# Question Answering models for information extraction
from perovskite materials science literature

M. Sipilä1, F. Mehryary2, S. Pyysalo2, F. Ginter2 and M. Todorović1
<EMAIL_ADDRESS>1Department of Mechanical and Materials Engineering,
University of Turku, Finland
2Department of Computing, University of Turku, Finland

###### Abstract

Scientific text is a promising source of data in materials science, with
ongoing research into utilising textual data for materials discovery. In this
study, we developed and tested a novel approach to extract material-property
relationships from scientific publications using the Question Answering (QA)
method. QA performance was evaluated for information extraction of perovskite
bandgaps based on a human query. We observed considerable variation in results
with five different large language models fine-tuned for the QA task. Best
extraction accuracy was achieved with the QA MatBERT and F1-scores improved on
the current state-of-the-art. This work demonstrates the QA workflow and paves
the way towards further applications. The simplicity, versatility and accuracy
of the QA approach all point to its considerable potential for text-driven
discoveries in materials research.

††preprint: APS/123-QED

##  I Introduction

New sustainable technologies and materials are urgently needed, and materials
design plays a key role in their development. In inorganic materials,
compositional engineering provides an effective route for designing new
materials and tuning functional properties towards intended applications. One
group of materials where this is especially utilised is perovskites. They
exhibit a promising combination of properties for photovoltaic applications,
including high power conversion efficiencies [1, 2, 3], which would allow
sustainable energy generation [4, 5, 6]. The stability and functional
properties of perovskites critically depend on the material composition.

The crystal structure of perovskites is characterised by the generic formula
ABX3, were A and X sites are typically populated by cations and anions and a
metal occupies the B site. This allows a very broad range of element
substitutions, where organic molecules can also serve as A-site cations
(hybrid perovskites). An entirely new range of functional properties can be
accessed via substitutional engineering or alloying, where multiple element
substitutions can be implemented on A, B, or X sites. The past decade has seen
a proliferation of compositional engineering studies of perovskites to tune
their properties, which resulted in a large number of publications across
disciplines and research areas [7, 8, 9, 10]. There is a need to consolidate
known properties of perovskites across different disciplines. This could be
achieved by extracting perovskite information from scientific literature with
language processing tools.

Natural language processing (NLP) is a field that relies on computation and
machine learning to process, generate and analyse human language. In recent
years, considerable advancements in NLP were achieved by transformer neural
networks and large language models (LLMs) [11]. They exploit transfer
learning, where a model is first pre-trained with a large dataset to learn
general relationships in text, and then fine-tuned further for a certain
specific task. Within NLP, we focus on the common task of information
extraction (IE): finding information of interest from non-structured textual
sources automatically.

In materials science, NLP was used to extract information from publications
with named entity recognition (NER) and relation extraction (RE). The aim of
NER is to identify entities of interest, such as names or properties of
materials from text. With RE, the aim is to determine entities in text, but
also their relationships to each other. These tasks can be carried out by
supervised machine learning algorithms or rule-based methods. The supervised
machine learning NER approach was deployed to extract materials synthesis
parameters [12], polymer names [13], general materials information [14] and
doping procedures of materials [15] from literature. Similar models were
applied to RE to extract synthesis parameters [16]. For rule-based methods,
the state-of-the-art is the multi-purpose toolkit ChemDataExtractor2 (CDE2)
[17], which is capable of RE for materials science. It combines grammar-based
parsing rules with the probabilistic Snowball algorithm and was used to create
databases about battery materials [18], thermoelectric materials [19] and
semiconductor bandgaps [20] among others.

Figure 1:  An example of question ”What is the numerical value of property of
material X?’, snippet and the answer, containing a value and units (extracted
from [21])

Supervised learning methods in NLP require manual annotation of entities to
train the models. Analogously, rule-based methods require manual input to
construct the syntactic rules by which the information is extracted from text.
This presents difficulties in applying the method to different properties and
materials: expert knowledge and human effort are needed to retrain the model
for each new purpose. Moreover, models tend to focus on processing single
sentences [17, 22]. This leads to loss of information where the relation
between entities crosses sentence borders.

Here we explore the potential of powerful LLMs, for IE in materials science.
We use LLMs with the Question Answering (QA) approach. QA is based on the LLMs
capability to exploit previous training and identify information that it has
not specifically been trained to extract. Starting with a pretrained language
model, one can fine-tune it with the general structure of questions and
answers to produce a QA model. This tool can then be applied to different
domain extraction tasks without need of retraining, in an unsupervised manner,
which sets it apart from the task-specific approaches of supervised machine
learning or rule-based RE and NER. Trained QA models can return answers to a
natural language question (human query) from a context document of arbitrary
length, crossing sentence borders. It returns the text span that is most
likely the correct answer, otherwise it returns an empty string. While QA
models are promising, this technique has not been applied to RE in materials
science to date and it is unclear if the QA performance would be satisfactory.

In this study, our objectives were to build a QA framework for extracting
information from perovskite materials science literature, evaluate its
performance and apply it to a large textual dataset. To implement QA, we
considered model choices that range from selecting the question, materials,
properties of interest and language models, to deciding what kind of context
documents were most appropriate. The context documents are materials science
literature segments, referred as snippets throughout this manuscript. Snippets
are computationally more efficient to use than full-text publications, which
contain a lot of unnecessary text and complex context, and could possibly lead
to retrieving erroneous information.

The task was to extract material-property relationships as [material,
property, value, unit] with the question is ’What is the numerical value of
property of material X?’. A typical text snippet with the question and
extracted answer is illustrated in Figure 1. For the purposes of testing the
model, the property of interest was the bandgap. Bandgap greatly affects the
optoelectrical properties of perovskites, which is important for the solar
cell research community. To demonstrate the method, we focused on five
different perovskite materials: three hybrid (MAPI, MAPB and FAPI) and two
inorganic halide (CsPbI3 and CsPbBr3) perovskites. The narrow focus allowed us
keep the initial study compact, evaluate the effect of different model choices
and pave the way towards further applications.

A key question was which pre-trained language model to use as the basis of the
QA model, since the base model may have a considerable effect on the model
performance. We compared five different transformer models pre-trained with
different datasets: base BERT [11], SciBERT [23] and three BERT models trained
with materials science texts [15, 24, 25]. Each of the models were fine-tuned
towards QA by training with the general domain dataset SQuAD2 [26]. This
state-of-the-art QA training dataset contains also empty answers, which allow
the model to correctly detect cases where the snippet does not contain any
relevant values. Although SQuAD2 is not targeted towards scientific questions,
we relied on the generic capability of the QA models to answer to the
questions. The performance of QA systems with different BERTs was compared to
the baseline method CDE2. We set out to establish whether the QA frameworks
were capable of extracting material-property relationships accurately from
scientific texts, and how model performance compared to CDE2.

Figure 2: The workflow for implementing and testing the QA method for an IE
task.

##  II Methods

Figure 2 illustrates the general workflow of our implementation of the QA
method for IE. First, the textual data, i.e. corpus of scientific
publications, was downloaded via publisher application programming interfaces
(APIs) and web scraping. Next, this textual data, which was obtained in
multiple formats, was converted to plain text and duplicated text was removed.
After dividing the text into snippets, we selected a suitable QA model and
applied it to the selected snippets. Finally, the QA extracted values were
postprocessed, used to evaluate the model and analysed.

Table 1: File formats of publications from different sources, numbers of downloaded texts and conversion tools employed.  Source | Format | Number | Conversion tool
---|---|---|---
Elsevier Article Retrieval API | XML | 117,044 | CDE2 ElsevierXmlReader [17]
Springer Nature API | XML | 34,395 | Conversion script
Core API | Json | 65,924 | Text selected by json-tag
arXiV API | PDF | 3,814 | CDE2 Document package [17]
Royal Society of Chemistry | HTML | 17,254 | CDE2 RscHtmlReader [17]

###  II.1  Data acquisition

The first step in IE was to create a dataset of scientific publications. We
downloaded publications from 5 sources: arXiV API [27], Elsevier Article
Retriever API [28], Core API [29, 30], Springer Nature API [31] and also web
scraped the Royal Society of Chemistry (RSC) articles [32]. Permission to
download publications was granted through the university licenses, but
permission to scrape publications through the RSC webpage was obtained
directly from the RSC. We used the query ’perovskite’ or ’perovskites’ when
downloading and web scraping publications: if a publication matched a word, it
was added to the corpus.

We downloaded a total of 238,431 scientific publications from different
sources combined. The number of articles obtained from each source is
summarised in Table 1. In addition to full texts, we collected the following
metadata: year of publication, source and article identifier. In most cases,
the article identifier was the digital object identifier (DOI), but 1,161
publications obtained via the arXiV API lacked DOIs, so they were labelled by
the arXiV id. Similarly, 12,843 articles from the Core API lacked a DOI and
were labelled by the Core id.

###  II.2  Data processing into snippet text segments

Data processing involved three stages: converting data to plain text, removing
duplicated publications and formatting snippets. We converted the scientific
publications to plain text because language models cannot process PDFs, HTML
or XML tags. We also removed tables and figures from the texts, since these
cannot be processed as normal text and, at present, there are no satisfactory
methods for addressing these formats. Figure captions were preserved.
Converting different publication formats to plain text is a challenging task
because the formats of the publications vary, so we used multiple software
tools. Table 1 describes the original format of publications and summarises
the range of conversion tools employed. Here, 89 articles could not be
converted with the available tools and were omitted. After conversion, the
plain text dataset totalled 195,872 publications.

It is important that the dataset contains only unique publications, to avoid
redundancy of extracted values. By using DOI as a unique identifier for
publications, we identified and removed 1,505 duplicate manuscripts. The
articles without DOI were checked against the entire dataset: we converted all
plain text articles to numerical vectors of token counts and compared the
vector representations to each other. This allowed us to remove 45
publications without DOI, where the token count vectors were near-identical to
another publication. The conversion process and duplicate removal is described
in detail in section S1 of the Supplementary Information (SI). After removing
duplicates there were 194,322 unique scientific publications in the dataset.

The plain text articles were segmented into text snippets that contained
valuable information. To keep the IE efficient and accurate, the remainder of
the manuscript text was discarded at this stage. First, all the articles
bearing this information of value were identified. We used the Elasticsearch
search engine [33] to query articles by keywords. The keyword list contained
the name of the material of interest, the word ’bangap’, and their multiple
synonyms and acronyms (see Table S2 in SI). From the entire corpus, we
retained for further processing only the manuscripts which contained the
keywords for material name and property. For example, with the material MAPI
we selected 11,193 publications with bandgap information. This step is
specific to material name and property of interest and was repeated for each
material.

The snippet approach was adopted to purge the text of information irrelevant
to the task, such as introductory passages, discussions of other materials or
references. Based on preliminary experiments, we set the length of a snippet
to seven sentences. Section S2 in SI describes related information analysis.
While most of the information could be found within one sentence (40.4 % of
cases), a substantial portion of snippets had information spanning two (17.8
%), three (12.7 %) or more (29.1 %) sentences. The seven-sentence snippet thus
ensures more context than a single sentence and allows to extract also
information divided between multiple sentences.

