Optical materials discovery and design with federated databases and machine
learning

Victor Trinquet,∗a Matthew L. Evans,∗ab Cameron J. Hargreaves,a Pierre-Paul De
Breuck,‡a and Gian-Marco Rignanese a

Combinatorial and guided screening of materials space with density-functional
theory and related approaches has provided a wealth of hypothetical inorganic
materials, which are increasingly tabulated in open databases. The OPTIMADE
API is a standardised format for representing crystal structures, their
measured and computed properties, and the methods for querying and filtering
them from remote resources. Currently, the OPTIMADE federation spans over 20
data providers, rendering over 30 million structures accessible in this way,
many of which are novel and have only recently been suggested by machine
learning-based approaches. In this work, we outline our approach to non-
exhaustively screen this dynamic trove of structures for the next-generation
of optical materials. By applying MODNet, a neural network-based model for
property prediction, within a combined active learning and high-throughput
computation framework, we isolate particular structures and chemistries that
should be most fruitful for further theoretical calculations and for
experimental study as high-refractive-index materials. By making explicit use
of automated calculations, federated dataset curation and machine learning,
and by releasing these publicly, the workflows presented here can be
periodically re-assessed as new databases implement OPTIMADE, and new
hypothetical materials are suggested.

##

††footnotetext: a UCLouvain, Institut de la Matiere Condensée et des
Nanosciences (IMCN), Chemin des Étoiles 8, Louvain-la-Neuve 1348,
Belgium††footnotetext: b Matgenix SRL, 185 Rue Armand Bury, 6534 Gozée,
Belgium††footnotetext: † Electronic Supplementary Information (ESI) available:
The structure and properties of the materials selected by the active learning
loops for consideration with density-functional perturbation theory have been
deposited to the Materials Cloud Archive
(10.24435/materialscloud:5p-vq).††footnotetext: $\ast$ These authors
contributed equally to this work.††footnotetext: ${\ddagger}$ Present address:
Ruhr-Universität Bochum, Universitätsstr. 150, 44801 Bochum, Germany

##  1 Introduction

The advent of robust quantum mechanical calculations of material properties
has expanded the opportunities for computational materials design. It is now
relatively commonplace for a single materials design study to consider vast
swathes of the space of known inorganic compounds ($10^{5}$ entries), as
curated in experimental 1, 2 and computational 3, 4 databases. In recent years
a large number of hypothetical materials have been suggested to be stable (for
some definition) by data-driven and machine-learning approaches 5, 6, 7, 8.
These hypothetical materials can eventually make their way into curated
databases, but often are released as static datasets that are hard to discover
programmatically. This new space is too large to study exhaustively, and the
feasibility of said hypothetical materials requires significant attention 9.
Data-driven screening methods must therefore be adapted to be able to
rationale the most efficient allocation of experimental resources within this
dynamic, growing, decentralised design space, especially when targeting
specific material properties.

In this work, we devise a framework to search for materials with strong linear
optical response, i.e., high-refractive-index (high-$n$) materials. These are
sought after for their application in waveguides, interference filters,
mirrors, sensors, and anti-reflective coatings for solar cells 10, 11, 12, 13,
14, 15, 16. Nonlinear optical materials are typically used in signal
processing, wavelength conversion, and quantum optics, but are more difficult
to find 17, 18, 19. As the linear optical response has been shown to be an
indicator of non-linear (higher-order) optical response 20, 21, high-$n$
semiconductors may be used as the starting point in the search for more exotic
nonlinear optical properties.

For materials with the desired linear optical response, the electronic band
gap, $E_{g}$, should be as wide as possible in order to guarantee transparency
over the spectral range of interest, which will also increase the range of
operating temperatures and voltages that a device can withstand 22. However,
the empirical trend is that the band gap is inversely proportional to the
refractive index 23, 10. These conflicting physical properties define a multi-
objective Pareto front which must be optimised 24. To overcome this challenge,
this work combines machine learning property predictions with high-throughput
screening of hypothetical materials databases to identify novel optical
materials that best optimise this trade-off in different regions of the
spectral range.

Previous high-throughput studies have used first-principles simulations 25, 10
to predict the refractive index of thousands of semiconductors present in the
Materials Project (MP) 3. The inverse trend linking $n$ and $E_{g}$ was
observed in both studies, and modeled by Naccarato et al. 10 with a two-
variable parametric relationship derived from properties of the electronic
structure. Their main aim was to provide public datasets that could
subsequently be screened to find promising high-$n$ compounds. Here, we build
on the dataset of 4,040 calculations provided in Naccarato et al. 10
(henceforth N19). The dataset and workflows of Petousis et al. 25 have been
integrated with the MP and now span $\sim$5,600 materials, here referred to as
P17 (with some overlap with N19).

As first-principles calculations are computationally demanding, optimising the
set of candidate materials that are studied, without lowering the hit-rate of
high-$n$ materials, is desirable. To this end, two recent studies have used
inexpensive machine-learned proxies for the refractive index to pre-screen
compounds.

Carriço et al. 26 trained a graph neural network on P17, to predict the
refractive index of 55,000 materials contained in the Alexandria database 27,
5. Materials whose predictions fell above the $(n,E_{g})$ Pareto front of the
N19 dataset were selected for validation. The refractive index was then
computed by first principles, and added to the training set for a second round
of this process. This approach successfully identified 2,431 compounds in the
target region of the $n$-$E_{g}$ distribution (henceforth C24).

Riebesell et al. 28 instead focused on high-$k$ dielectric properties, which
additionally consider the ionic contribution to the dielectric tensor. Using
P17, they trained two deep ensembles of independent Wren models 29, to predict
the ionic and electronic components of the dielectric tensor for all materials
in the MP and the WBM dataset of Wang et al. 30. An additional 100,000 trial
structures were generated by chemical substitution for screening with the
trained models. All the candidate materials were then ranked according to a
figure of merit (FoM) for miniming current leakage. Finally, first-principles
calculations were performed to compute the dielectric constant of the top-
ranked candidates, yielding 2,691 hypothetical materials with computed
dielectric tensors (referred to from now on as R24).

