had no formal training. The third group, consisting of
four legal experts, had specific legal expertise in social media advertisement
regulations and had participated in annotations before. We further split the
subgroups of annotators into two groups regarding annotation setup: one
without explanations, in which annotators only had access to the captions, and
one augmented with the generated explanations. One group of four annotators
labelled the posts in both setups: with and without explanations. To
summarise, our study includes three distinctive groups: novices with no prior
annotation experience, intermediate annotators with previous experience but no
formal training, and legal experts knowledgeable in social media regulations.

To select the 200 Instagram posts for our user study, we turned to a sample
previously labelled by law students in another annotation task. Although the
labels and definitions used in that task differed from ours, they provided a
way to identify which posts were undisclosed ads, allowing us to include them
in our study. We selected posts published between 2017 and 2020 by 66
different influencers based in the United States, with 62% being mega-
influencers and 38% being micro-influencers. We also included 15% of posts
with clear ad disclosures (such as the hashtag #ad) as an attention check to
ensure annotators noticed the disclosures. Based on the labels from the
previous annotations, we estimate that 65% (130) of the posts were likely
sponsored, and 50% (100) were likely undisclosed ads.

We set up the study using the open-source annotation platform
Doccano555https://github.com/doccano/doccano. Each participant had a unique
project, and although all annotators labelled the same 200 posts, the labels
were not shared, and each participant only had access to their annotations.
The annotation interface displayed the caption of the post and the two
possible labels (Sponsored and Non-Sponsored) as buttons. After the post
caption, we added the generated explanations with an explicit delimitation.

Accurately measuring inter-annotator agreement is crucial in data annotation
tasks, as it allows us to estimate the annotated data’s quality and the
decision-making process’s reliability. To assess inter-annotator agreement in
our study, we used three main metrics: Krippendorff’s Alpha ($\alpha$),
absolute agreement, and accuracy in detecting disclosed posts. Krippendorff’s
Alpha measures the degree of agreement among annotators, considering the level
of agreement expected by chance alone [20, 14]. The absolute agreement
indicates the proportion of annotations where all annotators agreed on the
same label. We also used accuracy in detecting disclosed posts as an attention
check mechanism, as it measures annotators’ ability to correctly identify
posts with clear disclosures as sponsored. This metric is crucial because
disclosures may not always be easily visible in posts [21]. We also analysed
additional metrics in some experiments, which we will introduce when
describing the specific experiments.

###  3.5 User-experience Survey

After the annotation, we conducted a user-experience survey to gather feedback
from annotators on their experience using the explanations to assist with
their decision-making process. The survey consisted of seven questions, with
five closed-ended and two open-ended questions. We describe all questions and
the rating scale used below:

  * •

“On a scale of 1 (not helpful) to 5 (extremely helpful), how helpful were the
explanations in identifying undisclosed advertisement partnerships?”

  * •

“How accurate, from 1 (extremely inaccurate) to 5 (extremely accurate), did
you think the explanations were?”

  * •

“How often, from 1 (0% of the time) to 5 (100% of the time), did you agree
with the AI explanations?”

  * •

“Did the AI explanations help you feel more confident in your decision-making
(Yes/No)?”

  * •

“What aspects of the AI explanations were most helpful for your decision-
making process?” This was a multiple-choice question with five options:
Reasoning, Identifying specific words or phrases, Clear examples, Other
(specify), and None.

  * •

“In what ways did the AI explanations improve your understanding of what
constitutes an undisclosed advertisement partnership?” Open-ended.

  * •

“How could the AI explanations be further improved to better support your
decision-making process? Did you find anything noticeable you want us to
know?” Open-ended.

The participants who received annotations augmented with explanations all
completed the questionnaires, and we ensured their anonymity by not collecting
any identifiable information. Additionally, we made it clear to the annotators
that their responses would be entirely anonymous.

##  4 Experimental Results

This section presents the main findings from the annotation task and user-
experience survey. Table 2 shows the metrics comparing the agreement between
annotators who labelled the posts with and without explanations. Seven
participants were in the No Explanations group (one with no experience, four
with some experience, and two legal experts). The With Explanations group had
eight people (three with no experience, three with some experience, and two
legal experts) – one participant from the no experience group and three from
some experience labelled in both settings. In addition to the metrics
presented in subsection 3.4, we also evaluate the proportion of posts with at
most one disagreement (1-Disag) and show the percentage of posts labelled as
sponsored (Sponsored). The last two rows present the absolute and relative
(normalised) differences in metrics between the groups. The relative
differences in metrics indicate the proportional change (in percentage).
Positive differences represent an increase in agreement.

Table 2: Agreement metrics comparing annotations with and without explanations. | $\alpha$ | Abs | 1-Disag | Acc | Sponsored
---|---|---|---|---|---
No Explanations | 54.98 | 46.50 | 69.50 | 90.62 | 54.64
With Explanations | 63.58 | 54.50 | 75.00 | 93.75 | 59.81
Absolute Diff | 8.61 | 8.00 | 5.50 | 3.12 | 5.17
Relative Diff | 15.65 | 17.20 | 7.91 | 3.45 | 9.46

Using explanations to enhance the annotations resulted in a consistent
improvement across all inter-annotator agreement metrics. Specifically, there
was a 15.65% increase in $\alpha$ and a 17.20% increase in absolute agreement.
However, the final values were still relatively low, typical of annotations in
complex decision-making tasks [9, 10, 27]. Accuracy in detecting disclosed
posts also improved by 3.45%, but the final result was not perfect, suggesting
that annotators still fail to identify all disclosure hashtags, even with
explanations highlighting them. Additionally, the proportion of posts labelled
as sponsored increased by 9.46%, indicating that explanations led annotators
to identify more as sponsored. We also analyse the agreement between all pairs
of annotators to measure the variation in agreement and ensure the reliability
of the annotations. Table 3 summarises the pairwise agreement metrics. The Min
and Max columns represent the lowest and highest agreement metric values among
the annotator pairs, respectively, and the $\pm$ column denotes the standard
deviation.

Table 3: Pairwise agreement comparing annotations with and without explanations. | Min Abs | Max Abs | $\pm$ | Min $\alpha$ | Max $\alpha$ | $\pm$
---|---|---|---|---|---|---
No Explanations | 66.00 | 88.50 | 5.28 | 30.81 | 77.04 | 10.83
With Explanations | 73.00 | 90.00 | 4.49 | 43.13 | 79.53 | 10.00
Absolute Diff | 7.00 | 1.50 | -0.79 | 12.31 | 2.48 | -0.82
Relative Diff | 10.61 | 1.69 | -14.98 | 39.96 | 3.22 | -7.62

The pairwise metrics reveal considerable variation in the agreement between
annotator pairs. For the No Explanation group, there was a substantial
difference of 46.23 in $\alpha$ between the pair with the lowest and highest
agreement, with a standard deviation of 10.83. This difference indicates that
some annotators are significantly less reliable than others. However, the
group With Explanations showed a consistent improvement, with less variation
between pairs. The standard deviation decreased by 14.98% for absolute
agreement and 7.62% for $\alpha$, indicating more reliable annotations. Even
the lowest-agreement pair showed significant improvement, with an increase of
10.61% for absolute agreement and 39.96% for $\alpha$. These results suggest
that using explanations to augment annotations led to a higher inter-annotator
agreement overall, improved consistency between pairs, and even increased
agreement among the least reliable annotators. To better understand the impact
of augmenting the annotation with explanations, we also investigated how it
affects different subgroups of annotators. We divided the subgroups into three
categories: legal experts, non-experts, and annotators who labelled in both
settings (with and without explanations) – this category does not include
legal experts. Table 4 presents the agreement metrics for each category in
both subgroups of annotators, as well as the relative difference between them.
# indicates the number of participants within the subgroup. For clarity, we
did not report the proportion of annotations with at most one disagreement
because some subgroups contain a single pair of annotators.