To build the snippets, we had to ensure they contained at least one mention of
the material name (for example, ’MAPI’), the property name (’bandgap’) and the
unit name (’eV’). Because the relevant terminology varies, we normalised the
text by standardising pertinent nouns to one chosen form. Here, the
previously-constructed list of synonyms was used to convert all materials and
property names to e.g. ’MAPI’ and ’bandgap’. For each article in the corpus
subset, we extracted seven-sentence snippets which contained mention of
material, property and unit. The snippets were extracted by moving a seven-
sentence text window down the manuscript text and saving the snippets which
contained the necessary information (see SI section S2). For example, for MAPI
material we extracted 7,281 snippets from the 11,193 articles parsed.

Figure 3:  A schematic figure of transfer learning that underpins the QA model
for materials science.

###  II.3  QA model implementation

The state-of-the-art methodology in the field of NLP are transformer neural
networks based on self-attention [34]. This enables the language model to
learn relationships between words and the context of the word, regardless of
the text length. In this study, we opted for the encoder-only transformer
model, since it is intended for analysing textual data (decoder-only and
encoder-decoder models are designed for text generation).

Transformers rely on transfer learning to acquire new model knowledge and
adapt to more specific tasks. This approach allows models, which learned
useful features, context or patterns from a previous dataset, to retain
previous information and add to it with the further training. Figure 3
illustrates the process of training a base language model in stages with a
carefully selected set of domain-specific texts. It is common to start with
the generic encoder-only language model BERT (Bidirectional Encoder
Representations from Transformers), trained on general-domain English language
texts, such as English Wikipedia, books and text scraped from multiple web
sources. [11]. This model can be trained with scientific texts to generate a
science English language model, which then can be further trained with
materials science texts. In our approach, the final language model was further
trained towards question answering tasks to produce the QA model.

Different base models may have a profound impact on the performance of the QA
model. Since there is no literature precedent, we tested five different
language models: two general ones and three materials science BERTs. We tested
the base BERT model, trained with general-domain English language text [11],
and the scientific language model SciBERT, base BERT trained with scientific
text from the computer science and biomedical domains [23]. The materials
science models were MatBERT [15], MatSciBERT [25] and MaterialsBERT [24].
MatBERT was trained from base BERT with randomly sampled two million
documents, mostly consisting of peer-reviewed materials scientific journal
articles [15]. On the other hand, MatSciBERT was trained from SciBERT [23]
with 153,978 scientific publications about inorganic materials. MaterialsBERT
was trained based on PubMedBERT [35] with 2.4 million scientific abstracts
from the multiple materials science sub-domains.

We fine-tuned the above language models for QA task with the SQuAD2 [26].
Implementation was based on the Huggingface transformers-library [36] version
4.20.1 and Pytorch version 1.12 [37]. All computing was performed on the
NVIDIA Volta V100 GPU. The SQuAD2 features 107,785 question-answer pairs, as
well as 53,775 unanswerable questions, on 526 Wikipedia articles. Section S3
in SI summarises the hyperparameter training procedure. In applications, we
accounted for statistics by using 5 different seeds for weight initialisation
in the training process to build and test 5 QA models for each BERT. The
average values and standard deviation were reported as the final performance
metrics. The trained QA model, when provided with a question, generates a
number of answers with a model confidence score associated with each one. By
default only the top answer is retrieved since it is most likely to be correct
(as illustrated in Figure 1), but the workflow can be adjusted to retrieve
multiple answers. Should the correct value be missing in the text, we would
expect an empty string to be returned, based on SQuAD2 training.

###  II.4  Performance evaluation

Before applying the QA model to the complete dataset, we benchmarked the
performance of the different language models in the QA framework to identify
the most suitable BERT. For this, it was necessary to postprocess model
answers, evaluate suitable peformance metrics and visualise the retrieved
data.

In applied NLP, answers are typically postprocessed to standardise them and
facilitate the computation of performance metrics. In response to our query
prompt ’What is the numerical value of bandgap of material X?’, the model
could return a number of answers. We expected the correct answer to contain
the correct numerical value of the bandgap and the corresponding unit ’eV’,
but text answers, numerical ranges and conjunction words were also possible
and valid. Several postprocessing schemes were devised and tested to
standardise correct results (see section S4 of the SI for details). For
minimal processing and optimal outcomes, we omitted from consideration all the
answers without any digits and with letters outside the set ’e, v, t, o, a, n,
d’. The rest of the results were evaluated against gold standard answers to
compute metrics. After this, any values denoting a range were averaged
(extracted range ’1.5-1.6’ eV was recorded as 1.55 eV) to ensure that the
extracted values would not carry bias towards the ends of the range simply
because of rounding practices.

Human opinion sets the gold standard in NLP tasks. From the complete dataset,
we manually annotated a subset to establish the correct answers. The 600 text
snippets were selected to evenly balance bandgap information on the five
materials considered. Six materials science experts performed the annotation
(including empty answers) so that each snippet was reviewed by two persons.
The process is described in section S5 of the SI. After agreement was
achieved, 175 snippets were found to contain 209 numerical bandgap values, and
425 featured none. Out of 175 texts, the annotators identified 151 texts with
a single bandgap value, and 24 snippets with two or more (up to 5) values.

The annotated dataset was used to compute classification accuracy metrics for
the QA models. A true positive was achieved if the QA result was either
identical to the manually annotated value or overlapping with it. Here, vast
majority of overlapping values had minor differences, for example ’1.5 eV’ and
’$\sim$1.5 eV’. Similarly, a true negative denoted a correctly returned empty
text span. These metrics were employed to compute precision (the fraction of
correct answers from all retrieved answers), and recall (the fraction of
correct returned answers compared to all golden standard answers). We used
F1-score, the harmonic mean between precision and recall, to identify the best
QA BERT model. Results were compared against the well-established CDE2 code
(version 2.1.2) for IE, which was applied to the annotated dataset following
previously established procedures [20].

Since the QA model returns the text span with the highest confidence score
(most likely answer), evaluating performance on snippets with a single bandgap
value was straightforward. However, it was unclear how to approach the 24
snippets with up to 5 bandgap values. An easy solution would have been to
acknowledge any correct value returned, but this would not have allowed us to
extract all possible answers, which is desirable in applications. For a more
complete outcome, we shifted from considering just one result with the highest
confidence score to considering several. From the many prospective answers
returned by the model, we retrieved 6 with the highest confidence scores to
ensure that all bandgaps could be extracted from each snippet in the annotated
dataset.

At this stage, we introduced the QA confidence threshold as a model parameter.
A closer examination of the confidence scores associated with the text answers
revealed that model certainty varies considerably for different snippets, and
this should be taken into account (because less confidence points to a likely
wrong answer). We examined model performance with different confidence
thresholds applied to the top 6 answers: 0.0125, 0.025, 0.05, 0.1, 0.2, 0.4
and 0.8. Here, all of answers with the confidence score above the threshold
were retained and performance metrics were computed (up to a maximum of 6),
while the rest were discarded. Low thresholds would permit most answers at the
expense of accuracy, which increases recall but lowers precision. The opposite
is true for high thresholds, so considering the entire range is a helpful test
to establish which confidence threshold best balances precision with recall,
optimising F1-scores.

All extracted answers comprised a distribution of bandgap values for a
particular material. We performed a statistical analysis for the range, mean
and median values. When extracting bandgap values with CDE2, only values
corresponding to the set of five perovskites were selected, in order to
realiably compare the QA method to CDE2.

##  III Results

Figure 4:  The evaluation metrics of the answers returned by different
language models trained towards QA task by confidence score threshold: (a)
F1-score, (b) precision and (c) recall. The grey dashed line denotes the CDE2
performance with a 48.8 F1-score, precision at 81.1 and recall at 34.9.

The predictive power of QA for bandgap extraction was tested with respect to
model choices, namely different BERTs. We first address the model quality
tests with different QA confidence thresholds. Performance metrics (precision,
recall and F1-score) were evaluated on the annotated dataset for all five
BERTs and compared to CDE2 values. Next, all QA models were applied to
limited-scale IE to examine the effect of model choices on the range the
extracted values. Based on tests results, we selected optimal QA model and
applied to the complete corpus to demonstrate QA-based IE.

Table 2: Evaluation metrics calculated by the first returned answer or two first returned answers. The results represent mean values from five models trained with different seeds, with the standard deviation in parentheses. Analogous tables for thresholds are in SI section S6. QA language model | First answer |  |  | Two first answers |  |
---|---|---|---|---|---|---
| F1-score | Precision | Recall | F1-score | Precision | Recall
BERT | 45.8 (± 3.0) | 62.4 (± 6.8) | 36.9 (± 6.5) | 42.9 (± 1.2) | 31.9 (± 1.0) | 65.7 (± 1.5)
SciBERT | 60.5 (± 1.6) | 58.1 (± 2.5) | 63.2 (± 2.8) | 48.3 (± 0.2) | 34.1 (± 0.3) | 77.5 (± 1.9)
MaterialsBERT | 59.4 (± 5.6) | 64.4 (± 4.0) | 56.1 (± 9.9) | 50.0 (± 0.9) | 37.4 (± 0.7) | 75.5 (± 2.2)
MatSciBERT | 64.4 (± 6.8) | 72.4 (± 3.2) | 59.3 (± 12.3) | 49.9 (± 0.2) | 36.5 (± 0.5) | 79.0 (± 1.2)
MatBERT | 64.6 (± 2.0) | 61.8 (± 5.0) | 68.0 (± 1.5) | 49.5 (± 0.6) | 35.9 (± 0.5) | 79.5 (± 1.5)

###  III.1  Evaluating QA model performance

The predictive power of the QA model on the annotated dataset is illustrated
in Figure 4. Precision, recall and F1-score were computed for all 5 BERTs as a
function of confidence threshold, and compared to the CDE2 results. When the
threshold increases, model precision rises, but recall decreases. This is
expected behaviour, because high thresholds enforce model certainty, and
simultaneously reduces the number of answers returned. Overall, the general
BERT achieves the poorest outcomes due to fewest answers returned (inferior
recall). The performance of other BERT models is very similar, as summarised
by the best F1-scores in Tables S5 and S7 in SI.

The optimal F1-score at 66.2 was observed with QA MatSciBERT (at threshold
0.1), which can be associated with consistently high precision of this
language model (Figure 4b). In contrast, MatBERT produced consistently high
recall (Figure 4c). MaterialsBERT was the least effective of the three
materials language models and performed similarly to SciBERT on this task. We
note that CDE2 favours precision over recall (by construction) and thus did
not reach F1-scores beyond 48.8. MatSciBERT and MatBERT outperformed CDE2 on
precision only at very high tresholds, but at the cost of much degraded recall
and F1-scores. Throughout the different thresholds, MatSciBERT exhibited high
standard deviations, which is evident from the data uncertainties in Figure 4
b) and c). Elsewhere the standard deviations are smaller, pointing to a
reduced dependence on initial data in model training.