The present work follows a similar methodology to the above studies. The aim
is to search for high-$n$ semiconductors by combining a machine-learned proxy
for the refractive index and first-principles calculations as validation of
promising candidates. This study differs by applying an extended active
learning (AL) scheme, in which the process of "training-predicting-selecting-
computing" is repeated multiple times, in contrast to the one or two
iterations that Riebesell et al. 28 and Carriço et al. 26 respectively
performed. Our candidate pool generation also leverages the possibilities
offered by OPTIMADE 31 to simplify the querying of curated databases and their
integration into the present workflow thanks to their standardised format. The
30+ databases with OPTIMADE APIs provide a design space of over 27 million
crystal structures which could be selected 32. Such a large, uncurated and
decentralised set of structures presents unique opportunities and challenges
for screening studies.

In this paper, we first introduce the computational workflow for computing the
static refractive index, the implementation of our active learning procedure
and the feedstock datasets used as the candidate pool. As the result of
several active learning campaigns, we computed the static refractive index of
around 2,000 materials. Our selected materials were then screened against
several additional criteria important for the realistic application of these
materials. Finally, we present the details of 21 of the most promising
high-$n$ materials uncovered by our procedure and find that many have already
been realised experimentally outside of the context of high-$n$ materials.

##  2 Methods

###  2.1 Refractive index workflow

In order to train a surrogate model via active learning, an “oracle” is
required to define the ground truth property value for a given material so
that the training dataset can grow over the AL iterations. In this work, the
electronic contribution to the dielectric tensor is computed in the framework
of density-functional perturbation theory (DFPT) 33, 34, 35, as implemented in
the Vienna Ab-Initio Simulation Package 36, 37, 38, 39 (VASP version 6.3.0).
Following Petousis et al. 25, the static refractive index, $n_{s}$ or simply
$n$, is given by:

$n_{s}=\sqrt{\varepsilon^{\infty}_{\text{poly}}}=\sqrt{\frac{\lambda_{1}+\lambda_{2}+\lambda_{3}}{3}},$
(1)

where $\varepsilon^{\infty}_{\text{poly}}$ is the average of $\lambda_{i}$,
the eigenvalues of the electronic contribution to the dielectric tensor.

The exchange-correlation energy is modeled using the Perdew-Burke-Ernzerhof
generalised-gradient approximation (GGA-PBE) 40, 41 with Projector Augmented
Wave pseudopotentials 42, 43, 44 (PBE_64). The self-interaction energy is
corrected with Hubbard $U$ values 45 recommended by the MP 46. The plane wave
energy cut-off was set to $680\text{\,}\mathrm{eV}$ and the $k$-point grid was
automatically generated with a density of 1,500 $k$-points per reciprocal
atom.

These parameters balance efficiency with accuracy, keeping the relative error
with respect to fully converged DFPT calculations below 10%, as reported in
Petousis et al. 47. This was confirmed on our dataset by computing $n_{s}$
with $k$-point grid densities of 1,500 and 3,000 on a representative set of
176 structures (with $\leq$ 5 atoms); the relative error on our high-$n$
biased dataset was observed to be much lower, less than 0.2%, expanding the
number of trials we could perform with a given compute budget.

The generation of inputs, the $k$-point grids, and the retrieval of outputs
for the computation of the dielectric constants were automated with the
DielectricMaker class of the _atomate2_ Python package 48, 49. It is important
to note that the crystal structures were not re-relaxed in this work, as all
candidates were drawn from previously relaxed PBE calculations.

###  2.2 Active learning

In our AL loop, the MODNet architecture 50 is used for the surrogate model.
MODNet consists of a relatively simple feed-forward neural network, with a
relevance-redundancy-based feature selection algorithm applied to the broad
set of featurisers implemented by the _matminer_ library 51. On the Matbench
benchmark suite 52, MODNet has been shown to excel on datasets with fewer than
10k entries, and retains fast training and inference times on community grade
hardware 53. By contrast, state-of-the-art graph models 54, 55, 56 can require
more data to attain sufficient accuracy, although transfer learning from more
exhaustive datasets has shown promise 57, 58, 6. Other approaches targeted at
small data, such as SISSO 59, excel at interpretability over performance by
constraining the inferred relationships via symbolic regression.

The EnsembleMODNet provides epistemic model uncertainty through ensemble
variance which is an essential factor when carrying forward active learning on
a predicted search space, as it enables a balance between exploration and
exploitation. Full exploration means that candidates with the highest
uncertainty are chosen for oracle evaluation at each step of the AL loop, to
be added to the training set in the following iteration. Under this regime,
the hope is that the ML model will improve whilst maximising data efficiency,
as it is successively trained on the instances it was least certain about.
Full exploitation means that at each AL iteration, the candidates with the
“best” predictions are chosen for validation. This means that computing
resources are exclusively reserved for materials deemed promising by the
model, but could overlook under-explored promising regions.

Our objective is to identify top candidates that exhibit both a high
refractive index, $n$, and a high band gap, $E_{g}$. Thus, we aim to discover
new materials that extend the Pareto frontier in the $(n,E_{g})$ space, with
the most efficient use of computational resources. To achieve this, we employ
a strategy that combines cost-effective machine learning predictions with a
more resource-intensive density-functional perturbation theory (DFPT) oracle.
A secondary aim is to devise a framework that can be iteratively reapplied in
a manner akin to online learning, as new hypothetical materials are suggested
in the literature and served through decentralised databases implementing
OPTIMADE APIs 32, with the aim of accelerating the suggestion of candidate
materials for specific applications, and their subsequent experimental
verification.

A pool-based active learning strategy is used, meaning that we sample from a
discrete domain of fixed materials, $\mathcal{P}$, spanned by the publicly
available OPTIMADE APIs that serve stability and band gap calculations. At
each learning cycle $i$, a MODNet ensemble is trained on training set $T_{i}$
in order to provide an estimate
$p_{i}(n|x)\sim\mathcal{N}(\mu_{n,i},\sigma_{n,i})$ for a material $x$ with
mean ensemble model prediction $\mu_{n,i}(x)$ and uncertainty
$\sigma_{n,i}(x)$. From this, an upper bound to the refractive index, $n_{U}$,
is computed:

$n_{U,i}(x)=\mu_{n,i}(x)+\lambda\sigma_{n,i}(x),\quad\lambda\geq 0,$ (2)

where $\lambda$ is a free parameter controlling the emphasis between
exploration ($\lambda\to\infty$) and exploitation ($\lambda\to 0$), which can
be tuned throughout the AL loop. Following the methodology of Riebesell et al.
28, a compromise is achieved by incorporating both the model prediction and
its uncertainty in the acquisition function used at each step ($\lambda=1$).