Table 4: Agreement metrics for different subgroups of annotators, aggregated according to their expertise level. | $\alpha$ | Abs | Acc | Sponsored | #
---|---|---|---|---|---
Legal Experts No Explanations | 52.11 | 76.50 | 96.88 | 57.25 | 2
Legal Experts With Explanations | 61.94 | 83.00 | 100.00 | 66.50 | 2
Relative Diff | 18.86 | 8.50 | 3.23 | 16.16 | -
Non-Experts No Explanations | 62.04 | 62.50 | 93.75 | 53.60 | 5
Non-Experts With Explanations | 64.89 | 59.50 | 93.75 | 57.58 | 6
Relative Diff | 4.59 | -4.80 | 0.00 | 7.43 | -
Labelled Both No Explanations | 66.74 | 70.00 | 96.88 | 53.12 | 4
Labelled Both With Explanations | 73.15 | 74.50 | 100.00 | 54.50 | 4
Relative Diff | 9.60 | 6.43 | 3.23 | 2.59 | -

The annotations augmented with explanations showed consistent improvements in
all subgroups, except for absolute agreement within the non-expert group.
Legal experts had the most significant improvement in $\alpha$ (18.86%).
Additionally, the proportion of posts labelled as sponsored increased
significantly (16.16%), with the subgroup Legal Experts With Explanations
having the highest value (66.5%). This subgroup and Labelled Both With
Explanations achieved 100% accuracy in detecting disclosed sponsored posts.
Labelled Both also had the highest $\alpha$ in both settings. It is important
to note that higher agreement does not necessarily imply higher accuracy in
correctly identifying sponsored posts. The metrics measure how much a subgroup
of annotators agree on the definitions they are applying to label; they could
be wrongly applying a consistent judgement. Therefore, we cannot reliably
conclude which group had the best performance. Moreover, the high agreement
within the subgroup Labelled Both could be influenced by the annotators
labelling the same posts twice in both settings. Although we randomly shuffled
the posts to reduce the likelihood of memorisation, repetition could still
affect agreement. Nevertheless, the high proportion of sponsored content and
absolute agreement for the annotation within Legal Experts With Explanations
indicate that experts agree that there are more sponsored posts than non-
experts tend to identify.

While explanations can improve the quality of annotations, they may also
introduce bias by influencing annotators to rely on specific cues presented in
the explanation; annotator bias is a common challenge in text annotation tasks
[1, 11]. To investigate potential bias introduced by explanations in our
study, we examine whether annotators tended to use the same label predicted by
GPT. Although we did not explicitly provide GPT’s prediction as part of the
explanation, the model’s reasoning and highlighted words and phrases might
imply the predicted label, leading to over-reliance on the model and
decreasing the accuracy of annotations. Thus, it is essential to analyse the
impact of GPT’s predictions on annotator behaviour to ensure the reliability
and fairness of the annotations. Specifically, we calculate two metrics – the
distribution of posts labelled as sponsored and the majority agreement with
GPT predictions – to compare the agreement between annotators who received
explanations and those who did not. We use majority agreement instead of
absolute to reduce the impact of low-agreement pairs and fairly compare all
groups. If the agreement with GPT predictions increased in the group with
explanations, it could indicate that annotators followed the model’s
predictions. We hypothesise that, for the Labelled Both group, an increase in
agreement with GPT predictions proportionally more than the percentage of
sponsored posts would suggest that annotators changed their judgements based
on the model’s cues. Table 5 summarises the results of this analysis.

Table 5: Proportion of posts labelled as sponsored and majority agreement with GPT predictions across subgroups of annotators. | Sponsored | Agreement
---|---|---
No Explanations | 54.64 | 85.50
With Explanations | 59.81 | 90.50
Relative Diff | 9.46 | 5.85
Legal Experts No Explanations | 57.25 | 77.50
Legal Experts With Explanations | 66.50 | 92.00
Relative Diff | 16.16 | 18.71
Non-Experts No Explanations | 53.60 | 81.00
Non-Experts With Explanations | 57.58 | 88.50
Relative Diff | 7.43 | 9.26
Labelled Both No Explanations | 53.12 | 78.50
Labelled Both With Explanations | 54.50 | 87.00
Relative Diff | 2.59 | 10.83

The majority agreement with GPT predictions is consistently high across all
subgroups, ranging from 77.5% to 92%. All subgroups that received explanations
had an increase in agreement with GPT predictions compared to the
corresponding No Explanations subgroup. Specifically, except for Labelled
Both, all subgroups showed proportional increases in both metrics, indicating
no clear bias for GPT predictions. However, the Labelled Both subgroup
demonstrated a significant increase in agreement with GPT predictions compared
to the proportion of sponsored posts, suggesting that the annotators changed
their decision-making process after having access to explanations. While this
result indicates a bias towards the model’s predictions, more experiments are
needed to determine its impact on data quality. Given the generally high
accuracy of GPT demonstrated in our classification experiments, relying on
them could improve annotation accuracy.

On the other hand, the difference in agreement with the predictions between
the Legal Experts subgroups adds uncertainty about the model’s accuracy. The
subgroup of legal experts with no explanations had the lowest agreement with
GPT predictions; in contrast, those with explanations had the highest. The
groups include different annotators, and Legal Experts No Explanations had low
inter-annotator agreement; therefore, we cannot effectively measure the
model’s accuracy. Although we found evidence of explanations biasing the
annotators, further research is needed to investigate how this result impacts
data quality.

Finally, we conducted a user-experience survey to gather feedback from
annotators on their experience using the explanations to assist with their
annotation process. All the responses are available online on
https://tinyurl.com/sponsored-annotation-survey. We ensured that the document
preserves the anonymity of all parties involved in the study.

The survey results showed that 87.5% of annotators felt more confident in
their decision-making with the help of explanations. Additionally, 62.5% rated
the explanations highly helpful and accurate (4 out of 5). Only one
participant rated them as unhelpful (2 out of 5). The average estimate of
agreement with the explanations was close to the agreement with GPT
predictions, with 62.5% of annotators estimating that they agreed with the
explanations between 80% and 100% of the time. Notably, all annotators
selected the words and phrases highlighted by the model explanations as a
helpful feature, while only 37.5% selected the reasoning behind the
predictions. This result indicates a preference for precise explanations.
Comparable explanations could be generated from any classifier using local-
explainability methods such as LIME [24]. This shows that the methodology
proposed and evaluated in our study does not rely on GPT’s capability of
generating longer text-based explanations and could be reproduced with simpler
models.

The open-ended questions revealed two clear trends among participants. First,
most participants found the highlighted words and phrases helpful in
identifying brands and context-relevant hashtags in the posts. Second,
participants suggested that adding the likelihood of a post being sponsored as
a feature would be a useful improvement to the explanations. Overall, these
results indicate that participants had a positive experience with the
explanations, found them helpful and accurate, and felt they improved their
decision-making.