The threshold range test revealed that all language models achieve best
F1-scores at relatively low values, 0.1 or 0.2 in confidence (Tables S5 and S7
in SI). This unexpected finding was clarified once we calculated the average
confidence scores of the top answers (Table S8 in SI). While the average first
answer confidence score was 0.7, the average second answer confidence score
was as low as 0.06. This means that simply selecting the 1-2 top answers could
produce good results, without the need to consider confidence score
thresholds.

Figure 5: Histograms of extracted bandgap values with QA models based on BERT,
SciBERT, MaterialsBERT, MatSciBERT and MatBERT for a) MAPI and b) FAPI hybrid
perovskites. The y axis was modified by $\mathrm{log}_{10}$ for clearer
comparison.

To verify this, we calculated evaluation metrics for all 5 BERTs when only 1
or 2 top answers were selected, regardless of model confidence. The results
are presented in Table 2, and we compared them to Tables S5 and S7 in SI. It
is clear that selecting just one top answer (the recommended QA approach)
produces F1-scores that are very similar to the best threshold results, within
a few points difference. For SciBERT and MatBERT, working with the top answer
resulted in a better F1-score, but for the others the performance was slightly
degraded. Extracting the top 2 answers notably reduced F1-scores, on account
of lower precision. The relative performance of different BERTs was the same:
MatBERT and MatSciBERT produced best results, while BERT was the poorest.
However, MatSciBERT demonstrated very high standard deviation in recall
($\Delta$=12.3), compared to MatBERT recall standard deviation $\Delta$=1.5,
indicating lower robustness. The tests above led us to conclude that although
QA confidence thresholds allow to extract multiple answers from a single
snippet, working with top QA answers is far simpler and produces similar
results on this dataset.

The performance metrics indicated differences between the BERTs tested on the
annotated dataset (600 snippets). Still, it is unclear to what extent this
influences IE in real applications, with thousands of context documents. To
explore the scale of this effect, we applied the 5 language models to our full
dataset, and extracted the top answer only. We focused on the two materials
with the most and least snippets available: MAPI had extensive literature
coverage (7,283 snippets) while FAPI was least featured (1,251 snippets). Here
we expect both precision and recall to play a role in the number of extracted
values and their distribution. On average, it required 2.7 h and 0.5 h to
extract MAPI and FAPI values respectively on a single NVIDIA Volta V100 GPU.

The bandgap distributions presented in Figure 5 were subject to statistical
analysis for the number of extracted values (Table S9 in the SI). In the case
of MAPI, all models recorded the mode of 1.55 eV and median of 1.55 or 1.56
eV. Base BERT bandgaps were most diverse with a range of 0.85 eV, while
MatBERT returned a narrow range of 0.35 eV. A broad range suggests many
retrived values: QA BERT extracted 1,465 MAPI bandgap values, and QA
MaterialsBERT achieved the best retrieval with 1,997 results. For FAPI, we
observed similar trends but the performance differences were much attenuated.
QA BERT obtained the maximum of 395 bandgap results in contrast to QA MatBERT
with 341 values. This was reflected in their ranges of 0.46 eV and 0.21 eV
respectively. Differences in the distribution did not affect accuracy: the
mode (and median) bandgap of 1.48 eV for FAPI was obtained for all models.

Figure 6:  Histograms of bandgap values extracted with MatBERT: a) MAPI, b)
MAPB, c) FAPI, d) CsPbI3 and e) CsPbBr3.

While a large number of retrieved values is always desirable, an overly broad
range of extracted bandgaps may not indicate quality. Large bandgaps over 2 eV
identified for MAPI by BERT and MaterialsBERT are unlikely and indicate lack
of precision in extraction. MatSciBERT and MatBERT exhibit the best
performance in this test. Here, we additionally clarified if the extremes of
the distribution ranges contained correct bandgap values from text, or
erroneous answers. We manually selected 10 random QA MatBERT retrieved MAPI
bandgaps from the high range (above 1.7 eV) and low range (below 1.49 eV) and
inspected the original snippets. In all 10 examples the extracted values were
correct bandgaps, but in two cases MAPI had been doped by Br, which resulted
in a bandgap shift. The latter observation made clear that many scientific
factors may produce a genuine difference in bandgaps reported in literature.

Based on the performance evaluation tests, we opted for the MatBERT in all
subsequent IE. It exhibited the highest F1-score in the Table 2, while
retaining low standard deviation. Additionally, based on Figure 5 and SI Table
S9, MatBERT extracted much information (1,845 MAPI bandgaps) from the most
narrow numerical range, allowing less scope for error. The second best QA
results were observed with MatSciBERT, but the sensitivity to the initial
training set indicated a variable model performance.

###  III.2 Information extraction of perovskite bandgap values from the full
dataset

Table 3: Number of extracted snippets (Ns), MatBERT extracted values (Nv) and mode of extracted values after postprocessing and literature references.  Material | Ns | Nv | Mode (eV) | Ref.
---|---|---|---|---
MAPI | 7,283 | 1,846 | 1.55 | [38, 39, 40]
MAPB | 1,646 | 412 | 2.30 | [41, 42, 43]
FAPI | 1,251 | 349 | 1.48 | [44, 45, 46]
CsPbI3 | 1,734 | 684 | 1.73 | [47, 48, 49]
CsPbBr3 | 2,029 | 706 | 2.30 | [50, 51, 52]

We applied QA MatBERT to extract bandgap values for five perovskite materials:
three hybrid and two inorganic ones. The number of snippets (Ns) and number of
extracted values (Nv) are presented in the Table 3. The resulting
distributions are depicted in histograms in Figure 6. We first verified the
overall accuracy of extracted data. The mode of the extracted values agrees
well with the experimental values found from the literature sources (Table 3).
MAPI data featured higher peaks at 1.5 and 1.6 eV in Figure 6a) (MAPI), which
can be explained by the trend of researchers reporting values in two
significant digits.

The statistical bandgap distributions for different materials were compared in
the Figure 7. Statistical analysis can be found in Table S10 of the SI. Most
of the observed distributions featured a broad spread and data bias. Bias
towards lower values and lack of low outliers was observed for the hybrid
perovskites (MAPI and FAPI). The behaviour for inorganic perovskites was less
clear: CsPbI3 and CsPbBr3 bandgap data presented long tails towards high
bandgap values.

The ranges of extracted bandgap values could vary considerably. Overall, the
ranges for hybrid perovskite bangaps were smaller than those for the
inorganics. We compared distribution spreads to the number of extracted values
in Table 3 to establish if more values simply leads to broader distributions.
However, this was not the case, with MAPI registering both a large number of
extracted data and a narrow spread. As previously noted, there could be many
factors behind the spread of extracted values, and this may not be a concern
in database applications as long as the averages values are accurate.

##  IV Discussion

We had set out to explore the potential of QA models for IE in materials
science, given the task to extract perovskite bandgaps from literature. We
found the QA models straightforward to implement, which is an advantage. The
only model training required was with the general QA dataset, which was easy
and fast. They are also versatile, and could be directed towards specific
knowledge domains via different base BERTs.

The different tests demonstrated that QA models are capable of IE from
materials texts with good accuracy. Overall, retrieving only top confidence
answers presented the optimal way of using the models. With this dataset,
fractionally higher accuracies could be achieved by allowing multiple answers
via confidence thresholds. The same approach would allow to tune QA models to
emphasise either precision or recall, depending on the task at hand. Still,
the optimal threshold settings vary with BERTs and datasets and would require
additional tuning to each new task. This approach is recommend only in cases
where the text might be dense with relevant information (multiple values in a
sentence).

Figure 7:  Comparison of the bandgap distributions extracted for different
perovskite materials.

The choice of the QA language model was most important for the accuracy of
information retrieval. In our tests, base BERT was least optimal, most likely
because it had not been trained with domain literature in materials science.
Nevertheless, we note that QA BERT precision is comparable to that of other
models, meaning that the BERT-extracted bandgaps were just as likely to be
correct. We expected the three materials BERTs to perform the best, but here
we also observed differences. The F1-scores of QA MaterialsBERT were slightly
lower and it extracted a broader range of numerical values in application
tests, which could lead to erroneous answers. Meanwhile, the QA MatSciBERT
exhibited dependence on initial training data and larger uncertainties in
performance. Despite its high precision scores, such model sensitivity could
produce less reliable results. The QA MatBERT demonstrated the most
consistently high performance on this task.

The differences between QA models can be explained by the different training
sets of the base language models. MatBERT was trained with 2 million
publications from a broad materials science domain, which is much more text
than the 2.4 million materials science abstracts used to train the
MaterialsBERT. Also, abstracts can lack some concepts or words which full text
articles contain. MatSciBERT training set was smaller (153,978 publications),
but the underlying SciBERT may have produced the high F1-score of the model by
contributing the capability to process scientific text. Further tests are
needed to establish if the comparative BERT performance carries over to
different QA extraction tasks and datasets.

It was surprising to discover that the F1-scores of all five BERT language
models exceeded the CDE2 baseline. We note, however, that CDE2 was optimised
towards a different IE task: to retrieve all materials properties for all
materials found in texts. This may render it less sensitive to a particular
property and material, a task made specific by the textual query we used to
guide QA extraction. The performance gap could be due to another practical
difference: although CDE2 operated on entire snippets, it was designed to
extract information from single sentences only. Text search analysis revealed
that all necessary information was contained in one sentence for only 40.4 %
of the snippets. That might explain lower overall F1-scores observed for CDE2.
In precision of extracted values, CDE2 still outperformed the QA models. In
the future, the QA method could also be compared to the IE capability of newly
available open-source generative models (GPT).

An advantage of the QA method is that it can be adapted to different
extraction tasks. The QA models were not explicitly trained to extract the
value of ’bandgap’, but were capable of doing so with relatively high
accuracy. The QA models could now be used to retrieve also other properties
without need to retrain the transformers, given a suitable text corpus. This
presents an attractive alternative to more established machine learning
methods for IE, where a large amount of manually annotated text is typically
required to retrain models for each new task. The QA framework can effectively
be used as an unsupervised machine learning tool. From the end user point-of-
view, QA models require only the question prompt and the context documents,
which makes it more accessible to non-expert users.

Information extraction of materials properties inevitably results in a
numerical distribution. With unsupervised use of QA, it is important to
critically evaluate the results. The extracted values might deviate from the
median for multiple reasons: experimental measurement conditions may vary, old
references may contain incorrect measurement values, and computational results
differ from the experimental ones. Previous work [53] has established that the
computational bandgap of PbI2 varies from 1.4 eV to nearly 3.6 eV depending on
the computational approach. Since PbI2 forms part of MAPI and CsPbI3, the
computational bandgap results might vary considerably. A broad distribution
does not necessarily mean poor performance. Still, if the range is very wide,
there is no means to differentiate the origin of the values without
supervision, so it is recommended to favour QA models that produce a narrower
range. While our demonstrated application was limited in scope to five
materials, it indicates the predictive power of the QA approach and will serve
to guide further extraction work.