Following the derivation of Naccarato et al. 10, we recast the multi-objective
Pareto front optimisation as the maximisation of a semi-empirical effective
frequency $\omega_{\text{eff}}$:

$\omega_{\text{eff}}(n,E_{g})=\left(n^{2}-1\right)^{\frac{1}{3}}\left(E_{g}+6.74-\frac{1.19}{E_{g}}\right)$
(3)

Although originally derived using only the direct band gap, here we
approximate it with the fundamental gap for ease of comparison across multiple
datasets. for The following acquisition function is then used to select
materials for oracle evaluation (with e.g., DFPT):

$\alpha_{i}(x)=\omega_{\text{eff}}(n_{U,i}(x),E_{g}(x)).$ (4)

The acquisition function $\alpha(x)$ is evaluated on the candidate pool of
materials, $\mathcal{P}$. Given some user-defined compute budget, the
materials that maximise the acquisition function are added to the next
training phase:

$\displaystyle T_{i+1}$
$\displaystyle=T_{i}\,\cup\,\underset{x\in\mathcal{P}}{\mathrm{argmax}}\
\alpha_{i}(x)$ (5)

A MODNet model is then trained on, $T_{i+1}$, and the loop can be repeated
until sufficient accuracy, search coverage or materials performance is
obtained.

Finally, it is worth pointing out the importance of correctly calibrating the
uncertainty, $\sigma$, such that $p_{i}(n|x)$ matches the observed empirical
distribution. As shown in previous works, ensembles of MODNet models tend to
give rather well-calibrated uncertainties 60. This was confirmed on this
dataset and displayed in Fig. A.1, which shows the calibration curve and
miscalibration area (MA) for our models on different AL iterations. One could
reduce this area by a _post hoc_ rescaling of $\sigma$ that minimises the MA
on a hold-out set. Experiments show that a simple prefactor rescaling of
$\sigma\to 1.3\sigma$ is sufficient here.

###  2.3 Training and candidate data

The active learning loop was initialised with a training set of 4,040 density-
functional perturbation theory (DFPT) calculations, $T_{0}$, containing the
static refractive index and band gap from Naccarato et al. 10 (N19). For each
entry in this dataset, the most recent structure was pulled from the Materials
Project OPTIMADE API, when available. As 352 MP entries have been deprecated,
3,688 entries from the original N19 set remained.

A baseline ensemble MODNet model was trained to predict the refractive index
on this dataset, using the Matminer2024FastFeaturizer preset implemented in
MODNet v0.4.3, with relevance-redundancy criteria used to rank the computed
features 50. A genetic algorithm was used for hyperparameter optimisation,
employed with 5-fold cross-validation to assess model performance 60. The
predictions of this baseline model are shown in 1(a). This baseline was used
as the starting point for the AL loop.

The first set of trial structures were taken from the MP, which consists
primarily of density-functional theory (DFT) relaxations of experimentally
verified 1, 2 crystal structures (as well as many other computed properties).
Filtering the MP for phases with predicted band gaps greater than
$0.05\text{\,}\mathrm{eV}$ and distances from the MP convex hull (as computed
by the latest mixed GGA+U/mGGA workflow 61) less than
$25\text{\,}\mathrm{m}\mathrm{eV}\mathrm{/}\text{atom}$ results in 33,087
compounds, which we now refer to as the MP-33k set. Any material with an
existing refractive index calculation in the MP 25 (around 5,000) was removed
from the training or selection set, due to overlap with the N19 dataset.
Whilst these phases are all accessible through the MP’s OPTIMADE API, the
functionality to filter by band gap has not yet been exposed, so these phases
were instead filtered using the MP’s native API and then converted into the
OPTIMADE structure format.

In addition to the MP, the AL loop was also able to explore a dataset
constructed from the OPTIMADE API of the Alexandria PBE database 27, 5. The
OPTIMADE client implemented within the _optimade-python-tools_ package 62 was
used to filter for hypothetical compounds lying close to the Alexandria convex
hull (again, $25\text{\,}\mathrm{m}\mathrm{eV}\mathrm{/}\text{atom}$) and with
a PBE-computed band gap greater than $0.05\text{\,}\mathrm{eV}$. This approach
provided an additional 104,860 structures to the design space which could be
selected by the AL loop, henceforth Alexandria-105k.

###  2.4 Implementation

The entire end-to-end workflow is implemented in a Python package named
re2fractive, available on GitHub at modl-uclouvain/re2fractive under the MIT
license. Despite its name, this package has been designed to perform the
entire active learning workflow on any physical property that MODNet can
predict.

Each active learning run is controlled by an re2fractive.Campaign object,
which is initialised with:

  * •

a programmatic description of an initial dataset that defines a set of
properties,

  * •

a MODNet model type 50, 53 and the chosen preset of _matminer_ featurisers 51,

  * •

a set of “oracles” for property values (e.g., _atomate2_ workflows 48),

  * •

an acquisition function comprised of the available properties,

  * •

a set of logistical settings controlling the AL procedure and the calculation
workflows (via Jobflow 49 and jobflow-remote 63).

The datasets can either be defined from a code pipeline which downloads,
cleans and featurises a dataset directly (e.g., the N19 dataset 10), or
generating dynamic datasets with the latest data from a live OPTIMADE query. A
distinction must be made between the initial template dataset which contains
the structure-property map (which can be empty), and the additional datasets
used to define the design space, which are simply a source of trial
structures. These trial structures may come from other databases queried via
OPTIMADE (or otherwise), but could be a pool generated by structure
prediction, generative models, or other means.

At each step, the Campaign caches the datasets used, feature stores, and
oracle results to disk, allowing the campaign to be restarted and extended
from any point, with the performance at each checkpoint being plotted and
recorded.

###  2.5 Unsupervised embeddings

Given the number of compounds in this study it is helpful to represent these
in a condensed format, such as a plot, to observe the distribution of
candidate materials in compositional space. The Element Movers Distance (ElMD)
64 is an established metric of chemical similarity, which uses optimal
transport to return a consistent measure of (dis)similarity between two given
compositions that aligns with chemical intuition. An embedding of datasets
with respect to this metric may be carried forward via kernel principal
component analysis (PCA) by performing PCA on the ElMD distance matrix to give
2-dimensional coordinates for each material. It has been previously
demonstrated that the Euclidean distances between resultant PCA embedded
points are accurate realisations of the metric distances given by the ElMD 64.