##  5 Summary

Our experiments show that inter-annotator agreement metrics consistently
improve when augmenting the annotation process with explanations. We observed
a 15.65% increase in $\alpha$ and a 17.20% increase in absolute agreement
among the general population of annotators. The accuracy in detecting
disclosed sponsored posts improved by 3.45%, and the proportion of posts
labelled as sponsored increased by 9.46%. These findings indicate that
explanations not only help annotators identify more sponsored content but also
enhance the reliability of annotations and reduce variation between annotator
pairs. Our user-experience survey shows that most annotators found the
explanations helpful and accurate, increasing their trust in decision-making.
Therefore, our proposed annotation framework could lead to higher-quality data
labelling and improve decision-makers’ experience in regulatory compliance
contexts. We made theids of posts in our dataset, along with all the labels
annotated by annotators and the GPT predictions, publicly available
666https://github.com/thalesbertaglia/chatgpt-explanations-sponsored-content/,
offering a valuable resource that could benefit research in the field.

Nevertheless, our study has some limitations. One potential issue is the bias
introduced by explanations, as annotators may rely on specific cues presented
in the explanation. While we found no clear bias for most subgroups, we note
that the group that labelled posts in both settings showed a significant
increase in agreement with GPT predictions compared to the proportion of
sponsored posts. Another area for improvement is the small sample size of
legal experts and the variation in agreement metrics among different
subgroups, which may impact the generalisability of our results.

Future research should investigate the impact of explanations on annotator
bias and data quality and explore open-source models with greater
transparency, such as LLaMA [26], instead of OpenAI’s GPT – which is a
privately-owned model with limited information regarding its training data.
Moreover, conducting experiments with larger and more diverse samples of
annotators, including more legal experts, could shed light on the role of
expertise in the annotation process. Expanding the study to other annotation
tasks and domains would also provide insights into the generalisability of our
findings, potentially benefiting a broader range of applications.

Despite these limitations, it is important to consider that digital
enforcement and market monitoring by authorities such as consumer agencies
will exponentially grow in the coming years. Thus, monitoring techniques must
consider transparency and explainability to avoid accuracy issues when
applying legal sanctions.

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Passive quenching, signal shapes, and space charge effects
in SPADs and SiPMs

Oxford]P. Windischhofer
CERN]W. Riegler

[Oxford]University of Oxford

In this report we study the dynamics of passive quenching in a single-photon avalanche diode.
Our discussion is based on a microscopic description of the electron-hole avalanche coupled to the
equivalent circuit of the device, consisting of the quench resistor and the junction capacitance.
Analytic expressions for the resulting signal shape are derived from this model for simple electric
field configurations, and efficient numerical prescriptions are given for realistic
device geometries.
Space charge effects arising from the avalanche are included using simulations.
They are shown to distort the signal shape, but alter neither its basic characteristics nor the
underlying quenching mechanism.

§ INTRODUCTION

Single-photon avalanche diodes (SPADs) and silicon photomultipliers (SiPMs) are Geiger-mode avalanche detectors with
direct single-particle sensitivity to photons and charged particles [1, 2, 3].
In these devices, the small amount of charge deposited by the primary particle triggers a self-sustaining
electron-hole avalanche which leads to the creation of a detectable electric signal.
The avalanche develops in a thin region with very high electric fields, created in a p-n diode which is
initially reverse-biased above its breakdown voltage.

The resulting exponential growth of the avalanche must be stopped (“quenched”) at a suitable moment so that all
charge carriers are removed from the junction and the detector can register the next event.
In passively quenched devices, the diode is decoupled from the bias supply through a series resistance.
The charge produced by the avalanche then reduces the electric field in the junction and suppresses the avalanche
Quenching occurs when the voltage across the diode drops below the breakdown limit.
Once all charges have drifted out of the junction, the original bias voltage is restored.

The resulting voltage and current waveforms form the output signal.
The signal properties are ultimately determined by the complicated interplay of impact ionisation,
the quenching circuit, and the diode capacitance.
The operating conditions are typically chosen such that the achieved signal charge is around $10^4$–$10^5$ electrons,
i.e. very large avalanches must be considered.

Existing analyses of the signal formation in passively quenched SPADs either model the avalanche discharge and quenching
purely in terms of an equivalent electrical circuit [4, 5, 6], or use a coarse
effective description of the avalanche development, e.g. in terms of rate equations [7, 8].
In this report, we construct a simple analytic quenching model which is directly based on the microscopic dynamics of electron-hole
It naturally connects to our previous treatment of avalanche statistics given in Ref. [9].
As a result, our model describes the detector signal in terms of the underlying material parameters and the device
It can therefore guide the evaluation of specific detector designs and allows direct comparisons to be made with measurements.

This paper is structured as follows.
Section <ref> briefly reviews the avalanche formation and summarises properties
of large electron-hole avalanches used to formulate the quenching model.
Analytic predictions for the signal shape and the spatial distributions of charge carriers in the high-field region
are derived in Section <ref>.
The impact of the space charge field of the avalanche on the quenching process is then studied in
Section <ref>.

§ PROPERTIES OF LARGE ELECTRON-HOLE AVALANCHES

The development of the electron-hole avalanche is determined by the transport of charge carriers of both polarities
through the multiplication region and their interactions with the material.
For sufficiently strong electric fields, both electrons and holes can cause impact ionisation and produce additional
electron-hole pairs.
The probability for a charge carrier to undergo a multiplication reaction may be parameterised in terms of the material
parameters and, in general, the past history of the carrier.
The drifting carriers in the avalanche induce a current $\Iind$ on the metallised device contacts, which
may be computed with the Ramo-Shockley theorem [10, 11].

Impact ionisation is a stochastic process.
The avalanche development is thus subject to fluctuations, which are particularly pronounced at early times when the total
number of participating charge carriers is still small.
For avalanches containing many charge carriers, these fluctuations average out and the further evolution appears deterministic.
(The continued evolution of an avalanche containing $N$ charges will generate relative fluctuations with a magnitude proportional to
$1/\sqrt{N}$, as discussed in Section 3.5 of Ref. [9].)
It is useful to introduce a current scale $\Idet$ to separate these two regimes, chosen such that avalanche fluctuations
are important whenever $\Iind \ll \Idet$ and a deterministic description of the avalanche is appropriate when $\Iind \gtrsim \Idet$.
The current $\Idet$ therefore defines the level of fluctuations that are deemed “important” in a particular situation.
It implements a purely semantic definition, and expressions for physical observables do not depend explicitly on $\Idet$.

(Colour online.) The grey lines represent typical simulated avalanche events, each starting
from a single primary electron.
For large avalanches, the evolution of $\Iind$ in each event becomes proportional
to the average current $\avg{\Iind}$, parameterised by $I_0$ and $S_1$, as indicated by the red line.

Our model of the quenching process is based on the following two general properties of large electron-hole avalanches
($\Iind \gtrsim \Idet$) developing in a time-independent electric field, illustrated in Fig. <ref>:

* The shape of the induced current $\Iind$ is identical for each avalanche event.
It is directly proportional to the average current $\avg{\Iind}$, where the average is taken over all avalanche evolutions
starting from the same initial conditions,

(t) = k (t).

The proportionality factor $k$ is a random variable.
The resulting fluctuations around the average evolution determine the time resolution, but are irrelevant for the study of the quenching process.