QA models seem to derive additional performance from the context found in
surrounding texts. This could be attributed to the latent knowledge conferred
by their LLM base models. While this is an advantage, working with snippets
can also present an application bottleneck. In this study, snippet generation
was specific to material name and property of interest (bandgap). This step
was designed to remove any irrelevant textual information, consequently a
search for any other materials property with the same snippet corpus would
have been inappropriate (and not as accurate). Applying QA models to other
extraction tasks may not require retraining the models, but it would require
generating new snippets from the full corpus. In future studies, snippet
generation with the same Elastic search approach could be automated to require
minimal human involvement. Additionally, several aspects of the workflow could
be parallelised for greater computational efficiency.

##  V Conclusions

We constructed a workflow to extract bandgaps from materials science articles
with the QA method, where a language model extracts information from the
context document based on the question asked. We collected a large textual
dataset of scientific publications on perovskites and trained five different
BERTs to perform the QA task. Models were compared on the task of extracting
materials bandgaps from the scientific literature. The MatBERT-based model
exhibited best performance on the manually annotated evaluation dataset and
the functional test, exceeding the performance of CDE2. This QA model was
applied to bandgap extraction from a database of 194,322 scientific articles
for five prototypical perovskites. The method was easy to apply and results
were accurate.

This study demonstrates the suitability of the QA method for extracting
material-property relationships from materials science literature. QA-based IE
was accurate and efficient with materials science literature snippets and
reached F1-score above the state-of-the-art. Moreover, QA allows extraction by
human language queries, which would lower the barriers for non-expert users
and promote diverse applications. Further tests are needed to explore the
performance of QA tools on different tasks. The versatility of the approach
indicates considerable potential of QA for NLP applications in the field of
materials science, which could be deployed to accelerate discovery and
materials design.

##  VI Data availability

The annotated dataset used for supervised evaluation is subject to
preparations for manuscript publication. The identifiers of publications in
the large 194,322 article dataset are available [54].

##  VII Code availability

All codes used in this study are publicly available [37].

##  VIII Author contributions

M.T., F.G., and S.P. conceived the original plan for the research and
supervised the work. M.S. performed all computational and analysis work and
drafted the manuscript. F.M. advised on the NLP training and application.
Everyone contributed to data analysis and writing the manuscript.

##  IX Competing interests

The authors declare no competing interests.

###### Acknowledgements.

The authors thank the CSC-IT Center for Science in Finland for high
performance computing resources. We acknowledge Mahboubeh Hadadian, Aleksi
Kamppinen, Christer Söderholm and Ransell D’Souza for participating in
evaluation dataset annotation and Emil Nuutinen for QA training scripts.
Research was funded by the Research Council of Finland through grant number
345698.

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# Achievable Rate Analysis of Intelligent Omni-Surface Assisted NOMA
Holographic MIMO Systems

Qingchao Li, Graduate Student Member, IEEE,
Mohammed El-Hajjar, Senior Member, IEEE,
Yanshi Sun, Member, IEEE,
Ibrahim Hemadeh, Senior Member, IEEE,
Yingming Tsai, Member, IEEE,
Arman Shojaeifard, Senior Member, IEEE,
and Lajos Hanzo, Life Fellow, IEEE (Corresponding author: Lajos Hanzo.)
Qingchao Li, Mohammed El-Hajjar and Lajos Hanzo are with the School of
Electronics and Computer Science, University of Southampton, Southampton SO17
1BJ, U.K. (e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>lh@ecs.soton.ac.uk). Yanshi Sun is with the School of Computer Science and
Information Engineering, Hefei University of Technology, Hefei, 230009, China.
(email:sys@hfut.edu.cn). Ibrahim Hemadeh, Yingming Tsai and Arman Shojaeifard
are with InterDigital, London EC2A 3QR, U.K. (e-mail:
<EMAIL_ADDRESS><EMAIL_ADDRESS>Arman.Shojaeifard@interdigital.com).

###### Abstract

An intelligent omni-surface (IOS) assisted holographic multiple-input and
multiple-output architecture is conceived for $360^{\circ}$ full-space
coverage at a low energy consumption. The theoretical ergodic rate lower bound
of our non-orthogonal multiple access (NOMA) scheme is derived based on the
moment matching approximation method, while considering the signal distortion
at transceivers imposed by hardware impairments (HWIs). Furthermore, the
asymptotically ergodic rate lower bound is derived both for an infinite number
of IOS elements and for continuous aperture surfaces. Both the theoretical
analysis and the simulation results show that the achievable rate of the NOMA
scheme is higher than that of its orthogonal multiple access counterpart.
Furthermore, owing to the HWIs at the transceivers, the achievable rate
saturates at high signal-to-noise ratio region, instead of reaching its
theoretical maximum.

###### Index Terms:

Holographic multiple-input and multiple-output, intelligent omni-surfaces,
non-orthogonal multiple access, hardware impairments.

##  I Introduction

Holographic multiple-input and multiple-output (HMIMO) systems are expected to
evolve towards an intelligent software reconfigurable paradigm in support of
improved spectral efficiency. They can be realized by harnessing a large
number antennas for constructing a spatially near-continuous aperture [1].
However, it is infeasible to realize HMIMO schemes relying a large number of
conventional radio frequency (RF) chains and antennas due to the excessive
power consumption.

In [2, 3], Deng et al. proposed a reconfigurable holographic surface (RHS),
consisting of multiple feeds and a large number of metamaterial based
radiation elements. They constructed a hybrid beamforming scheme, where the
digital beamformer and the holographic beamformer are employed at the base
station (BS) and the RHS, respectively. To be specific, the digital beamformer
relies on the state-of-the-art zero-forcing transmit precoding method, while
the holographic beamformer relies on the coefficient configuration of the
amplitude-controlled RHS radiation elements. The digital beamformer and the
holographic beamformer are alternatively optimized for maximizing the
achievable sum-rate. The simulation results showed that the RHS assisted
hybrid beamformer achieves higher sum-rate than the state-of-the-art massive
MIMO hybrid beamformer based on phase shift arrays. Furthermore, Hu et al. [4]
proposed a holographic beamformer based on amplitude-controlled RHS elements
having a finite resolution. They demonstrated that the holographic beamformer
associated with as few as 2-bit quantization approached the sum-rate
associated with unquantized values.

To further increase the energy efficiency, Zeng et al. in [5] conceived a
single-RF based HMIMO architecture, where a reconfigurable refractive surface
(RRS) illuminated by a single feed is employed at the BS for beamforming by
optimizing the coefficient of each RRS element. Both the theoretical analysis
and the simulation results demonstrate that at the same data rate as the
conventional MIMO systems relying on phased arrays, the RRS-based HMIMO
systems have higher energy efficiency. Nevertheless, in the RRS-assisted HMIMO
architecture, only the users in the $180^{\circ}$ half-plane can be supported.
In [6], [7], [8], the technique of intelligent omni-surface (IOS), also termed
as simultaneously transmitting and reflecting reconfigurable intelligent
surface, was introduced to realize $360^{\circ}$ full-space coverage.
Furthermore, it was demonstrated that the outage probability performance of
the IOS assisted non-orthogonal multiple access (NOMA) systems outperforms
that of the IOS assisted orthogonal multiple access (OMA) systems. However,
idealized perfect hardware quality was assumed at the RF-chains of
transceivers in [2], [3], [4], [5], [6], [7], [8], ignoring the signal
distortion resulting from practical hardware impairments (HWIs). To deal with
these issues, in this paper we propose an IOS assisted HMIMO architecture,
while considering the effect of HWIs, where the modulated signal can be both
reflected and refracted from the IOS, to realize $360^{\circ}$ full-space
coverage. Our contributions in this compact letter are presented as follows:

  * •

We conceive an IOS assisted HMIMO architecture having low energy consumption
and $360^{\circ}$ full-space coverage. The theoretical lower bound of the
achievable ergodic rate of our NOMA scheme relying on the popular successive
interference cancellation (SIC) based reception is derived based on the moment
matching approximation method. Additionally, the distortion of both the signal
emission at the transmitters and the signal reception at the receivers imposed
by the HWIs is also taken into account. Furthermore, the asymptotic ergodic
rate lower bound is derived both when the number of IOS elements tends to
infinity and for continuous aperture surfaces.

  * •

The theoretical analysis and simulation results show that the achievable rate
of our NOMA scheme is higher than that of its OMA counterpart in the IOS
assisted HMIMO systems. Furthermore, owing to the HWIs of transceivers, the
achievable rate saturates at high signal-noise ratios (SNR) instead of
reaching its theoretical maximum.

Notations: $\jmath=\sqrt{-1}$, $|a|$ represents the amplitude of the complex
scalar $a$, $\mathcal{CN}\left(\mu,\sigma^{2}\right)$ is a circularly
symmetric complex Gaussian random variable with the mean $\mu$ and the
variance $\sigma^{2}$, $\mathbb{E}\left[x\right]$ represents the mean of the
random variable $x$, $\mathop{\sum_{(n_{1},n_{2},\cdots,n_{q})=1}^{N}}$
represents
$\mathop{\sum_{n_{1}=1}^{N}\sum_{n_{2}=1}^{N}\cdots\sum_{n_{q}=1}^{N}}\limits_{n_{1}\neq
n_{2}\neq\cdots\neq n_{q}}$.

##  II System Model

The downlink of our holographic MIMO system is illustrated in Fig. 1, where an
intelligent omni-surface illuminated by a single RF-chain feed is used at the
BS for beamforming in support of two single-antenna user equipment (UE) at
different sides of the IOS. In contrast to [5], where a reconfigurable
refractive surface is employed at the BS and only the users at one side of the
surface can be supported, the intelligent omni-surface considered achieves
$360^{\circ}$ full-space coverage. We denote the user located within the
reflecting area as UE-1 and the user in the refracting area as UE-2.

###  II-A Intelligent Omni-Surface

As shown in Fig. 1, the $xoy$ plane coincides with the IOS and the origin is
located at the center of the IOS. We assume that a total of $N=N_{x}\times
N_{y}$ IOS elements are compactly placed in a uniform rectangular planar
array, with $N_{x}$ and $N_{y}$ elements in the $x$ axis direction and $y$
axis direction, respectively. Each IOS element has the size of $\delta_{x}$
and $\delta_{y}$ in the $x$ axis direction and $y$ axis direction,
respectively. To arrange for the intelligent surface to both reflect and
refract the impinging signal simultaneously, the energy splitting protocol of
[9] is employed. Specifically, each IOS element can split the total energy
between the reflection and refraction modes. The coefficient of the $n$th IOS
element in the reflection mode and refraction mode can be represented as
$\zeta_{1,n}=\beta_{1,n}\mathrm{e}^{\jmath\theta_{1,n}}$ and
$\zeta_{2,n}=\beta_{2,n}\mathrm{e}^{\jmath\theta_{2,n}}$, respectively, where
it must be satisfied that $\beta_{1,n}^{2}+\beta_{2,n}^{2}=1$. Referring to
[10], we can set $\beta_{1,1}=\beta_{1,2}=\cdots=\beta_{1,N}=\beta_{1}$ and
$\beta_{2,1}=\beta_{2,2}=\cdots=\beta_{2,N}=\beta_{2}$ for reasons of hardware
simplicity. Thus, for the $n$th IOS element we have:

$\displaystyle\zeta_{i,n}=\beta_{i}\mathrm{e}^{\jmath\theta_{i,n}},\quad
i\in\\{1,2\\},$ (1)

subject to $\beta_{1}^{2}+\beta_{2}^{2}=1$.