##  3 Results

###  3.1 Learning procedure

(a)

(b)

Fig. 1: The refractive index-band gap space spanned by the N19 dataset 10. The
left-hand panel, (a), shows the first iteration of model predictions on the
“design space” datasets of MP-33k and Alexandria-105k. Only the top 50
predictions per $0.5\text{\,}\mathrm{eV}$ band gap bin are displayed. The
right-hand panel, (b), shows the DFT computed refractive index and band gap
values for the compounds selected after 8 AL iterations by the
$\omega_{\text{eff}}$ criteria described in subsection 2.2.

Several _ad hoc_ active learning runs were performed as described in
subsection 2.2 to select candidate structures on which to perform DFPT
calculations. At each selection step, an additional constraint was applied to
have fewer than 100 sites in the unit cell, but structures exceeding this
limit are still kept in the ML prediction pool. All DFT calculations performed
as part of these AL runs were saved in a global database such that future AL
runs could make use of these results, without recomputing these properties.

We first quantified the advantage of active learning on the P17 dataset,
compared to random and static learning strategies; the latter referred to as a
single fitted model screening for candidates. Figures A.5, A.6 illustrate that
a similar Top-100 score on P17 can be achieved with a lower oracle budget
through active learning, thanks to iterative improvements made to the model
during the search process. For further details, see Appendix A.2.

In our production runs, the model ensemble is trained on 80% of the N19
dataset before making a selection, leaving a global holdout set for
validation. This initial ensemble achieved already an excellent accuracy on
this holdout set, with an mean absolute error on the refractive index of 0.066
and mean absolute uncertainty of 0.038, consistent with the miscalibration
area discussed previously. The ensemble is then applied to the candidate pools
spanned by the aforementioned MP-33k and Alexandria-105k datasets, as shown in
1(a). A number (initially 200, then 500) of the top candidates across these
two sets were selected at each iteration $i$ for further DFPT computations,
using the $\omega_{\text{eff}}(n_{U,i}(x),E_{g}(x))$ acquisition function
introduced in Equation 4. The AL loop was repeated 8 times to select the
candidates comprising the final dataset shown in 1(b). The N19 holdout error
was not seen to decrease with further iterations, indicating that the selected
materials were sampled from underexplored regions. When evaluating the models
_post hoc_ on the P17 and C24 sets (appropriately filtered and de-duplicated
for any overlap with our selected candidates), we observe a much higher MAE
around 0.2 and 0.4, respectively.

Fig. 2: The $(n,E_{g})$ space spanned by the N19 10 (blue), C24 26 (orange)
and VT-2k datasets (green). The numbered black crosses correspond to the
materials that pass each of the filtering steps described in subsection 3.1,
and listed in Table 2. Figure A.3 shows the result of applying the same
filters to the C24 and R24 datasets.

The culmination of this work is a new, publicly available, dataset of static
refractive index calculations for 2,413 materials, VT-2k. The dataset of
structures, band gaps and refractive indices computed with the workflow
described above is made available on the Materials Cloud Archive 65 alongside
the final model. Figure 2 depicts the full dataset superimposed on top of N19,
the starting training set, and C24 as it was also designed to target high-$n$
materials. As hoped, both VT-2k and C24 are less disperse than N19 in the
$(n,E_{g})$ space. This implies that computational resources were used more
efficiently in regard to the goal of identifying new high-$n$ compounds. A
similar representation with respect to P17 is shown in Figure A.2.

This sampling of high-$n$ materials is even more visible in Figure 3 via the
distribution of $\omega_{\text{eff}}$ (see Equation 3) for the different
datasets. The upper panel shows that the pre-screening with a machine learned
proxy has enabled to further push the Pareto front of high-$n$ materials when
compared with the usual screening performed by Petousis et al. 25 and
Naccarato et al. 10. By comparison with C24, VT-2k displays a narrower
distribution (lower panel), which may result from using a higher accuracy
proxy to select an appropriate pool of trial structures for this target. This
can also be said for the comparison against R24 to some extent, although their
goal was high-$k$ dielectrics instead of high refractive index compounds.
Figure 4 provides an ElMD embedding of this dataset, demonstrating the
chemical diversity present. VT-2k is comprised of $\sim$43% oxides (or at
least, oxygen-containing compounds), with sulfur and selenium present in 16%
and 11% of the entries respectively. The dataset includes both simple and more
complex structures, with $\sim$42% of the computed entries having more than 20
atoms in their primitive unit cell.

Fig. 3: Comparison of the distributions of the effective frequency,
$\omega_{\text{eff}}$, across the DFT datasets considered in this work.

###  3.2 Filtering the resulting materials

One of the goals of the present dataset is to offer a pool of materials for
various screening procedures with emphasis on optical applications. Since the
present work is not focused on a particular energy range, we illustrate a
naive filtering by binning VT-2k with respect to the band gap (30 bins) and
choosing the compound with the largest effective frequency in each bin. Table
1 lists the resulting candidates, which span a wide range of the
electromagnetic spectrum and possess the best compromise between band gap and
refractive index. However, out of the 30 structures, only 11 of them have been
observed experimentally, and 9 of them contain lead, which is known for being
toxic. This means that this method of selection lead to an oversampling of
(potentially) unsynthesisable and toxic compounds, since VT-2k in aggregate
consists of $\sim 43$% of experimentally observed materials and $\sim 8$% of
lead-containing compositions. This simple procedure illustrates that, although
the theoretical optimisation portion of this work treats the problem as a
2-objective optimisation, other factors must be considered when screening for
realistic materials applications. In order to identify the most promising
high-$n$ materials for further investigation, the VT-2k compounds were
screened by heuristics for the elemental sustainability, cost and
availability, their capacity to be potentially synthesised in the future, and
the figure of merit with respect to the targeted application.