* The average current $\avg{\Iind}$ evolves exponentially as

(t) = I_0 exp(S_1 t),

and is parameterised by the dimensionful quantities $I_0$ and $S_1$.
The current $I_0$ depends on the number and the spatial distribution of the primary charges initiating the avalanche.
The parameter $S_1$ represents the asymptotic net growth rate of the avalanche, i.e. it takes into account the generation of charges in
the high-field region and their outflow across its boundaries.
It depends on the device geometry and the reverse bias voltage $V_d$ applied across the junction, i.e. $S_1 = S_1(V_d)$.
In the following, the breakdown voltage $\Vbr$ is defined such that $S_1(\Vbr) = 0$.
This implies that $S_1(V_d) > 0$ for $V_d > \Vbr$, i.e. exponentially growing avalanches are possible if the
device is operated “above breakdown”.
For a bias voltage below the breakdown limit, $V_d < \Vbr$, impact ionisation cannot sustain the discharge.
The avalanche current then decays exponentially, i.e. $S_1(V_d) < 0$.

For a given device and electric field configuration, the parameters $S_1$ and $I_0$ are always accessible through Monte Carlo (MC)
simulations of the avalanche development (such as the one in Fig. <ref>); efficient numerical prescriptions
for quasi-one-dimensional situations are referenced below.
The above properties arise as explicit predictions of the avalanche model studied analytically in a one-dimensional geometry
in Ref. [9]. It is briefly reviewed in the following.
We use this model to illustrate our results in Sections <ref> and <ref>.
However, our description of the quenching process is more general and applies to all scenarios in which the avalanche
obeys the properties (i)–(ii) above, including realistic three-dimensional device geometries.

One-dimensional memoryless avalanche model

[]0.02      []0.02    []0.025

subfig:bounded_avalanche_region Electrons and holes drift in opposite directions with velocities
$v_e$ and $v_h$ in the electric field $\vE(x)$ of the p-n junction, inducing a current $\Iind$.
subfig:impact_ionisation Both charge carrier species can undergo impact ionisation and produce
additional electron-hole pairs, parameterised by the impact ionisation coefficients $\alpha$ and $\beta$.
subfig:quenching_circuit Equivalent circuit used for the analysis of the quenching process,
consisting of the diode capacitance $C_d$ and the quench resistor $R_q$.
The current source represents the induced current $\Iind$.

The diameter $D$ of a typical SPAD pixel is significantly larger than the longitudinal thickness $d$ of the high-field
region, typically $D / d \gtrsim 10$.
Close to the centre of a pixel, this results in a quasi-one-dimensional field geometry through which electrons and holes drift in opposite directions
with velocities $v_e$ and $v_h$, respectively.
The drift direction is taken to be along the $x$-axis (Fig. <ref>).

The one-dimensional model of Ref. [9] describes the avalanche development in this geometry.
In this model, impact ionisation is parameterised by the impact ionisation coefficients $\alpha$ and $\beta$,
which represent the probability per unit length for an electron or hole to undergo a multiplication reaction
(Fig. <ref>).
This probability is assumed to depend only on the local electric field strength $\vE(x)$, i.e. $\alpha(x) = \alpha(\vE(x))$
and $\beta(x) = \beta(\vE(x))$.
Any dependence on the history of the charge carriers is neglected, i.e. the avalanche is assumed to be a memoryless
stochastic process.

For general field profiles $\vE(x)$, the growth parameter $S_1$ may be obtained from the numerical solution of the
differential equations describing the evolution of the average charge content of the avalanche, Eqs. 30 and 31 in Ref. [9].
These equations also determine the spatial carrier densities in the junction.
As shown in Appendix B of Ref. [3], the breakdown condition $S_1 = 0$ may be expressed in terms of the
impact ionisation coefficients as

∫_0^d dx   α(x) exp[-∫_0^x dx'   (α(x') - β(x'))] = 1.

It is thus equivalent to the breakdown condition of Ref. [12],
which evaluates the point at which the gain for a constant current injected into the junction diverges.

For constant fields, $S_1$ may be expressed as the product of the effective carrier drift velocity
$v^* = 2 v_e v_h / (v_e + v_h)$ and the growth factor $\gamma_1$, i.e. $S_1 = \gamma_1 v^*$.
The latter is defined in Eq. 35 in Ref. [9].
The drift velocities for electrons and holes, $v_e$ and $v_h$, are themselves dependent on the local electric field.
The spatial carrier densities for electrons and holes, $n_e(x, t)$ and $n_h(x, t)$, are proportional to the functions $f_{\lambda_1}^e(x, t)$ and $f_{\lambda_1}^h(x, t)$, respectively.
Their analytic expressions are given in Eqs. 36 and 37 of Ref. [9].
The numerical parameter $\lambda_1$ is related to the time constant $S_1$ and defined in Eq. 38 of Ref. [9].

The proportionality factor $k$ in Eq. <ref> is approximately distributed according to a
gamma distribution with shape parameter $A$ and scale parameter $A^{-1}$ (cf. Section 3.6 of Ref. [9]),

p(k) dk = k^A-1 e^-k/A^-1/Γ(A)A^-A dk,

where $\Gamma(z)$ is the gamma function, and the avalanche parameter $A$ is defined as

A = αv_e N_e^0 + βv_h N_h^0/αv_e + βv_h

for position-independent electric fields.
In Eq. <ref>, $N_e^0$ and $N_h^0$ label the numbers of primary electrons and holes, respectively.
For general field profiles $\vE(x)$, the avalanche parameter may be computed as shown in Eq. 79 of Ref. [9].

§ PASSIVE QUENCHING IN THE ABSENCE OF SPACE CHARGE EFFECTS

To describe the quenching process, the avalanche evolution summarised in Section <ref>
is coupled to the electrical equivalent circuit of a single SPAD (Fig. <ref>).
In this circuit, the p-n junction is represented by its capacitance $C_d$, and the current $\Iind$ acts as
a parallel current source.
(Note that the dynamic resistance $R_d$ of the diode is automatically included through the dependence of the carrier
drift velocities $v_e$ and $v_h$ on the electric field and is thus not present as an explicit circuit component.)
In the absence of free charge carriers in the junction, $\Iind = 0$ and $V_d = \Vsupply$.
The excess voltage $\Vex$ is defined as $\Vex = \Vsupply - \Vbr > 0$, where the breakdown voltage $\Vbr$ is determined
by Eq. <ref>.

The time-dependent voltage $V_d(t)$ determines the electric field  $\vE(x,t) = \vE(x, V_d(t))$ which is relevant for the development of the avalanche.
To ensure quenching, the decoupling resistor $R_q$ must be chosen such that the recharging time constant $R_q C_d$
is much longer than the time scale $1 / S_1(\Vsupply)$ on which the avalanche develops.
The initial phase of the quenching process thus remains unchanged as $R_q\rightarrow\infty$.

In this dynamical situation, the instantaneous growth rate of the avalanche, $S_1(t)$, becomes itself time-dependent.
Provided that $V_d(t)$ changes on time scales that are long compared to the internal time constants of the avalanche,
the changing growth rate may be approximated as $S_1(t) \approx S_1(V_d(t))$, where $S_1(V_d)$ is the function introduced in
Section <ref> for static situations.
This “adiabatic approximation” is shown below to be applicable to practically relevant situations.
For $R_q\rightarrow \infty$ and $\Iind \gtrsim \Idet$, the combined deterministic system of avalanche and equivalent circuit is
then described by the equations

d /dt    =    S_1(V_d) ,

d V_d/dt    =    -/C_d.

More complicated equivalent circuits that include additional parasitic elements or finite values of the
quench resistance $R_q$ are readily accommodated using the same steps.
Eq. <ref>, which describes the evolution of the avalanche, must then be solved together with
the corresponding circuit equations.
For the situation discussed here, Eqs. <ref>–<ref> may be solved numerically for
general $S_1(V_d)$.
This is similar to the rate model constructed in Ref. [7], but in contrast to the calculations
performed there automatically includes boundary effects.