Figure 1: System model of the IOS assisted HMIMO architecture.

###  II-B Channel Model

####  II-B1 Channel link spanning from the feed to the IOS

Since the feed is near the IOS, the near-field channel model is employed for
the link spanning from the feed to the IOS. We assume that the feed is located
on the normal of the IOS through the origin with the distance of $d_{0}$.
Hence the coordinates of the feed is $\left(0,0,-d_{0}\right)$. Besides, we
denote the coordination of the center point of the $n$th IOS element as
$\left(x_{n},y_{n},0\right)$. Therefore, the distance between the feed and the
center point of the $n$th IOS element can be expressed as

$\displaystyle d_{n}=\sqrt{x_{n}^{2}+y_{n}^{2}+d_{0}^{2}},$ (2)

while the energy impinging on the $n$th IOS element can be represented as [5]

$\displaystyle\gamma_{n}=\iint_{\mathcal{D}_{n}}\frac{\left(\alpha+1\right)d_{0}^{\alpha+1}}{2\pi}\left(d_{0}^{2}+x^{2}+y^{2}\right)^{-\frac{\alpha+3}{2}}\mathrm{d}x\mathrm{d}y,$
(3)

where $2\left(\alpha+1\right)$ is the gain of the feed, and the integration
region is
$\mathcal{D}_{n}=\\{\left(x,y\right)\in\mathbb{R}^{2}:x_{n}-\frac{\delta_{x}}{2}\leq
x\leq x_{n}+\frac{\delta_{x}}{2},y_{n}-\frac{\delta_{y}}{2}\leq y\leq
y_{n}+\frac{\delta_{y}}{2}\\}$. According to (2) and (3), we can describe the
channel link spanning from the feed to the $n$th IOS element, denoted as
$g_{n}$, as

$\displaystyle
g_{n}=\sqrt{\gamma_{n}}\mathrm{e}^{-\jmath\frac{2\pi}{\lambda}d_{n}}=\sqrt{\gamma_{n}}\mathrm{e}^{-\jmath\frac{2\pi}{\lambda}\sqrt{x_{n}^{2}+y_{n}^{2}+d_{0}^{2}}},$
(4)

where $\lambda$ is the carrier wavelength.

####  II-B2 Channel link spanning from the IOS to UEs

We consider the far-field Nakagami-$m$ fading channel model for the channel
link spanning from the IOS to the UEs. The channel links spanning from the
$n$th IOS element to UE-1 and UE-2 are given by

$\displaystyle
h_{i,n}=\sqrt{\varrho_{i}}q_{i,n}\mathrm{e}^{\jmath\psi_{i,n}},\quad
i\in\\{1,2\\},$ (5)

where $\varrho_{i}$ represents the large-scale fading between the IOS to
UE-$i$, and $q_{i,n}\mathrm{e}^{\jmath\psi_{i,n}}$ is the small-scale fading.
Specifically, in the small-scale fading, the phase shift $\psi_{i,n}$ obeys
the identical uniform distribution within the support interval of $[0,2\pi)$.
The amplitude $q_{i,n}$ follows the Nakagami distribution having the
probability density function of
$f_{i}(x)=\frac{2m_{i}^{m_{i}}}{\Gamma\left(m_{i}\right)}x^{2m_{i}-1}\mathrm{e}^{-m_{i}x^{2}}$,
where $\Gamma\left(\cdot\right)$ represents the gamma function while $m_{i}$
is the shape parameter in the corresponding link.

####  II-B3 Equivalent channel from the feed to UEs

According to (1), (4) and (5), we can characterize the equivalent channel
spanning from the feed to UE-$i$ as
$h_{i}=\sum_{n=1}^{N}g_{n}\zeta_{i,n}h_{i,n}=\sqrt{\varrho_{i}}\beta_{i}\sum_{n=1}^{N}\sqrt{\gamma_{n}}q_{i,n}\mathrm{e}^{-\jmath\frac{2\pi}{\lambda}\sqrt{x_{n}^{2}+y_{n}^{2}+d_{0}^{2}}}\mathrm{e}^{\jmath\theta_{i,n}}\mathrm{e}^{\jmath\psi_{i,n}}$.
To maximize the channel gain of the equivalent channels, the phase shift of
the IOS element can be configured as
$\theta_{i,n}=\frac{2\pi}{\lambda}\sqrt{x_{n}^{2}+y_{n}^{2}+d_{0}^{2}}-\psi_{i,n}$.
Therefore, the equivalent channel between the feed and UE-1 as well as UE-2
are given by

$\displaystyle
h_{i}=\sqrt{\varrho_{i}}\beta_{i}\sum_{n=1}^{N}\sqrt{\gamma_{n}}q_{i,n},\quad
i\in\\{1,2\\}.$ (6)

###  II-C NOMA Design

Since the NOMA principle can serve additional far-field users by exploiting
the spatial beams precoded for the legacy near-field users [7], we focus our
attention on the NOMA architecture, where the BS uses superposition coding for
non-orthogonal multiplexing and the UEs employ the classic SIC algorithm for
recovering the information [11]. We denote the signal for UE-1 and UE-2 as
$s_{1}$ and $s_{2}$ respectively, and the transmit signal at the feed can be
formulated as $s=\sqrt{\kappa_{1}}s_{1}+\sqrt{\kappa_{2}}s_{2}$, where
$\kappa_{1}$ and $\kappa_{2}$ are the power allocation coefficients satisfying
$\kappa_{1}+\kappa_{2}=1$. Thus, the signals received by UE-1 and UE-2 are
[12]

$\displaystyle y_{1}=$
$\displaystyle\underbrace{\sqrt{\rho\kappa_{1}\varepsilon_{u^{(1)}}\varepsilon_{v}}h_{1}s_{1}}_{\text{Desired
signal of
$s_{1}$}}+\underbrace{\sqrt{\rho\kappa_{1}\varepsilon_{u^{(1)}}\left(1-\varepsilon_{v}\right)}h_{1}v_{1}}_{\text{BS
HWI distortion on $s_{1}$}}$
$\displaystyle+\underbrace{\sqrt{\rho\kappa_{1}\left(1-\varepsilon_{u^{(1)}}\right)}h_{1}u^{(1)}_{1}}_{\text{UE-1
HWI distortion on
$s_{1}$}}+\underbrace{\sqrt{\rho\kappa_{2}\varepsilon_{u^{(1)}}\varepsilon_{v}}h_{1}s_{2}}_{\text{Desired
signal of $s_{2}$}}$
$\displaystyle+\underbrace{\sqrt{\rho\kappa_{2}\varepsilon_{u^{(1)}}\left(1-\varepsilon_{v}\right)}h_{1}v_{2}}_{\text{BS
HWI distortion on
$s_{2}$}}+\underbrace{\sqrt{\rho\kappa_{2}\left(1-\varepsilon_{u^{(1)}}\right)}h_{1}u^{(1)}_{2}}_{\text{UE-1
HWI distortion on $s_{2}$}}$ $\displaystyle+\underbrace{w_{1}}_{\text{Additive
noise at UE-1}},$ (7)

and

$\displaystyle y_{2}=$
$\displaystyle\underbrace{\sqrt{\rho\kappa_{1}\varepsilon_{u^{(2)}}\varepsilon_{v}}h_{2}s_{1}}_{\text{Desired
signal of
$s_{1}$}}+\underbrace{\sqrt{\rho\kappa_{1}\varepsilon_{u^{(2)}}\left(1-\varepsilon_{v}\right)}h_{2}v_{1}}_{\text{BS
HWI distortion on $s_{1}$}}$
$\displaystyle+\underbrace{\sqrt{\rho\kappa_{1}\left(1-\varepsilon_{u^{(2)}}\right)}h_{2}u^{(2)}_{1}}_{\text{UE-2
HWI distortion on
$s_{1}$}}+\underbrace{\sqrt{\rho\kappa_{2}\varepsilon_{u^{(2)}}\varepsilon_{v}}h_{2}s_{2}}_{\text{Desired
signal of $s_{2}$}}$
$\displaystyle+\underbrace{\sqrt{\rho\kappa_{2}\varepsilon_{u^{(2)}}\left(1-\varepsilon_{v}\right)}h_{2}v_{2}}_{\text{BS
HWI distortion on
$s_{2}$}}+\underbrace{\sqrt{\rho\kappa_{2}\left(1-\varepsilon_{u^{(2)}}\right)}h_{2}u^{(2)}_{2}}_{\text{UE-2
HWI distortion on $s_{2}$}}$ $\displaystyle+\underbrace{w_{2}}_{\text{Additive
noise at UE-2}},$ (8)

where $\rho$ denotes the total transmit power at the BS. Furthermore,
$v_{1}\sim\mathcal{CN}\left(0,1\right)$,
$u^{(1)}_{1}\sim\mathcal{CN}\left(0,1\right)$ and
$u^{(2)}_{1}\sim\mathcal{CN}\left(0,1\right)$ are the distortion of the
information symbol $s_{1}$ due to hardware impairments, resulting from power
amplifier non-linearities, amplitude/phase imbalance in the In-
phase/Quadrature mixers, phase noise in the local oscillator, sampling jitter
and finite-resolution quantization in the analog-to-digital converters [13],
at the BS, UE-1 and UE-2, respectively. Similarly,
$v_{2}\sim\mathcal{CN}\left(0,1\right)$,
$u^{(1)}_{2}\sim\mathcal{CN}\left(0,1\right)$ and
$u^{(2)}_{2}\sim\mathcal{CN}\left(0,1\right)$ are the distortion of the
information symbol $s_{2}$ due to hardware impairments at the BS, UE-1 and
UE-2, respectively. Finally, $\varepsilon_{v}$, $\varepsilon_{u^{(1)}}$ and
$\varepsilon_{u^{(2)}}$ represent the hardware quality factor of the BS, UE-1
and UE-2, respectively, satisfying $0\leq\varepsilon_{v}\leq 1$,
$0\leq\varepsilon_{u^{(1)}}\leq 1$ and $0\leq\varepsilon_{u^{(2)}}\leq 1$.
Explicitly, a hardware quality factor of 1 indicates that the hardware is
perfectly ideal, while 0 means that the hardware is completely inadequate
[14]. Furthermore, $w_{1}\sim\mathcal{CN}\left(0,\sigma_{w_{1}}^{2}\right)$
and $w_{2}\sim\mathcal{CN}\left(0,\sigma_{w_{2}}^{2}\right)$ represent the
additive noise at UE-1 and UE-2, respectively.