Identifier | Formula |  $E_{g}$ (eV) | $n_{s}$ |  $\omega_{eff}$ (eV) | Spacegroup | Exp. observed | MP ID
---|---|---|---|---|---|---|---
mp-1028576 |  Te8MoW3 | 4.011 | 3.006 | 20.940 | 156 | - | -
mp-1218922 |  SnTe(PbS)4 | 0.555 | 7.592 | 19.781 | 160 | - | -
mp-571169 |  InBi2Se4Br | 1.295 | 4.741 | 19.778 | 12 | ✓ | -
mp-1030335 |  Te4MoW | 3.172 | 3.028 | 19.206 | 156 | - | -
mp-1195735 |  Te3As2 | 0.699 | 6.022 | 18.814 | 11 | ✓ | -
mp-1028594 |  Te4MoW | 2.936 | 3.028 | 18.669 | 164 | - | -
agm002332658 |  NbCoGe | 1.074 | 4.726 | 18.596 | 216 | - | -
agm002233584 |  PRhS | 1.512 | 4.049 | 18.570 | 29 | - | -
mp-1212561 |  Ge19(AsBr)4 | 1.425 | 4.106 | 18.415 | 218 | - | -
agm003202390 |  YZn4Ir | 0.956 | 4.879 | 18.294 | 216 | - | -
mp-1030319 |  Te2Mo | 2.596 | 3.104 | 18.215 | 164 | - | -
mp-1212555 |  Ge19(PCl)4 | 1.650 | 3.752 | 18.067 | 218 | - | -
mp-4891 |  HoAlO3 | 5.836 | 2.018 | 17.987 | 62 | ✓ | -
agm003231828 |  DyF3 | 8.062 | 1.680 | 17.897 | 165 | - | mp-1212889
mp-4434 |  TbAlO3 | 5.598 | 2.044 | 17.825 | 62 | ✓ | -
mp-1035108 |  LiMg14GaO16 | 6.873 | 1.810 | 17.682 | 47 | - | -
mp-21872 |  PrLuO3 | 4.876 | 2.124 | 17.284 | 62 | ✓ | -
mp-1212158 |  Hf3GeO8 | 4.550 | 2.189 | 17.197 | 121 | - | -
agm002175849 |  Hf3GeO8 | 4.352 | 2.207 | 16.989 | 215 | - | -
agm003214191 |  Pr2PbS4 | 2.031 | 3.147 | 16.966 | 122 | - | mp-675638
mp-33406 |  La2PbSe4 | 1.828 | 3.290 | 16.957 | 122 | - | -
agm003209963 |  PbICl | 2.691 | 2.753 | 16.844 | 62 | ✓ | mp-23053
mp-36538 |  La2PbS4 | 2.155 | 3.030 | 16.809 | 122 | - | -
mp-674363 |  Zr3Sc4O12 | 3.947 | 2.237 | 16.492 | 1 | - | -
mp-1190186 |  ZrO2 | 3.579 | 2.344 | 16.481 | 61 | ✓ | -
mp-22997 |  PbBrCl | 3.438 | 2.385 | 16.456 | 62 | ✓ | -
mp-1192803 |  Lu2PbS4 | 2.302 | 2.822 | 16.279 | 62 | ✓ | -
mp-669414 |  HfPbO3 | 3.199 | 2.431 | 16.258 | 32 | ✓ | -
agm002171412 |  PrGaO3 | 3.816 | 2.209 | 16.097 | 167 | - | -
mp-1200793 |  Ga4Ge(PbS3)4 | 2.367 | 2.740 | 16.065 | 114 | ✓ | -
Table 1: List of the 30 best materials based on the effective frequency of
Equation 3. Crystal structures with corresponding entries in the ICSD 1, 2 are
marked as “experimentally observed” and the MP ID is indicated for Alexandria
materials matching an entry of the MP based on composition and spacegroup.
Fig. 4: An ElMD embedding of the N19 10 dataset (blue) and VT-2k dataset 65
(green, this work) in the composition space spanned by 2M structures across
the OPTIMADE consortium 32 (grey). The materials in Table 1 are overlaid in
black stars, with a selection of these compounds labelled.

For the first criterion, the sustainability of each element can be
characterised by the Herfindahl-Hirschman Indices (HHIs) 66. These measure
market concentration which, when applied to the geological and geopolitical
availability of the element 67, provides insight into possible future market
abuse and subsequent price spikes of the materials. Using the values from
Gaultois et al. 67 and following Kim et al. 68, we only consider elements with
an HHI lower than 6000 for both production and reserves***Namely Li, B, C, N,
O, F, Na, Mg, Al, Si, P, S, Cl, Ca, Zn, Ga, Ge, As, Se, Sr, Cd, In, Sn, Te, I,
Ba, Hg, Pb. Compounds containing mercury (Hg), lead (Pb), cadmium (Cd), or
arsenic (As) are also removed from this list due to their inherent toxicity.
It should be noted that HHIs are not static, and the values from Gaultois et
al. 67 are derived using geological surveying data from 2009-2011. As
resources become more economically valuable, the number of geological surveys
accelerate and new reserves are discovered, leading to oscillations in each of
these indices.

To assess the potential synthesisability of a hypothetical material,
typically, the main consideration is the predicted thermodynamic stability. By
including only compounds with a predicted formation energy close to the global
convex hull, i.e., whether the structure is at or close to the global minimum
for that composition 69, materials which are unlikely to synthesise are
excluded. This is a reasonably strong, but imperfect, indicator of the
synthesisability of a hypothetical material, as the experimental ground truth
is heavily affected by kinetics, entropic considerations and the availability
of precursor materials 9. In an attempt to model the difference between (zero
temperature) DFT thermodynamic stability and realistically achievable
synthesisability, recent works have created global reaction networks 70 to
find likely thermodynamic pathways, and machine learning models to directly
predict relative synthesisability using semi-supervised or positively-
unlabelled learning 71, 72, 73.

In this work, we take the Composition Graph Neural Fingerprint (CGNF) atomic
embedding provided by Kim et al. 73 (as trained on the MP 3 and OQMD 4) to
predict the probability of synthesisability of the hypothetical materials in
VT-2k. From these predictions, the third quartile of synthesisability
predictions across VT-2k was used as a lower threshold for a material to be
considered potentially synthesisable. As a relative threshold is used, this
approach is applicable to any dataset, and ensures that the materials in the
upper tail of the synthesisability score distribution are selected.