In many practically relevant situations, $\Vex / \Vbr \ll 1$, i.e. the voltage $V_d$ remains close to $\Vbr$ during the full
evolution of the system.
In this case, the function $S_1(V_d)$ in Eq. <ref> may be replaced by its linear expansion around
With the breakdown condition $S_1(\Vbr) = 0$, this gives

S_1(V_d) ≈d S_1/d V_d|_ ·(V_d - ) =: ·(V_d - ).

The parameter $\Sbr$ may be easily obtained through finite differences provided that $S_1(V_d)$ is known
around $V_d \approx \Vbr$.
An analytic expression for $\Sbr$ for a memoryless avalanche and position-independent fields is given in
Eq. <ref> below.

If the primary charge deposit occurs at $t=0$, Eqs. <ref>–<ref> must be solved for
$t \geq \tdet$, where $\tdet$ is the time at which the avalanche current first exceeds $\Idet$.
The time $\tdet$ is a random variable whose statistics are further discussed below.
Defining $V_d(\tdet) = \Vdet$, the relevant initial condition is $\Iind(\tdet) = \Idet$, leading to the following
analytic solutions,

(t)    =    C_d / [1 -

V_d(t)    =    - [tanh(t--/) +

= (-) √(1 + 2 /C_d (-)^2),

the quenching time constant $\tauQ$,

= 2/,

and the time $\deltaT$,

= (- /).

The quantity $\tauQ$ parameterises the time scale on which $V_d$ changes as the avalanche is quenched.
For Eqs. <ref>–<ref> to be a valid description of the quenching process,
i.e. for the adiabatic approximation to be applicable, $\tauQ$ must be larger than the transit time $d/v^*$.
The latter approximates the time scale on which the spatial charge distribution in the junction, and therefore also the
instantaneous growth rate $S_1(t)$, react to changes of $V_d$.

The general expressions in Eqs. <ref>–<ref> can be further simplified.
For practical devices, the diode capacitance $C_d$ typically amounts to a few femtofarads.
With the elementary charge $e_0$, around $C_d \Vex / e_0 \sim 10^4$ charge carriers need to be produced by the avalanche
in the high-field region before any significant discharge of $C_d$ occurs, i.e. any avalanche fluctuations remaining during
the quenching process amount to a few percent at most.
Working to this accuracy, $V_d(\tdet) = \Vdet \approx \Vsupply$ and $\Vdet - \Vbr \approx \Vex$; furthermore, $\Idet$ can be
selected so that $\Idet \ll C_d \Sbr (\Vdet - \Vbr)^2$.
In this limit, Eq. <ref> simplifies to $\Veff \approx \Vex$ and the quenching time constant is directly given in terms of
the excess voltage,

and the time $\Delta t$ defined in Eq. <ref> becomes

≈/2 log2 C_d ^2/ ≈1/S_1() log2 C_d ^2/.

With this, $\deltaT / \tauQ \gg 1$ and
Eqs. <ref>–<ref> simplify to

(t)    =    C_d / [1 - tanh^2(t--  /)],

V_d(t)    =    - [1 + tanh(t-- /)],

and the adiabatic approximation is valid provided that $\Sbr \Vex \lesssim 2 v^* / d$.
The current signal $\Iind$ attains a peak value of $\Iindmax = C_d \Vex / \tauQ$ at time $\tmax = \tdet + \deltaT$.
If $t=0$ is instead regarded as the position of the peak, the signal shape reads

(t)    =    C_d / [1 - tanh^2(t/)],

(t)    =    - [1 + tanh(t/)].

Its evolution is symmetric around this peak with a FWHM of

() = 2  1/√(2) ≈1.76  .

The observable voltage step $\Delta V_d = \Vsupply - \lim_{t\rightarrow\infty} V_d(t)$

ΔV_d = 2 ,

and $V_d$ falls from $\Vsupply - 0.1 \Delta V_d$ to $\Vsupply - 0.9 \Delta V_d$ within
about $2.2\,\tauQ$.
The total signal charge is

Q = ∫_-∞^∞ dt  (t) = 2 C_d .

The signal charge is thus directly proportional to the excess voltage at which the junction is operated and
does not depend on the primary charge deposit which initiated the avalanche.
The physical origin of the result in Eq. <ref> is easily understood.
As long as $V_d(t) > \Vbr$, impact ionisation is self-sustaining and the avalanche grows exponentially.
The avalanche, and the induced current $\Iind$, consequently attains its maximum size when $V_d(\tmax) = \Vbr = \Vsupply - \Vex$.
Impact ionisation continues to occur for $V_d(t) < \Vbr$, but is no longer self-sustaining.
The current $\Iind$ then decays exponentially, reducing $V_d$ below $\Vbr$ and increasing the output voltage
step $\Delta V_d$ beyond $\Vex$.
In situations where the approximation in Eq. <ref> is appropriate, the collapse of the
avalanche exactly mirrors its initial growth; the evolution of $V_d(t)$ is then symmetric around $\Vbr$ and
so $\Delta V_d = G\, \Vex$ with $G = 2$.
Note that this conclusion deviates from other commonly-used quenching models, which take
the avalanche discharge to stop already at $V_d(t) \approx \Vbr$ and thus lead to $G \approx 1$
[4, 5, 6].
These differences deserve a detailed comparison and discussion, which we carry out at the end of this Section.

The timing of the detector signal also carries important information.
The average time of maximum current,

=  + ,

is experimentally accessible in a setup where the primary charge deposit occurs at a well-defined time, e.g. where charges are
deposited by means of a laser pulse or a charged-particle beam.
It contains information about the early evolution of the avalanche (through $\avg{\tdet}$), as well as the quenching process itself
(through $\deltaT$).
Eqs. <ref> and <ref> result in the following expression for $\avg{\tdet}$,

= 1/S_1()(log/I_0 - logk),

which, together with Eq. <ref>, gives for $\avg{\tmax}$

= 1/S_1()(log2 C_d ^2/I_0 - logk).

The logarithmic expectation $\avg{\log k}$ depends on the magnitude of the avalanche fluctuations;
an analytic expression valid for the memoryless avalanche model is derived in Eq. <ref> below.

Quenching for one-dimensional memoryless avalanche

[]    []

subfig:gamma_field Dependence of the growth parameter $\gamma_1$ on the electric field and the thickness $d$,
for the case of silicon and the impact ionisation coefficients of Ref. [13].
The limit of an unbounded multiplication region ($d\rightarrow\infty$) is also shown.
subfig:gamma_voltage The parameter $\gamma_1$ is expressed in terms of the voltage $V_d$ close to
the breakdown point.

We illustrate these results for the case of a simplified SPAD, for which we approximate the field distribution in the
multiplication region with a position-independent, one-dimensional electric field.
The field is thus taken to be $\vE(x, V_d) = -\ex \, V_d / d$, where $\ex$ is the unit vector along $x$
(cf. Fig. <ref>).
In this case, the breakdown voltage is related to the breakdown field $\Ebr$ as $\Vbr = \Ebr d$.
The effective drift velocity $v^*$ typically saturates in the high-field region, i.e. becomes independent of $V_d$.
Using the memoryless avalanche model from Section <ref>, $S_1(V_d) = \gamma_1(V_d) v^*$ and

= v^*/dd γ_1/d E|_ =
(' + ')v^*/2 d +
(sinh2- 2) ('+ ' ) v^*/2(cosh2- 1 ) - d (sinh2- 2) (+ ),

where $\alphabr = \alpha(\Ebr)$, $\betabr = \beta(\Ebr)$, $\alphabr' = d\alpha/dE\rvert_{\Ebr}$, $\betabr' = d\beta/dE\rvert_{\Ebr}$, and
$\kappabarbr = d |\alphabr - \betabr| / 2$.
In the adiabatic approximation, predictions for the charge carrier densities of electrons and holes are proportional to the functions
$f^e_{\lambda_1(V_d)}$ and $f^h_{\lambda_1(V_d)}$.
The linear expansion in Eq. <ref> describes $\gamma_1(V_d)$ to excellent precision for typical
values of $\Vex$.
This is visualised in Fig. <ref> for silicon, using the parameterisations of the impact ionisation coefficients from
Ref. [13].