At the user side, we employ the SIC algorithm for recovering information.
Without loss of generality, we assume that the channel gain of UE-1 is higher
than that of UE-2, i.e.
$\frac{\left|h_{1}\right|^{2}}{\sigma_{w_{1}}^{2}}>\frac{\left|h_{2}\right|^{2}}{\sigma_{w_{2}}^{2}}$.
In this case, UE-2 extracts its signal $s_{2}$ from the received composite
signal $y_{2}$ by regarding the signal of UE-1 as interference. By contrast,
UE-1 firstly detects the signal of $s_{2}$, and then subtracts its remodulated
version from the received composite signal $y_{1}$, so that UE-1 can detect
its own signal $s_{1}$ without interference from UE-2. Therefore, according to
(II-C) and (II-C), the achievable rate of UE-1 and UE-2 can be formulated as
$R_{1}=\log_{2}\big{(}1+\frac{\rho\kappa_{1}\varepsilon_{u^{(1)}}\varepsilon_{v}\left|h_{1}\right|^{2}}{\rho\kappa_{1}(1-\varepsilon_{u^{(1)}}\varepsilon_{v})\left|h_{1}\right|^{2}+\rho\kappa_{2}(1-\varepsilon_{u^{(2)}}\varepsilon_{v})\left|h_{1}\right|^{2}+\sigma_{w_{1}}^{2}}\big{)}$
and
$R_{2}=\log_{2}\left(1+\frac{\rho\kappa_{2}\varepsilon_{u^{(2)}}\varepsilon_{v}\left|h_{2}\right|^{2}}{\rho\kappa_{2}\left(1-\varepsilon_{u^{(2)}}\varepsilon_{v}\right)\left|h_{2}\right|^{2}+\rho\kappa_{1}\left|h_{2}\right|^{2}+\sigma_{w_{2}}^{2}}\right)$,
respectively.

For comparison, the achievable rate of UE-1 and UE-2 in the OMA protocol can
be expressed as
$R_{1}^{\prime}=\kappa_{1}^{\prime}\log_{2}\Big{(}1+\frac{\rho\varepsilon_{u^{(1)}}\varepsilon_{v}\left|h_{1}\right|^{2}}{\rho\left(1-\varepsilon_{u^{(1)}}\varepsilon_{v}\right)\left|h_{1}\right|^{2}+\sigma_{w_{1}}^{2}}\Big{)}$
and
$R_{2}^{\prime}=\kappa_{2}^{\prime}\log_{2}\Big{(}1+\frac{\rho\varepsilon_{u^{(2)}}\varepsilon_{v}\left|h_{2}\right|^{2}}{\rho\left(1-\varepsilon_{u^{(2)}}\varepsilon_{v}\right)\left|h_{2}\right|^{2}+\sigma_{w_{2}}^{2}}\Big{)}$,
respectively, where $\kappa_{1}^{\prime}$ and $\kappa_{2}^{\prime}$ are the
orthogonal time/frequency resource ratio allocated to UE-1 and UE-2
respectively, satisfying $\kappa_{1}^{\prime}+\kappa_{2}^{\prime}=1$.

##  III Performance Analysis

In this section, we theoretically derive the ergodic rate of the NOMA scheme.

The ergodic rate of UE-1 and UE-2 can be formulated as

$\displaystyle\mathbb{E}\left[R_{1}\right]=\mathbb{E}\Bigg{[}\log_{2}\Bigg{(}1+$
$\displaystyle\frac{\rho\kappa_{1}\varepsilon_{u^{(1)}}\varepsilon_{v}\left|h_{1}\right|^{2}}{\rho\kappa_{1}\left(1-\varepsilon_{u^{(1)}}\varepsilon_{v}\right)\left|h_{1}\right|^{2}+\rho\kappa_{2}\left(1-\varepsilon_{u^{(2)}}\varepsilon_{v}\right)\left|h_{1}\right|^{2}+\sigma_{w_{1}}^{2}}\Bigg{)}\Bigg{]}$
$\displaystyle=$
$\displaystyle\mathbb{E}\left[\log_{2}\left(1+\frac{\rho\kappa_{1}\varepsilon_{u^{(1)}}\varepsilon_{v}}{\rho\left(1-\kappa_{1}\varepsilon_{u^{(1)}}\varepsilon_{v}-\kappa_{2}\varepsilon_{u^{(2)}}\varepsilon_{v}\right)+\sigma_{w_{1}}^{2}\frac{1}{\left|h_{1}\right|^{2}}}\right)\right],$
(9)

and

$\displaystyle\mathbb{E}\left[R_{2}\right]$ $\displaystyle=$
$\displaystyle\mathbb{E}\Bigg{[}\log_{2}\Bigg{(}1+\frac{\rho\kappa_{2}\varepsilon_{u^{(2)}}\varepsilon_{v}\left|h_{2}\right|^{2}}{\rho\kappa_{2}\left(1-\varepsilon_{u^{(2)}}\varepsilon_{v}\right)\left|h_{2}\right|^{2}+\rho\kappa_{1}\left|h_{2}\right|^{2}+\sigma_{w_{2}}^{2}}\Bigg{)}\Bigg{]}$
$\displaystyle=$
$\displaystyle\mathbb{E}\left[\log_{2}\left(1+\frac{\rho\kappa_{2}\varepsilon_{u^{(2)}}\varepsilon_{v}}{\rho\left(1-\kappa_{2}\varepsilon_{u^{(2)}}\varepsilon_{v}\right)+\sigma_{w_{2}}^{2}\frac{1}{\left|h_{2}\right|^{2}}}\right)\right],$
(10)

respectively. Based on (III) and (III), as well as the Jensen’s inequality,
the lower bound of the achievable ergodic rate of UE-1 and UE-2, denoted as
$\overline{R}_{1}$ and $\overline{R}_{2}$ respectively, become

$\displaystyle\overline{R}_{1}=$ $\displaystyle\log_{2}\Bigg{(}1+$
$\displaystyle\frac{\rho\kappa_{1}\varepsilon_{u^{(1)}}\varepsilon_{v}}{\rho\left(1-\kappa_{1}\varepsilon_{u^{(1)}}\varepsilon_{v}-\kappa_{2}\varepsilon_{u^{(2)}}\varepsilon_{v}\right)+\sigma_{w_{1}}^{2}\mathbb{E}\left[\frac{1}{\left|h_{1}\right|^{2}}\right]}\Bigg{)},$
(11)

and

$\displaystyle\overline{R}_{2}=\log_{2}\left(1+\frac{\rho\kappa_{2}\varepsilon_{u^{(2)}}\varepsilon_{v}}{\rho\left(1-\kappa_{2}\varepsilon_{u^{(2)}}\varepsilon_{v}\right)+\sigma_{w_{2}}^{2}\mathbb{E}\left[\frac{1}{\left|h_{2}\right|^{2}}\right]}\right),$
(12)

respectively. Therefore, to theoretically derive $\overline{R}_{1}$ and
$\overline{R}_{2}$, our objective is to determine the values of
$\mathbb{E}\left[\frac{1}{\left|h_{1}\right|^{2}}\right]$ and
$\mathbb{E}\left[\frac{1}{\left|h_{2}\right|^{2}}\right]$, which can be
derived based on the moment matching approximation method as follows.

######  Lemma 1.

According to (6), the first and second moment of $\left|h_{i}\right|^{2}$,
denoted as $\mathbb{E}[\left|h_{i}\right|^{2}]$ and
$\mathbb{E}[\left|h_{i}\right|^{4}]$ respectively, are

$\displaystyle\mathbb{E}[\left|h_{i}\right|^{2}]=$
$\displaystyle\mathbb{E}\left[\left(\beta_{i}\sqrt{\varrho_{i}}\sum_{n=1}^{N}\sqrt{\gamma_{n}}q_{i,n}\right)^{2}\right]$
$\displaystyle=$
$\displaystyle\varrho_{i}\beta_{i}^{2}\left(\mu_{i,2}\sum_{n=1}^{N}\gamma_{n}+\mu_{i,1}^{2}\mathop{\sum_{(n_{1},n_{2})=1}^{N}}\sqrt{\gamma_{n_{1}}\gamma_{n_{2}}}\right)$
$\displaystyle=$
$\displaystyle\varrho_{i}\beta_{i}^{2}\Big{(}\mu_{i,2}A_{2}+\mu_{i,1}^{2}\left(A_{1}^{2}-A_{2}\right)\Big{)},$
(13)

and

$\displaystyle\mathbb{E}\left[\left|h_{i}\right|^{4}\right]=\mathbb{E}\left[\left(\beta_{i}\sqrt{\varrho_{i}}\sum_{n=1}^{N}\sqrt{\gamma_{n}}q_{i,n}\right)^{4}\right]$
$\displaystyle=$
$\displaystyle\varrho_{i}^{2}\beta_{i}^{4}\Bigg{(}\mu_{i,4}\sum_{n=1}^{N}\gamma_{n}^{2}+4\mu_{i,3}\mu_{i,1}\mathop{\sum_{(n_{1},n_{2})=1}^{N}}\gamma_{n_{1}}^{\frac{3}{2}}\gamma_{n_{2}}^{\frac{1}{2}}$
$\displaystyle+3\mu_{i,2}^{2}\mathop{\sum_{(n_{1},n_{2})=1}^{N}}\gamma_{n_{1}}\gamma_{n_{2}}+6\mu_{i,2}\mu_{i,1}^{2}\mathop{\sum_{(n_{1},n_{2},n_{3})=1}^{N}}\gamma_{n_{1}}\gamma_{n_{2}}^{\frac{1}{2}}\gamma_{n_{3}}^{\frac{1}{2}}$
$\displaystyle+\mu_{i,1}^{4}\mathop{\sum_{(n_{1},n_{2},n_{3},n_{4})=1}^{N}}\gamma_{n_{1}}^{\frac{1}{2}}\gamma_{n_{2}}^{\frac{1}{2}}\gamma_{n_{3}}^{\frac{1}{2}}\gamma_{n_{4}}^{\frac{1}{2}}\Bigg{)}$
$\displaystyle=$
$\displaystyle\varrho_{i}^{2}\beta_{i}^{4}\Big{(}\mu_{i,4}A_{4}+4\mu_{i,3}\mu_{i,1}\left(A_{3}A_{1}-A_{4}\right)+3\mu_{i,2}^{2}\left(A_{2}^{2}-A_{4}\right)$
$\displaystyle+6\mu_{i,2}\mu_{i,1}^{2}\left(A_{2}A_{1}^{2}-2A_{3}A_{1}-A_{2}^{2}+2A_{4}\right)+$
$\displaystyle\mu_{i,1}^{4}\left(A_{1}^{4}-6A_{2}A_{1}^{2}+3A_{2}^{2}+8A_{3}A_{1}-6A_{4}\right)\Big{)},$
(14)