Finally, to filter materials based on the figure of merit (FoM) for the
intended application, an expressive metric that captures the specific utility
of a material is required. One could aim to minimise the current leakage
($\max{(\varepsilon_{poly}\cdot E_{g})}$) or maximise the energy storage
($\max{(\sqrt{\varepsilon_{poly}}\cdot E_{g})}$), however these FoMs would be
more relevant in searching for high-$k$ dielectrics 25, 28. For high
refractive index materials, an appropriate quantity to maximise is the
effective frequency, $\omega_{\text{eff}}$, which was used during our AL
selection process. In order to generalise our method for future uses, the most
promising materials can simply be defined as being those closest to the Pareto
front of the properties to optimise. In a 2-objective optimisation, these can
be selected by binning the data over one objective (the band gap) and
selecting all compounds above the third quartile of the second objective (the
refractive index) for each bin. This approach enforces the selection of
promising materials over a broad spectral range, which would not necessarily
be the case when using the effective frequency FoM alone.

As each filter uses a relative cut-off to screen the most promising materials,
these operations are likely to be non-commutative, and the order the different
filters are applied will influence the final choice of materials, which is
illustrated in Figure A.4. As presented, the selection of synthesisable and
Pareto-optimal compounds leads to the exclusion of 75% of the dataset. The
selection of low-HHI materials is even more drastic as 90% of the dataset is
discarded. At the final stage of each filter permutation, around 0.5% of the
dataset ($\sim$13 entries) is considered as promising and worth investigating.
The union of structures screened by each permutation of the filters comprises
21 materials; these are listed in Table 2. It should be noted that this set
already excludes any material covered in N19 or P17, which include most of the
already known and stable optical materials.

###  3.3 Promising materials

| Identifier | Formula |  $E_{g}$ (eV) | $n_{s}$ |  $\omega_{\text{eff}}$ (eV) | Spacegroup | Exp. observed | MP ID
---|---|---|---|---|---|---|---|---
1 | mp-1198754 |  ZnSnP14 | 1.193 | 3.368 | 15.111 | 62 | ✓ | -
2 | mp-1195188 |  Ca3(Si4P7)2 | 1.236 | 3.314 | 15.101 | 14 | ✓ | -
3 | mp-29817 |  GaP2I9 | 1.711 | 2.074 | 11.550 | 61 | ✓ | -
4 | mp-1193105 |  Ba4In2Te2Se5 | 1.843 | 2.631 | 14.361 | 127 | ✓ | -
5 | mp-1193666 |  Ba4In2Te2S5 | 2.064 | 2.511 | 14.351 | 127 | ✓ | -
6 | mp-568803 |  Mg(B6C)2 | 2.140 | 2.759 | 15.621 | 74 | ✓ | -
7 | mp-1228652 |  B8O | 2.370 | 2.556 | 15.225 | 6 | - | -
8 | mp-1227978 |  BaGa2SiSe6 | 2.389 | 2.609 | 15.512 | 1 | - | -
9 | mp-758800 |  B84O11 | 2.578 | 2.521 | 15.496 | 1 | - | -
10 | mp-655591 |  LiB13C2 | 2.645 | 2.617 | 16.101 | 74 | ✓ | -
11 | mp-14425 |  Sr(GaS2)2 | 2.797 | 2.280 | 14.698 | 70 | ✓ | -
12 | agm003284058 |  Ca(BS2)2 | 2.894 | 2.476 | 15.905 | 205 | ✓ | mp-30958
13 | mp-849286 |  Ba(GaS2)2 | 2.907 | 2.311 | 15.072 | 205 | ✓ | -
14 | mp-1198976 |  Ba3GaS4Cl | 3.169 | 2.190 | 14.870 | 62 | ✓ | -
15 | mp-2983 |  Ba(PN2)2 | 3.811 | 2.145 | 15.697 | 205 | ✓ | -
16 | mp-6404 |  BaCa2(PN2)6 | 4.073 | 2.135 | 16.063 | 205 | ✓ | -
17 | mp-567486 |  BaSr2(PN2)6 | 4.154 | 2.120 | 16.097 | 205 | ✓ | -
18 | agm003230381 |  ZnB4O7 | 5.538 | 1.807 | 15.840 | 63 | ✓ | mp-558690
19 | agm003283778 |  Sr(BO2)2 | 6.407 | 1.800 | 16.959 | 205 | ✓ | mp-8878
20 | agm003251671 |  BaB4O7 | 6.412 | 1.815 | 17.101 | 62 | ✓ | mp-556974
21 | agm003272662 |  BaAlF5 | 7.495 | 1.511 | 15.300 | 19 | ✓ | mp-4376
Table 2: List of the most promising materials based on our screening with
respect to the HHI, the synthesisability, and the quality as a high-$n$
compound, sorted by ascending $E_{g}$. Crystal structures with corresponding
entries in the ICSD 1, 2 are marked as “experimentally observed” and the MP ID
is indicated for Alexandria materials matching an entry of the MP based on
composition and spacegroup.

Of the final set of 21 materials listed in Table 2, 18 have been previously
observed experimentally. By comparison, just 43% of the broader VT-2k set have
been experimentally observed. This is an indicator that the crude ML-based
synthesisability criteria is perhaps too biased towards known materials, or
alternatively, that the hypothetical materials found predominantly in
Alexandria-105k are too distant from the set of known materials to be
considered plausible (without new directed synthesis campaigns). The promising
materials span a wide band gap range from $1.27.5$ whilst retaining a high
$\omega_{\text{eff}}>14.5\text{ eV}$.

In the low gap region ($1.21.8$ ), ZnSnP14–$Pnma$ (1) 74,
Ca3(Si4P7)2-$P2_{1}/c$ (2) 75 and GaP2I9-$Pbca$ (3) 76 are all previously
isolated compounds with weakly bound structures and relatively large unit
cells (50, 64 and 98 sites respectively), which may explain why they were
excluded from previous high-throughput screenings. The reported colours of
experimental powders in their initial synthesis reports 74, 75, 76 are
consistent with the expected underestimated values predicted by DFT.

Two isostructural quaternary chalcogenides Ba4In2Te2$Q$5 ($Q=\text{S, Se}$) (4
and 5) crystallising in the $P4/mbm$ space group 77 are suggested with
$\omega_{\text{eff}}$ around $14\text{\,}\mathrm{eV}$. They consist of layered
[In2Te2$Q$4]^6- covalent units, and UV-vis measurements of their respective
band gaps ($2.3\text{\,}\mathrm{eV}$ and $1.8\text{\,}\mathrm{eV}$) are
broadly consistent with calculations, indicating that the computed refractive
indices of $\sim$2.5 are feasible. The similar hypothetical quarternary barium
chalcogenide BaGa2SiSe6-$P1$ (8) (hull distance
$14\text{\,}\mathrm{m}\mathrm{eV}\mathrm{/}\text{atom}$) is also predicted to
have a similar refractive index in this band gap range, though has not yet
been isolated and is instead expected to decompose to the simpler Ba(GaSe2)2
phase (and SiSe2) even without further entropic considerations.