With the distribution of $k$ in Eq. <ref>, $\avg{\log k} = \psi(A) - \log(A)$ with the digamma function $\psi(z)$,
and the expression for $\avg{\tmax}$ in Eq. <ref> becomes

= 1/S_1()(log2 C_d ^2/I_0 + log(A) - ψ(A)),

and $I_0$ may be computed for a given initial charge deposit as shown in the paragraph following Eq. 46
in Ref. [9].

The analytic results derived above are compared to a MC implementation of the memoryless avalanche model,
coupled to the quenching circuit of Fig. <ref>.
The avalanche simulation is performed as described in the Appendix, using the impact ionisation coefficients
for silicon from Ref. [13] and saturated drift velocities $v_e=v_h=v^*=\micron{0.1}/\mathrm{ps}$ [14].
The circuit equations are integrated using the forward Euler method.
The induced current $\Iind$ is computed assuming the weighting field of a parallel-plate geometry with
an electrode spacing of $d$ for the multiplication region.
A dedicated conversion layer, common in SPADs optimised for infrared photon detection, is not included.
The full simulation is referred to as “memoryless avalanche MC + circuit” (MAMC + circuit).
For the comparisons shown here, the thickness $d$ of the high-field region is chosen as $d = \micron{0.5}$ and its diameter is $D = \micron{10}$.
Assuming a parallel field geometry and using $\eps_r = 11.7$ [15], the junction capacitance is $C_d\approx\fF{16}$.
A quench resistor with $R_q=\kohm{200}$ is used.
The breakdown voltage for this geometry is $\Vbr \approx 20.34$ V
and with Eq. <ref> the model parameter $\Sbr$ evaluates to $\Sbr \approx 0.105\,\left(\text{V ps}\right)^{-1}$.
The quenching time constant is $\tauQ \approx \ps{9.5}$ for $\Vex = 2$ V and $\tauQ \approx \ps{6.3}$ for
$\Vex = 3$ V.
The transit time $d/v^*$ is 5 ps, i.e. the adiabatic approximation is valid for excess voltages not
significantly above 3 V.

[]    []    (Colour online.) The grey lines represent the induced current $\Iind$ for several avalanche events simulated with the MAMC + circuit model
for the simplified silicon device described in the main text.
Each avalanche is initiated by a single primary electron placed at $x_0 = 0$.
The dashed lines compare the analytic expression for $\Iind$ from Eq. <ref> to one avalanche event,
for $\Vex = 2$ V in subfig:avalanche_sample_2V and $\Vex = 3$ V in subfig:avalanche_sample_3V.

[]  []
(Colour online.)
Comparison between the MC simulation (thick lines) and the adiabatic approximation
in Eqs. <ref> (dashed lines), for excess voltages of $\Vex=2$ V (orange) and $\Vex=3$ V (green).
The induced current $\Iind$ is shown in subfig:Iind_without_space_charge and the voltage
$V_d - \Vbr$ in subfig:Vd_without_space_charge.

Fig. <ref> shows the induced current $\Iind$ for several avalanche events and different excess voltages, obtained
with the MAMC + circuit simulation model.
Fluctuations in the early development of the avalanche lead to a stochastic displacement of the deterministic current waveform from
Eq. <ref>.
The simulated time of maximal current $\avg{\tmax}$ is $\ps{55.9}$ ($\ps{40.8}$) for an excess voltage $\Vex$ of $2$ V ($3$ V).
These values are well reproduced by Eq. <ref>, which evaluates to $\avg{\tmax} \approx \ps{56.1}$ ($\ps{40.9}$)
for $\Vex = 2$ V ($3$ V).

Evolution of the average charge carrier densities with time, for $\Vex=2$ V
in subfig:evolution_wo_space_charge_Vex_2_first–subfig:evolution_wo_space_charge_Vex_2_last
and $\Vex=3$ V
in subfig:evolution_wo_space_charge_Vex_3_first–subfig:evolution_wo_space_charge_Vex_3_last.
The thick lines show the densities obtained from the MAMC + circuit simulation, and the dashed lines correspond to the predictions from the
adiabatic model.
All parameters are identical to those used in Fig. <ref>.

Fig. <ref> compares the shapes of the approximate solutions in
Eqs. <ref>–<ref> to the result from the MAMC + circuit simulation
model around the time $\tmax$, demonstrating very good agreement between the analytic result and the simulation.
Small deviations arise as $V_d$ crosses the breakdown voltage, resulting in a slight underestimation of $\Delta V$ for large
excess voltages.
This is caused by additional dynamic effects originating from changes in the shapes of the spatial carrier densities
of electrons $n_e(x,t)$ and holes $n_h(x,t)$ that are not included in the adiabatic model.
The temporal evolution of these densities is compared in Fig. <ref>.
(For the simulation, the average densities $\avg{n_e}$ and $\avg{n_h}$ are shown to ensure a meaningful comparison with the
adiabatic model even at early times where the number of charge carriers is small.)
Charge carriers of both polarities are present throughout the entire multiplication region.
The highest densities are attained at the corresponding downstream ends, i.e. at $x=d$ for electrons and at $x=0$ for holes.
The shape of the distributions, and in particular the curvature of $n_h(x)$, depends on $V_d$.

Discussion and comparison with other quenching models

The relation $\Delta V_d = G\,\Vex$ with $G = 2$ is derived in Eq. <ref> as an exact analytic result that holds in
situations where the linear expansion in Eq. <ref> is appropriate.

Equivalent circuit used in Refs. [4, 5, 6] to describe a passively quenched SPAD,
consisting of the quench resistor $R_q$, the diode capacitance $C_d$ and the dynamic diode resistance $R_d$.
The switch $S$ controls the state of the avalanche.

This conclusion differs from predictions made with simpler, widely-used, quenching models [4, 5, 6].
In these models, the avalanche is taken to directly transition the device from its initial inactive configuration ($V_d = \Vsupply$ and $\Iind = 0$)
into a quasi-time-independent, active state where $V_d = \Vbr$ and a persistent current $I_p = \Vex / R_q$ flows through the quench resistor.
If $R_q$ is chosen sufficiently large, the current $I_p$ is small enough so that avalanche fluctuations may spontaneously quench
the discharge after a brief period of time (the probability per unit time for this to happen is exponentially suppressed in $I_p$ [6]).
This leads to a voltage step of $\Delta V_d = \Vex$, i.e. $G=1$.
The central proposition is that this behaviour can be modelled by the equivalent circuit shown in Fig. <ref>, where the
diode resistance $R_d$ is explicitly included and the avalanche is represented by a switch $S$ whose state marks the presence (closed) or absence (open)
of an avalanche discharge.