respectively, where the $k$th ($k=1,2,3,4$) moment of $q_{i,n}$ is given by
$\mu_{i,1}=\mathbb{E}\left[q_{i,n}\right]=\frac{\Gamma\left(m_{i}+\frac{1}{2}\right)}{\Gamma(m_{i})}\left(\frac{1}{m_{i}}\right)^{\frac{1}{2}}$,
$\mu_{i,2}=\mathbb{E}\left[q_{i,n}^{2}\right]=1$,
$\mu_{i,3}=\mathbb{E}\left[q_{i,n}^{3}\right]=\frac{\Gamma\left(m_{i}+\frac{3}{2}\right)}{\Gamma(m_{i})}\left(\frac{1}{m_{i}}\right)^{\frac{3}{2}}$,
and $\mu_{i,4}=\mathbb{E}\left[q_{i,n}^{4}\right]=1+\frac{1}{m_{i}}$.
Furthermore, we have
$A_{1}=\left(\frac{\delta_{x}\delta_{y}}{\lambda^{2}}\right)^{-\frac{1}{2}}\iint_{\mathcal{D}}\omega^{\frac{1}{2}}\mathrm{d}x\mathrm{d}y$,
$A_{2}=\iint_{\mathcal{D}}\omega\mathrm{d}x\mathrm{d}y$,
$A_{3}=\left(\frac{\delta_{x}\delta_{y}}{\lambda^{2}}\right)^{\frac{1}{2}}\iint_{\mathcal{D}}\omega^{\frac{3}{2}}\mathrm{d}x\mathrm{d}y$
and
$A_{4}=\left(\frac{\delta_{x}\delta_{y}}{\lambda^{2}}\right)\iint_{\mathcal{D}}\omega^{2}\mathrm{d}x\mathrm{d}y$,
with
$\omega=\frac{\left(\alpha+1\right)d_{0}^{\alpha+1}}{2\pi}\left(d_{0}^{2}+x^{2}+y^{2}\right)^{-\frac{\alpha+3}{2}}$
and the integration region is given by
$\mathcal{D}=\sum_{n=1}^{N}\mathcal{D}_{n}=\\{\left(x,y\right)\in\mathbb{R}^{2}:-\frac{N_{x}}{2}\delta_{x}\leq
x\leq\frac{N_{x}}{2}\delta_{x},-\frac{N_{y}}{2}\delta_{y}\leq
y\leq\frac{N_{y}}{2}\delta_{y}$}.

###### Proof:

(1) and (1) can be derived based on the polynomial theorem and on the
independence of the channel links spanning from each IOS element to the UEs. ∎

Based on the moment matching approximation method, the random variable of
$\left|h_{i}\right|^{2}$ approximately follows the Gamma distribution having
the shape parameter
$\nu_{i}=\frac{(\mathbb{E}[\left|h_{i}\right|^{2}])^{2}}{\mathbb{E}[\left|h_{i}\right|^{4}]-(\mathbb{E}[\left|h_{i}\right|^{2}])^{2}}$
and the scale parameter
$\varsigma_{i}=\frac{\mathbb{E}[\left|h_{i}\right|^{4}]-(\mathbb{E}[\left|h_{i}\right|^{2}])^{2}}{\mathbb{E}[\left|h_{i}\right|^{2}]}$,
respectively. Therefore, $\frac{1}{\left|h_{i}\right|^{2}}$ follows the
inverse gamma distribution associated with the shape parameter
$\nu_{i}^{\prime}=\nu_{i}=\frac{(\mathbb{E}[\left|h_{i}\right|^{2}])^{2}}{\mathbb{E}[\left|h_{i}\right|^{4}]-(\mathbb{E}[\left|h_{i}\right|^{2}])^{2}}$
and the scale parameter
$\varsigma_{i}^{\prime}=\frac{1}{\varsigma_{i}}=\frac{\mathbb{E}[\left|h_{i}\right|^{2}]}{\mathbb{E}[\left|h_{i}\right|^{4}]-(\mathbb{E}[\left|h_{i}\right|^{2}])^{2}}$.
Thus, the means of $\frac{1}{\left|h_{i}\right|^{2}}$ are given by

$\displaystyle\mathbb{E}\left[\frac{1}{\left|h_{i}\right|^{2}}\right]=\frac{\varsigma_{i}^{\prime}}{\nu_{i}^{\prime}-1}=\frac{\mathbb{E}[\left|h_{i}\right|^{2}]}{2(\mathbb{E}[\left|h_{i}\right|^{2}])^{2}-\mathbb{E}[\left|h_{i}\right|^{4}]},$
(15)

with $\mathbb{E}\left[\left|h_{i}\right|^{2}\right]$ and
$\mathbb{E}\left[\left|h_{i}\right|^{4}\right]$ given in (1) and (1),
respectively. Furthermore, by substituting (15) into (III) and (12), the
theoretical values of $\overline{R}_{1}$ and $\overline{R}_{2}$ can be
formulated as

$\displaystyle\overline{R}_{1}=\log_{2}\left(1+\frac{\rho\varrho_{1}\beta_{1}^{2}\kappa_{1}\varepsilon_{u^{(1)}}\varepsilon_{v}}{\rho\varrho_{1}\beta_{1}^{2}\left(1-\kappa_{1}\varepsilon_{u^{(1)}}\varepsilon_{v}-\kappa_{2}\varepsilon_{u^{(2)}}\varepsilon_{v}\right)+\eta_{1}\sigma_{w_{1}}^{2}}\right),$
(16)

and

$\displaystyle\overline{R}_{2}=\log_{2}\left(1+\frac{\rho\varrho_{2}\beta_{2}^{2}\kappa_{2}\varepsilon_{u^{(2)}}\varepsilon_{v}}{\rho\varrho_{2}\beta_{2}^{2}\left(1-\kappa_{2}\varepsilon_{u^{(2)}}\varepsilon_{v}\right)+\eta_{2}\sigma_{w_{2}}^{2}}\right),$
(17)

respectively, where $\eta_{1}$ and $\eta_{2}$ are given by

$\displaystyle\eta_{i}=\frac{\iota_{i,1}^{2}A_{1}^{2}+\iota_{i,2}A_{2}}{\iota_{i,1}^{4}A_{1}^{4}-2\iota_{i,2}\iota_{i,1}^{2}A_{2}A_{1}^{2}-\iota_{i,2}^{2}A_{2}^{2}-4\iota_{i,3}\iota_{i,1}A_{3}A_{1}-\iota_{i,4}A_{4}},$
(18)

with $i\in\\{1,2\\}$ and $\iota_{i,1}=\mu_{i,1}$,
$\iota_{i,2}=\mu_{i,2}-\mu_{i,1}^{2}$,
$\iota_{i,3}=\mu_{i,3}-3\mu_{i,2}\mu_{i,1}+2\mu_{i,1}^{3}$,
$\iota_{i,4}=\mu_{i,4}-4\mu_{i,3}\mu_{i,1}-3\mu_{i,2}^{2}+12\mu_{i,2}\mu_{i,1}^{2}-6\mu_{i,1}^{4}$.

We derive the theoretical achievable rate’s lower bound, when the transmit
power tends to infinity as follows.

######  Theorem 1.

When the transmit power obeys $\rho\rightarrow\infty$, the theoretical lower
bound of UE-1 and UE-2, denoted as $\overline{R}_{1}^{(\rho=\infty)}$ and
$\overline{R}_{2}^{(\rho=\infty)}$, are given by

$\displaystyle\overline{R}_{1}^{(\rho\rightarrow\infty)}=\log_{2}\left(1+\frac{\kappa_{1}\varepsilon_{u^{(1)}}\varepsilon_{v}}{1-\kappa_{1}\varepsilon_{u^{(1)}}\varepsilon_{v}-\kappa_{2}\varepsilon_{u^{(2)}}\varepsilon_{v}}\right),$
(19)

and

$\displaystyle\overline{R}_{2}^{(\rho\rightarrow\infty)}=\log_{2}\left(1+\frac{\kappa_{2}\varepsilon_{u^{(2)}}\varepsilon_{v}}{1-\kappa_{2}\varepsilon_{u^{(2)}}\varepsilon_{v}}\right),$
(20)

respectively, which cannot increase boundlessly when $\rho\rightarrow\infty$
due to the limitation of HWI at the transceiver.

###### Proof:

(19) and (20) can be derived upon substituting $\rho\rightarrow\infty$ into
(16) and (17). ∎

Furthermore, we derive the high SNR slope, which is a key parameter
determining the scaling law of the ergodic rate in high SNR region [10], as
follows.

######  Remark 1.

According to (16), the high SNR slopes of UE-1 is
$S_{1}=\lim_{\rho\rightarrow\infty}\frac{\overline{R}_{1}}{\log_{2}\rho}=1$
when $\varepsilon_{u^{(1)}}=\varepsilon_{u^{(2)}}=\varepsilon_{v}=1$, and
$S_{1}=\lim_{\rho\rightarrow\infty}\frac{\overline{R}_{1}}{\log_{2}\rho}=0$
when $\varepsilon_{u^{(1)}}<1$ or $\varepsilon_{u^{(2)}}<1$ or
$\varepsilon_{v}<1$. By contrast, according to (17), the high SNR slopes of
UE-2 is
$S_{2}=\lim_{\rho\rightarrow\infty}\frac{\overline{R}_{1}}{\log_{2}\rho}=0$
for any values of $\varepsilon_{u^{(1)}}$, $\varepsilon_{u^{(2)}}$ and
$\varepsilon_{v}$. It means that the ergodic rate of the near-filed user, i.e.
UE-1, at the high-SNR region is just limited by the HWIs at the transceivers,
while that of the far-filed user, i.e. UE-2, at the high-SNR region is limited
by both the HWIs at the transceivers and the date rate of the near-filed user.

######  Theorem 2.

We evaluate the theoretically achievable rate’s lower bound in the case of
infinitely many IOS elements as follows. When the number of IOS elements tends
to infinity, the theoretical lower rate bound of UE-1 and UE-2, denoted as
$\overline{R}_{1}^{(N\rightarrow\infty)}$ and
$\overline{R}_{2}^{(N\rightarrow\infty)}$ respectively, tend to constant
values, given by

$\displaystyle\overline{R}_{1}^{(N\rightarrow\infty)}=\log_{2}\Bigg{(}1+$
$\displaystyle\frac{\rho\varrho_{1}\beta_{1}^{2}\kappa_{1}\varepsilon_{u^{(1)}}\varepsilon_{v}}{\rho\varrho_{1}\beta_{1}^{2}\left(1-\kappa_{1}\varepsilon_{u^{(1)}}\varepsilon_{v}-\kappa_{2}\varepsilon_{u^{(2)}}\varepsilon_{v}\right)+\eta_{1}^{(N\rightarrow\infty)}\sigma_{w_{1}}^{2}}\Bigg{)},$
(21)

and

$\displaystyle\overline{R}_{2}^{(N\rightarrow\infty)}=\log_{2}\Bigg{(}1+\frac{\rho\varrho_{2}\beta_{2}^{2}\kappa_{2}\varepsilon_{u^{(2)}}\varepsilon_{v}}{\rho\varrho_{2}\beta_{2}^{2}\left(1-\kappa_{2}\varepsilon_{u^{(2)}}\varepsilon_{v}\right)+\eta_{2}^{(N\rightarrow\infty)}\sigma_{w_{2}}^{2}}\Bigg{)},$
(22)

respectively, with $\eta_{1}^{(N\rightarrow\infty)}$ and
$\eta_{2}^{(N\rightarrow\infty)}$ given in (23).