The $2.02.6$ range is populated by boron-based materials, from the weakly
bound B12 icosehedra in Mg(B6C)2-$Imma$ (6) 78, and LiB13C2-$Imma$ (10) 79
have both been previously isolated and, in the case of LiB13C2 with an
excellent $\omega_{\text{eff}}$ value of $16.1\text{\,}\mathrm{eV}$, has been
observed to be colourless. Two hypothetical boron suboxides B8O-$Pm$ (7) and
B84O11-$P1$ (9), derived from B6O-$R\bar{3}m$ 80, have previously shown
promise for their mechanical properties as composites, but are hard to
isolate.

Between $2.7\text{\,}\mathrm{eV}$ and $3.2\text{\,}\mathrm{eV}$, the selected
materials are dominated by relatively well-known sulphide materials, including
the thiogallates Sr(GaS2)2-$Fddd$ (11), Ba(GaS2)2-$Pa\bar{3}$ (13),
Ba3GaS4Cl-$Pnma$ (14) and the related derivative Ca(BS2)2-$Pa\bar{3}$ (12) 81
(seemingly only accessible via high-pressure synthesis). These compounds have
previously been studied as phosphor materials 82 for their response under
ultraviolet light.

In the wide gap range above $3.8\text{\,}\mathrm{eV}$, three barium
nitrodophosphates Ba(PN2)2-$Pa\bar{3}$ (15), BaCa2(PN2)6-$Pa\bar{3}$ (16) and
BaSr2(PN2)6-$Pa\bar{3}$ (17) 83 have all been previously observed with
relatively large unit cells containing 84 atoms. These are the final three
materials considered that would be transparent in the visible range, whilst
still retaining a refractive index greater than 2.

Finally, three very wide-gap oxides ZnB4O7-$Cmcm$ (18), Sr(BO2)2 (19) and
BaB4O7 (20) are suggested with excellent $\omega_{\text{eff}}$ values, driven
by their band gaps which exceed $5.5\text{\,}\mathrm{eV}$. These materials
have only ever been isolated at high pressures 84, 85, 86 but possess
relatively low predicted hull distances (below
$25\text{\,}\mathrm{m}\mathrm{eV}\mathrm{/}\text{atom}$. Beyond these, the
BaAlF5-$P2_{1}2_{1}2_{1}$ (21) 87 phase with a band gap of
$7.5\text{\,}\mathrm{eV}$ has the widest gap considered here, and whilst its
refractive index is just 1.5, this still corresponds to a competitive
$\omega_{\text{eff}}$ of $15.3\text{\,}\mathrm{eV}$.

As previously mentioned, these results must be caveated by the fact that GGA-
based DFT typically underestimates band gaps in an uncontrolled and
unsystematic way; this will also be reflected in the overestimation of the
refractive index. However, we believe this set of materials comprise a
promising set to consider for linear (and potentially nonlinear) optical
applications, which may motivate further attempts to synthesise those that
have not yet been isolated.

##  4 Conclusions

In recent years, the field of materials informatics has evolved from curating
databases of calculations on known materials ($\sim 10^{5}$) 3, to directly
predicting crystal structures at a given composition ($\sim 10^{3}$) 88, 29,
to using machine learning approaches to suggest stable hypothetical materials
($\sim 10^{6}$) 89, 7, increasingly targeted towards particular properties 90,
91, 8. As curated materials datasets grow and new hypothetical materials are
suggested, it is imperative that we are able to effectively screen these
compounds to guide experimental work towards the most fruitful areas. We
stress the importance of a decentralised approach to data sharing following
the model of OPTIMADE, that complements the work done in curating high-quality
databases, as this enables materials informatics studies to screen our best
and latest guess of the global phase diagram in a machine-actionable manner.
With this in mind, there are three directions in which the present work will
be extended.

Firstly, the procedure for screening high-$n$ materials will be applied to
additional datasets as they become available with OPTIMADE APIs automatically
31. This takes advantage of the decentralised nature of the OPTIMADE consortia
to encompass not only the growing number of structures available in curated
experimental databases such as the ICSD 1, 2 (and their corresponding
calculations in the MP 3), but also hypothetical structures provided in small
datasets 92, ensuring that each screening step is given a broad exposure to
the landscape of hypothetical materials. This work will be used in part to
motivate the ongoing development of the OPTIMADE stability namespace 32, which
will enable data unification across the providers that serve crystal
structures with corresponding thermodynamics calculations. This would enable
our AL campaign to run indefinitely and autonomously, adding to the dataset of
validated materials based on a simple multi-provider OPTIMADE query.†††Such an
OPTIMADE query could be _stability_hull_distance <= 0.05 AND last_modified >=
"2024-09-10-00:00" All of the results generated by this online procedure will
be made available themselves via an OPTIMADE API, to allow other human and
machine agents to make use of our predictions. Additionally, generative
approaches will be investigated that make use of the final AL-trained model as
a discriminator for high-$n$ materials. These generative trials can then be
used to iterate much more rapidly, without needing to wait for the results of
external materials discovery campaigns to be made available.

Secondly, the best-performing materials screened in this investigation will be
examined with more scrutiny before it is possible to propose them for costly
experimental synthesis and characterisation. This will include incorporating
calculations of band gaps at higher levels of theory, which may then re-enter
future AL campaigns via multi-fidelity learning 60. The filtering procedure
described in subsection 3.1 may be further refined (e.g., via improvements to
the synthesisability filter, or by relaxing some of the constraints on HHI
based on the latest data).

Finally, the resulting models and datasets created here for high-$n$ materials
will be used to initialise a campaign searching for materials with more exotic
physical properties, such as nonlinear optical response. We believe the
re2fractive AL approach is particularly well suited for this case, where there
are significantly fewer calculations available, and each individual
calculation (or oracle evaluation) requires significantly more computational
resources to execute. In this sense, the entire AL campaign performed in this
work will form part of the selection criteria of a broader campaign, and the
re2fractive framework outlined in subsection 2.4 will be expanded to
accommodate this stacked approach.