The microscopic analysis conducted here indeed confirms the existence of this active equilibrium state:
Eq. <ref> implies $d\Iind/dt = 0$ at $V_d = \Vbr$, which reveals $\Iind = I(R_q) = I_p$ as a valid configuration of
the circuit in Fig. <ref>.
This shows that the simple model in Fig. <ref> can adequately describe the possible active (switch closed) and inactive
(switch open) equilibrium configurations of the device.

There is, however, no a-priori reason to expect this passive circuit to correctly model the fast transition between these two states.
In particular, the avalanche current $\Iind$ in Fig. <ref> decays exponentially as $V_d$ approaches
$\Vbr$, and $V_d \geq \Vbr$ throughout the entire quenching process.
This is apparently inconsistent with the definition of $\Vbr$ as the voltage above which each existing carrier produces a diverging
number of secondary charges [6].
To the best of our knowledge, the existing literature does not address (or resolve) this contradiction for switch-based models.

The microscopic treatment reveals that it is not the absolute value of the avalanche current (or, equivalently, the total number
of charges in the avalanche) that vanishes close to the breakdown point, but rather its growth rate $S_1$, thus
giving rise to a different transient behaviour and a proportionality factor $G = 2$.
A similar factor $G \approx 2$ has also been obtained in Refs. [7, 8] through the numerical solution of a system of
rate equations describing the avalanche, and in Ref. [16] through a microscopic MC simulation similar to the one presented
Refs. [7, 8] furthermore report measurements of $\Delta V_d$ and find $G > 1$.

§ SPACE CHARGE EFFECTS

For sufficiently large avalanches, the space charge field $\vEsc(\vx, t)$ caused by the charge carriers in the avalanche
may become comparable to the field $\vEext(x, V_d(t))$ produced as a result of the externally applied voltage $V_d$.
The electric field profile $\vE$ that is relevant for the development of the avalanche is now the sum of both components,
i.e. $\vE = \vEext + \vEsc$.
A full description of the effects of space charge (SC) on the quenching process requires a three-dimensional simulation of the avalanche together with the computation
of a self-consistent solution for the space charge field at every time step.
In particular, $\vEsc$ depends on the spread of the avalanche in the direction transverse to the externally impressed field $\vEext$.

For the computation of the space charge field, the high-field region is taken to be delimited by two conducting plates of
infinite transverse extent, located at $x=0$ and $x=d$.
Charge carriers are constrained to move along the $x$-axis. This charge distribution is extended into a three-dimensional
charge density $\rho(r, x, t)$, illustrated here in the plane $x=x_0$.
The field $\vEsc(r, x, t)$ is evaluated on the $x$-axis as the electric field generated by $\rho(r, x, t)$.
The external field $\vEext(t)$ also contributes to the total electric field $\vE(x, t)$ relevant for the development
of the avalanche.

The situation is studied here in a simplified way as illustrated in Fig. <ref>.
The evolution of the avalanche continues to be simulated by the MAMC + circuit simulation model, with impact ionisation
resulting from the drift of charge carriers along the $x$-axis.
A parameterised model is used to describe the development of the avalanche in the transverse $yz$-plane, defining a radially symmetric
charge profile $\rho(r, x, t)$.
Different physical effects may be included in this parameterisation, e.g. charge carrier diffusion as well as drift due to
nonvanishing radial electric field components.
The prescribed charge distribution $\rho$ is then used to compute the space charge field $\vEsc$ along the $x$-axis,
which, by symmetry, only has a nonvanishing $x$-component, $\vEsc(r=0,x,t) = \Esc(x, t) \, \ex$.
This (one-dimensional) field distribution is used to determine the impact ionisation coefficients $\alpha(x,t)$ and $\beta(x,t)$.
The finite transverse extent of the avalanche is thus taken into account when computing its space charge field, but
it is neglected when evaluating the backaction of this field on the further evolution of the avalanche.
This is analogous to the “1.5-dimensional” model of space charge effects in resistive plate chambers (RPCs) discussed in
Ref. [17].
As before, the field $\vEext$ is assumed to be position-independent, i.e. $\vEext = -\ex \, V_d / d$.
The resulting simulation model is referred to as “MAMC + circuit + SC”.

For the results presented here, only transverse diffusion is considered for the construction of $\rho(r,x,t)$.
This simplified model is likely to overestimate the true effect of space charge on the development of the avalanche.
First, the space charge field $\vEsc$ itself is overestimated: the $x$-component of the field originating from the chosen
radially symmetric charge density $\rho(r,x,t)$ attains a maximum at $r=0$ (cf. Eqs. <ref> and <ref> below).
All drifting charge carriers are thus subject to the maximal field, while regions of lower field strengths at $r>0$ are not probed
by the avalanche.
Second, the transverse size of the avalanche is underestimated by only considering diffusion, but ignoring carrier drift.
This leads to a narrower distribution of charges and also enhances the computed space charge field.

Transverse spread of the avalanche and space charge field

Charge carrier diffusion is modelled by prescribing a normal distribution for the transverse distribution of charge carriers, while
the longitudinal distribution continues to be given by the densities $n_e(x, t)$ and $n_h(x, t)$.
The densities $\rho_e$ and $\rho_h$ that are relevant for the computation of the space charge field are thus

ρ_e(r, x, t)    =    -e_0 n_e(x, t)/2πσ_e^2(t) exp[-r^2/2σ_e^2(t)],

ρ_h(r, x, t)    =    e_0 n_h(x, t)/2πσ_h^2(t) exp[-r^2/2σ_h^2(t)].

The transverse extent is governed by the transverse diffusion coefficients $D_T$ for electrons and holes,
i.e. $\sigma_e(t) = \sqrt{2 D_{T,e} t}$ and $\sigma_h(t) = \sqrt{2 D_{T,h} t}$.
Values of $D_{T,e} = 15\,\mathrm{cm}^2/\mathrm{s}$ [18] and
$D_{T,h} = 11\,\mathrm{cm}^2/\mathrm{s}$ [19] are used in the following.
To the best of our knowledge, no measurements of the transverse diffusion coefficients are available in the literature for
the very strong longitudinal fields $|\vEext|>\Vum{20}$ that occur above breakdown.
The selected values correspond to a field of $\Vum{5}$ in silicon.
The transverse diffusion coefficients depend only relatively mildly on the longitudinal electric field,
and it is expected that these values give at least an approximate description of diffusion in the relevant regime.
This is sufficient for the qualitative assessment of space charge effects attempted here.

The electrostatic Green's function for the parallel-plate geometry of Fig. <ref>
derived in Refs. [20, 21] is used to compute the space charge field $\vEsc$.

Quenching dynamics with space charge

subfig:density_w_space_charge_first–subfig:density_w_space_charge_last Evolution of the
average charge carrier densities with time, when space charge effects are included, for $\Vex = 2$ V.
The electric field configurations corresponding to these time steps are shown in
The induced current $\Iind$ is shown in subfig:Iind_w_space_charge and the voltage
$V_d$ in subfig:Vd_w_space_charge.
In all plots, the thick lines correspond to results from the MAMC + circuit + SC simulation, and dashed lines
denote predictions from the adiabatic model.
All parameters are identical to those used in Fig. <ref>.

Fig. <ref> shows the evolution of the carrier densities, the electric field, and the
induced signal.
The same simulation parameters and material properties as in Section <ref> are used.
At early times, space charge effects are of subleading importance, and the resulting charge carrier densities are described by the adiabatic
model in good approximation.
The separation of charge caused by $\vEext$ leads to a space charge field which reduces the electric field in the centre of the gain
layer and enhances it close to its boundaries.
(Space charge cannot alter the voltage $V_d$ measured across the junction contacts, i.e. $\int_0^d d\vs \cdot \vEsc = 0$.)
This results in increased impact ionisation rates and carrier densities near the edges of the multiplication region.
For the chosen material parameters and device geometry, this effect overcompensates the suppressed avalanche growth in regions with
reduced field.
The signal charge and the voltage step $\Delta V_d$ increase by about 50% compared to the situation without space charge effects,
but the signal shape is not significantly altered.