$\displaystyle\eta_{i}^{(N\rightarrow\infty)}=\frac{\left(\frac{\delta_{x}\delta_{y}}{\lambda^{2}}\right)^{-1}8\pi\left(\alpha+1\right)\left(\alpha-1\right)^{2}d_{0}^{2}\iota_{i,1}^{2}+\iota_{i,2}}{\frac{64\pi^{2}\left(\alpha+1\right)^{2}\left(\alpha-1\right)^{4}d_{0}^{4}\iota_{i,1}^{4}}{\left(\frac{\delta_{x}\delta_{y}}{\lambda^{2}}\right)^{2}}-\frac{16\pi\left(\alpha+1\right)\left(\alpha-1\right)^{2}d_{0}^{2}\iota_{i,2}\iota_{i,1}^{2}}{\frac{\delta_{x}\delta_{y}}{\lambda^{2}}}-\iota_{i,2}^{2}-\frac{16\left(\alpha+1\right)^{2}\left(\alpha-1\right)}{3\left(\alpha+\frac{5}{3}\right)}\iota_{i,3}\iota_{i,1}-\frac{\left(\frac{\delta_{x}\delta_{y}}{\lambda^{2}}\right)\left(\alpha+1\right)^{2}}{4\pi\left(\alpha+2\right)d_{0}^{2}}\iota_{i,4}}$
(23)

###### Proof:

When the number of IOS elements tends infinity, $A_{1}$, $A_{2}$, $A_{3}$ and
$A_{4}$ can be formulated as

$\displaystyle A_{1}=$
$\displaystyle\left(\frac{\delta_{x}\delta_{y}}{\lambda^{2}}\right)^{-\frac{1}{2}}\int_{\infty}^{\infty}\int_{\infty}^{\infty}\omega^{\frac{1}{2}}\mathrm{d}x\mathrm{d}y$
$\displaystyle=$
$\displaystyle\left(\frac{\delta_{x}\delta_{y}}{\lambda^{2}}\right)^{-\frac{1}{2}}\sqrt{8\pi}\left(\alpha+1\right)^{\frac{1}{2}}\left(\alpha-1\right)d_{0},$
(24) $\displaystyle
A_{2}=\int_{\infty}^{\infty}\int_{\infty}^{\infty}\omega\mathrm{d}x\mathrm{d}y=1,$
(25) $\displaystyle A_{3}=$
$\displaystyle\left(\frac{\delta_{x}\delta_{y}}{\lambda^{2}}\right)^{\frac{1}{2}}\int_{\infty}^{\infty}\int_{\infty}^{\infty}\omega^{\frac{3}{2}}\mathrm{d}x\mathrm{d}y$
$\displaystyle=$
$\displaystyle\left(\frac{\delta_{x}\delta_{y}}{\lambda^{2}}\right)^{\frac{1}{2}}\sqrt{\frac{2}{9\pi}}\left(\alpha+1\right)^{\frac{3}{2}}\left(\alpha+\frac{5}{3}\right)^{-1}d_{0}^{-1},$
(26)

and

$\displaystyle A_{4}=$
$\displaystyle\left(\frac{\delta_{x}\delta_{y}}{\lambda^{2}}\right)\int_{\infty}^{\infty}\int_{\infty}^{\infty}\omega^{2}\mathrm{d}x\mathrm{d}y$
$\displaystyle=$
$\displaystyle\left(\frac{\delta_{x}\delta_{y}}{\lambda^{2}}\right)\frac{1}{4\pi}\left(\alpha+1\right)^{2}\left(\alpha+2\right)^{-1}d_{0}^{-2},$
(27)

respectively. Then $\eta_{i}^{(N\rightarrow\infty)}$ in (23) can be derived
upon substituting (III), (25), (III) and (III) into (18). ∎

######  Theorem 3.

For a continuous-aperture surface having $\delta_{x}\delta_{y}\ll\lambda^{2}$
and $N\rightarrow\infty$, the theoretical lower bounds of $\overline{R}_{1}$
and $\overline{R}_{2}$, denoted as
$\overline{R}_{1}^{(\delta_{x}\delta_{y}\ll\lambda^{2},N\rightarrow\infty)}$
and
$\overline{R}_{2}^{(\delta_{x}\delta_{y}\ll\lambda^{2},N\rightarrow\infty)}$,
are given by

$\displaystyle\overline{R}_{1}^{(\delta_{x}\delta_{y}\ll\lambda^{2},N\rightarrow\infty)}=\log_{2}\Bigg{(}1+$
$\displaystyle\frac{\rho\varrho_{1}\beta_{1}^{2}\kappa_{1}\varepsilon_{u^{(1)}}\varepsilon_{v}}{\rho\varrho_{1}\beta_{1}^{2}\left(1-\kappa_{1}\varepsilon_{u^{(1)}}\varepsilon_{v}-\kappa_{2}\varepsilon_{u^{(2)}}\varepsilon_{v}\right)+\frac{\left(\frac{\delta_{x}\delta_{y}}{\lambda^{2}}\right)\sigma_{w_{1}}^{2}}{8\pi\left(\alpha+1\right)\left(\alpha-1\right)^{2}d_{0}^{2}\mu_{1,1}^{2}}}\Bigg{)},$
(28)

and

$\displaystyle\overline{R}_{2}^{(\delta_{x}\delta_{y}\ll\lambda^{2},N\rightarrow\infty)}=\log_{2}\Bigg{(}1+$
$\displaystyle\frac{\rho\varrho_{2}\beta_{2}^{2}\kappa_{2}\varepsilon_{u^{(2)}}\varepsilon_{v}}{\rho\varrho_{2}\beta_{2}^{2}\left(1-\kappa_{2}\varepsilon_{u^{(2)}}\varepsilon_{v}\right)+\frac{\left(\frac{\delta_{x}\delta_{y}}{\lambda^{2}}\right)\sigma_{w_{2}}^{2}}{8\pi\left(\alpha+1\right)\left(\alpha-1\right)^{2}d_{0}^{2}\mu_{2,1}^{2}}}\Bigg{)},$
(29)

respectively.

###### Proof:

According to (23), we can show that when $\delta_{x}\delta_{y}\ll\lambda^{2}$,
the value of $\eta_{i}^{(N\rightarrow\infty)}$ tends to
$\frac{\frac{\delta_{x}\delta_{y}}{\lambda^{2}}}{8\pi\left(\alpha+1\right)\left(\alpha-1\right)^{2}d_{0}^{2}\iota_{i,1}^{2}}$,
and (3) and (3) can be derived by substituting them into (2) and (22),
respectively. ∎

Figure 2: Theoretical lower bound and simulation results of the achievable
ergodic rate comparison for UE-1 and UE-2 with different number of IOS
elements.

(a) $\varepsilon_{u^{(1)}}=\varepsilon_{u^{(2)}}=\varepsilon_{v}=1$

(b) $\varepsilon_{u^{(1)}}=\varepsilon_{u^{(2)}}=\varepsilon_{v}=1-10^{-4}$

(c) $\varepsilon_{u^{(1)}}=\varepsilon_{u^{(2)}}=\varepsilon_{v}=1-10^{-2}$

Figure 3: Theoretical analysis and simulation results of the geometric-mean
rate $R_{\mathrm{RM}}$ with different hardware quality factors.

##  IV Numerical and Simulation Results

In this section, the theoretical and simulation results of the ergodic rate of
the NOMA scheme are presented. They are also compared to those of the OMA
scheme. The simulation parameters are: $\lambda=0.3\mathrm{m}$,
$d_{0}=10\lambda$, $\alpha=2$, $\mu=1$. Furthermore, referring to [2], [3],
[4], the physical size of each IOS element equals one quarter wavelength, i.e.
$\delta_{x}=\delta_{y}=\frac{\lambda}{4}$. The equal power allocation for
reflecting and refracting the impinging signals on the IOS is employed [15],
i.e. $\beta_{1}^{2}=\beta_{2}^{2}=0.5$.

Both the theoretical lower bound (theo.) and the simulation results (simu.) of
the ergodic rate of UE-1 and UE-2 are shown in Fig. 2 for different number of
IOS elements in both the NOMA and OMA scheme. In this figure we have the ideal
hardware quality factors of
$\varepsilon_{u^{(1)}}=\varepsilon_{u^{(2)}}=\varepsilon_{v}=1$. The received
SNRs of UE-1 and UE-2 are set to
$\frac{\rho\varrho_{1}}{\sigma_{w,1}^{2}}=20\mathrm{dB}$ and
$\frac{\rho\varrho_{2}}{\sigma_{w,2}^{2}}=-20\mathrm{dB}$, respectively. The
achievable rate of UE-1 and UE-2 can be adjusted by controlling the parameters
$\kappa_{1}$ and $\kappa_{2}$ in the NOMA scheme and $\kappa_{1}^{\prime}$ and
$\kappa_{2}^{\prime}$ in the OMA scheme. Fig. 2 shows that the achievable rate
is improved upon increasing the number of IOS elements $N$. Furthermore, when
the number of IOS elements tends to infinity, observe in Fig. 2 that the
achievable rates of UE-1 and UE-2 saturate at a constant value instead of
increasing boundlessly due to the assumption of near-field condition for the
link spanning from the feed to the IOS. This agrees with (2) and (22).
Furthermore, the achievable rate of the NOMA scheme is higher than that of the
OMA scheme.

It is meaningful to explore maximizing the geometric-mean-rate
$R_{\mathrm{GM}}=\sqrt{R_{1}R_{2}}$, since it shows a substantially improved
rate-fairness amongst the users [16]. Fig. 3 shows the theoretical analysis
and simulation results of the geometric-mean rate $R_{\mathrm{GM}}$ for
different hardware quality factors. The power allocation coefficients
$\kappa_{1}$ and $\kappa_{2}$ in the NOMA scheme are optimized based on the
popular bisection method, while the geometric-mean-rate in the OMA scheme is
maximized when the orthogonal time/frequency resource ratio is
$\kappa_{1}^{\prime}=\kappa_{2}^{\prime}=0.5$. The channel parameters are
$\frac{\varrho_{1}}{\sigma_{w,1}^{2}}=40\mathrm{dB}$ and
$\frac{\varrho_{2}}{\sigma_{w,2}^{2}}=0\mathrm{dB}$. Fig. 3 shows that the
optimized geometric-mean-rate in the NOMA scheme is better than that in the
OMA scheme. When the hardware quality is ideal, i.e.
$\varepsilon_{u^{(1)}}=\varepsilon_{u^{(2)}}=\varepsilon_{v}=1$, the
geometric-mean-rate $R_{\mathrm{GM}}$ is improved without limit upon
increasing the transmit power $\rho$. By contrast, when the hardware quality
is non-ideal, the geometric-mean-rate $R_{\mathrm{GM}}$ becomes saturated due
to the HWIs, agrees with (19) and (20).

##  V Conclusions

We conceived a HMIMO architecture based on the IOS and theoretically analyzed
the ergodic rate of our NOMA scheme relying on the popular SIC detector. Both
the theoretical analysis and the simulation results show that the achievable
rate of the NOMA scheme is higher than that of the OMA scheme. Furthermore,
the hardware impairments at the transceiver limit the achievable rate in the
high-SNR region.

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