## Author Contributions

V.T.: Conceptualisation, Methodology, Software, Validation, Investigation,
Data curation, Writing - Original Draft, Writing - Review and Editing,
Visualisation. M.L.E.: Conceptualisation, Methodology, Software, Validation,
Investigation, Data curation, Writing - Original Draft, Writing - Review and
Editing, Visualisation. C.J.H.: Conceptualisation, Writing - Original Draft,
Writing - Review and Editing, Visualisation. P-P.D.B.: Conceptualisation,
Methodology, Validation, Writing - Original Draft. G-M.R.: Conceptualisation,
Resources, Writing - Review and Editing, Supervision, Funding acquisition.

## Conflicts of interest

G.-M.R. is a shareholder and Chief Innovation Officer of Matgenix SRL.

## Acknowledgements

Computational resources have been provided by the supercomputing facilities of
the Université catholique de Louvain (CISM/UCL) and the Consortium des
Équipements de Calcul Intensif en Fédération Wallonie Bruxelles (CÉCI) funded
by the Fond de la Recherche Scientifique de Belgique (F.R.S.-FNRS) under
convention 2.5020.11 and by the Walloon Region. The present research benefited
from computational resources made available on Lucia, the Tier-1 supercomputer
of the Walloon Region, infrastructure funded by the Walloon Region under the
grant agreement n°1910247. V.T. acknowledges the support from the FRS-FNRS
through a FRIA Grant. M.L.E. thanks the BEWARE scheme of the Wallonia-Brussels
Federation for funding under the European Commission’s Marie Curie-Skłodowska
Action (COFUND 847587). C.J.H. thanks the FRS-FNRS for their support as part
of the Fish4Diet project.

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##  Appendix A Appendix

###  A.1 Supplementary figures

Fig. A.1: Calibration curves for the MODNet model used in the different
iterations of the active learning campaign Fig. A.2: The $(n,E_{g})$ space
spanned by the P17 25 (purple), and VT-2k datasets (green). The black crosses
correspond to the materials that pass each of the filtering steps described in
subsection 3.1. They are also listed in Table 2.

Fig. A.3: The filters described in subsection 3.1 applied to the external
datasets of C24 26 (orange crosses, upper panel) and R24 28 (red crosses,
lower panel), in comparison with the filtered materials from VT-2k (black
crosses). Exceptional outlier materials (Te-W-Mo and Sn-Te containing
compounds) from each dataset are labelled explicitly. Fig. A.4: Effect of the
order of the filters on the final screening of promising high-$n$ materials.
The y-axis shows the relative quantity of selected materials with respect to
the present dataset (2,413 entries). The different orders of filters are
situated along the horizontal axis where ’S’ stands for the criterion on
synthesisability, ’P’ for the restriction to the Pareto neighbourhood, and ’H’
for the exclusion of high HHI elements.

###  A.2 Active versus static learning

The advantage of active learning with respect to other common search methods
is demonstrated using the P17 dataset. Initially, a subset of 1,000 labeled
samples from P17 is designated as the starting set $\mathcal{L}$ (simulating a
known dataset), while the remaining $\sim$ 5,000 data points form the
candidate pool $\mathcal{P}$ and remain unlabeled (simulating known structures
with unknown characteristics). Different strategies are then employed to query
$b$ points (as defined by the budget, up to 1,000) from $\mathcal{P}$. These
points are subsequently labeled using the P17 dataset (simulating an oracle).
The quality of the selected points is quantified by computing the top-$k$
score over $\omega_{\text{eff}}$:

$\text{top}_{k}(\omega_{\text{eff}})=\frac{1}{k}\sum_{i=1}^{k}\omega_{\text{eff},(i)},$

where $\omega_{\text{eff},(i)}$ represents the $i$-th highest score among the
newly labeled samples. This metric is calculated over the $k$ samples with the
highest $\omega_{\text{eff}}$, where $k$ is an adjustable parameter.

Fig. A.5: Top-$k$ score as a function of $k$ for different search budgets,
comparing AL (Active Learning) and SL (Static Learning) algorithms.

As a baseline strategy, one might randomly select points from $\mathcal{P}$.
The P17 dataset presents average values of $n=2.29\pm 0.86$, $E_{g}=2.44\pm
1.67$, and $\omega_{\text{eff}}=11.91\pm 3.61$ ($\pm$ denoting the standard
deviation). For a random search baseline, the expected value of the top-$k$
score, calculated over the highest $k$ scores from randomly selected points,
would approximate the mean of $\omega_{\text{eff}}$, i.e., 11.91. The variance
of this score would decrease with increasing $k$, scaling approximately as
$1/k^{2}$.

A more commonly used approach is to train a surrogate machine learning (ML)
model on $\mathcal{L}$, and then use it to screen $\mathcal{P}$ for promising
candidates, selecting the $b$ candidates with the highest predicted
$\omega_{\text{eff}}$. We refer to this method as static learning (SL). In
contrast, active learning (AL) iteratively updates its ML model while
exploring $\mathcal{P}$. Both strategies are allowed to query $b$ new samples
from $\mathcal{P}$, as defined by the budget.

Figure A.5 represents the top-$k$ score as a function of $k$, for different
budget sizes. It is observed that AL systematically outperforms SL. Although
the advantage is small, it is noteworthy that with a budget of 800, AL
achieves a similar or better top-$k$ score than SL with a budget of 1000,
across almost the entire range of $k$. This represents a non-negligible saving
of resources of 20%.

Fig. A.6: Top-100 score as a function of iterations (i.e., search budget) for
AL (Active Learning) and SL (Static Learning) search algorithms. Error bars
represent the standard deviation over 5 repeated experiments.

Figure A.6 displays the Top-100 score for various budgets, while also
depicting the observed standard deviation over repeated experiments. A slight
yet consistent advantage is observed for active learning, with a notably
smaller variance among selected candidates. Notably, a budget of 1000 is
necessary for static learning to achieve a Top-100 score comparable to that
reached by active learning with just a budget of 400.

Finally, Figure A.7 shows the Mean Absolute Error (MAE) on a hold-out test set
as a function of the AL iterations (i.e., search budget). As expected, it
demonstrates that in AL, the model progressively improves with each search
iteration. In contrast, for SL, the accuracy remains constant, as the model
does not update during the search process.

Fig. A.7: Mean Absolute Error (MAE) of the models used in Active Learning (AL)
and Static Learning (SL) on a hold-out test set, as a function of iterations
(i.e., search budget).