This is very different from the space charge effects in RPCs studied in detail in Ref. [17].
Also in the RPC geometry the field integral throughout the sensitive volume remains unchanged so there will be regions with reduced
field and regions with increased field.
Since in a gas detector there is however only electron multiplication and no ion multiplication, the electrons end up only in these
low field regions and the gain is suppressed by many orders of magnitude.

Measurements of the voltage step $\Delta V_d$ in SPADs can thus serve as a direct probe of the importance of space charge effects
for the avalanche development.

§ CONCLUSIONS

In a passively quenched SPAD, the diverging avalanche discharges the detector capacitance, which in turn
lowers the electric field in the multiplication region and reduces the probability for impact ionisation.
The dynamics of this process is here described quantitatively based on a microscopic model of the avalanche
development, applicable as long as space charge effects are negligible.
The main conclusions of this treatment are:

* The duration of the avalanche discharge scales with the quenching time constant $\tauQ$ (cf. Eq. <ref>).
It is determined in terms of the excess voltage $\Vex$ and the parameter $\Sbr$ (cf. Eq. <ref>).
The latter may be extracted starting from the field profile $\vE(x)$ around the breakdown point.
It is known analytically for position-independent fields (cf. Eq. <ref>).

* The avalanche attains its maximum size as the voltage $V_d$ measured across the diode crosses the breakdown voltage.
The total voltage step $\Delta V_d$ caused by the discharge is therefore $\Delta V_d = 2 \Vex$.
The total signal charge is $Q = 2 \Vex C_d$, with the junction capacitance $C_d$ (cf. Eqs. <ref> and <ref>).

* Analytic expressions for the evolution of $V_d(t)$ and the current $\Iind(t)$ induced by the avalanche are available
in Eqs. <ref> and <ref>.
Analytic formulae for the spatial charge carrier densities also exist.

The space charge field produced by the charge carriers in the avalanche enhances the electric field
(and therefore impact ionisation) close to the boundaries of the multiplication region and reduces it in
its interior.
Charge carriers are present throughout the entire junction, and the avalanche growth may, in principle,
get enhanced or suppressed by the altered field configuration.
A simplified MC simulation of a representative silicon device geometry shows a modest enhancement
of the signal charge as a result of space charge.
This is in stark contrast to gas detectors such as RPCs, where space charge considerably reduces the gas gain.

Our results in Eqs. <ref> and <ref> deviate from the prevalent description of passive quenching, in which
the avalanche discharge is modelled as a bistable switch.
We encourage further experimental work to investigate and clarify this discrepancy.

§ APPENDIX

To simulate the development of the avalanche for equal carrier drift velocities, $v_e = v_h = v^*$, the avalanche region
$x \in [0, d]$ is partitioned into $N$ equidistant bins of width $\Delta x = d / N$.
The units are chosen such that charges drift a distance of $\Delta x$ during the simulation time step
$\Delta t = \Delta x / v^*$.
The bin centre of bin $j$ is located at coordinate $x^j$ and the simulation time steps are indexed as $t^i = i \Delta t$.

The electric field distribution $\vE(x, t) = -E^x(x, t)\, \ex$ and the resulting impact ionisation coefficients $\alpha(x, t)$
and $\beta(x, t)$ are discretised into $E^j(t^i) = E^x(x^j, t^i)$, $\alpha^j(t^i) = \alpha(x^j, t^i)$, and
$\beta^j(t^i) = \beta(x^j, t^i)$.
The charge carrier densities $n_e(x,t)$ and $n_h(x,t)$ are represented by the number of electrons $N_e^j(t^i)$ and holes $N_h^j(t^i)$
in each bin.
These carriers are assumed to be uniformly distributed within the bin.

[]0.5mm        []
(Colour online.)
Illustration of the update step $t^i\rightarrow t^{i+1}$ for the evolution of electrons in subfig:e_update_rule
and for the evolution of holes in subfig:h_update_rule.
The number of electrons $N_e^j$ and holes $N_h^j$ contained in a bin $j$ are represented by coloured bars within the respective bin.
A number of exemplary drift lines are shown to illustrate the movement of charge carriers.
Impact ionisation events are indicated by black dots.

In the absence of impact ionisation, the $\Nej$ holes present in bin $j$ at time $t^i$ drift into bin $j+1$ at $t^{i+1}$, and the
$\Nhj$ holes move into bin $j-1$.
Impact ionisation is simulated as illustrated in Fig. <ref>.
During the time interval $\Delta t$, the $\Nej$ electrons ($\Nhj$ holes) in bin $j$ produce $\Neehj$ ($\Nhehj$) additional
electron-hole pairs.
These numbers are drawn from a Poisson distribution with rate parameters $\alpha^j \Delta x$ and $\beta^j \Delta x$, respectively,

(t^i)    ∼    ((t^i); α^j(t^i)Δx),

(t^i)    ∼    ((t^i); β^j(t^i)Δx).

Impact ionisation occurs uniformly along $\Delta x$.
The $\Neehj$ electrons created in bin $j$ at $t^i$ drift into bin $j+1$ at $t^{i+1}$, while the $\Neehj$ holes spread across
multiple bins: $\NeehjL$ holes reach bin $j-1$, $\NeehjC$ remain in bin $j$, and $\NeehjR$ enter bin $j+1$.
The same consideration holds for the $\Nhehj$ electrons created from hole-initiated impact ionisation events, leading to
$\NhehjL$, $\NhehjC$, and $\NhehjR$ electrons in bins $j-1$, $j$, and $j+1$, respectively.
These numbers are drawn from a multinomial distribution with event probabilities of $p_{j\rightarrow j-1} = 0.25$,
$p_{j\rightarrow j} = 0.5$, and $p_{j\rightarrow j+1} = 0.25$, respectively,

{(t^i), (t^i), (t^i)}    ∼    ((t^i); {p_j→j-1, p_j→j, p_j→j+1}),

{(t^i), (t^i), (t^i)}    ∼    ((t^i); {p_j→j-1, p_j→j, p_j→j+1}).

The updated charge carrier distributions are then computed as

(t^i+1)    =    N_e^j-1(t^i) + N_e→eh^j-1(t^i) + N_h→eh^j+1→j(t^i) + N_h→eh^j→j(t^i) + N_h→eh^j-1→j(t^i),

(t^i+1)    =    N_h^j+1(t^i) + N_h→eh^j+1(t^i) + N_e→eh^j+1→j(t^i) + N_e→eh^j→j(t^i) + N_e→eh^j-1→j(t^i).

Assuming a uniform weighting field $E_w / V_w = 1 / d$, the current $\Iind(t^i)$ induced by the drifting charges is computed as

\begin{equation*}
\Iind(t^i) = \frac{e_0 v^*}{d} \sum_j\left(N_e^j(t^i) + N_h^j(t^i)\right),
\end{equation*}

where $e_0$ is the elementary charge.

In the continuum limit where $\alpha\cdot\Delta x  \rightarrow 0$ and $\beta\cdot\Delta x \rightarrow 0$, this simulation model converges to
the avalanche model described in Section <ref>.
For the results presented in the main body, the avalanche region is divided into $N = 500$ bins.

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